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University of Southern California Dissertations and Theses
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Emerging Jordan forms, with applications to critical statistical models and conformal field theory
(USC Thesis Other)
Emerging Jordan forms, with applications to critical statistical models and conformal field theory
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EmergingJordanforms,withapplications to critical statistical models and conformal field theory by Lawrence Liu ADissertation Presented to the FACULTYOF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA InPartial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) May 2023 Copyright2023 LawrenceLiu Dedication Universispreceptoribusmeis,praeteritum,presentibus et futuris, scolarem ad uitam remanere ii Acknowledgments Muchsupportduringthisresearchprojectcamefromfamilyandfriends. Adebtofgratitudehasbeenpaid, inpart,byincludingtheminthededication page, since they are certainly among that group. SpecialthanksgotoHubertSaleur,forentrustingmewithchallengingproblems,toL’InstitutdePhysique Théorique at CEA Saclay and its community for their hospitality, and to Jesper Jacobsen for comments on earlierdraftsofthiswork. I am also grateful for the musicians whose works filled my offices and other working spaces. Non- exhaustively, they include: Isaac Albéniz, Charles-Valentin Alkan, George Antheil, Johann Sebastian Bach, Mily Balakirev, Ludwig van Beethoven, Aleksandr Borodin, Sergei Bortkiewicz, Lili Boulanger, Jo- hannes Brahms, Ferruccio Busoni, Fryderyk Chopin, Mikhail Glinka, Leopold Godowsky, Enric Granados, Joseph Haydn, Adolf von Henselt, Nikolai Kapustin, György Ligeti, Ferenc Liszt, Gustav Mahler, Nikolay Medtner, Felix Mendelssohn, Olivier Messiaen, Moritz Moszkowski, Wolfgang Amadeus Mozart, Sergey Prokofiev,SergeiRachmaninoff,EinojuhaniRautavaara,Jean-HenriRavina,ChristopherRouse,FranzSchu- bert,AleksandrScriabin,KarolSzymanowski,PyotrTchaikovsky,SigismondThalberg,IannisXenakis. The cognoscentiwillrecognizetheobviousbias that appears in this list. iii TableofContents Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Outlineofthiswork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Chapter1 Mathematics: basicconcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1 Innerproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Thespectraltheoremfornormal operators . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 TheJordancanonicalform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Representationtheoryofalgebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter2 Mathematics: representation theory of various Temperley–Lieb algebras . . . . . . 10 2.1 GeneratorsandrelationsforTemperley–Lieb algebras . . . . . . . . . . . . . . . . . . . 10 2.2 Cellularalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Standardandco-standardmodules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 0, ±2 andindecomposability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5 Cellmodulesforroot-of-unity values of. . . . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter3 Physics: basicconcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1 Quantumandstatisticalphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Conformalfieldtheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 Twoconcretephysicsproblems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3.1 ThequantumHalleffect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3.2 Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Chapter4 Physics: conformalfieldtheory on the lattice . . . . . . . . . . . . . . . . . . . . . 28 4.1 Discretizationofcontinuumtheories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1.1 Necessityofdiscretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.1.2 Identificationoffieldsonthe lattice . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.2 Pottsmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2.1 Generaldescription. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2.2 DiscreteVirasoroalgebrain the Potts model . . . . . . . . . . . . . . . . . . . . . . 32 4.2.3 AnoteontheXXZrepresentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2.4 Thechoicesofmetric,and duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.3 ℓ(2|1) superspinchain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Chapter5 Computationalmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.1 Matrixdiagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2 Arnoldimethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.3 Numericalcomputationofthe Jordan canonical form . . . . . . . . . . . . . . . . . . . 41 iv II Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Chapter6 EmergingJordanforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.1 The measure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.2 RatesofconvergenceandtheJordan coupling . . . . . . . . . . . . . . . . . . . . . . . 48 6.2.1 Rank2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.2.2 Rank3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.2.3 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.3 Nextsteps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.3.1 Furtherstudyofemerging Jordan blocks . . . . . . . . . . . . . . . . . . . . . . . 53 6.3.2 Possibleapplicationsofemerging Jordan blocks. . . . . . . . . . . . . . . . . . . . 53 Chapter7 BiorthogonalanddualJordan quantum physics. . . . . . . . . . . . . . . . . . . . 55 7.1 Biorthogonalprojectionoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.2 Anotherdiagnosticforemerging Jordan blocks. . . . . . . . . . . . . . . . . . . . . . . 57 7.3 DualJordanprojectionoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.3.1 SingleJordanblock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7.3.2 MultipleJordanblocksfor a single eigenvalue. . . . . . . . . . . . . . . . . . . . . 59 7.3.3 ThegeneralJordancanonical form. . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.4 Nextsteps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 III Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Chapter8 TheactionoftheVirasoro algebra in the two-dimensional Potts and loop models. . 64 8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 8.2 Vir⊕Vir modulesinthePottsmodel CFT . . . . . . . . . . . . . . . . . . . . . . . . . 65 8.2.1 Thenon-degeneratecase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 8.2.2 Thedegeneratecase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 8.3 EvidencefromthelatticeviaKoo–Saleur generators . . . . . . . . . . . . . . . . . . . . 69 8.3.1 Scaling-weakconvergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.3.2 Numericalresultsfor 0, ±2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.3.3 Numericalresultsfor 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.4 Observationofsingletstatesin the loop model. . . . . . . . . . . . . . . . . . . . . . . 74 8.4.1 = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8.4.2 = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 8.4.3 = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.5 Parityandthestructureofmodules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Chapter9 NumericalstudyofJordan blocks in the dense loop model CFT . . . . . . . . . . . 80 9.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 9.2 NumericalamplitudesandJordan blocks. . . . . . . . . . . . . . . . . . . . . . . . . . 80 9.3 Modulesfortheloopmodelin the degenerate case: the OPE point of view . . . . . . . . 84 9.4 Measurementofindecomposability parameters via emerging Jordan blocks. . . . . . . . 88 9.4.1 EmergingJordanblocksin 11 at generic . . . . . . . . . . . . . . . . . . . . . . 90 9.4.2 EmergingJordanblocksin 21 at generic . . . . . . . . . . . . . . . . . . . . . . 92 9.5 Thelimit→ 0: Jordanblocks between different modules . . . . . . . . . . . . . . . . . 95 9.5.1 Gluedcellrepresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9.5.2 AbsenceorpresenceofJordan blocks at finite size. . . . . . . . . . . . . . . . . . . 96 9.5.3 Measurementof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Chapter10 The periodic alternating ℓ(2|1) superspin chain and its scaling-weak convergence toalogarithmicconformal field theory . . . . . . . . . . . . . . . . . . . . . . . . 100 10.1 IdentificationoffieldsandJordan structure at finite size . . . . . . . . . . . . . . . . . . 100 10.2 Scaling-weakconvergenceofconformal identities . . . . . . . . . . . . . . . . . . . . . 103 10.3 Actionoftheℓ(2|1)superalgebra on singlet states . . . . . . . . . . . . . . . . . . . . . 110 v Chapter11 Mixingofconformalfieldsat finite size . . . . . . . . . . . . . . . . . . . . . . . . 112 11.1 Mixingof andΦ 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 11.2 Theregion 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 11.3 Thefield Φ 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Conclusionandoutlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 AppendixA Dynamicmultiplicityadjustment in the implicitly restarted Arnoldi method . . . . 137 AppendixB EfficientMathematicaimplementation of lattice field theories . . . . . . . . . . . . 139 B.1 Commondefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 B.2 Loopmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 B.2.1 Constructionofthebasis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 B.2.2 Translationoperatoranddiagonalization . . . . . . . . . . . . . . . . . . . . . . . 141 B.2.3 ConstructionoftheTemperley–Lieb generators . . . . . . . . . . . . . . . . . . . . 142 B.2.4 Innerproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 B.2.5 Thecontractionparameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 B.3 ℓ(2|1) superspinchain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 B.3.1 Constructionofthebasis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 B.3.2 Translationoperatoranddiagonalization . . . . . . . . . . . . . . . . . . . . . . . 148 B.3.3 ConstructionoftheTemperley–Lieb generators . . . . . . . . . . . . . . . . . . . . 149 B.3.4 Innerproducts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 AppendixC DirectcalculationofthesubquotientstructureofTemperley–Liebalgebrarepresen- tations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 AppendixD Correctionstosometables in Koo and Saleur (1994). . . . . . . . . . . . . . . . . . 152 AppendixE Additionalcalculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 E.1 Anotherloopmodelscalarproduct. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 E.2 Loopmodelcontractionparameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 E.3 Illustration of the absence of Jordan blocks for =±1, and the -independence of the measurementof at= 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 AppendixF Additionaltablesandfigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 F.1 SupplementarydataforSection 8.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 F.2 SupplementarydataforSection 10.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 F.3 SupplementalfiguresforChapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 vi ListofTables 8.1 Thevaluesof∥ −1 ∥ 2 forvariouslengths andparameters. isthefieldinthe = 0sector withconformalweights(ℎ 11 ,ℎ 11 )=(0,0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 8.2 The values of∥Π (1) −1 ∥ 2 for various lengths and parameters . Π (1) is a projection to the state of lowest energy within the = 0, = 1 sector. This is a state that has conformal weights(ℎ 1,−1 ,ℎ 11 )=(1,0). The extrapolation is obtained by fitting the last five data points toafourth-orderpolynomialin1/. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.3 Thevaluesof∥ −1 Φ 11 − −1 Φ 11 ∥ 2 /∥ −1 Φ 11 ∥ 2 for various lengths and parameters. . . . 73 8.4 Thevaluesof∥Π (2) ( −1 Φ 11 − −1 Φ 11 )∥ 2 /∥Π (2) −1 Φ 11 ∥ 2 forvariouslengths andparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.5 Thevaluesof∥ 12 Φ 12 − 12 Φ 12 ∥ 2 /∥ 12 Φ 12 ∥ 2 for various lengths and parameters. . . . 74 8.6 Thevaluesof∥Π (4) ( 12 Φ 12 − 12 Φ 12 )∥ 2 /∥Π (4) 12 Φ 12 ∥ 2 forvariouslengths andparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8.7 Conformalspectruminthesector 11 for= 1/2. . . . . . . . . . . . . . . . . . . . . . . . . 76 8.8 Conformalspectruminthesector 2, 2 for= 1/2. The label corresponds to 2 =(−1) . 77 8.9 Conformalspectruminthesector 3, 2 for= 1/2. The label corresponds to 2 = e 2i/3 . 77 9.1 Amplitudes of the correlation function S corresponding to selected fields within 21 , in finite size . The distance between the two points within each group is taken as =⌊/2⌋. Thelinesofthetablearelabeled,asinTable8.8,bytheindex 13 . Thesevaluesaregivento8 significantfiguresinGrans-Samuelsson et al. [13]. . . . . . . . . . . . . . . . . . . . . . . . . 82 9.2 Amplitudes of the correlation function S corresponding to selected fields within 21 , in finite size . The distance between the two points within each group is now chosen the smallest possible,= 1. These values are given to 8 significant figures in Grans-Samuelsson etal.[13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 9.3 Differentmeasurementsfor =⟨|⟩ in 0, ±2+ 11 + 21 . . . . . . . . . . . . . . . . . . . . 99 10.1 Jordan structure of the lowest 125 of 711 eigenvalues of 0 on = 10 sites, in the vacuum sectoratmomentum0. isthealgebraicmultiplicityoftheeigenvalueonline. = Í =1 is the running dimension. In the “Jordan structure” column,× means rank- Jordan blocksappearforthateigenvalue,and≡ 1×.(,)Φmeansalevel-(,)descendantof Φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 10.2 Testsof⟨| † |⟩= 0 and⟨| † |⟩= 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 10.3 Testsof⟨| 1 −1 |⟩= 0 and⟨| 1 −1 |⟩= 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 10.4 Thenorm∥Π () (−)∥ 2 forvariousprojectorranks andsystemlengths. Inthistable “0*”meansanumberthatislessthanabout2×10 −6 . Iestimatetheuncertaintyinthenonzero numbers to be about 10 −6 . “—” means that the projector of a given rank at length is ill-definedduetodegeneracies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 10.5 The norm∥Π () (−)∥ 2 /∥Π () ∥ 2 for various projector ranks and system lengths. Inthistable“*”meansahighlyvariablenumberoforder1. Theyarelikelytheresultofa0/0 since the corresponding values in Table 10.4 are so small. “0*” means a number that is less than 10 −6 . I estimate the uncertainty in the nonzero numbers to be about 10 −6 . “—” means thattheprojectorofagivenrank at length is ill-defined due to degeneracies. . . . . . . 105 vii 10.6 The norm∥Π () (−)∥ 2 for various projector ranks and system lengths. In this table “0*”meansanumberthatislessthan10 −5 . Iestimatetheuncertaintyinthenonzeronumbers to be about 10 −4 . “—” means that the projector of a given rank at length is ill-defined duetodegeneracies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 10.7 The norm∥Π () (−)∥ 2 /∥Π () ∥ 2 for various projector ranks and system lengths . In this table “0*” means a number that is less than 10 −6 . I estimate the uncertainty in the nonzero numbers to be about10 −5 . “—” means that the projector of a given rank at length isill-definedduetodegeneracies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 11.1 Thedomainsinwhichthevaluesofthegivenparameterslie. iRmeanspurelyimaginary.ℂ ℝ meansthat10 < Re/Im < 1000,ℂ ℝ ∗ meansRe/Im≥ 1000,ℂ iℝ means10 < Im/Re < 1000,andℂ iℝ ∗ meansIm/Re≥ 1000. InnocasedidIobservetherealandimaginaryparts tobethesameorderofmagnitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 11.2 Parametersintheguess|˜ 1 /˜ 2 |= 0 − 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 11.3 Parametersintheguess|/|= 2 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 11.4 Parametersintheguess|/|= 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 11.5 Bestfitparametersfor ∥Φ 21 ∥ 2 =|| − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 D.1 Numericalvaluesof|⟨| 1 ⟩| inthe = 0 sector with = 0. . . . . . . . . . . . . . . . . . . 153 D.2 Numericalvaluesof|⟨| 1 ⟩| inthe = 0 sector with =−2/(+1). . . . . . . . . . . . . 153 F.1 Thevaluesof∥ −1 Φ 11 ∥ 2 forvarious lengths and parameters. . . . . . . . . . . . . . . . 159 F.2 Thevaluesof∥Π (2) −1 Φ 11 ∥ 2 forvarious lengths and parameters. . . . . . . . . . . . . . 159 F.3 Thevaluesof∥ 12 Φ 12 ∥ 2 forvarious lengths and parameters. . . . . . . . . . . . . . . . 160 F.4 Thevaluesof∥Π (4) 12 Φ 12 ∥ 2 forvarious lengths and parameters. . . . . . . . . . . . . . 160 F.5 Jordan structure of the lowest 155 of 3991 eigenvalues of 0 on = 12 sites, in the vacuum sectoratmomentum0. isthealgebraicmultiplicityoftheeigenvalueonline. = Í =1 is the running dimension. In the “Jordan structure” column,× means rank- Jordan blocksappearforthateigenvalue,and≡ 1×.(,)Φmeansalevel-(,)descendantof Φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 F.6 Jordan structure of the lowest 131 of 23391 eigenvalues of 0 on = 14 sites, in the vacuum sectoratmomentum0. isthealgebraicmultiplicityoftheeigenvalueonline. = Í =1 is the running dimension. In the “Jordan structure” column,× means rank- Jordan blocksappearforthateigenvalue,and≡ 1×.(,)Φmeansalevel-(,)descendantof Φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 F.7 Jordanstructureofthelowest119of143073eigenvaluesof 0 on = 16sites,inthevacuum sectoratmomentum0. isthealgebraicmultiplicityoftheeigenvalueonline. = Í =1 is the running dimension. In the “Jordan structure” column,× means rank- Jordan blocksappearforthateigenvalue,and≡ 1×.(,)Φmeansalevel-(,)descendantof Φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 F.8 The norm∥Π () (−)∥ 2 for various projector ranks and system lengths, in the loop model representation. In this table “0*” means a number that is less than about 2×10 −8 . I estimatetheuncertaintyinthenonzero numbers to be about10 −7 . . . . . . . . . . . . . . . . 164 F.9 The norm∥Π () (−)∥ 2 /∥Π () ∥ 2 for various projector ranks and system lengths, in the loop model representation. In this table “*” means a highly variable number of order 1. Theyarelikelytheresultofa0/0sincethecorrespondingvaluesinTableF.8aresosmall. “0*”meansanumberthatislessthan10 −6 . Iestimatetheuncertaintyinthenonzeronumbers tobeabout10 −6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 F.10 The norm∥Π () (−)∥ 2 for various projector ranks and system lengths, in the loop modelrepresentation. Inthistable“0*”meansanumberthatislessthan10 −5 . Iestimatethe uncertaintyinthenonzeronumbers to be about10 −4 . . . . . . . . . . . . . . . . . . . . . . . 166 viii F.11 Thenorm∥Π () (−)∥ 2 /∥Π () ∥ 2 forvariousprojectorranks andsystemlengths,in the loop model representation. In this table “0*” means a number that is less than 10 −6 . I estimatetheuncertaintyinthenonzero numbers to be about10 −5 . . . . . . . . . . . . . . . . 167 ix ListofFigures 3.1 TheHalleffect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 ThequantumHalleffect. FromTong [37]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Aconfigurationillustratingbondpercolation. FromDiFrancesco,Mathieu,andSénéchal[33]. 26 3.4 Percolationofwaterthroughanetworkoffirmlypacked,finelygroundcoffee. PhotobyScott Schiller,usedunderCCBY2.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 8.1 Comparison of lattice results using projectors of different of rank, illustrating the concept of scaling-weak convergence for = 1. The horizontal axis is 1/. The vertical axis is ∥Π () ( −1 Φ 11 − −1 Φ 11 )∥ 2 /∥Π () −1 Φ 11 ∥ 2 . The tags on the graphs indicate the rank of the projectorΠ () . The dotted lines are fourth-order polynomial fits (in 1/) to the five leftmost datapoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.2 Comparison of lattice results using projectors of different of rank, illustrating the concept of scaling-weak convergence for = 1. The horizontal axis is 1/. The vertical axis is ∥Π () ( 12 Φ 12 − 12 Φ 12 )∥ 2 /∥Π () 12 Φ 12 ∥ 2 . The tags on the graphs indicate the rank of the projectorΠ () . The dotted lines are third-order polynomial fits (in 1/) to the four leftmost datapoints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 9.1 Ratio 1 / 2 between the amplitudes of the two singlet fields (see Table 9.2), corresponding to the lines with 13 = 24 and 13 = 35 (see Table 8.8), plotted against 1/. The curve is a second-orderpolynomialfittothelast three data points. . . . . . . . . . . . . . . . . . . . . 83 9.2 The formation of a Jordan block between two singlets in 11 . The legend indicates the value of = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 9.3 Themeasurementof (1) 11 between and. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 9.4 Themeasurementof (2) 11 between and. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 9.5 Thetwomeasurementsof 11 compared with the expected value. . . . . . . . . . . . . . . . 92 9.6 The formation of a Jordan block between two singlets in 21 . The legend indicates the value of = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 9.7 The measurementof (1) 12 between and. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 9.8 Themeasurementof (2) 12 between and. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9.9 Thetwomeasurementsof 12 compared with the expected value. . . . . . . . . . . . . . . . 94 11.1 Plotoftheexponent 1,s asafunction of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 11.2 Plotoftheexponent 2,s asafunction of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 11.3 Plotoftheexponent s asafunction of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 11.4 Plotsoftheexponents 1,s , 2,s ,and s as functions of. . . . . . . . . . . . . . . . . . . . . . 118 11.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 11.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 11.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 11.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 11.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 11.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 11.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 x 11.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 11.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 11.14Theratio|˜ 1 /˜ 2 | closeto= 0,along with curves from Table 11.2. . . . . . . . . . . . . . . . 125 11.15The ratio|/| close to = 0, along with curves from Table 11.3. For comparison is the straightline/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 11.16Theratio|/| closeto= 0,along with curves from Table 11.4. . . . . . . . . . . . . . . . . 126 11.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 11.18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 11.19MeasurementsoftheeigenvalueofΦ 21 ,givingthetotalconformaldimension. Theblackline istheexactvalue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 F.1 Plotoftheexponent 1 asafunction of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 F.2 Plotoftheexponent 2 asafunction of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 F.3 Plotoftheexponent asafunction of. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 F.4 Plotsoftheexponents 1 , 2 , as functions of. . . . . . . . . . . . . . . . . . . . . . . . . . 170 F.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 F.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 F.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 F.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 F.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 F.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 F.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 F.12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 F.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 xi Abstract Twonovelframeworksforhandlingmathematicalandphysicalproblemsareintroduced. Thefirst,emerging Jordanforms,generalizestheconceptoftheJordancanonicalform,awell-establishedtooloflinearalgebra. The second, dual Jordan quantum physics, generalizes the framework of quantum physics to one in which the hermiticity postulate is considerably relaxed. These frameworks are then used to resolve some long- outstanding problems in theoretical physics, coming from critical statistical models and conformal field theory. I describe these problems and the difficulties involved in finding satisfactory solutions, then show how the concepts of emerging Jordan forms and dual Jordan quantum physics are naturally suited to overcoming these difficulties. Although the applications of these frameworks in this work are limited in scope to rather specific problems, the frameworks themselves are completely general, and I describe ways in which they may be used in other areas of mathematics and physics. Several appendices close the work, whichincludeimprovementstoawidelyusedcomputationalalgorithmandcorrectionstosomepublished data. xii Introduction “Wewhocutmerestonesmustalwaysbeenvisioning cathedrals.”—medieval quarry workers’s creed [1, 2]. * * * Twonovelframeworksforhandlingmathematicalandphysicalproblemsareintroduced. Thefirst,emerging Jordanforms,generalizestheconceptoftheJordancanonicalform,awell-establishedtooloflinearalgebra. The second, dual Jordan quantum physics, generalizes the framework of quantum physics to one in which the hermiticity postulate is considerably relaxed. These frameworks are then used to resolve some long- outstanding problems in theoretical physics, coming from critical statistical models and conformal field theory. I describe these problems and the difficulties involved in finding satisfactory solutions, then show how the concepts of emerging Jordan forms and dual Jordan quantum physics are naturally suited to overcoming these difficulties. Although the applications of these frameworks in this work are limited in scope to rather specific problems, the frameworks themselves are completely general, and I describe ways in which they may be used in other areas of mathematics and physics. Several appendices close the work, whichincludeimprovementstoawidelyusedcomputationalalgorithmandcorrectionstosomepublished data. * * * Theresultsdiscussedinthisworkbeganfromaprojectwhosepurposewastogainunderstandingofthe action of conformal symmetry on lattice models, and to use this understanding to determine multi-point correlation functions in a variety of systems of physical interest. As will be evident, a thorough study of these problems requires a mixture of new techniques that draw inspiration from ideas in mathematics, physics,and numericalanalysis,manyofwhich came about while carrying out this undertaking. The topic of conformal field theory has grown manifold since its beginnings in the early 1990s, and is relevant now to condensed matter physics as well as high-energy physics and mathematics, in two dimensions as well as higher numbers of dimensions. Despite this impressive progress, some of the very questionsthatpromptedthetopicinthefirstplaceremainunsolved. Thereisafundamentalreasonforthis failure: most of the successes of conformal invariance so far have been based on “top-down” approaches, where results were largely guessed based on symmetry considerations, and then checked to be correct. Such top-down approaches work well in the context of ordinary quantum field theories, but fail for many problemsofphysicalinterestincondensedmatterphysics,likepolymers,percolation,ordisorderedsystems. Intechnicalterms,thisisbecausethefieldtheoriesdescribinglong-distancepropertiesofstatisticalproblems at criticality are not necessarily unitary when considered as one-dimensional quantum field theories. This technicalproblemtranslatesintothefactthattheactionoftheconformalsymmetry—thatis,oftheVirasoro algebra—is not necessarily simple: modules can appear which are not fully reducible, or nontrivial zero- norm-squarestatescanoccurthatareinfactphysical. Theassociatedmathematicalproblemsaredaunting when considered abstractly: the representation theory of the Virasoro when the constraint of unitarity is removedistechnicallywild,andhopeforprogress using a top-down approach is very limited. The goal then is to solve the difficulty via a “bottom-up” approach, by moving from the usual abstract 1 symmetryapproachtoamuchmoreconcreteonebystudyingdirectlytheactionoftheconformalsymmetry on lattice models. Using a mix of analytical and numerical techniques, one can determine what kinds of modules of the Virasoro algebra are relevant to, for instance, the percolation problem, and what kinds of degeneracy equations physical observables (such as the probability to belong to a certain cluster) may satisfy. This knowledge will be combined with the direct determination of the spectrum of percolation in the and channelsoffour-pointcorrelationfunctions,usingrepresentationtheoryoftheassociatedlattice algebra, the Temperley–Lieb algebra. Finally, the information thus gained will be used as starting points forabootstrapapproach,where,byimplementingcrossingsymmetryinparticular,four-point(andhigher) correlationfunctionswillbebuiltassums over conformal blocks. Whilethe projectis ambitious,it willbe based on significant achievements of the last few years. On the onehand,theideaofstudyingtheconformalsymmetryonthelatticebybuildingdiscretizedversionsofthe Virasorogenerators,whichwaspioneeredbyKooandSaleur[3],hasattractedmuchattentionrecently,both inthemathematicscommunity[4],andinthequantuminformationcommunity[5]. Excitingnewtechnical possibilities (based in part on the matrix product states technique, as well as the form-factors technique) have appeared that should be exploited very quickly. On the other hand, significant progress has also occurred both in the abstract bootstrap approach [6] and in the determination of spectra of geometrical models[7]. Thetopicseemsripeforabreakthrough, which, it is hoped, this work will describe. Apart from fundamental progress in our understanding of geometrical problems, the completion of this program would also constitute a significant step forward in the solution of nonunitary conformal field theories such as those occurring in the description of critical points in different universality classes of topological insulators (like the plateaux transition in the integer quantum Hall effect). In a different direction, the issue of discretizing the Virasoro algebra is of importance in quantum computing, where manyresearchersaretryingtoimaginequantumcomputersandalgorithmsabletosimulatequantumfield theories [8]. Finally, any studyof quantum field theory based on lattice discretizations is of high interest to mathematicians,sincequantumfieldsandpath integrals prove so difficult to define rigorously. Over the course of carrying out this program, on the lattice side, my work and ideas, and those of manyofmycollaborators,seemedtoleadinexorablytothetwoframeworkspresentedhere,particularlythe emergingJordanblock. Atfirst,theemergingJordanblocksimplyservedapurposetoallowustocompute numbers where we otherwise could not. Unexpectedly, the results of some of these measurements were sosuccessful(inFigure9.5,forinstance,themeasured (1) 11 andthetheoretical 11 nearlyoverlapexactlyfor < 1/2), that we wondered whether there was something more to our method. Considerable effort was expended in figuring out some of the more important properties of and structures within these emerging Jordan blocks, and along the way we found counterexamples to some of our earlier conjectures, leading to theirrefinementandfurtherinsight. Wehaveassembledourobservationsintothebeginningsofacoherent mathematical structure that we genuinely believe to be of general practical use, subject to the validity of Conjectures 1 and 2. And when we look back at our earlier attempts at our problems, we essentially see that the emerging Jordan block came looking for us. To me, this is what the interplay between physics and mathematicsisabout. Outlineofthiswork Thisworkisdividedintothreeprimaryparts,correspondingtobackgroundinformation,noveltheoretical frameworks,andfinallyapplicationsthatput together the first two parts. PartIbeginswithtwochaptersonmathematics. Thefirstoftheseconsistsofacollectionofconceptsfrom mathematics,allwell-establishedinthemathematical canonandpartof thecoreeducationfor mathemati- cians. They are written, however, using the conventions most often encountered in physics, which include departuresinterminologyandnotations. AsecondchapterontherepresentationtheoryofTemperley–Lieb algebras consists of more recent and specialized results, but which have a bearing on the physical models studiedinChapter4andPartIII.Thetwochaptersonphysicsfollowthesamescheme,withthesecondone pertaining to the study of conformal field theory using discrete lattice models. Finally, Chapter 5 discusses some aspects of matrix computations. Ordinarily used as a black-box tool, sophisticated numerical algo- rithms and software packages typically produce accurate and reliable results without needing fine-tuning from the user. However, overcoming some of the shortcomings of these computational tools produced 2 insightsthathelpedtodevelopapproachesforhandlingsomeofthephysicsproblems. Ithereforedescribe algorithms for the most frequently used matrix computations in this work, as they are an integral part of a thorough study of the models contained herein, and to provide context for an improvement described in AppendixA. Manytreatiseswithasubstantialsegmentdevotedtobackgroundmaterialclaimtoincludethesedetails so that the work is essentially self-contained, assuming sufficient mathematical and logical sophistication onthepartofthereader. Ihaveyettoseethisexecutedinaconvincingway,exceptpossiblythefirstvolume of Bourbaki’s Elements of Mathematics series [9, 10] and Whitehead and Russell’s Principia Mathematica [11, 12]. Let me then disclaim the self-contained nature of this work, and state instead that the purposes of the background chapters of Part I are to provide context for the new techniques developed in Part II and their applications inPartIII,andtoestablishnotations and terminology used throughout. New methods to handling the mathematical and physical questions studied are described in Part II. While the development of these methods took place in the context of studying the problems of Chapters 2 and 4 and Part III, they are easily abstracted and apply very generally. The purpose of this part is to present them as self-contained and coherent frameworks for studying problems within their domains of application. Chapter6considersthelimitofasequenceofdiagonalizablelinearoperators(orthelimitofa [continuous]functionwhosecodomainisaspaceoflinearoperators)andthequestionofwhetherthelimitis diagonalizable. Whiletheanswerturnsouttobeverysimple,mycollaboratorsandIweresurprisedtofind that the approach described therein had not been tried before, as far as we could find. Chapter 7 develops a systematic method to compute eigenspace projection operators for operators that are not normal. In the standard formulation of quantum physics the Hamiltonian operator is postulated to be hermitian (with respect to a positive-definite inner product), thus normal, and its eigenstates, assumed to form a complete set, furnish a convenient basis in terms of which to express a general state. The components of a general state in this basis are easily computed—formally, one uses the eigenspace projection operators constructed fromtheeigenstates. Thischapterthereforerepresentstheanalogueofthisbasicprocedureforaframework ofquantumphysicsinwhichtheassumption of hermiticity is lifted. Assembling all of these admittedly elaborate pieces, I finally turn to new results and implications in physics in Part III. I demonstrate that the sophistication of these new methods pays off when insight is gained in long-outstanding problems whose progress has stalled in recent years. I also hope that the readerseesthatthesemethodswerenecessary,andessentiallyunavoidableinanypathtowardsacomplete solution—they are tailor-made to overcome the difficulties encountered in previous attempts at solution. Chapter8describeshowindecomposablestructures,andhenceJordanblocks,ariseinthecontinuumlimit oftheloopformulationofthePottsmodel,andverifiessomesignaturesofthatindecomposability. Chapter 9thendescribesattemptstoobservethisJordanblockstructuredirectly—firstthroughcorrelationfunctions and OPEs, then via emerging Jordan blocks. Chapter 10 describes the application of the new projection operators to the ℓ(2|1) spin chain. While many problems remain unresolved, the results presented there demonstratetheinternalconsistencyoftheconstructionofthedualJordanprojectionoperators. InChapter 11, I describe how some lattice quantities do not directly correspond to a clear continuum analogue, and myattempts atdisentanglingthem. Six appendices close the work. Appendix A describes what I call “dynamic multiplicity adjustment,” a simple enhancement to the widely used implicitly restarted Arnoldi method that drastically improves convergence for highly degenerate problems, like those encountered in this work. Appendix B contains efficient code in Mathematica that the reader may use as a starting point to replicate the numerical results describedhere. AppendixCdescribesanalgorithmforverifyingsubquotientstructuresinindecomposable modules,whichhasnotyetbeenusedinthestudyoftheproblemsdescribedinthiswork. AppendixDlists corrections to the data in the first two tables of Koo and Saleur [3], so that they may serve as a useful check forthoseattemptingtoreplicatetheresultsdescribedthere. Finally,AppendicesEandFcontainadditional calculations,tables,andfiguresthatsupplement the main text, but are not otherwise essential. Parts of the material in Chapters 2, 4, 8, and 9 are adapted from Grans-Samuelsson et al. [13, 14] (in which “et al.” includes myself, and where the latter version contains another appendix added post- publication). However, one important change that I apply uniformly compared to the published article is that for consistency with the remainder of the present work, I retain the notations(·,·)and(·|·)for the standard complex scalar product, and⟨·,·⟩and⟨·|·⟩for the conformal (loop) scalar product. I also modify other notations based on uniformity and personal preference. Sorry for the possible confusion, but this is 3 myterritory. SomeofthedescriptionoftherepresentationtheoryoftheaffineTemperley–Liebalgebraand the two Jones–Temperley–Lieb algebras comes from [15]. I have likewise rewritten much of the exposition, madeeditorialchangesandcorrections,and changed notations. All errors are mine. Some of the research leading to the findings in Chapter 8 and its subsequent publication [13, 14] was supportedbyaChateaubriandFellowshipofferedbytheOfficeforScienceandTechnologyoftheEmbassy of France in the United States. The Center for Advanced Research Computing (CARC) at the University of Southern California (https://carc.usc.edu) provided computing resources that have contributed to the researchresultsreportedwithinthiswork. 4 PartI Background 5 Chapter1 Mathematics: basicconcepts 1.1 Innerproducts I assume the reader is already familiar with vector spaces and inner products. This purpose of this section istoestablishtheconventionsusedfortheremainderofthework,asthereareanumberofcompetingones usedthroughoutmathematicsandphysics. Definition 1. Let be a vector space over, where =ℝ orℂ. An inner product is a map :×→ such thatforall,, ∈ and ∈ , (,+)= (,)+ (,), (1.1a) (,)= (,), and (1.1b) (,)= (,) ∗ . (1.1c) If,additionally,foreverynonzero∈, (,) > 0, (1.1d) iscalledapositive-definiteinnerproduct . Finally, aninner product issaid to benon-degenerateif (,)= 0 forall∈ impliesthat= 0. NotethatIfollowthephysicsconventionoflinearityinthesecondargument,andthataninnerproduct is not explicitly required to satisfy (1.1d). Nevertheless, I will occasionally use the term “indefinite” inner producttocallattentiontothefactthat(1.1d)isviolated. Thesquarednormof∈ is (,). Byproperty (1.1c), this number is real, though not necessarily positive. Occasionally, in the physics literature, “norm” is used to mean the squared norm (,), and in particular, “negative norm” means (,) < 0 (used, for instance, in the discussion of ghosts in string theory [16, 17]). But to avoid such imprecision, (,) and equivalent notations in this work will always be called “squared norm,” “norm squared,” or some unambiguousvariant. Alternativenotationsfor (,),whichIwilluseexclusivelyhereafter,are(,),(|),⟨,⟩,and⟨|⟩. Mathematicians may be willing to forgive the use of “inner product” for the general indefinite inner product, which they would call an “hermitian form.” But “squared norm” for a possibly negative quantity isundoubtedlyamoredifficultpropositiontoaccept. Ioffersomeappeasementbyhavingused todenote afield,althoughIcannotdiscussfieldextensions here (except to note that ℂis a field extension of ℝ). 1.2 Thespectraltheoremfornormaloperators Throughoutthiswork,“operator”always means “linear operator.” Given a non-degenerate inner product, to every linear operator there corresponds its adjoint (or hermitianadjoint,orhermitianconjugate) † , such that for any,∈, ( † ,)=(,). (1.2) 6 It is necessary that the inner product be non-degenerate for † to be well-defined and unique [18, p. 533]. Thereaderisfamiliarwiththestandardinnerproductforafinite-dimensionalvectorspace(insomebasis), (,) = Í ∗ , for which the hermitian adjoint is the conjugate-transpose, † = ∗ . A linear operator suchthat= † iscalledself-adjoint,orhermitian. Theconceptofadjointmaybeextendedtovectors. For a vector, its adjoint is an element of the dual space, the linear functional † such that † =(,) for any ∈. A standard theorem of linear algebra states that every hermitian operator is unitarily diagonalizable, withrealeigenvalues. Thatis,thereexists a basis{ } such that = , (1.3a) ( , )= , (1.3b) for a set of eigenvalues{ }⊂ℝ. The machinery of the standard formulation of quantum mechanics rests upon this result (see Section 3.1). Nevertheless, its proof relies on property (1.1d), and fails in the more generalcontext. A number of physical problems involve indefinite inner products which arise naturally (i.e., the correct physicsobtainswhencalculationsarecarriedoutusingtheseinnerproducts),whoseHamiltonianoperators arehermitianwithrespecttotheseinnerproducts,butcannotbeunitarilydiagonalized. Iwillthusreserve theterm“hermitian”tomeanhermitianwithrespecttosomepositive-definiteinnerproducthenceforth. A non-hermitianoperatoristhereforenothermitian with respect to any positive-definite inner product. Igivetwostatementsofthespectraltheoremasusedinthiswork. Theyarestatedfornormaloperators , which satisfy[, † ] = 0, and therefore subsume hermitian operators. Its analogue for non-normal finite-dimensionaloperatorswithrealspectra is treated in Chapter 7. Theorem1(Spectraltheoremintermsofeigenvectors). Ifisanormaloperatoronafinite-dimensionalcomplex inner product space, where the inner product is positive-definite, then has an orthonormal basis consisting of eigenvectorsof. Therefore,anormaloperator on such a space is diagonalizable. Theorem2(Spectraltheoremintermsofprojectionoperators). Letbeanormaloperatoronafinite-dimensional complexinnerproductspace,wheretheinnerproductispositive-definite,and { 1 ,..., }itsdistincteigenvalues. Then = Ê =1 , (1.4) where istheeigenspaceassociatedtoeigenvalue . IfΠ is the hermitian projection operator onto , then Õ =1 Π =, (1.5a) Π Π = Π , and (1.5b) = Õ =1 Π . (1.5c) WhileIfocusexclusivelyondiscretespectra,anoperator,necessarilyinfinite-dimensional,withadiscrete andcontinuousspectrumcanbetreatedcompactlyusingmeasure-theoreticideas[19]. Forexample,toeach self-adjointoperatorΩ,thereexistsaunique spectral family{ } such that Ω= ∫ ∞ −∞ d . (1.6) Thespectralfamily{ } isaone-parameter family of orthogonal projection operators such that 1. thefamilyismonotonicallyincreasing: ′ = ′ = when ′ ≥ ; 2. thefamilyisright-continuous: lim →0 + + = ; 3. lim →−∞ = 0 andlim →∞ =. Theintegralcanberestrictedtotherealline because the self-adjoint property implies a real spectrum. 7 1.3 TheJordancanonicalform Ingeneral,notalloperatorsarediagonalizable,evenoverthefieldofcomplexnumbers. Theclosestonecan get for such operators is the following theorem, whose statement is adapted from Hoffman and Kunze [20, p.222]. Theorem3. Letbealinearoperatoronthefinite-dimensionalcomplexvectorspace . Thenthereisadiagonalizable operator on andanilpotentoperator on suchthat=+ and =. Thediagonalizableoperator andthenilpotentoperator areuniquelydetermined and each of them is a polynomial in. The simplest explicit matrix representation of the operator decomposition in Theorem 3 is the Jordan canonicalform[Id.atSection7.3](also“Jordan normal form”). Theorem 4 (Jordan canonical form). Let be a linear operator on the finite-dimensional complex vector space . Thereexistsa basis of suchthatthematrix of in this basis is block diagonal: = © « 1 2 . . . ª ® ® ® ® ¬ , (1.7a) where 1 ,..., arematrices. Each isofthe form = © « 1 2 . . . ª ® ® ® ® ¬ , (1.7b) whereeach isanelementaryJordanblockwith eigenvalue : = © « 1 1 . . . . . . 1 ª ® ® ® ® ® ¬ . (1.7c) By the primary decomposition theorem [Id. at p. 220], the vector space = É =1 is a direct sum of invariant subspaces, where is the null space of(− ) , being the multiplicity of as a root of the minimalpolynomialfor. Furthermore, is the rank of the largest elementary Jordan block in . TheJordancanonicalformsubsumesthediagonalizationofdiagonalizableoperators. Whenanoperator is diagonalizable, all of the elementary Jordan blocks are 1× 1 matrices, with no space for off-diagonal elements, and the Jordan form is a diagonal matrix. If all of the elementary Jordan blocks have rank 1, then the Jordan canonical form will be called trivial. The process of finding the Jordan form of an operator—its elementary Jordan blocks and a basis in which the operator has such a form—will still be called diagonalization,forbrevity. The general expression given for an elementary Jordan block has ones above the main diagonal. This means that to each elementary Jordan block , there is a proper eigenvector 1 such that 1 = 1 andasetofgeneralizedeigenvectors,{ |2≤ ≤ },where isthedimensionorrankof ,suchthat = + ,−1 . Bydefining 0 = 0, the preceding equation can be used with1≤ ≤ . By a trivial rearrangement of the basis vectors, the ones in the elementary Jordan block can be moved belowthemaindiagonal,whichleadstoanequivalentstatementoftheJordancanonicalform. Wewillhave use for both of these versions, so when the distinction is important, I will call a matrix in Jordan canonical form with elementary Jordan blocks of the form in Eq. (1.7c) upper Jordan and lower Jordan when the ones arebelowthemaindiagonal—thetranspose of Eq. (1.7c). 8 The fact that the off-diagonal elements are equal to 1 in particular is not terribly important. Each off- diagonalelementcanbechangedtoanarbitrarynonzeroscalarbyrescalingtheelementsofthebasis. With thisinmind,anymatrixoftheformdescribedbyEq.(1.7)oritstranspose,wheretheoff-diagonalelements are not necessarily equal to1, will be considered to be in Jordan form. I will generically refer to the nonzero off-diagonalelementsas couplingsorJordancouplings,andIwillreservetheterm“Jordancanonicalform”for thespecialcaseinwhichalloftheJordancouplings are equal to1. Onemorepieceofterminology: thecollectionofeigenvectors(properandgeneralized)thatformabasis for an elementary Jordan block (of rank) will be called a tower (of height), especially when the rank of theJordanblockis3orgreater. Althoughthisterminologyisnotstandardinthiscontext,Ihavefoundthat itgivestherightintuitionwhenthinkingabout the action of an operator with a nontrivial Jordan form. WhileIfocusalmostexclusivelyonoperatorsactingonfinite-dimensionalvectorspaces,someinteresting subtletiesariseintheinfinite-dimensionalcase. Forinstance,theeigenvaluesofafinite-dimensionaloperator are found on the main diagonal of its Jordan form. In the infinite-dimensional case, this may not be true. The most familiar examples of infinite Jordan towers to physicists, though not usually thought of as such, comefromthecreationandannihilationoperators † andfortheharmonicoscillator. Inthebasisofenergy eigenstates|⟩ =( † ) / √ !|0⟩, is upper Jordan and † is lower Jordan, both with only zeros appearing on the main diagonal. The annihilation operator has infinitely many eigenstates parametrized by ∈ℂ, the coherent states|⟩= e † |0⟩, while the creation operator has no eigenstates (here, I am allowing infinite linear combinations). This may be contrasted with, say, the raising and lowering operators + and − = † + in finite-dimensional highest-weight representations of (2). Another example is the differentiation map actingonthespaceofpolynomials[](with afieldofcharacteristiczero),which,inthebasis { /!},is representedbyaninfiniteelementaryJordan block with eigenvalue 0. 1.4 Representationtheoryofalgebras Representation theory is a huge enterprise of its own [21, 22], and so I will not even attempt to give an introductiontothissubject. Ionlyusethe present section to state some definitions as I use them. Definition 2. Let be a field. A -algebra is a ring with identity such that has a -vector space structure compatible with the multiplication operation of the ring. A left-module is a pair(,·)consisting of a -vector space and a multiplication·: ×→ such that·is compatible with the vector space operations in (left andrightdistributivity,associativity,unitality). A representationof is a left-module. Definition 3. Let be an algebraically closed field, let be a-algebra, and let be an-module. is simple or irreducible if is nonzero and the only submodules of are the zero module and . is semisimple if is a direct sum of simple-modules. is indecomposable if is nonzero, is not simple, and has no module direct sum decomposition 1 ⊕ 2 in terms of nonzero-modules. The -algebra is semisimple if every -moduleissemisimple. NotethatIhaveadoptedtheconventionthatanindecomposablemoduleisnotirreducible(notsimple). In standard terminology, an irreducible or simple module is indeed indecomposable. However, because so much discussion centers on modules that are indecomposable but not irreducible, I have specifically excludeditfrommydefinitiontoavoidoverusingthephrase“indecomposablebutnotirreducible.” (Indeed, Icountover25usesoftheterminthiswork.) “-algebra”willbeshortenedto“algebra”(intheremainderof this work, =ℝ orℂ), “-module” will be shortened to “module” when the algebra is clear from context, and“module”willalwaysmean“leftmodule.” Otherwise,thereshouldbenothingcontroversialaboutthe definitions,adaptedfromAssem,Skowronski, and Simson [22, Chapter 1]. Unfortunately, “representation” is such a ubiquitous word that many instances of it in this work do not refer to the object of Definition 2, but also the correspondence between mathematical symbols and abstract orconcreteobjects. 9 Chapter2 Mathematics: representationtheoryof variousTemperley–Liebalgebras 2.1 GeneratorsandrelationsforTemperley–Liebalgebras Intermsofgeneratorsandrelations,variousalgebrastowhichthename“Temperley–Lieb”(TL)isassociated beginwith 2 = , (2.1a) ±1 = , and (2.1b) = , (|−| > 1) (2.1c) where the indices take the values 1,...,−1. These relations are typically illustrated using diagrams for the generators. On two horizontal rows of points each, connect the neighboring pairs at and+1, and verticallymatchallotherpairs: = 1 ··· +1 ··· (2.2) When two generators are multiplied, the corresponding diagram is obtain following certain rules: stack the diagrams, replace closed loops with a numerical factor , and straighten out the curves that connect vertically. The ambiguity in the last rule can be avoided by defining elements of the algebra as equivalence classesofsuchdiagramsunderisotopy,butwewillnotusethisdescription. Inanydiagramofthealgebra, the lines that connect from top to bottom (rather than between two sites in the same horizontal row) are calledthrough-lines,andthesitestowhichtheyareconnectedarecalledfree. Theidentityelement1consists ofadiagramwithall verticalpairsofsitesconnectedbystraightthrough-lines. Itcanbeshownformally that this algebra of diagrams is isomorphic to the abstract Temperley–Lieb algebra defined by generators andrelations[23]. Asimplegeneralizationistoadjoinagenerator andsite+1,andidentifysite+1withsite1,thereby obtaining a periodic Temperley–Lieb algebra. The sites in the top and bottom rows can be thought of as beingplacedontheinnerandouterboundariesofanannulus,evenlyspaced,andmultiplicationconsistsof nesting two annuli after rescaling one so that its inner boundary matches the outer boundary of the other. Inbothcasestheaccessiblediagramsconsistofplanarpairingsofpointsusingcurvesthatdonotintersect. Asbefore,thecurvesthatjoinpointsontheinnerboundarytopointsontheouterboundaryoftheannulus arecalledthrough-lines. With periodic boundary conditions, it is natural to introduce a translation . In diagrammatic terms, usingthelabelsoftherighthandsideofEq.(2.2),consistsofthediagramwheresite ofthebottomrowis 10 connectedtosite+1(mod )ofthetoprowfor= 1,...,. Thefollowingadditionaldefiningrelations arethenobeyeddiagrammatically: −1 = +1 , and (2.3a) 2 −1 = 1 ··· −1 . (2.3b) Moreover, ± arecentralelements. Thealgebrageneratedbythegenerators and ±1 togetherwiththese relationsisusuallycalledtheaffineTemperley–Lieb algebra (). One may restrict to even powers of. This requires = 2 to be an even number, which defines , and in the remainder of this work we consider only this situation (i.e., even). Formally, the elements ±2 are adjoinedtothealgebrainsteadof ±1 ,and Eq. (2.3a) is replaced by 2 −2 = +2 . (2.4) ForboththeaffineTemperley–Liebalgebraandthepresentcontext,itispossiblefornoncontractibleloopsto begeneratedthatcircletheannulus,andthatcannotbeshrunktoapointviaahomotopywithoutcolliding with the inner boundary of the annulus. In the present setting, we assign these noncontractible loops the same weight—they are replaced by a factor of . Finally, we take a quotient by the ideal generated by − 1. In terms of annular diagrams, one annular diagram becomes equivalent to another obtained from the first by rotating the inner boundary through a full rotation, with the sites and attached curves followingalongwithit. Theresultingobjectwiththemodificationsdescribedinthisparagraphiscalledthe augmentedJones–Temperley–Liebalgebra (),anditisfinite-dimensional[15]. ()isslightlylarger than the Jones–Temperley–Lieb algebra (), introduced by Read and Saleur [24] and further studied by Gainutdinov, Read, and Saleur [25]. The difference is entirely in the ideal with zero through-lines—i.e., the annular diagrams in which points of the outer boundary are paired together and points of the inner boundaryarepairedtogether(again,usingcurvesthatdonotintersect),andnopointoftheinnerboundary ispairedwithapointoftheouterboundary. In (),allsuchdiagramsareallowed. In (),only diagrams that can be drawn without crossing the periodic “boundary” between both pairs of sites and 1 are allowed—in other words, the diagrams that can just as well be drawn on the (nonperiodic) rectangle. Thisideal(thesubalgebraof ()withzerothrough-lines)isalsoknownastheorientedJonesannular subalgebrain theBrauer algebra[26]. Formally, there is a covering homomorphism (surjection) of algebras : ()→ (), which acts nontrivially only in the zero through-lines subalgebra (of ()). Itsactionisbestunderstoodthroughanexample: : ↦→ (2.5) In Eq. (2.5) and what follows, dotted “framing” rectangles are understood to have their left and right boundaries identified, as if they were cut from an annulus. The homomorphism essentially (uniquely) redraws the curves that keep the same pairs matched, but without crossing the boundary of the framing rectangle. Thealgebra () hasdimension dim( ())= /2 2 + /2 Õ =1 /2− 2 , (2.6) while () hasdimension dim( ())= /2 − /2−1 2 + /2 Õ =1 /2− 2 , (2.7) 11 2.2 Cellularalgebras A unified framework for discussing the representation theory of Temperley–Lieb and other algebras is the concept of a cellular algebra. Cellular algebras were first introduced by Graham and Lehrer [27] in the courseoffindinga“systemicunderstandingofthenon-semisimple[specializations]ofHeckealgebrasand ofavarietyofotheralgebraswithgeometricconnections”—toincludesomeofthevariousTemperley–Lieb algebrasofSection2.1. Definition4 (GrahamandLehrer[27,Eq.(1.1)]). Let beacommutativeringwithidentity. Acellularalgebra over isanassociative(unital)algebra,together with cell datum(Λ,,,∗)where 1. Λ is a partially ordered set and for each ∈ Λ,() is a finite set (the set of “tableaux of type ”) such that : Ý ∈Λ ()×()→ isaninjective map with image an-basis of. 2. If ∈ Λ and ,∈ (), write (,) = ∈ . Then∗ is an -linear anti-involution of such that = . 3. If∈Λ and,∈() thenforanyelement∈ we have ≡ Õ ∈() (,) (mod (< )) (2.8) where (,)∈ is independent of and where (< ) is the -submodule of generated by{ | < ;,∈()}. This definition of cellular algebras appears as it was introduced by Graham and Lehrer [ Id.], with some notations exchanged to correspond to that of Gainutdinov et al. [15] and the rest of the present work. The algebras () and () arecellular algebras, but () is not. Aparticularsetofrepresentationsofagivencellularalgebraexistasanaturalconsequenceoftheaxioms. Definition 5 (Graham and Lehrer [27, Eq. (2.1)]). Let be a cellular algebra with cell datum(Λ,,,∗). For each ∈ Λ define the (left) -module () as follows: () is a free -module with basis{ |∈ ()} and -actiondefinedby = Õ ∈() (,) (∈,∈ ()) (2.9) where (,) is the element of defined in Definition 4. () is called the cell representation, or cell module of correspondingto∈Λ. Similarly, co-cell representations or co-cell modules f () are duals to the cell modules—the spaces of linearfunctionalson () withthenaturally induced action of the algebra. 2.3 Standardandco-standardmodules We specialize now to the various Temperley–Lieb algebras, beginning first with the affine Temperley–Lieb algebra (). Although it is not cellular, we are interested in some of finite-dimensional representations, whichturnouttobethecellmodulesof ()and (). Define ,aquantumgroupparameter,via =+ −1 . For the generic case where is not a root of unity, there are irreducible representations ,e i parametrizedbytwonumbers. (Thenumberofsites isassumedfixedanddoesnotappearinthenotation for the representations ,e i, although, strictly speaking, there is a different representation for each value of.) In terms of diagrams, the first is the number of through-lines 2, with = 1,...,; recall that = 2 is assumed even (Section 2.1, discussion surrounding Eq. (2.4)). Consider a particular diagram with 2 through-lines within this representation. Anticipating that the parameter will be involved in the set of 12 weights Λ, whenever the natural action of the algebra—the stacking or nesting of diagrams in Section 2.1—decreases the number of through-lines, we stipulate that the result is zero; i.e., whenever the action contractstwoormorefreesites. ComparewithDefinition5,whereallofthebasiselementsofacellmodule havethesameweight. Furthermore,foragivenvalueof,itispossible,usingtheactionofthealgebra,tocyclicallypermutethe freesites: thisgivesrisetotheintroductionofapseudomomentum,whichisparametrizedby. Whenever adiagramhas2 through-lineswindingcounterclockwisearoundtheannulus times,thisdiagrammaybe replaced by the same with the through-lines unwound, multiplied by a numerical factore i . Stated more simply, there is a phase e i/2 for each complete counterclockwise winding of a through-line; however, it is impossibletounwindasinglethrough-lineonitsownwithoutcollidingwithotherthrough-lines. Similarly, for 2 lines winding clockwise times, the numerical factor is e −i when they are unwound, or e −i/2 per through-linepercompletewinding. We will often use an equivalent representation, where the phase is evenly graduated over the course of unwinding through-lines, instead of abruptly applying it to each complete winding. Instead of a phase e ±i/2 when a through-line makes a complete winding, we instead apply a phase e ±i/2 for each site a through-linemovesrightorleft. Thisformulation preserves invariance under the translation operator. A slightly more convenient formulation of this representation, ,e i, can be obtained via the following consideration. Since the free sites are not allowed to be contracted, the pairwise connections between non-free sites on the inner boundary cannot be changed by the action of the algebra. This part of the diagrammatic information is thus extraneous and must be omitted to avoid overcounting the basis, and to ensure the representation is irreducible. It suffices to focus on the (isotopically distinct) upper halves of the affine diagrams, obtained by cutting the affine diagrams across its 2 through-lines, and discarding the bottom half. Each upper half is then called a link state, and for simplicity the “half” through-lines attached to the free sites on the outer boundary (or top boundary of the framing rectangle) are still called through- lines. The phase e ±i/2 is now attributed each time one of these through-lines moves through the periodic boundaryconditionoftheframingrectangleintherightwardorleftwarddirection,or,equivalently,e ±i/2 for each step right or left, as before. With these conventions, we will use this formulation to study the representations ,e i, with the Temperley–Lieb algebra action obtained by stacking the affine diagrams on topofthelinkstates. Thedimensionsofthesemodules ,e i aretheneasilyfoundbycountingthelinkstates. Theyaregiven by ˆ = /2+ . (2.10) Note that these dimensions do not depend on , but representations with different e i are not isomorphic. These cell modules ,e i are also known as standard modules. (Again, we have abused terminology by callingthemcellmodulesinthesenseofDefinition5eventhough ()isnotacellularalgebra—theyare, however,cellmodulesof ()and (). We will thus use “standard module” henceforth.) Thestandardmodules ,e i areirreducibleforgenericvaluesofand. However,degeneraciesappear wheneverthefollowing resonancecriterion is satisfied [28, 29]: ∃∈ℕ : e i = 2+2 . (2.11) The representation , 2+2 then becomes reducible, and contains a submodule isomorphic to +, 2. The quotient , 2+2/ +, 2 isgenericallyirreducible, with dimension ≡ ˆ − ˆ + . (2.12) Therearealsostandardrepresentationsfor= 0—i.e.,nothrough-lines. Thereisnopseudomomentum, but representations are still characterized by a second parameter, which now specifies the weight given to noncontractible loops, which are not possible for > 0. Parametrizing this weight as + −1 , the corresponding standard module is denoted 0, 2. This module is isomorphic to 0, −2, so we denote both henceforthby 0, ±2. Ifweidentify= e i/2 , the resonance criterion of Eq. (2.11) still applies. 13 It is natural to require that + −1 = , so that contractible and noncontractible loops get the same weight. Imposing this requirement leads to the module 0, ±2, which is reducible even for generic . Indeed, Eq. (2.11) is satisfied with = 0, = 1, and hence 0, ±2 contains a submodule isomorphic to 11 . Takingthequotient 0, ±2/ 11 leadstoasimple module for generic, denoted by 0, ±2. It has dimension 0 = ˆ 0 − ˆ 1 = /2 − /2+1 , (2.13) inagreementwithEqs.(2.10)and(2.12),so that we may extend the definitions of both to = 0. The difference between 0, ±2 and 0, ±2 is the analogous to the difference between () and (). In 0, ±2,curvesthatpairsitesinthelinkstatescancrosstheperiodic“boundary,”whilein 0, ±2 they cannot. (A curve pairing sites in 0, ±2 cannot cross the boundary more than once [on net]; otherwise the curve would intersect itself. The same is true for the subalgebra of () with zero through-lines.) There exists a projection mapping defining the quotient whose action on link states in 0, ±2 is analogous tothatofEq.(2.5)ondiagramsin (). Someconcreteexamplesoflinkstatesare, for = 4: (2.14) In 0, ±2,thelattertwodiagramsareidentified. Two examples of link states for = 6 are: (2.15) At this point I introduce the following convenient notation for link states, useful both as a compact notation and for computation (Appendix B). A link state is represented by a collection of ordered pairings andsingletons. Thesingletonsareoptional,andasingleton()representsathrough-lineatsite. Apairing () or(,) denotes a curve that starts from site and goes forward to in the rightward direction. When > such a curve starts from site, goes past the last site and crosses the boundary, returns periodically frombehindsite1,thenterminatesatsite. Forthesixlinkstatesexhibitedgraphically,theirrepresentations in my notation are(12)(3)(4),(1)(2)(34),(23)(41),(14)(23),(14)(23)(5)(6), and(1)(23)(45)(6). Since singletons areoptional,theymaywellbedenoted(12),(34),(23)(41),(14)(23),(14)(23),and(23)(45). Ofcourse, must be understood from context to avoid ambiguities, such as the two instances of(14)(23), but I will point out that exactly the same ambiguity arises in the representations of elements of the symmetric group by disjoint cycles when fixed points are omitted. And, as with , the order of the pairings and singletons is immaterial, though the order of sites within a pairing is important. In this notation, the action of on a state with singletons()(+1) is ()(+1)(···)=(,+1)(···) , where(···)stands for all other pairs, and (,+1)(···)=(,+1)(···) . Theactionof onageneralstatecanbereducedtoseveralcases(Appendix B). A drawback of this notation is that it is difficult to account for the winding of through-lines. It is thus most useful in 1 , where winding through-lines incur no phase when unwound, and in 0, ±2 and 0, ±2, wheretherearenothrough-lines. 2.4 0, ±2 andindecomposability The standard module 0, ±2 most simply illustrates the indecomposability we study in detail in Chapters 8 and 9. Consider the case = 2—i.e., the loop formulation of the Potts model for a two-site system (see Section 4.2.1), in the sector with no through-lines and with noncontractible loops given the same weight =+ −1 ascontractibleones. Let us first write the two elements of the Temperley–Lieb algebra in the basis of the two link states 1 =(12)and 2 =(21)(pictorially, and ): 1 = 1 1 0 0 , and (2.16a) 2 = 0 0 1 1 . (2.16b) 14 Clearly, 1 ( 1 − 2 )= 2 ( 1 − 2 )= 0. Meanwhile, at = 2 the action of 1 and 2 on the single state(1)(2) (pictorially, ) in 11 is zero by definition, since the number of through-lines would decrease. Thus we see that 0, ±2 admits a submodule, generated by 1 − 2 , that is isomorphic to 11 . Diagrammatically, usingwhatistechnicallycalleda Loewydiagram, we have 0, ±2 : 0, ±2 11 . (2.17) In such a diagram, the bottom module is a submodule, while the top module is a quotient module. The arrow indicates that within the standard module 0, ±2 a state in 11 can be reached from a state in 0, ±2 throughtheactionoftheTemperley–Liebalgebra, but the opposite is impossible. 2.5 Cellmodulesforroot-of-unityvaluesof When is a root of unity, there are infinitely many solutions to Eq. (2.11), leading to a complex pattern of degeneracies. Now specializing to the Jones–Temperley–Lieb (JTL) algebras () and (), the rule that winding through-lines can simply be unwound (from the quotient by − 1) means that the pseudomomentummustsatisfy[26] ≡ 0 (mod 2). (2.18) All possible values of the parameter ±2 = e ±i are thus th roots of unity, satisfying 2 = 1. The kernel of the homomorphism described by Eq. (2.5) acts trivially on the standard modules if > 0. The standard modules ,e i and 0, ±2 arethusthecell modules of () and () for generic. In the terminology of cellular algebras, the basis consists of the annular diagrams, and the anti- involution∗isaninversionoftheannulus(orareflectionoftheframingrectangleaboutitshorizontalaxis). ThesetofweightsΛ consistsofpairs(,) satisfying Eq. (2.18) and(0, 2 ): Λ={(0, 2 )}∪{(,e 2i/ )|1≤ ≤,0≤ ≤ −1}. (2.19) The finite sets () and the basis of the cell representations can be identified with the link states with 2 through-lines,where=(,)(toinclude= 0). Thereisapartialorder⪯ onΛduetoGrahamandLehrer [29]: first, ( 1 , 1 )⪯( 2 , 2 ) if 1 ≤ 2 asintegers and 1 = 2 2 and 2 = 2 1 , (=±1) (2.20) then⪯ isextendedtoapartialorderbytransitivity. Since not all pairs of weights inΛ are comparable, the partial order⪯ induces equivalence classes in Λ—two weights are in the same equivalence class if and only if they are comparable. A result of Graham andLehrer[Id.] isthatthereexistnontrivialhomomorphismsonlybetweencellmoduleswithweightsfrom the same equivalence class. There are nontrivial classes containing two or more weights only when is a root of unity. Then the cell modules ,e i whose weights belong to a nontrivial class are indecomposable, except for weights that are maximal with respect to⪯. Denote the top simple subquotient—the quotient by its maximal submodule—by[,e i ]. Using the partial order, the simple-module content of these cell modulescan bededuced[Id.]: , ≃ Ê ( ′ , ′ )∈Λ (,)⪯( ′ , ′ ) [ ′ , ′ ], (2.21) where≃denotesequalityasvectorspaces,butnotas ()modules,astheactionofthealgebraconnects simplemoduleswithdifferingweights. () is thus non-semisimple. 15 Hereafterinthiswork,wewillonlybeconcernedwiththetwoJones–Temperley–Liebalgebras (1) and (1), with= 1. (For the affine Temperley–Lieb algebra, we consider the range ∈(−2,2].) With = 1,= e i/3 ,andwemaynowdrawthe corresponding Loewy diagram for 0, 2: 0, 2 : [0, 2 ] [1,1] [2,1] [3, 2 ] [3, −2 ] [4,1] [5,1] [6, 2 ] [6, −2 ] . . . . . . (2.22) Of course, for finite , the tower terminates. The structure depends on the value of mod 3, and these cases are exhibited separately by Gainutdinov et al. [15] (in which may also be found detailed diagrams of thepartialorder⪯ thatleadstotheaboveLoewydiagram). Othermodules ,e i whoseweightsappearas simple modules[,e i ] in Eq. (2.22) are given by diagrams that “emanate” from the corresponding simple modules, and which are identical in structure (except for the length of the tower) but with different labels ateachnode. 16 Chapter3 Physics: basicconcepts This chapter summarizes the basic concepts of physics relevant to the rest of the work, and exhibits two physics problems to motivate the theoretical study of the models therein. Most of the general material is assumedtobefamiliartothereader,andservestoprimethereaderfortheremainderofthework. Although I am rather concise in this section, nothing presented here should be controversial. The reader will have their own preferred sources for this material, and I draw from Shankar [30], Nielsen and Chuang [31], Kardar[32],DiFrancesco,Mathieu,andSénéchal [33], and Cardy [34]. 3.1 Quantumandstatisticalphysics To oversimplify matters, quantum physics is a framework for physical theories that attempt to explain the behavior of physical systems at a small scale where classical physics fails. There are several equivalent formulationsofthepostulatesofquantum physics. * * * Postulatesofquantumphysics(Schrödinger formulation) 1. The state of a system is described by an elementΨ in a Hilbert space with a positive-definite inner product. 2. Observables are represented by hermitian operators. For quantum observables and that have classical analogues and ,[,]= iℏ{,}, where[,] is the commutator of and, and{,} istheclassicalPoissonbracketof and. 3. The measurement of an observableΩ yields one of its the eigenvalues, . If the system is in the state Ψ, then()=⟨Ψ|Π |Ψ⟩, whereΠ is the projection operator onto the eigenspace associated to , istheprobabilityofobtainingthevalue if ispartofthediscretespectrumofΩ,ortheprobability densityat if ispartofthecontinuous spectrum. 4. Ψevolvesintimeaccordingto iℏ Ψ =Ψ, (3.1) where istheHamiltonianoperator for the system. Postulatesofquantumphysics(Heisenberg formulation) 1. The state of a system is described by an elementΨ in a Hilbert space with a positive-definite inner product. 17 2. Observables are represented by hermitian operators. For quantum observables and that have classical analogues and ,[,]= iℏ{,}, where[,] is the commutator of and, and{,} istheclassicalPoissonbracketof and. 3. The measurement of an observableΩ yields one of its the eigenvalues, . If the system is in the state Ψ, then()=⟨Ψ|Π |Ψ⟩, whereΠ is the projection operator onto the eigenspace associated to , istheprobabilityofobtainingthevalue if ispartofthediscretespectrumofΩ,ortheprobability densityat if ispartofthecontinuous spectrum. 4. TheoperatorsΩ evolveintimeaccording to iℏ Ω =[Ω,], (3.2) where istheHamiltonianoperator for the system. Postulatesofquantumphysics(vonNeumann formulation) 1. The state of a system is described by a positive hermitian density operator acting on a Hilbert space withapositive-definiteinnerproduct. It satisfies tr= 1. 2. Observables are represented by hermitian operators. For quantum observables and that have classical analogues and ,[,]= iℏ{,}, where[,] is the commutator of and, and{,} istheclassicalPoissonbracketof and. 3. If an observable Ω is measured many times in the state , the average (expected) value of these measurementsisE[Ω]= tr(Ω). 4. Thedensityoperator evolvesintime according to iℏ =[,], (3.3) where istheHamiltonianoperator for the system. * * * Which formulation one ultimately chooses as an analytical and computational framework is largely a matterofconvenience. Forinstance,theHeisenbergformulation islargely usedinquantum fieldtheoryas the only state one really treats explicitly is the ground state, and the von Neumann formulation is tailor- made for mixed ensembles. Note, however, that in each of these formulations the postulate of observables beingrepresentedbyhermitianoperatorsremainsthesame. Arichseamofphysicsisuncoveredwhenthis assumptionislifted,andmuchoftherestofthisworkisdevotedtothebeginningsofasystematictreatment ofquantumphysicswiththispostulateappropriately replaced. Aunifyingelementforallofthethreeformulationspresentedisthepropagator. Usingthepropagator toadvancearbitrarystatesforwardintime, Ψ()=()Ψ(0), (3.4) itself satisfies Eq. (3.1) with initial condition (0) = . For a time-independent Hamiltonian, which encompassesmanycasesofinterest,andan assumption retained throughout this work, one has ()= e −i/ℏ . (3.5) By considerations involving the independence of⟨Ψ()|Ω()|Ψ()⟩ and tr[Ω()()] across different formu- lationsofquantumphysics,wehavesimilarly Ω()= † ()Ω(0)() and (3.6) ()=()(0) † () (3.7) 18 intheirrespectiveformulations. Eachoftheseexpressionsissimplifiedwhen isexpressedintermsofthe projectionoperatorsassociatedwiththespectral decomposition= Í Π : ()= Õ e −i /ℏ Π . (3.8) Thesefactsshowthattheprimarytaskinanyquantummechanicalproblemisessentiallytofindthespectral decompositionoftheHamiltonian. Finally, an alternative expression for the propagator is given in terms of Feynman’s path integral (or functional integral). If generically represents the degrees of freedom for the system, then the matrix elementsofthepropagatorare ⟨ ()| (0)⟩=⟨ |()| ⟩= ∫ ( ,) ( ,0) [d]e i[]/ℏ . (3.9) The integration is over all paths in phase space starting at , ending at , and taking place over the time interval[0,];[] is the (classical) action associated with such a parametrized path. The integration measuremaybewritten [d]= lim →∞ r 2iℏ © « −1 Ö =1 r 2iℏ d ª ® ¬ , (3.10) where =(/). Quantum field theory is a quantum mechanical description of systems whose basic degrees of freedom are fields , rather than point particles. Once a Lagrangian is specified, the path integral provides the quickestgeneralizationforthetransitionamplitudes: ⟨ (, )| (, )⟩= ∫ [d(,)]e i[] . (3.11) Inquantumfieldtheorytheterms“field”and“operator”areusednearlysynonymously. Inthepathintegral formalism,thefieldsappearinginapathintegraltakeonallallowedclassicalfieldconfigurations,possibly subjecttoconstraintsintheintegrationmeasure. Inthecanonicalquantizationoffields,whichmoreclosely parallels the postulates discussed at the beginning, the fields become operator-valued distributions. Both formalisms are useful and provide differing theoretical perspectives, and so I will also use these terms interchangeablyinthecontextswherethey are commonly used. A primary quantity of interest is the correlation function, as it is directly related to the experimentally measurablescatteringamplitudes. Forapoint particle, the-point correlation function is ⟨( 1 )( 2 )···( )⟩=⟨0|T[( 1 )( 2 )···( )]|0⟩. (3.12) Inthisequation,|0⟩ isthegroundstate,andTthetime-orderingoperator. Thetime-orderingoperatormay beavoidedbyusingthepathintegralformalism: ⟨( 1 )( 2 )···( )⟩= lim →0 ∫ [d]( 1 )( 2 )···( )e i []/ℏ ∫ [d]e i []/ℏ , (3.13) where is the action obtained by replacing by(1−i). This “ prescription” is essential in proving the equality of the two expressions given for the-point function. It is useful to push this prescription further, and define correlation functions entirely in imaginary time =−i (∈ℝ). In doing so, the action in real time becomes the Euclidean action in imaginary time: i[(→−i)]= [()]. The change to imaginary timealsocausesthekineticenergytermintheLagrangiantochangesign,becominginsteadthe(real-time) 19 Hamiltonian. ThisEuclideanformalismisusedhereafter,switching backto anddroppingthesubscript . Consider a correlation function⟨ ( ) ( )Φ⟩, whereΦ denotes an arbitrary product of operators far away from and . Invoking ideas of locality and completeness of the operators, the operator product expansion (OPE)postulatesthatthiscorrelation function may be replaced by a sum of the form ⟨ ( ) ( )Φ⟩= Õ ( − )⟨ ( )Φ⟩. (3.14) Importantly, the coefficients do not depend on the arbitrary productΦ. An OPE such as this one is frequentlywrittenwithouttheexpectation value and the arbitrary fields as ( ) ( )= Õ ( − ) ( ) or (3.15a) ( ) ( )∼ Õ ( − ) ( ), (3.15b) it being understood that both sides of an OPE appear within a correlation function, possibly multiplied by other fields. The equalities do not make sense on their own, since, for instance, in the path integral formalism,thefieldsmustbeallowedtotakeallpossiblevaluesindependently. WhiletheexistenceofOPEs is axiomatic for generic quantum field theories, they may be formalized in special cases via vertex algebras andvertexoperatoralgebras,particularly for two-dimensional conformal field theory. * * * Statistical physics is a framework of physics that attempts to infer macroscopic properties of complex physicalsystemswithonlyknowledgeoftheHamiltonianintermsofmicroscopicdegreesoffreedom. From Boltzmann’s ergodic hypothesis and Laplace’s suggestion that one should apply a uniform distribution to unknowneventsaccordingtothe“principleofinsufficientreason,”oneobtainstheBoltzmanndistribution forasysteminthermalcontactwithitssurroundingsattemperature—theprobabilitythatthemicroscopic degreesoffreedomareinthestate isproportional toe − : = e − , (3.16) where istheenergyofthesysteminconfiguration and= 1/. Asthetotalprobabilitythatthesystem befoundinanystateis1,thenormalizing factor, the partition function, must be = Õ e − . (3.17) The sum may be discrete or continuous according to the nature of the states describing the system. The preceding expression holds whether the system is described using classical or quantum physics. For a quantumsystem,aconvenientexpression that compactly incorporates the sum is = tre − . (3.18) Theprimarygoalofstatisticalphysicsisthecalculationofthepartitionfunction,asallmacroscopicquantities maybederivedfromit. Suppose our system is quantum, with degrees of freedom. The trace in the partition function may be writtenas = ∫ d⟨|e − |⟩= ∫ (,ℏ) (,0) [d]e −[()] . (3.19) 20 Here, we have used the fact that the propagator = e −i/ℏ becomes e −/ℏ in the Euclidean formalism, allowingustoidentify with/ℏ andusethe path integral. This analogy may be generalized to a system with a continuum of degrees of freedom. In all instances, the partition function of a-dimensional quantum system may be written as a path integral where time is imaginaryandtakesonthecharacterofaspatialvariable,thusgivingrisetoa(+1)-dimensionalclassical system. This equivalence underlies the study of two-dimensional classical problems, such as percolation andthetwo-dimensionalPottsmodel,using one-dimensional quantum systems. 3.2 Conformalfieldtheory Aconformalfieldtheory(CFT)isaquantumfieldtheorywithconformalsymmetry. Systemswithconformal symmetry are invariant, or covariant, under conformal transformations. Conformal transformations are essentially a scale transformation combined with a rotation, with scale factors and rotations that may vary smoothly from point to point. They thus preserve the angles between two arbitrary curves crossing at some point. In two dimensions, conformal invariance takes a new meaning. Any analytic mapping of the complex plane onto itself is conformal, and thus furnishes a local conformal transformation. It is this local conformal invariance that allows for exact solutions to two-dimensional CFTs, which will be the only CFTs we consider hereafter. Because of the close connection between analytic and conformal mappings, we use complexcoordinates and onthecomplex plane. To each field, we may associate a holomorphic conformal dimension (or holomorphic conformal weight) ℎ andanantiholomorphicconformaldimension(antiholomorphicconformalweight)ℎ. Alternativeterminology for“holomorphic”includes“left”and“chiral,”andalternativeterminologyfor“antiholomorphic”includes “right” and “antichiral.” If, under an invertible conformal transformation→() and→(), a field transformsas (,)→ d d −ℎ d d −ℎ (,), (3.20) iscalledquasi-primary. If transformsthesamewayunderarbitraryconformalmaps,itiscalledprimary. Thisbehaviorfullydeterminestheformof two- and three-point functions: ⟨ 1 ( 1 , 1 ) 2 ( 2 , 2 )⟩= 12 2ℎ 12 2ℎ 12 ℎ 1 = ℎ 2 = ℎ and ℎ 1 = ℎ 2 = ℎ 0 otherwise , (3.21a) ⟨ 1 ( 1 , 1 ) 2 ( 2 , 2 ) 3 ( 3 , 3 )⟩= 123 ℎ 1 +ℎ 2 −ℎ 3 12 ℎ 2 +ℎ 3 −ℎ 1 23 ℎ 3 +ℎ 1 −ℎ 2 13 ℎ 1 +ℎ 2 −ℎ 3 12 ℎ 2 +ℎ 3 −ℎ 1 23 ℎ 3 +ℎ 1 −ℎ 2 13 , (3.21b) where(ℎ ,ℎ ) are the left and right conformal weights of , = − , and = − . The coefficient 123 is closely associated to the function 123 (in these notations) from the OPE in Eq. (3.15). Four-point functionsarenotsouniquelyconstrained,sincethereareanharmonicratiosoffourpointsthatareconformally invariant. Ingeneral,thefour-pointfunction has the form ⟨ 1 ( 1 , 1 ) 2 ( 2 , 2 ) 3 ( 3 , 3 ) 4 ( 4 , 4 )⟩= 12 34 13 24 , 12 34 13 24 4 Ö ,=1 < ℎ/3−ℎ −ℎ ℎ/3−ℎ −ℎ , (3.22) whereℎ= Í 4 =1 ℎ andℎ= Í 4 =1 ℎ . Four-point functions may be reduced to a sum of three-point functions using the OPE; one frequently encounters four-point functions given by “sums over conformal blocks,” which encapsulates this process. If one assumes that the conformal blocks are constrained by “crossing symmetry,”anaturalassumption,thentheycanbefound,inprinciple. Themethodofcomputingconformal blocksbysimplyassumingcrossingsymmetryiscalledthebootstrapapproach. Thebootstrapapproachmust be validated by checking self-consistency conditions, since it may potentially over-constrain the theory. 21 Numerical studies may also be used to corroborate the conclusions of the bootstrap approach (Chapters 4, 8,and9). Ward identities express the consequence of symmetries on the correlation functions of field theories. ThemostimportantWardidentityinvolvesthestress–energytensor(orenergy–momentumtensor) andits antiholomorphiccounterpart (moreprecisely, and arefields,butnormalizedcomponentsofagenuine rank-2tensor),herewrittenasOPEs: () = Õ =1 ℎ (− ) 2 + − +reg. and (3.23a) () = Õ =1 " ℎ (− ) 2 + − # +reg., (3.23b) where = 1 ( 1 , 1 )··· ( , ) and “reg.” stands for a holomorphic function of (antiholomorphic function of ). An infinitesimal conformal map → + (), → + () gives the following conformal Wardidentity: , =− 1 2i ∮ d()()+ 1 2i ∮ d()(), (3.24) where the contour encloses all of the points( , ). This expression does not require the fields in to beprimary,merelylocal. The conformal Ward identity establishes and as the generators of conformal transformations of quantumfields. Theymaybeexpandedinmodes: ()= ∞ Õ =−∞ −−2 (3.25a) ()= ∞ Õ =−∞ −−2 . (3.25b) Themodegenerators obeytheVirasoroalgebra: [ , ]=(−) + + 12 ( 2 −1) ,− , (3.26a) [ , ]= 0, (3.26b) [ , ]=(−) + + 12 ( 2 −1) ,− . (3.26c) It is a direct sum of two identical algebras, the summands of which are also called the Virasoro algebra. To be explicit, we will call the algebra with generators and obeying Eq. (3.26) Vir⊕ Vir. The bar on the rightsummanddoesnotimplyamodification,butstandsasareminderthatitsignifiestheantiholomorphic counterparttotheleftone. Thegenerators −1 and −1 areoftendenotedby= and= ,astheyhave thesameeffectwithinOPEsandcorrelationfunctions. Thereisasubalgebraof Virspannedby −1 , 0 ,and 1 ;theygeneratetheinvertibleconformal transformations. The Virasoro algebra is the unique central extension of the Witt algebra. The coefficient in the central termiscalledthe centralcharge,anditisgiven by the following OPEs between and: ()()= /2 (−) 4 + 2() (−) 2 + () − , (3.27a) ()()= 0, (3.27b) ()()= /2 (−) 4 + 2() (−) 2 + () − . (3.27c) 22 Thus and arenotprimary. However,theyarequasi-primary,withconformaldimensions(2,0)and(0,2) (bycomparisonwithEq.(3.23)). Here, I collect some parameters and notations used ubiquitously in generic CFTs. The central charge that describes a given CFT comes from the central term in the Virasoro algebra commutation relations, or fromthemostsingulartermintheOPEof with itself. A parameter,, is defined via = 1− 6 (+1) . (3.28) Theinverserelation,validfor−∞ < < 1, is given by =− 1−+ p (1−)(25−) 2(1−) . (3.29) Theserelationswillbeusedfrequentlytoconvertbetween and asindependentvariableswithoutexplicit mention. TheKacformula, ℎ = [(+1)−] 2 −1 4(+1) , (3.30) uses two labels to describe conformal weights. Particularly for integers and , the Kac formula gives a compactrepresentationofmanyconformal weights important to a given theory (specified by ). Highest-weight representations of Vir are constructed by fixing a highest-weight state |ℎ⟩ satisfying 0 |ℎ⟩= ℎ|ℎ⟩ and |ℎ⟩= 0 for > 0,then declaring the descendant states − 1 − 2 ··· − |ℎ⟩ (1≤ 1 ≤···≤ ) (3.31) tobelinearlyindependent. Thedescendantstatesabovecorrespondtoaneigenvalueℎ ′ = ℎ+ 1 + 2 +···+ of 0 (or conformal weight ℎ ′ ), and a linear combination of descendant states all with conformal weights ℎ+ iscalledadescendantatlevel. These highest-weight representations are often calledVerma modules. AninnerproductmaybedefinedontheVerma module by ⟨ℎ|ℎ⟩= 1 and † = − . ArepresentationofVir is unitaryifithas no negative norm squared states. From ⟨ℎ| − |ℎ⟩= 2ℎ+ 12 ( 2 −1) (3.32) it follows that representations with < 0 or highest weight ℎ < 0 are nonunitary (these conditions are sufficient,butnotnecessary). The structure of Verma modules over the Virasoro algebras has been determined by Feigin and Fuchs [35]. For≤ 1, a condition that will hold for the remainder of the work, a reducible Verma module either admitsafiltrationbysimplemodules,orhasabraid-typesubmodulestructure. For ≥ 2apositiveinteger, theVermamoduleshowsthebraid-typestructure (,) (,2−) (+−1,−) (,2+) (+2(−1),) . . . . . . (,+(−1) +[1−(−1) ]/2) (+(−1),(−1) +[1−(−1) ]/2) . . . . . . (3.33) 23 ρ xx ρ xy B Figure 3.1: The Hall effect. where each vertex(,) represents a Verma module built on the highest-weight ℎ , and an arrow→ means is a submodule of , with transitivity implied. To obtain the corresponding Loewy diagram, replace the module(,) by its simple subquotient, obtained by taking a quotient by the sum of its two maximal submodules indicated by the arrows. This structure may be compared with that of Eq. (2.22), showingthattheJTLalgebraisusefultostudymanystructuresofthecontinuumCFTthathaveanalogues atfinitesize—thesubjectofChapter4. The modules(,) at each vertex are generated by singular vectors, which are particular descendants of the field of highest weight that may be constructed systematically. They imply a (hypergeometric-type) differential equation for the undetermined function in the four-point function, whose solution may also be usedasacheckforthebootstrapapproach. 3.3 Twoconcretephysicsproblems 3.3.1 ThequantumHalleffect Here I give the briefest possible introduction to the quantum Hall effect, which is now a huge enterprise in physics [36]. I follow Tong [37] with the presentation condensed and inessential details swept under the rug. Fromclassicalelectromagnetism,wearefamiliarwithOhm’slaw(upgradedfromthehumble =), J=E, (3.34) where is the resistivity tensor, J is the current density, and E is the electric field. Now focus on two dimensions, the -plane, and put a magnetic field B = ˆ z through the system. A calculation in the frameworkofclassicalelectromagnetismpredicts that = 2 and (3.35a) = , (3.35b) where is the electron mass, is the density of free electrons, is a parameter that measures the average scatteringtimeforelectrons,and istheelementaryelectriccharge. Thisistheessenceofthe(classical)Hall effect: an electric field in the -direction gives rise to a current with a component in the -direction in the presence of a magnetic field perpendicular to the system. The transverse resistivity increases linearly with the strength of the magnetic field and remains constant. Graphically, we expect something like Figure3.1. At low temperatures and strong magnetic fields, quantum effects become important. Experimental measurements yield a graph that is shown in Figure 3.2. The staircase represents and the noisy peaks are . Thisisthe(integer)quantumHall effect. 24 Figure3.2: The quantum Hall effect. From Tong [37]. The most obvious feature is the existence of plateaux, intervals of magnetic field strengths which give thesameresistivitywitheach. Theresistivity plateaux have the values = ℎ 2 , (3.36) whereℎ isPlanck’sconstantand hasbeenmeasuredtobeanintegertoanaccuracyof1partinabout10 9 . Atthecenterofeachoftheseplateaux,the magnetic field is = ℎ . (3.37) Perhaps counterintuitively, the underlying mechanism for the existence of the plateaux is disorder. In the limiting case of zero disorder, corresponding to a perfectly clean sample, the plateaux disappear and we return to Figure 3.1. (There is an intermediate case where plateaux appear at some rational values , appropriately termed the fractional quantum Hall effect.) In fact, at increasing disorder (up to a point) the plateauxstabilizeandgrowwider. The physics challenge here is to explain how the presence of strong disorder gives rise to something as exactandpureasaninteger,andthenecessity of this disorder. 3.3.2 Percolation Thereareanumberofformulationsofpercolation. Asthesimplestcase,Idescribebondpercolationonthe two-dimensionalsquarelattice. Asquarelattice(say,aconnectedsubsetofℤ 2 )hasbonds,orlinks,thatcan independently be occupied with probability or empty with probability1− (Figure 3.3). The most basic question regarding this setup is whether there is a path along occupied bonds from one side to the other. For the limiting case of the infinite square lattice ℤ 2 , it is known that there is a critical value = c = 1/2, abovewhich theprobabilityofsuchapath existing is 1, and below which the probability is 0 [38]. A physical realization of percolation in three dimensions is the extraction of espresso (Figure 3.4). At the top, highly pressurized water (15 bar) makes its way through densely packed, finely ground coffee, emergingatthebottom. Thegoalinthisscenarioistoevenlysaturatethegrounds,allowingforafull,even extraction. Aside from the basic presence or absence of a given bond, there are a number of more sophisticated observables one can consider [15]. Cluster connectivities can be defined as the probability that a given set of points belongs to the same cluster. For two points, it is known that this probability decays with their 25 Figure3.3: Aconfigurationillustratingbond percolation. From Di Francesco, Mathieu, and Sénéchal[33]. Figure 3.4: Percolation of water through a network of firmly packed, finely ground coffee. Photo by Scott Schiller,usedunderCCBY2.0. 26 separation as −5/24 ,for= c . Itisalsopossibletoconsiderrefinedconnectivities,suchastheprobability thatasetofpointsbelongstothesameclusterandalsoareconnectedviatwonon-intersectingpathsonthis cluster. Ofinterestlaterwillbepercolationhulls,theprobabilitythattwopointsbelongtotheboundaryof the same cluster, which decays as −1/2 . This can be generalized to observables where cluster boundaries cometogether,withtheprobabilitydecaying as −(4 2 −1)/6 [39]. * * * These two problems are, in fact, closely related. We understand that the existence of plateaux is due to disorder, and the integers are in fact Chern numbers, arising from topological considerations [37]. On the other hand, our analytical understanding of the plateaux transition is still lacking [40]. It is expected that this transition is described by some nonunitary conformal field theory, a candidate of which has been proposed [41]. At a jump, a percolating path develops and the electrons undergo a localization– delocalization transition. The essential features of this quantum percolation transition are summarized in the Chalker–Coddington random network model, which has resisted analytical treatment. However, the Chalker–Coddington model may be systematically truncated and studied, leading to a series of two- dimensionalloopmodelsoveraTemperley–Liebalgebra[42]. Theloopmodelisdiscussedasaformulation ofthe-statePottsmodelinSection4.2,and its algebraic aspects are studied there and in Part III. 27 Chapter4 Physics: conformalfieldtheoryonthe lattice 4.1 Discretizationofcontinuumtheories 4.1.1 Necessityofdiscretization Numericalstudyoffieldtheoriesonacomputernecessarilyinvolvesreducingthecontinuumofdegreesof freedom to a finite number of them. At the same time, the degrees of freedom that are retained must be sufficient to probe the properties of the field theory accurately. For local field theories, at least, a natural choice is the collection of field amplitudes at a discrete set of positions. The problem of percolation is nonlocal,butitmaybetradedforalocalformulation (at the cost of nonunitarity) [43]. Evenoutsideoftheparticularcontextoflatticemodels,notethatsignaturesofdiscretizationarepresent inthefieldtheoriesthemselves. Forinstance,thepathintegralmeasure,Eq.(3.10),isdefinedviaalimiting processinwhichthetimeinterval[0,]istreatedasasequenceofdiscretevalues/,with0≤ ≤ and →∞. 4.1.2 Identificationoffieldsonthelattice Inordertostudyacontinuumtheoryasalimitofalattice-discretizedtheory,wemustidentifythoseobjects inthecontinuumtheorythathaveacorresponding realization in the discrete model. Thefirstquestionishowonecanrealizethecornerstoneoftwo-dimensionalconformalfieldtheory,the Virasoro algebra, using the algebra generators available in the lattice model. For generators obeying the Temperley–Liebrelations,ithasbeenconjectured [3] that the Koo–Saleur generators =− 4 F Õ =1 e 2i/ − ∞ + i F [ , +1 ] + 24 0 , (4.1a) =− 4 F Õ =1 e 2i/ − ∞ − i F [ , +1 ] + 24 0 , (4.1b) furnish representations ofVir andVir. in the limit→∞. While the precise nature of the convergence is subtle and under active study (Section 8.3, see also Milsted and Vidal [5]), numerical and analytical checks havedemonstratedmanyoftheexpectedproperties of the Virasoro algebra [3] (see also Appendix D). Thus, an eigenstate of 0 ( 0 ) with eigenvalue ℎ() (ℎ()) can be viewed as a lattice representation on sites of a field with left (right) conformal dimension ℎ≡ lim →∞ ℎ() (ℎ≡ lim →∞ ℎ()). The scaling dimensionsΔ() = ℎ()+ℎ() are obtained by diagonalizing 0 = 0 + 0 . The precise identification of conformal fields is a subtle process because the convergence ℎ()→ ℎ and ℎ()→ ℎ can be slow, 28 because these values are discretely parametrized by rather than a continuous parameter, making them moredifficulttotrack,andbecausethenumberofstatesproliferatesrapidlywithincreasing . Tolabelthe scalingstatescorrectly,onemustcarefullyfollowsequencesofeigenvaluesℎ()withincreasing,paying attention to other characteristics such as momentum and parity. The general methodology is explained by Jacobsen and Saleur [7, Appendix A.5]. For our present purposes, we will assume that this assignment of scaling fields can be done, and we will check that our identifications are self-consistent by showing that the lattice versions have properties expected of their corresponding continuum fields. Examples of these scalingfieldassignmentsappearinSection8.4. 4.2 Pottsmodel 4.2.1 Generaldescription The -state Potts model is central to much of this work. It may be defined on an arbitrary graph as a collectionof spins livingon thevertices of the graph, labeled by. The spins take (at first) integer values from1 to. For a graph with vertices the number of possible configurations is , and the energy of a particularconfigurationis []=− Õ ()∈ , (4.2) where is the set of edges in the graph and are edge-dependent coupling constants. With all couplingsequal,thepartitionfunction(Eq. (3.17)) reads = Õ {} Ö ()∈ e , (4.3) with=−≡− . Nowwritee = 1+(e −1) ,expandtheproduct,andperformthesumover allspinstoobtain = Õ ⊂ (e −1) || () , (4.4) where the sum is over all 2 | | subsets of , and|| is the cardinality of. () denotes the number of connected components (also called Fortuin–Kasteleyn clusters or FK clusters) in the subgraph with the same vertices as but only the edges. is now written as a polynomial in, and extending it to values of thatarenotanintegerdefinesthepartition function for the Potts model at arbitrary . The simplest case is to take the vertices of the graph to be a regular lattice, with edges going between only nearest neighbors, and to take all couplings to be equal. Henceforth we take to be the square lattice intwodimensions. Underthesefurtherassumptions,thereisacriticalvalueof, c ,givenbye c −1= √ , sothatthemodelisconformallyinvariantin the continuum limit. Anequivalentformulationof whenisthetwo-dimensionalsquarelatticeisgivenbytheloopmodel on the medial lattice =( , ). The vertices of are situated at the midpoints of the original edges ,andtwoverticesin areconnectedbyanedgein whenevertheformerstandonedgesin that are incident on a common vertex from. In particular, when is a square lattice, is just another square lattice, tilted through an angle/4 and scaled down by a factor of √ 2. There is a bijection between edge subsets⊂ and completely-packed loops on . The loops are defined so that they turn around theFKclustersandtheirinternalcycles. One has then, using the Euler relation, = ||/2 Õ ⊂ e −1 √ || ()/2 , (4.5) where () is the number of loops as defined by the bijection. The loop fugacity is √ ≡ + −1 , which definesthequantumgroupparameter . 29 Theloop–clusterformulationgivesrisetoarepresentation—inthetechnicalsenseofarepresentationof an associative algebra—of , as we now explain. In practice, states in the transfer matrix must be defined so as to allow the bookkeeping of the nonlocal quantities() or(). In the cluster picture, a state is a set partition of the sites in a row, with two vertices belonging to the same block in the partition if and only if they are connected via the part of the FK clusters seen below that row. Equivalently, in the loop picture, a stateisapairwisematchingof = 2 medialsitesinarow,witheachsiteseeingeitheravertexof onits leftandadualvertexof onitsright,orviceversa. Thebijectionbetweenclusterandloopconfigurations provides as well a bijection between the corresponding cluster and loop states. The transfer matrix evolves the loop states by the relations of the affine Temperley–Lieb algebra given in Eqs. (2.1) and (2.3), and to matchtheloopweightsbetweenEqs.(2.1a) and (4.5) we must identify = p . (4.6) To account also for the computation of correlation functions, a few modifications must be made. The caseoffour-pointfunctionshasbeenexpoundedbyJacobsenandSaleur[7],butinthepresentdiscussionit sufficestoconsiderthesimplercaseoftwo-pointfunctions. Thesecanbecomputedinthecylindergeometry byplacingonepointateachextremityofthecylinder. Theissueisthenensuringthepropagationofdistinct clusters between the two extremities in a setup compatible with the transfer matrix formalism. This can be done, on the one hand, in the cluster picture by letting the states be -site set partitions including markedblocks,and,ontheotherhand,inthelooppicturebylettingthestatesbe-sitepairwisematchings including 2 defect lines—precisely the through-lines already encountered in the discussion of . The sum over states must then be restricted so as to ensure that the marked clusters or defect lines propagate all along the cylinder. Moreover, it turns out to be necessary to keep track of the windings of either type of marked object around the periodic direction of the cylinder. Fortunately, in the loop picture, these considerationsleaddirectlytothedefinitionofatypeofrepresentation—theaffineTemperley–Liebstandard module—whichiswell-knowninthealgebra literature, and which has been covered in Chapter 2. Aspecialpointmustbemadeinconnectionwiththepresentdiscussion: thereissometimesaconfusion related to the type of object one may wish to consider as part of “the” Potts model CFT. By such a CFT we meanherethefieldtheorydescribinglong-distancepropertiesofobservableswhicharebuiltlocallyinterms of Potts spins for integral, then continued to real using the FK expansion. Examples include the spins themselvesbutalsotheenergyandmanymoreobservablesasdiscussed,forinstance,byVasseur,Jacobsen, and Saleur [44], Vasseur and Jacobsen [45], and Couvreur, Jacobsen, and Vasseur [46]. Other objects have been defined and studied in the literature, in particular those describing the properties of domain walls, boundariesofdomainswherethePottsspinstakeidenticalvalues[47,48]. Thesearenotlocalwithrespect to the Potts spin variables. Whether there is a “bigger” CFT containing all of these observables at once remainsanopenquestion—seeVasseurand Jacobsen [49] for an attempt in this direction. TohaveabetterideaoftheobservablespertainingtothePottsmodelCFTforgeneric,onecanstartwith thetoruspartitionfunction,whichhasbeen determined inthe continuum limit[50, 51, 52]. Parametrizing p = 2cos +1 , (∈(0,∞]) (4.7) thecentralchargeis = 1− 6 (+1) , (4.8) whiletheKacformulareads ℎ = [(+1)−] 2 −1 4(+1) . (4.9) Thecontinuum-limitpartitionfunctionisthen given by = 0, ±2+ −1 2 0,−1 + Õ >0 ˆ ′ 0 1 + Õ >0,>1 | Õ 0<< ∧=1 ′ ,/ ,e 2i/, (4.10) 30 where∧ isthegreatestcommondivisorof and. Thecoefficients ˆ ′ canbethoughtofas“multiplic- ities,” although for generic, they are not integers. Their interpretation in terms of symmetries is beyond thescopeofthisdiscussion[15,53]. Theyare given by ˆ ′ = 1 −1 Õ =0 e 2i (,∧), (4.11) (∧0= bydefinition)and (,)= 2 + −2 + −1 2 (i 2 +i −2 )= 2 + −2 +(−1) (−1). (4.12) The ,e i aregivenbysums, ,e i = −/24 −/24 ()() Õ ∈ℤ ℎ − ,− ℎ − , , (4.13) inwhich ()= ∞ Ö =1 (1− ) (4.14) and = /2. As usual, and are the modular parameters of the torus. (), the inverse generating functionforthepartitionnumbers,canbewrittenintermsoftheDedekindetafunctionas()= −1/24 (). Equations (4.10) and (4.13) encode the operator content of the-state Potts model CFT. The conformal weightsarisingfromthelastterminEq.(4.10) are of the form (ℎ −/, ,ℎ −/,− ). (∈ℤ) (4.15) Thefirsttwotermsmustbehandledslightly differently. Using the identity 0, ±2− 11 = ∞ Õ =1 1 1 ≡ 0, ±2 (4.16) withtheKaccharacter = ℎ −/24 1− () , (4.17) weseethatwegetthesetofdiagonalfields (ℎ 1 ,ℎ 1 ). (∈ℕ ∗ ) (4.18) Thepartitionfunctioncanthenberewritten as = 0, ±2+ −1 2 0,−1 + 11 + Õ >0 ˆ ′ 0 1 + Õ >0,>1 | Õ 0<< ∧=1 ′ ,/ ,e 2i/. (4.19) We notice now that ˆ ′ 10 = 2 + −2 −(− 1) = − 2−(− 1) =−1. Hence 11 disappears, in fact, fromthepartitionfunction. 11 correspondsgeometricallytotheso-calledhulloperator[39]—relatedtothe indicator function that a point is at the boundary of an FK cluster—with corresponding conformal weights (ℎ 01 ,ℎ 01 ). Therefore, this operator is absent from the partition function. We will, nevertheless, continue to consider 11 , since this module does appear in related models, such as the “ordinary” loop model or the “()” model [51]. We note meanwhile that the higher hull operators—related to the indicator function 31 that > 1 distinct hulls come close together at the scale of the lattice spacing—with conformal weights (ℎ 0 ,ℎ 0 ) in 1 doappearinthepartitionfunction, including the Potts case. The decomposition of the Potts-model partition function in Eq. (4.19) for generic is, in fact, in one-to- one correspondence with an algebraic decomposition of the Hilbert spaceℋ in terms of modules of the affineTemperley–Liebalgebrathatisexactin finite size [24]. This decomposition formally reads ℋ = 0, ±2+ −1 2 0,−1 + 11 + Õ >0 ˆ ′ 0 1 + Õ >0,>1 | Õ 0<< ∧=1 ′ ,/ ,e 2i/. (4.20) Eq. (4.20) is only formal in the sense that, for generic, the multiplicities are not integers, andℋ cannot beinterpretedasapropervectorspace. Bycontrast,themodules ,e i arewell-definedspaceswithinteger dimension independent of, as discussed in Section 2.3. Note that one must take into account the fact that thesumsmustbeproperlytruncatedfora finite lattice system. ThetoruspartitionfunctionofEq.(4.10) is obtained by a trace overℋ , = tr ℋ e − e −i , (4.21) wheretherealparameters > 0and determinethesizeofthetorus,while and denoterespectively thelatticeHamiltonianandmomentumoperators. Introducing the (modular) parameters = exp[− 2 ( +i )], (4.22a) = exp[− 2 ( −i )], (4.22b) wehave,inthelimitwherethesizeofthesystem→∞,with , →∞suchthat and remainfinite, tr ,e i e − e −i → ,e i. (4.23) The nature of the convergence of this final expression is subtle and will be discussed in great detail in Chapter8. 4.2.2 DiscreteVirasoroalgebrainthePottsmodel While the Potts model is often defined as an isotropic lattice model on the square latte, a point of view we shareinSection4.2.1,itiswellknownthatthecorrespondinguniversalityclassextendstoacriticalmanifold with properly related horizontal and vertical couplings. The case of an infinitely large vertical coupling (theverticaldirectionistakenasimaginarytime—seeSection3.1)leadstotheHamiltonianlimitwherethe modeldynamicsisdescribedbyaHamiltonianinsteadofatransfermatrix. Werestricttothislimitinwhat follows, in order to match as closely as possible the lattice model to the formalism of radial quantization of thecontinuumCFT. TheHamiltoniandescribingthe-statePottsmodelcanbeexpressedusingTemperley–Liebgenerators [3]as =− 1 F Õ =1 ( − ∞ ) (4.24) for even . Here, the prefactor F , the Fermi velocity (sometimes sound velocity), is chosen to ensure relativistic invariance at low energy. Its value is F = sin/, where ∈[0,) is defined via = e i , so that = √ ∈(−2,2]. ∞ is a constant energy density added to cancel out extensive contributions to the ground-stateenergy. Itsvalueisgivenby ∞ = sin ∫ ∞ −∞ sinh[(−)] sinh()cosh() d. (4.25) 32 In Eq. (4.24), the generators can be taken to act on different representations of (). The original representation, used for integer , uses matrices of dimension × , corresponding to a chain of = /2 Potts spins. The FK formulation of the Potts model for real can be obtained by using the loop formulationinstead. Note that when taking one of the standard modules ,e i as the representation of choice, the energy density ∞ isindependentof . It is also known that the XXZ or vertex model representation of could be used instead of the loop representationwith“verysimilarresults.” Thispointhastobeconsideredwithcaution,however: whilethe algebra isalwaysthesame,therepresentations(i.e.,usingloops/clustersorspins/arrowsinthetransfer matrix)arenotnecessarilyisomorphic. Section 4.2.3 discusses this point in more detail. Following Eq. (4.24), we define the Hamiltonian density as ℎ =− / F . From the Hamiltonian density we then construct a lattice momentum density = i[ℎ ,ℎ −1 ] =−i/ 2 F [ −1 , ] using energy conservation [5]. Wecanthenintroduceamomentumoperator as =− i 2 F Õ =1 [ , +1 ]. (4.26) (Thisnotationwillnotbeusedinthesequel, where we reserve for parity.) Fromℎ and webuildcomponentsof a discretized stress tensor as = 1 2 (ℎ + ), (4.27a) = 1 2 (ℎ − ), (4.27b) from which we construct discretized versions of the Virasoro generators as the Fourier modes [Id.]. This constructiongivesrisetotheKoo–Saleurgenerators =− 4 F Õ =1 e 2i/ − ∞ + i F [ , +1 ] + 24 0 , (4.28a) =− 4 F Õ =1 e 2i/ − ∞ − i F [ , +1 ] + 24 0 , (4.28b) whichwerefirstderivedviaothermeansbyKooandSaleur[3](seealsoAppendixD).Thecrucialadditional ingredientisthecentralcharge,givenbyEq.(4.8). Notethattheidentificationofthecentralchargeisactually asubtlequestion,andmaybeaffectedbyboundaryconditions,asdiscussedfurtherbyGrans-Samuelsson, Jacobsen,andSaleur[54]. 4.2.3 AnoteontheXXZrepresentation IntheXXZrepresentation,thegenerators act on(ℂ 2 ) ⊗ as =− − + +1 − + − +1 − cos 2 +1 − isin 2 ( − +1 )+ cos 2 , (4.29) where the operators are the usual Pauli matrices, so the Hamiltonian is that of the familiar XXZ spin chain = 1 2 F Õ =1 [ +1 + +1 +cos( +1 −1)+2e ∞ ], (4.30) with anisotropy parameter Δ = cos. In the usual computational basis where(1,0) (read as a column vector) corresponds to spin up in the direction at a given site, the Temperley–Lieb generator acts on 33 spins and+1 (withperiodicboundary conditions) as =⊗···⊗⊗ © « 0 0 0 0 0 −1 −1 0 0 −1 0 0 0 0 0 ª ® ® ® ¬ ⊗⊗···⊗. (4.31) It is also possible to introduce twisted boundary conditions in the spin chain without changing the HamiltonianofEq.(4.24),bymodifyingtheexpressionoftheTemperley–Liebgeneratoractingbetweenthe first and last spins with a twist parametrized by . In terms of the Pauli operators, this twist imposes the boundaryconditions +1 = 1 and ± +1 = e ∓i ± 1 . Inthegenericcase,theXXZmodelwithmagnetization = Í =1 = and twist e i provides a representation of the module ,e i. This is not true in the non-genericcase. The XXZ and loop representations share many common features. Most importantly, the value of the ground-state energy is the same for both, and so is the value of the Fermi velocity determining the correct multiplicativenormalizationoftheHamiltonian(cf.Eqs.(4.24)and(4.30)). Thisoccursbecausetheground state is found in the same module ,e i for both models, or in closely related modules for which the extensive part of the ground-state energy (and thus, the constant ∞ ) is the same. In general, the XXZ and looprepresentationsinvolvemostlydifferentmodules . FortheXXZchain,themodulesappearinginthespin chain depend on the twist angle . For the loop model, the modules depend on the rules one adopts in treatingnon-contractibleloops,orlineswindingaroundthesystem. Ifeverythingwerealwaysbothgeneric and non-degenerate, a study of the physics in each irreducible module ,e i would suffice to answer all questions about all () models (as well as the corresponding Virasoro modules obtained in the scaling limit—see Section 8.2). However, it turns out that degenerate cases are always relevant to the physical problemsathand,andthemodulescan“break up” or “get glued” differently. To illustrate the latter point, we consider the XXZ representation with = 0 and twisted boundary conditions,withtwistparametere i = −2 . A basis of this sector is=|↑↓⟩ and =|↓↑⟩. We have then 1 = −1 −1 −1 , 2 = − 2 − −2 −1 . (4.32) We find that 1 (+ −1 ) = 2 (+ −1 ) = 0, while 1 (−) = (+ −1 )(−) and 2 (−) = (+ −1 )(−)+( 3 + −1 )(+ −1 ). Now consider the module 11 , which is the spin = 1 sector with no twist, where 1 = 2 = 0. By comparison, we see that+ −1 generates a module isomorphic to 11 . Meanwhile,− does not generate a submodule, since 2 acting on this vector yields a component along + −1 . However,ifwequotientbythesubspacegeneratedby+ −1 ,wegetaone-dimensionalmodule where 1 and 2 act as+ −1 , which is precisely the module 0, ±2. We thus get the same result as in the loopmodel—i.e.,thestructure(2.17)ofthe standard module. Consideringinsteade i = 2 ,wehave 1 = −1 −1 −1 , 2 = − −2 − 2 −1 . (4.33) Weseethat 1 (−)= 2 (−)=(+ −1 )(−),while 1 (+ −1 )= 0and 2 (+ −1 )=(− −3 )(−). Hencethistimewegetapropersubmoduleisomorphicto 0, ±2,whileweonlyget 11 asaquotientmodule. Thecorrespondingstructurecanberepresented as f 0, ±2 : 0, ±2 11 . (4.34) ObservethattheshapesinEqs.(2.17)and(4.34)arerelatedbyinvertingthe(uniqueinthiscase)arrows;the module in Eq. (4.34) is called co-standard, and we indicate this duality by placing a tilde on top of the usual notationforthestandardmodule. 34 In summary, from this short exercise we see that while in the generic case the loop and spin represen- tations are isomorphic, this equivalence breaks down in the non-generic case, where is such that the resonance criterion (Eq. (2.11)) is met. Only standard modules are encountered in the loop model while in the XXZ spin chain both standard and co-standard modules are encountered. This feature extends to larger [54]. We note that in the case where is also a root of unity, the distinction between the two representations becomes even more pronounced: the modules in the XXZ chain are no longer isomorphic tostandard orco-standardmodules. Thisidea will be further explored in a subsequent work [55]. 4.2.4 Thechoicesofmetric,andduality TheXXZchaincanbeconsideredinaprecisewayasalatticeanalogueofthetwistedfreebosontheory[54]. It is well known in the latter case that two natural scalar products can be defined. The first one—which is positivedefinite—correspondstothecontinuumlimitofthe“native”positive-definitescalarproductforthe spinchain,and,intermsofthefreebosoncurrentmodes,correspondstochoosingthehermitianconjugate ∗ = − . A crucial observation is that for this scalar product ∗ ≠ − . This means that squared norms of descendants cannotbeobtainedusingVirasoro algebra commutation relations. The second scalar product corresponds to one where the hermitian conjugate of the Virasoro generator is simply given by † = − . This “conformal scalar product” is known [56, 57, 58] to correspond, on the lattice, to a modified scalar product in the XXZ spin chain where is treated as a formal, self-conjugate parameter[59]. The loop model can be naturally equipped with two scalar products as well. Choosing basic link states to be mutually orthogonal and of unit squared norm defines a “native” positive-definite scalar product for which the Temperley–Lieb generators, the transfer matrix, and the Hamiltonian are not self-adjoint, while forthelatticeVirasorogenerators ∗ ≠ − . We will denote this scalar product by(·|·) . Meanwhile,wecanalsointroducethe“loopscalarproduct”⟨·|·⟩withinastandardmodule,obtainedby gluing the mirror image of one link state on top of the other and evaluating the result according to certain rules that we now describe. First, unless all through-lines connect from top to bottom the result is zero. We also take into account the weight of straightening the connected through-lines, using the idea of the graduatedphase: athrough-linethathasmovedtotheright(left)isassignedtheweighte i/2 (e −i/2 )for eachstep. Eachcontractibleloopcarriestheweight=+ −1 ,whileeachnon-contractibleloopcarriesthe weighte i/2 +e −i/2 . To illustrate this scalar product we take the following examples, where the solid lines around the rightmost diagrams signify that we assign them a value according to these rules (the notation onthelefthandsideisdefinedinSection2.3): ⟨(12)(3)(4)|(1)(2)(34)⟩= D E = = 0, (4.35a) ⟨(14)(23)(5)(6)|(1)(23)(45)(6)⟩= = = e 4i/2 , (4.35b) ⟨(23)(14)|(14)(23)⟩= = =(e i/2 +e −i/2 ). (4.35c) Thisloopscalarproductisthenextendedbysesquilinearitytothewholespaceofloopstates. Theadjoint † of a word in the Temperley–Lieb algebra can be defined similarly by reflecting the diagram representing itaboutahorizontalline: † = . (4.36) From this definition it is clear that the generators are themselves self-adjoint with respect to this inner product,andconsequently † = − . Itiswell known that theloop scalar product is invariant withrespect 35 totheTemperley–Liebaction:⟨|⟩=⟨ † |⟩. Theloopscalarproductis,ofcourse,notpositivedefinite. It is not degenerate, however, provided ≠ 0. Moreover, it is known to go over to the conformal scalar productinthecontinuumlimit[57]. For a given module , we can define the dual (conjugate) module f by the map ↦→⟨|·⟩; i.e., by taking mirror images. In general, we have an isomorphism f ,e i ,e −i. When ,e i is reducible but indecomposable,thecorrespondingLoewydiagramhasitsarrowsreversed,asillustratedbyacomparison ofEqs.2.17and4.34. Themodules 1 are self-dual. An important point is that, if a Temperley–Lieb module is self-dual, then since the Hamiltonian itself is, as well as the definition of scaling states, the action of the continuum limit of the Koo–Saleur generators shoulddefineanactiononthescalinglimitofthemodulethatisalsoinvariantunderdualityintheCFT.If boththeTemperley–LiebmoduleandtheVir⊕Virmoduleareirreducible,thishasnousefulconsequences. We will see (Chapters 8 and 9), however, that the modules 1 , while irreducible, have a continuum limit which is not so. Their self-duality implies invariance of the Loewy diagrams for the continuum limit with respecttoreversaloftheVir⊕Vir arrows, with very interesting consequences. 4.3 ℓ(2|1)superspinchain The periodic ℓ(2|1) superspin chain with alternating fundamental and conjugate fundamental represen- tations is closely related to the properties of the hulls of percolation clusters. It can be argued to have a conformally invariant continuum limit, which must have central charge = 0 and be logarithmic [15]. It is most conveniently defined in terms of creation and annihilation operators on a periodic lattice of length = 2. Each site carries a bosonic space of dimension 2 along with a fermionic space of di- mension 1, with the restriction that each site has only one particle. That is, the Hilbert space on site is span{ † 1 |0⟩, † 2 |0⟩, † |0⟩}≡ span{|0⟩,|1⟩,|2⟩}. Inthissectionandanyotherinvolvingthisparticularmodel, |0⟩ may variously denote the vacuum state, the first basis vector † 1 |0⟩, or the ground state, and it will be clear in context what is meant. By analogy with qubits or two-level systems, this basis will be called the computationalbasis. On each site the representations alternate. In terms of the commutation relations of the creation and annihilationoperators,theyread [ , † ]= , (4.37a) { , † }=(−1) . (4.37b) Theliteraturesometimesuses † 1 , † 2 , † , 1 , 2 ,and todenotecreationandannihilationoperatorsforthe fundamentalrepresentationonevensitesand † 1 , † 2 , † , 1 , 2 ,and fortheconjugaterepresentationon odd sites. I will just use the unbarred versions, keeping in mind the factor(−1) in Eq. (4.37b) to remember thatoddsitescarrytheconjugaterepresentation. Because of the sign in the fermionic anticommutation relations, the bilinear form⟨·|·⟩is indefinite. One cancalculatethesquarednormofabasisvectorbywritingouttheproductwithitshermitianconjugateand usingthecommutationrelationstoannihilate thevacuum. The shortcut is tolocate the fermions, andthen the squared norm is Î (−1) , where is a site where a fermion is located. For example, on = 4 sites, ⟨2200|2200⟩=(−1) 1 ×(−1) 2 =−1,and⟨2222|2222⟩=(−1) 1 ×(−1) 2 ×(−1) 3 ×(−1) 4 = 1. Asintheloopmodel, one may also naturally consider a positive-definite inner product, the standard inner product of ℂ where differentcomputationalbasisstatesaredeclared to be orthogonal and of unit norm squared: (|)= . Ifwedefine =[ † 1 † +1,1 + † 2 † +1,2 +(−1) † † +1 ][ 1 +1,1 + 2 +1,2 −(−1) +1 ], (1≤ ≤ ) (4.38) weobtainarepresentationoftheTemperley–Liebalgebrawithloopweight= 1. (Periodicityimpliesthat site+1shouldbeidentifiedwithsite1.) Inthebasis {|00⟩,|01⟩,...,|21⟩,|22⟩}oftheHilbertspaceofsites 36 and+1,thisgeneratorhasthematrixrepresentation = © « 1 0 0 0 1 0 0 0 (−1) +1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 (−1) +1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 (−1) 0 0 0 (−1) 0 0 0 −1 ª ® ® ® ® ® ® ® ® ® ® ® ® ¬ . (1≤ ≤ −1) (4.39) In the same basis of the Hilbert space of sites and 1, in that order, the very last generator has the matrix form = © « 1 0 0 0 1 0 0 0 (−1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 (−1) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −(−1) 0 0 0 −(−1) 0 0 0 −1 ª ® ® ® ® ® ® ® ® ® ® ® ® ¬ , (4.40) where is the number of fermions in the rest of the sites. For example,|0021⟩,|0022⟩, and|2022⟩ all have = 1sincethefermionsinthefirstandlastsitesaren’tcounted,and |0220⟩ has = 2. Afewmoreexamples shouldclarifytheactionof on = 4 sites: |2022⟩=−|0020⟩−|1021⟩−|2022⟩, (4.41a) |2012⟩=|0010⟩+|1010⟩−|2012⟩, (4.41b) |0000⟩=|0000⟩+|1001⟩−|2002⟩= |1001⟩= |2002⟩, (4.41c) |0020⟩=|0020⟩+|1021⟩+|2022⟩= |1021⟩. (4.41d) Adjoiningthetranslationoperatorby2sites, 2 , satisfying = 1, we obtain a representation of (1). Aswiththeloopmodel,weobtaintheHamiltonian =− 1 F Õ =1 ( − ∞ ). (4.42) In this case, since this model only furnishes a representation of the TL algebra for = 1, we have = 0, = 2,andevaluatingtheintegralofEq.(4.25), ∞ = 1. commuteswith and ,where = Õ =1 † + (−1) 2 ( † 1 1 + † 2 2 ) , (4.43a) = Õ =1 (−1) 2 ( † 1 1 − † 2 2 ). (4.43b) The allowed quantum numbers are−/4≤ ≤ /4 in steps of 1/2, and−/2+||≤ ≤ /2−|| in unit steps. The subspace where = = 0 is of primary importance, as it contains the ground state. On four sites, this sector has dimension 15, and is spanned by|0000⟩,|0011⟩,|0022⟩,|0110⟩,|0220⟩,|1001⟩, |1100⟩,|1111⟩,|1122⟩,|1221⟩,|2002⟩,|2112⟩,|2200⟩,|2211⟩, and|2222⟩. For = 6 it has dimension 93, for 37 = 8 it has dimension 639, and for = 10 it has dimension 4653. In general, the dimension is given by [60] dim()= /2 Õ =0 /2 2 2 . (4.44) This dimension grows asymptotically as dim()∼ (3 /). In fact, I computed dim()≈ 0.8269933× 3 / by computing these quantities for = 10 7 . It turns out also that F / = 3 √ 3/2 = 0.8269933..., where F is as defined after Eq. (4.24), which may be a fortuitous coincidence or may signal something important. Itisoftenhardtotellatfirstglance. By representing every integer from 0 to 3 −1 as length base 3 strings, it is simple via Mathematica (or any reasonable computing language) to pick out the strings corresponding to a basis of any sector corresponding to some combination of and (see Appendix B). Since the action of replaces pairs of adjacentmatchingdigitswithalinearcombinationofthesame(seeEq.(4.38)),itisalsosimpletoconstruct theactionofthe asmatrices,whetherjustinaparticularsector(conservedbyallofthe operators)oron the whole Hilbert space. requires some care because of the fermion number condition, but keeping this in mind, it is also doable. Then one simply takes their sum to form the Hamiltonian, and the Koo–Saleur generators. The idea that the action of is to replace adjacent pairs with a linear combination of pairs gives a way to generate the basis vectors of the smallest TL submodule containing the ground state. For symmetry reasons, weexpect theground state tocontain a component along the vector|0··· 0⟩, consisting of atype-1 bosononeachsite. Startwiththisstate,andgeneratenewbasisvectorsbyactingwithallofthe operators until no new states are generated. For an example on four sites, beginning with|0000⟩, the action of 1 to 4 generates|1100⟩,|2200⟩,|0110⟩,|0220⟩,|0011⟩,|0022⟩,|1001⟩, and|2002⟩. Repeating the procedure on all of these new states gives|1111⟩,|1122⟩,|2211⟩,|2222⟩,|2112⟩,|1221⟩, and no new states are generated after that. In practice it takes fewer iterations when starting with|0··· 0⟩,|1··· 1⟩, and|2··· 2⟩ as the seed. Note that this does not generate the full= = 0 sector, but rather an invariant submodule of it, which will be called the identity module or vacuum sector, as it contains the ground state, corresponding to the identity field. The dimension of this submodule on sites is the number of walks of length on the 3-regulartreebeginningandendingatsomefixedvertex[61]. Numerically,Ifindthatthedimensiongrows like 9.5746( √ 8) / 3/2 ≈ 12 p 2/( √ 8) / 3/2 . This asymptotic behavior should follow from the generating function ()= 4/(1+3 √ 1−8). Because the Hamiltonian in Eq. (4.42) exhibits Jordan blocks at finite size [15], it is a useful model to studytheindecomposabilityparametersof logarithmic conformal field theory (Chapter 10). 38 Chapter5 Computationalmethods 5.1 Matrixdiagonalization Matrixdiagonalizationisaprocesstofindabasisinwhichagivenmatrixisdiagonal(wherepossible—see Theorem 3). If is a matrix, one seeks a matrix and a diagonal matrix such that −1 = . This is often written as = , to easily account for partial diagonalizations, which are not invertible. For a diagonalizable matrix, the problem may be solved by computing its eigenvalues and eigenvectors—if is the matrix whose columns are the eigenvectors of , then is diagonal with its elements being the eigenvaluesof. Completediagonalizationrequiressolvingthecharacteristicequationofcompletely,thendetermining the eigenvectors by setting up a system of linear equations for each eigenvalue. Although straightforward in principle, this procedure is computationally expensive for large matrices, and subject to compounded numericalerrorsandinstabilities. Fortunately, in practice, one does not require the complete spectrum of , but only some information about a few of its extremal eigenvalues, or a small subset of the spectrum determined by some other criteria. Thisrelaxedrequirementopensuppossibilitiesforothertechniquesofcalculatingthesesubsetsof eigenvaluesandeigenvectors,whichIrefer to generically as partial diagonalization. Thesimplestexampleforcalculatingthesinglelargesteigenvalue(bymagnitude)ofisthepowermethod. Suppose 1 and 2 are the two largest eigenvalues of, and| 1 | >| 2 |. By taking an arbitrary vector 0 andrecursivelydefining +1 = /∥ ∥,then ∗ isapproximatelyequalto 1 witharelativeerror of order| 2 / 1 | (unless 0 has no component along the corresponding eigenvector, a possibility with vanishing probability). Here, the norm is the 2-norm and∗ the conjugate-transpose. If, furthermore, is hermitian,onemayextendthismethodtocalculateafewmoreeigenvectors,takingintoaccountproperties such as orthogonality and parity (and other techniques used in the variational method). The smallest eigenvalues and their corresponding eigenvectors may be accomplished (at least in principle, if not always efficiently) by applying the power method to −1 . If is not invertible because of a zero eigenvalue, then onemay“shiftandinvert,”andconsider(−) −1 withappropriatelychosen. Thistechniquealsoworks tofindtheeigenvaluesclosestto ,astheycorrespondtothelargesteigenvaluesoftheshiftedandinverted matrix. Asidefromthedifficultiesindealingwithmorethanoneeigenvector,adownsidetothepowermethod isthattheinformationobtainedisrefinedonlythroughacomputationinvolvingmultiplicationby ,which can be expensive if is large. Furthermore, the sequence 0 ,..., −1 may contain useful information, but thatinformationiseffectivelydiscardedasearlier iterations are not expressly taken into account. 5.2 Arnoldimethods For large sparse matrices, the implicitly restarted Arnoldi method is the dominant algorithm of choice to determinethedesiredpartofthespectrumandtheirassociatedeigenvectors. Moreprecisely,itdetermines abasisforthesubspacespannedbythese eigenvectors and the matrix form of restricted to this subspace 39 and in this basis. This latter matrix, much smaller than the initial matrix, can then be diagonalized using moretraditionalorblack-boxtools. The implicitly restarted Arnoldi method was developed by Sorensen, and a careful mathematical expo- sition showing how it produces a reliable output can be found in Sorensen [62] and Sorensen [63]. Here, I have gathered the appropriate definitions and algorithms that suffice to allow the reader to implement them. Furthermore,Ihavemodifiedthealgorithmsfoundthereintoworkforcomplexmatrices;asgivenin the published articles, they only work for real matrices. Throughout this section,∗ denotes the conjugate- transpose. If isan× complexmatrix,thena-step Arnoldi factorization of is a relation of the form = + , (5.1) where is an × matrix with orthonormal columns, ∗ = 0, and is a × upper Hessenberg matrix. (An upper Hessenberg matrix is upper triangular, except that it also allows nonzero elements on the first subdiagonal.) The idea is that if the residual vector is (nearly) zero, then the columns of span an invariant (or nearly invariant) subspace and is the projection of to that subspace. The-step ArnoldifactorizationisaccomplishedwithAlgorithm1. Theinputvector isarbitraryandcanbegenerated randomly. In line 1, 1 =( 1 ) means to initiate an ×1 matrix 1 with column entries given by 1 , and 1 =( 1 ) means to initiate a 1×1 matrix 1 with entry 1 . The notation(,) of lines 4 and 8 mean to adjoinacolumnto,withentriesgivenby, and, similarly, ˆ is given by adjoining a row to . Algorithm1:-stepArnoldifactorization Data: × matrix,-dimensionalcolumn vector,≤ Result: , , asinEq.(5.1) 1 1 =/∥∥; = 1 ; 1 = ∗ 1 ; 1 =− 1 1 ; 1 =( 1 ); 1 =( 1 ); 2 for= 1 to−1 do 3 =∥ ∥; +1 = / ; 4 +1 =( , +1 ); ˆ = ; 5 = +1 ; 6 ℎ= ∗ ; +1 =− +1 ℎ; 7 optional: = ∗ +1 +1 ; +1 ← +1 − +1 ;ℎ← ℎ+; 8 +1 =( ˆ ,ℎ); 9 end The-stepArnoldifactorizationisthefirststeptoapartialdiagonalization. However,theresultingmatrix does not yet reflect the criteria set for the eigenvalues sought. In a nutshell, the idea of the implicitly restarted Arnoldi algorithm is to extend the Arnoldi factorization, and use the additional data, along with some input from the user, to filter out “unwanted” eigenvalues and keep the “wanted” eigenvalues. The implicitlyrestartedArnoldimethodisgiveninAlgorithm2. Thedescriptor“implicit”referstothefactthat upon each iteration, an Arnoldi factorization is extended from steps to steps, then truncated back to steps,incontrastwith“explicitly”restartedArnoldimethodsthatgeneratenewstartingvectorsfromwhich tocomputeArnoldifactorizationsfromscratch. For Algorithm 2, the QR factorization of line 8 is a standard algorithm that is built into many comput- ing languages, and can be found in many libraries and packages. It refers to the fact that any complex square matrix can be written as a product of a unitary matrix and an upper triangular matrix . The convergence of line 13 is determined by the residual∥ ∥ being smaller than some set tolerance. In line 5, the spectrum of , ( ), furnishes an approximation to a subset of the spectrum of . Sort- ing ( ) ={ 1 , 2 ,..., , +1 ,..., + } by “importance,” a natural choice for the shifts in line 5 is { 1 ,..., } ={ +1 ,..., + }, which are “filtered” out of and . For example, if one wants the largest eigenvalues in magnitude of, then{ 1 ,..., } are the smallest elements of ( ) in magnitude. Finally,thenotation(1 :,1 :)referstothesubmatrixof consistingofitsfirst rowsand columns. 40 Algorithm2:ImplicitlyrestartedArnoldi method Data: × matrix,-dimensionalcolumn vector,, such that+≤ Result: , , asinEq.(5.1) 1 =+; 2 computea-stepArnoldifactorization using Algorithm 1; 3 repeat 4 beginningwiththe-stepArnoldifactorization( , , ), apply additional steps of the Arnoldiprocesstoobtainan-step Arnoldi factorization( , , )[i.e., run Algorithm 1, lines2–9,withline2replacedby“for= to+−1 do”]; 5 compute ( ) andselectasetof shifts 1 , 2 ,..., based on ( )or other information; 6 = ; 7 for= 1 to do 8 ( , )= QR( − ); 9 ← ∗ ;← ; 10 end 11 ← ; = +1 ; 12 ←( +1 )+ ( + ); ← (1 :,1 :); ← (1 :,1 :); 13 until convergence; 5.3 NumericalcomputationoftheJordancanonicalform Because of the instability of the Jordan canonical form with respect to arbitrarily small perturbations, a computationalalgorithmtofinditmustnecessarilyadaptthetheoryoftheJordanformtoafinite-precision situation. IhavefoundthealgorithmofKågströmandRuhe[64]tobefairlyreliablefortheproblemsstudied in Part III. In this section I summarize their algorithm and give some brief remarks about my particular implementation. However, the reader looking to implement this algorithm will probably need to refer to theoriginalarticle. 1. Suppose is a matrix to be diagonalized (which itself often comes from the output of Algorithm 2). The algorithm of Kågström and Ruhe [64] first calls for to be upper triangularized: = −1 , where is upper triangular. The authors use a sequence of library routines; I use a pre-built Schur decompositionalgorithmtoaccomplish this step. 2. Sortthediagonalelementsof sothatmultipleeigenvaluesareadjacent. Thesortingisaccomplished using a sequence of unitary matrices that swap two adjacent diagonal elements. The result is = 1 ∗ ,where 1 isstilluppertriangular, but its diagonal elements have been sorted. 3. Decide which groups of eigenvalues along the main diagonal correspond to numerical multiple eigenvalues. Ifthereare distinctnumerical eigenvalues, partition the matrix 1 as 1 = © « 11 12 ··· 1 22 ··· 2 . . . . . . ª ® ® ® ® ¬ , (5.2) witheachdiagonalblock correspondingtoadifferentnumericaleigenvalue. Supposegroup has elements,sothat isa × matrix. 4. Useeliminationmatricestoeliminatetheelementsabovethemainblockdiagonal—theblocks with < ,workingupwardsinrowsand then from left to right in columns. The result is 2 = −1 1 = diag{ 11 , 22 ,..., }. (5.3) 41 Theresultthusfarmaybesummarized as = 2 −1 , (5.4) with=. Partition the matrix into groups of columns, corresponding to the diagonal blocks : = . (1≤ ≤ ) (5.5) 5. For each , write = , where has orthonormal columns and is square and upper triangular—i.e.,theQRdecomposition for non-square matrices. 6. ′ = −1 is the restriction of to the invariant subspace spanned by the columns of (of dimension ). It is supposed to correspond to one numerical multiple eigenvalue, which is taken to betheaverageofthediagonalelements: = 1 tr ′ . (5.6) Iteratedsingularvaluedecompositionof ′ anditssubmatricesdeterminesthenumberoftowersand their heights, and from this a nilpotent matrix is constructed that is unitarily similar to ′ − uptosomenumericalerror. isstrictly upper triangular. 7. Gaussianeliminationon finallygivestheJordanform,uptoapermutationofthebasisvectors. The basis vectors—the eigenvectors and generalized eigenvectors—are the columns of the total similarity transformation obtained by accumulating all of the steps. In general, the resulting Jordan form does not have unit couplings, but it is simple enough to rescale the basis vectors to obtain the Jordan canonicalform. Themostimportantconsiderationsinvolvedecidingwhentwonumericalvaluesonthediagonalof in step 2 represent the same multiple eigenvalue. The simplest choice is to declare two diagonal elements and to represent the same multiple eigenvalue if|−| < for some eigenvalue tolerance . The choice of dependsonthenumericalprecisionofthecalculation. Aperturbationtoamatrixelementoforder in a Jordan block of rank 2 causes an order 1/2 shift in the exact eigenvalues (see beginning of Section 6.3.2). More generally, a perturbation of order in a Jordan block of rank shifts the eigenvalues by an amount of order 1/ . Thus, the eigenvalue tolerance will in part depend on the ranks of the Jordan blocks one is expecting. For a computation with working precision ∼ 10 −16 and Jordan blocks of rank at most 2, I have found 10 −7 ≲ ≲ 10 −6 to work well—knowing also that the correct eigenvalues have a separation greater thanthis. Similarly,whenJordanblocksof rank 3 are expected to appear, I take10 −5 ≲ ≲ 10 −4 . 42 PartII Theory 43 Chapter6 EmergingJordanforms This chapter describes how the formation of Jordan blocks can be observed in certain limits, when they are absent except possibly in the limit. More precisely, suppose an operator () parametrized by is generically diagonalizable, except at = 0 . Can we infer properties of( 0 ) without studying it directly, butinsteadbylookingatthelimitlim → 0 ()? Themathematicalframeworktobedescribedinthischaptercameaboutwhileinvestigatingtheformation ofJordanblocksintheHamiltoniansandtransfermatricesoflatticeregularizationsoflogarithmicconformal fieldtheoriesasthenumberofsitesonthelatticeincreased. Forsomerepresentations,Jordanblockscanbe found in the Hamiltonian or transfer matrix of a finite-size lattice, and computations could be carried out by working directly with the eigenvectors and generalized eigenvectors. In this situation many questions of interest can be answered on the lattice and then extrapolated to the continuum theory, as described in Chapter4. There are also indecomposable structures known or conjectured to appear in the scaling limit that are notmanifestonthelattice. Clearly,thescalinglimitisnotdirectlyaccessiblecomputationally. Wemustthen hopethattheJordanblockinthecontinuumtheory“buildsup”insomesensefromthelatticemodel,rather than appearing suddenly and discontinuously, and if so, whether there are signatures of this build-up that canbedetected. As we will see, such hopes are auspicious, and this kind of approach is possible. One strategy is to investigate the effect of an indecomposable structure on four-point functions and measure the amplitudes that would be affected by the presence or absence of such a structure. I describe how this strategy was appliedsuccessfullytotheloopmodelinSection 9.2. Nevertheless,onemayreasonablyaskwhetheradirectsignatureofanemergingJordanblockexists—one that would hold in a much more general setting, and not just limited to four-point functions in logarithmic conformalfieldtheories. Asitturnsout,theanswerisaffirmative,andalmostembarrassinglysimple,while being completely general. While there are some involved algebraic expressions, nothing in this chapter is conceptually complicated. Yet neither I nor my collaborators have been able to find this framework in the mathematicalliterature. Thelessonhereis that it pays to ask exactly the right questions. Because of the generality of the framework, I describe the theory of emerging Jordan blocks in a fully mathematicalsetting,devoidofanyphysical context, except for some references to its application. 6.1 The measure Consider a one-parameter continuous family of linear operators or matrices, (), acting on a finite- dimensionalvectorspace overℝorℂ. Theeigenvaluesofagenericoperatorarealldistinct,andthussuch anoperatorisdiagonalizable. Wheneveranoperatorisdiagonalizable,ithasabasisofeigenvectors. Asthe operator changes continuously, it is possible to choose a basis that changes continuously with it, so long as the operator remains diagonalizable. Even if one encounters a situation of non-diagonalizability, the limit of the continuously parametrized set of basis vectors is well-defined, although it may not be a basis, and it willnotbea basisofthelimitingmatrix. 44 Letusconcretelyconsidertwoeigenvaluesof(),()and(),withassociatedeigenvectors()and (),andsupposethatat= 0 ,( 0 )= ( 0 ). Does( 0 )remaindiagonalizable? Ifso,thenthegeometric eigenspace associated to the eigenvalue( 0 )= ( 0 ) has dimension 2, and is spanned by( 0 ) and( 0 ). In particular, ( 0 ) and( 0 ) remain linearly independent. If, however, ( 0 ) is not diagonalizable, with ( 0 ) being a defective eigenvalue, then the geometric eigenspace associated to( 0 ) has dimension 1, yet it is still spanned by( 0 ) and( 0 ). This means that( 0 ) and( 0 ) lie along the same line in, or that theyareparallel(orantiparallel). For a such a pair of eigenvectors{,} (hereafter we do not explicitly notate the dependence on), one canconsider thequotient (,)= (|) p (|)(|) , (6.1) where(·|·)is any positive-definite inner product. For simplicity, we will take it to be the standard inner product. Note that by the Cauchy–Schwarz inequality,|(|)| 2 ≤(|)(|) implies|(,)|≤ 1. The proof oftheCauchy–Schwarzinequalityalsoshowsthatitisanequalityifandonlyifthetwovectorsareparallel: = . If (,) approaches a number of unit modulus, e i , we are approaching a non-diagonalizable matrix,sinceabasisisbydefinitionalinearlyindependentset,whichcannotcontainparallelvectors. Thus Iwillconsider|(,)|→ 1as→ 0 tosignifythataJordanblock(ofrank2)formsinthatlimit,butisnot otherwisepresent. Tobesure, mustbeappliedtoapairofeigenvectorsofforwhichtheeigenvaluesarestilldistinct,and studiedinthelimitastheeigenvaluesbecomeequal. Ifitisappliedtotwoindependenteigenvectorswiththe sameeigenvalue,thenthetestisnotmeaningfulbecausethetwoeigenvectorscanberedefinedbychoosing adifferentspanningsetfortheeigenspace. Ifitisappliedtoaneigenvector/generalizedeigenvectorpairfor a defective eigenvalue, then the two are not parallel. Furthermore, the generalized eigenvector can always beredefinedbyaddingascalarmultipleofthe eigenvector, thus changing the value of . Finally, it seems that the diagnostic||→ 1 does not tell us anything about the generalized eigenvector; both and convergetothesamepropereigenvector,uptoascale. Wewillseehowthisinformationcomes about. Letmeconcretelyillustratetheforegoing discussion on a test example. The matrix ()= 0 1 0 (6.2) has(unnormalized)eigenvectors(1,0)and(1,). As→ 0,theybecomeequal,andaJordanblockobviously formsinthelimitingmatrix(0). If and are the two eigenvectors, then (,)= (|) p (|)(|) = 1 √ 1+ 2 , (6.3) whichisclearlylessthan1 for≠ 0 and1 for= 0. Next,Iturntotheconstructionofthegeneralizedeigenvectorat= 0,whichis(0,1). Actually,itcanbe anything of the form(,1), and I have made the choice that is orthogonal. One way to construct the gener- alized eigenvector from the proper eigenvectors as→ 0 is to perform Gram–Schmidt orthogonalization. For example, if we start with the pair(,), subtract off the component of along, and renormalize, we obtainand˜ =(0,1),whichisexactlywhatwewant. Ofcourse,forgeneric,˜ isnolongeraneigenvector. Because both and converge to the same vector at = 0, we should be able to treat them on an equal footing. IfweapplytheGram–Schmidtprocedure to the ordered pair(,), the result is (,) GS →(,˜ )≡ (1,), || √ 1+ 2 ,− sgn √ 1+ 2 , (6.4) andwehave lim →0 (,˜ )=((1,0),(0,∓1)), (6.5) 45 dependingonwhetherthelimitisapproachedfromaboveorbelow. Eitherway,werecovertheeigenvector andthegeneralizedeigenvectoruptoasign, which is easily corrected. Giving a label to the preceding situation, it seems natural to say that() is an emerging Jordan block of rank-2inthe limit→ 0. Inasense,theemergingJordanblockwiththeGram–Schmidt-orthogonalizedvectoristheonlypossible outcome if the two eigenvectors become parallel. Consider two eigenvectors with distinct eigenvalues that become parallel in some limit. They span a two-dimensional subspace⊂ . If we track this subspace, as the limit is taken, it continues to remain two-dimensional, but we seem to have only one basis vector for it as the eigenvectors have converged to become parallel. Equally well, we can take as a basis of one of theeigenvectorsandanorthonormalizedoneconstructedfromtheother. Thesecondvectorisnolongeran eigenvector,butitremainswell-definedasthelimitistaken. Itdoesnotbecomeapropereigenvector,either, because if we had two linearly independent eigenvectors in the limit then they would not have become parallel. The limiting subspace can only be the eigenspace associated to a single eigenvalue, because of the primary decomposition theorem (Section 1.3). The Gram–Schmidt vector must then be a generalized eigenvector,andtheoperatorrestrictedto this subspace has a nontrivial Jordan form. * * * Letusnowconsidertherank-3case. Consider the matrix (,)= © « 0 1 0 0 1 0 0 ª ® ¬ , (6.6) forwhichweconsiderthelimit,→ 0. The eigenvectors are =(1,0,0), (6.7a) =(1,), (6.7b) =(1,,(−)), (6.7c) correspondingtoeigenvalues0,,and. The values are (,)= 1 √ 1+ 2 , (6.8a) (,)= 1 p 1+ 2 + 2 (−) 2 , (6.8b) (,)= 1+ p (1+ 2 )(1+ 2 + 2 (−) 2 ) . (6.8c) As,→ 0,,, →(1,0,0) and become parallel, all three measures become1, and we obtain a rank-3 Jordanblockfortheeigenvaluezero. Wecan also consider separately the three cases where: 1. → 0, with fixed. 2. → 0,with fixed. 3. →,with fixedtoanonzerovalue. Case1(resp. 2,3)leadsto(,)→ 1(resp. (,)→ 1,(,)→ 1),withtheothertwotendingtoalimit that is less than 1. The limiting matrix then displays a rank-2 Jordan block for the eigenvalue0 (resp. 0,), andasingleeigenvalue (resp. ,0)thathas a geometric eigenvector. ApplyingtheGram–Schmidtprocedure to the ordered triple(,,), we obtain the standard basis (,,) GS →((1,0,0),(0,1,0),(0,0,1)), (6.9) 46 whichisexactlythegeometriceigenvectoralongwiththetwogeneralizedeigenvectorsoftheJordanblock. If we consider any other ordering for the three eigenvectors and take the limit ,→ 0, the same basis results,uptosomesignsinthevectors,whichareeasilycorrected. ThattheresultingbasisgivestheJordan canonical form of (0,0) is fortuitous; we will see how values of the Jordan couplings emerge in Section 6.2. Inparticular,normalizationtounityintheGram–Schmidtprocedureisnotnecessary,solongasnorms remainnonzerointhelimit. Asbefore,Iwillsaythat(,)isanemergingJordanblockofrank3inthelimit,→ 0. Butwecan also see that(,) contains an emerging Jordan block of rank 2 in any of the three limits→ 0, → 0, or→. Theseobservationsnaturallysuggestthe following conjecture. Conjecture1(EmergingJordanblockofrank). Let :→()beamapwhosecodomainisthespaceoflinear operators on the-dimensional vector space (overℝ orℂ), with a metric space. Suppose() is diagonalizable for ∈ , with distinct eigenvalues{ 1 (),..., ()} and associated eigenvectors{ 1 (),..., ()}. Let 0 ∈ ⊂ , where is the boundary of and the closure of. The domain of may be extended to, since () iscomplete. Suppose,as→ 0 , 1. Asubsetoftheeigenvalues,{ 1 (),..., ()}, all tend to the same limit; 2. forallpairs{,}⊂{1,...,},lim → 0 |( (), ())|= 1; 3. forallpairs{,}⊂{1,...,} suchthat≤ < ≤ ,lim → 0 |( (), ())| < 1. Then( 0 ) contains a Jordan block of rank for the eigenvalue . A basis of the generalized eigenspace is given by performing the Gram–Schmidt orthonormalization procedure on the set{ 1 (),..., ()}, in any order, and taking thelimit→ 0 . Thematrixof( 0 )restrictedtothiseigenspace,inthebasisofthelimitingGram–Schmidtvectors, is upper triangular. We say that() contains an emerging Jordan block of rank- in the limit→ 0 , or if=, then() isanemergingJordanblockinthesame limit. As formal as this conjecture sounds, the assumptions are not overly restrictive, and merely abstract the two concrete examples I exhibited in the most general terms I could imagine. In this formal setting, the rank-2 example corresponds to =ℝ\{0}, the punctured real line, and the rank-3 example corresponds to = ℝ 2 \{(0,0)}, the punctured plane. I will prove this for the cases = 2 and 3 with = ℝ , and give expressionsthatsuggesthowtoapproachtheproblemandproveitforallpositiveintegers inSection6.2. I also note that the Gram–Schmidt procedure must produce orthonormal vectors (in particular, no zero vectors)sincethestartingvectorsarelinearly independent, being eigenvectors with distinct eigenvalues. All of the examples thus far have dealt with finite-dimensional vector spaces, and these most directly inform Conjecture 1. Certainly many subtleties exist in infinite-dimensional vector spaces that make many conclusionshardertoestablishorfalse. Butitseemsnaturalheretoboldlyproclaimanobviousgeneraliza- tion. Conjecture 2 (Emerging Jordan blocks in infinite dimensions) . Conjecture 1 holds when suitably generalized to infinite-dimensional vector spaces and arbitrary-rank emerging Jordan blocks. The dimension may be countably infiniteoruncountable. TosomeextentsomeofConjecture2mustbetrueifwearetostudyseparableHilbertspaces,thedomains of our conformal field theories, using a sequence of finite-dimensional vector spaces to extrapolate their propertiesaccurately. Theframeworkdescribedinthissectionmayalsobeappliedtoasequenceofoperators(()) ∞ =1 inthe limit →∞, thinking of the eigenvalues and measures at discrete parameter values being smoothly interpolatedbetweenthediscretevalues,even if the operator itself does not exist for noninteger. * * * In general, an operator may have multiple emerging Jordan blocks as a limit is taken. The limiting operator will then have a Jordan form that contains multiple blocks, all emerging simultaneously. Thus an appropriatenamefortheformalismasawhole is that of the emerging Jordan form. 47 6.2 RatesofconvergenceandtheJordancoupling TheexamplesoftheprevioussectionmayhaveseemedartificialinthatIbeganwithamatrixwhoselimiting form was obviously a Jordan block, and then proceeded to verify that fact. In fact, those observations hold in a more general setting where the conclusion is not evident in the setup, as I demonstrate for ranks 2 and 3. Unfortunately, the generalization to arbitrary is not as straightforward as allowing all indices to run from1 to; we will see what complications arise in the rank-3 case compared to the rank-2 case, and what itmightmeanforthegeneralrank- case. Theprimaryrecurringthemeofthissectionisthatthewayinwhich dependsonthedifferencesamong theeigenvalueshasimplicationsfortherank and form of the emerging Jordan blocks. I believe the propositions and conjectures in this chapter are of quite general validity. However, to deal with the simplest cases, in this section all vector spaces are real. Certainly the reader should be able to imagine that a generalization exists for complex vector spaces, although the expressions will pick up complex conjugates in some places, and signs become phases. But the real case, which I have studied the most thoroughly (especially in the applications of Part III), should suffice to validate the ideas proposed here. 6.2.1 Rank2 Considermoregenerallythematrix (,)= 0 . (6.10) (The conclusions do not change if = (,) depends continuously on the parameters.) By changing the sign of one of the basis vectors, we may assume without loss of generality that ≥ 0. The eigenvectors correspondingtoeigenvalues and arenow=(1,0) and =(,−). Their measure is ≡(,)= p 2 +(−) 2 . (6.11) The measure correctly identifies that as long as ≠ 0, (,) is an emerging Jordan block of rank 2 as →. Takingseriesexpansionsin−, we find that = 1− (−) 2 2 2 +((−) 4 ), (6.12a) p 2(1−)= |−| +((−) 3 ). (6.12b) These expressions are invariant under rescaling of the matrix (up to a sign). If → then the new eigenvaluesareand,whiletheeigenvectorsremainthesame,and,forinstance, p 2(1−)=|−|/, where is the new off-diagonal matrix element. Turning this line of thought around, the relative rates of convergence of→ 1 and the difference of the eigenvalues to 0 may be able to tell us about the Jordan coupling. This hunch turns out to be true, as the following calculation shows. Suppose we have an operator andtwovectorssuchthat =, (6.13a) =, (6.13b) andwearegiventhat (|) p (|)(|) =, (6.14) 48 whereeverythingispotentiallydependenton and,butnotdenotedexplicitly. Notealsothatwehavenot assumed to act on a 2-dimensional vector space. This situation obtains, for instance, when one partially diagonalizesalargermatrixbutdoesnotknowaprioritheformof whenrestrictedtothesetwovectors. Bychangingthesignof,ifnecessary,wemayassumethat≥ 0. Createthenewvector(byorthogonalizing from,thennormalizingtheresult), ˜ = − (|) (|) s − (|) (|) − (|) (|) = ˆ −ˆ p 1− 2 . (6.15) Byroutinecalculationwefindthat ˜ =˜ + (−) p 1− 2 ˆ . (6.16) Therefore,inthebasis=(ˆ ,˜ ),thematrix of the operator is = © « (−) p 1− 2 0 ª ® ¬ , (6.17) Iflim → < 1,then lim → (−) p 1− 2 = 0. (6.18) This means that if the limiting eigenvectors and are not parallel, the limiting matrix representing is diagonalizable, with = 2 . If, on the other hand, lim → = 1, then, assuming analyticity, we may write = 1− (−) 2 2 2 +···, (6.19) with > 0. The linear term disappears since must have a maximum at = . The limiting off-diagonal matrixelementisthen lim → (−) p 1− 2 = lim → − p 1− 2 / =, (6.20) and we obtain a rank-2 Jordan block for the eigenvalue, with Jordan coupling . The central expression can be interpreted as the ratio of the difference in eigenvalues to a function of . To order 2 , it may be checkedthat p 2(1−)= p 1− 2 /,sothereisnoconflictwithEq.(6.12). Thus,theJordancouplingisgiven bytherelativeratesofconvergence. If islargethen→ 1faster,meaningthatthevectorsbecomeparallel faster. Sotheterm“coupling”isparticularly appropriate. Turning things around yet again, given , , , , , and as in Eqs. (6.13) and (6.14), we may define the emerging Jordan coupling as =(−)/ p 1− 2 . In these terms, is an emerging Jordan block of rank 2 if and only if the emerging Jordan coupling tends to a nonzero value. This last characterization is basis-independent, and depends only on knowledge of the eigenvalues and eigenvectors in any basis from whichitispossibletocalculate. We have found, in an arbitrary operator , an emerging rank-2 Jordan block based solely on the diagnostic that → 1. With some simple calculations, we are also able to extract the emerging Jordan coupling. Isummarizetheprecedingdiscussion in the following result. Proposition1(Characterizationsofemergingrank-2Jordanblocks). Let bealinearoperatorand and two ofitseigenvectors,with and beingtheirassociated eigenvalues. In the limit→, the following are equivalent: 49 1. (,)→ 1. 2. ≡(−)(,)/ p 1−(,) 2 tends to a nonzero value. 3. containsanemergingJordanblockof rank 2 for the eigenvalue with emerging Jordan coupling. All of the concepts of this section are fully illustrated on a concrete, less artificial example coming from physics in Section 9.5.2, using the matrix in Eq. (9.55). The reader may turn to that example now, which endswiththeparagraphcontainingEq.(9.64), as it is also written in a purely mathematical context. 6.2.2 Rank3 Let us try the same exercise with three vectors and try to obtain the corresponding result for an emerging rank-3Jordanblock. Sincetheresultingexpressionsarenotthoseonewouldimmediatelyguessbyanalogy with the rank-2 case, it is worth seeing this case written out in detail. Throughout this section I suppress explicitnotationoffunctionaldependencies on the real parameters,,, 1 , 2 , and 3 . Beginwiththematrix = © « 1 0 2 0 0 3 ª ® ¬ . (6.21) Byadjustingthesignsofthethreebasisvectors, we can make the three off-diagonal elements nonnegative. Theeigenvectorsare 1 =(1,0,0), (6.22a) 2 =(, 2 − 1 ,0), (6.22b) 3 =(+( 3 − 2 ),( 3 − 1 ),( 3 − 1 )( 3 − 2 )), (6.22c) correspondingtoeigenvalues 1 , 2 ,and 3 . The values ≡( , ) are 12 = p 2 +( 2 − 1 ) 2 , (6.23a) 13 = +( 3 − 2 ) p 2 2 + 2 ( 3 − 1 ) 2 + 2 ( 3 − 2 ) 2 +( 3 − 1 ) 2 ( 3 − 2 ) 2 +2( 3 − 2 ) , (6.23b) 23 = 2 +( 3 − 2 )+( 3 − 1 )( 3 − 2 ) p ( 2 +( 2 − 1 ) 2 )( 2 2 + 2 ( 3 − 1 ) 2 + 2 ( 3 − 2 ) 2 +( 3 − 1 ) 2 ( 3 − 2 ) 2 +2( 3 − 2 )) . (6.23c) Itmaybecheckedthatas 2 , 3 → 1 ,allthree valuesapproach1ifandonlyiftheJordancanonicalform of the limiting matrix has a rank-3 Jordan block. Whether this situation obtains depends on the values of , , and . It may also be checked that in the limit of one pair of eigenvalues becoming equal, with the third fixed to a different value, the value of the respective pair of eigenvectors tends to 1 if and only if the Jordancanonicalformofthelimitingmatrixhasarank-2Jordanblock. Theverificationofthesestatements requiresatediousexaminationofcasesand is best done by computer algebra. Nowsupposethedifferencesineigenvaluesareallsmall( 2 − 1 =(), 3 − 2 =(), 3 − 1 =()), andforconvenienceassume 3 > 2 > 1 . Taking series expansions gives 12 = 1− ( 2 − 1 ) 2 2 2 +( 4 ), (6.24a) 13 = 1− ( 3 − 1 ) 2 2 2 + ( 3 − 1 ) 2 ( 3 − 2 ) 3 +( 4 ), (6.24b) 23 = 1− ( 3 − 2 ) 2 2 2 + ( 3 − 2 ) 2 ( 3 − 1 ) 3 +( 4 ). (6.24c) 50 Perhapssurprisingly,alloftheleadingnontrivialcoefficientsonlygiveinformationabout . Togettheratio /,wemustusethethird-orderterm,andtogetthevaluesof and separatelyweneedthefourth-order term (not shown here). Based on the expressions, it does not seem productive to fiddle around with the three measures and guess at the functions of them that yield the three parameters,, and, especially whentheyarisemorereadilyinanupcoming analysis. However, the fact that the leading terms have the same dependence on yields a useful test. Suppose, among a set of eigenvectors, we are looking for three of them that form an emerging rank-3 Jordan block. Insteadofstudyingeverysubsetofthreevectorsindetail,onemaysimplycalculatethe measuresbetween allpairs,andextractthe(approximate)valueofforthatpair(forinstance,bycomputing( − ) / q 1− 2 , asinEq.(6.20)). Allthreepairsamongthecandidatevectorsforminganemergingrank-3Jordanblockmust displaythesamecoefficient. Conversely,supposewearegivenanoperator and three vectors such that 1 = 1 1 , (6.25a) 2 = 2 2 , (6.25b) 3 = 3 3 , (6.25c) and suppose we know the values ≡ ( , ). We assume the are distinct and study the situation in the limit where they all tend to the same value. Since the vector spaces are assumed real, = . Let us orthonormalizetheeigenvectorsfollowing the Gram–Schmidt procedure, and obtain 1 =ˆ 1 , (6.26a) 2 = ˆ 2 − 12 ˆ 1 √ 2 , (6.26b) 3 = (1− 2 12 )ˆ 3 −( 13 − 12 23 )ˆ 1 −( 23 − 12 13 )ˆ 2 √ 2 3 , (6.26c) where 2 = 1− 2 12 and 3 = 1− 2 12 − 2 13 − 2 23 +2 12 13 23 . In this basis,=( 1 , 2 , 3 ), the operator has thematrixrepresentation = © « 1 ( 2 − 1 ) 12 √ 2 ( 3 − 1 )( 13 − 12 23 )+( 3 − 2 )( 23 − 12 23 ) 12 √ 2 3 0 2 ( 3 − 2 )( 23 − 12 13 ) √ 3 0 0 3 ª ® ® ® ® ® ¬ . (6.27) Thedifferencesintheeigenvaluesagainplay an important role. It may be checked that substitution of Eq. (6.23) into Eq. (6.27) yields Eq. (6.21), when 3 > 2 > 1 . Lifting this assumption, we still obtain the same matrix, up to possible signs appearing in front of,, and . The verification of this assertion is yet again another tedious exercise in algebra that is best turned over toacomputer. 6.2.3 Rank Asofthiswriting,thegeneralformalismforemergingrank- blockshasyettobecompletedinasystematic manner. Instead,Icollectsomefactsandconsiderationsthatseemunavoidableintheprocessofcompleting thistheory. In the rank-2 and rank-3 cases I adjusted the signs of the eigenvectors so that the off-diagonal matrix elementswouldcomeouttobenonnegative. Thereasonisthatthroughvariouscomputationalexplorations, itseemedthatonlytheabsolutevaluesoftheoff-diagonalelementswererelevant,appearinginexpressions as 2 or||,forinstance. Thiswasalwayspossiblebecausethenumberofoff-diagonalelements, (−1)/2, was at most the dimension of the space. For≥ 4, it does not seem immediately obvious that the same assumptionconstitutesnolossofgenerality,andthatnonnegativeoff-diagonalelementsarealwayspossible 51 through adjusting signs. At the same time, it is not obvious that there is an easy counterexample, either. There exist constraints among the values, so that not all of them are independent. For example, it is intuitivelyobviousthat | |= 1 =⇒ | |=| |, (6.28) whichfollowseasilyfromthedefinitionsand the Cauchy–Schwarz inequality. Onesuchconstraintfollowsfromtheunremarkableobservationthat ≡ cos isjustthecosineofthe angle between and in the geometry of the space determined by the inner product, where 0≤ ≤ . Fromthegeometricfact + ≥ ≥| − | it is not hard to show 1 2 q (1+ )(1+ )+ q (1− )(1− ) ≥ r 1+ 2 ≥ 1 2 q (1+ )(1+ )− q (1− )(1− ) . (6.29) Suppose we begin with an operator and eigenvectors such that = . Following the steps of the rank-2 and rank-3 analyses, the first step is to create an orthonormal basis. The result of the Gram– Schmidtprocesscanbewritteninclosedform,usingformaldeterminants. Startingwiththeorderedsetof vectors( 1 ,..., ),theresultis = 1 p Δ −1 Δ det © « ( 1 | 1 ) ( 2 | 1 ) ··· ( | 1 ) ( 1 | 2 ) ( 2 | 2 ) ··· ( | 2 ) . . . . . . . . . . . . ( 1 | −1 ) ( 2 | −1 ) ··· ( | −1 ) 1 2 ··· ª ® ® ® ® ® ¬ , (6.30a) with Δ = det © « ( 1 | 1 ) ( 2 | 1 ) ··· ( | 1 ) ( 1 | 2 ) ( 2 | 2 ) ··· ( | 2 ) . . . . . . . . . . . . ( 1 | ) ( 2 | ) ··· ( | ) ª ® ® ® ® ¬ . (6.30b) Formaldeterminantscontainingvectorsaredefinedbycofactorexpansion. TheresultoftheGram–Schmidt procedure is unaffected by rescaling of all vectors by positive constants. We may therefore replace with ˆ andobtain = 1 p −1 det © « 11 21 ··· 1 12 22 ··· 2 . . . . . . . . . . . . 1,−1 2,−1 ··· ,−1 ˆ 1 ˆ 2 ··· ˆ ª ® ® ® ® ® ¬ , (6.31a) with = det( ) ,=1 , (6.31b) using the fact that(ˆ |ˆ ) = . As a check, the first three values give the unit vectors of Eq. (6.26), using symmetryand = 1. Because isasumofterms,eachofwhichcontainsexactlyoneˆ ,theactionof on isgivenbythe sameexpressionfor withˆ replacedby ˆ : = 1 p −1 det © « 11 21 ··· 1 12 22 ··· 2 . . . . . . . . . . . . 1,−1 2,−1 ··· ,−1 1 ˆ 1 2 ˆ 2 ··· ˆ ª ® ® ® ® ® ¬ (6.32) Toseethis,imagineperformingthecofactorexpansionalongthebottomrow,applying,thenputtingthe resultbackintoformaldeterminantform. 52 6.3 Nextsteps 6.3.1 Furtherstudyofemerging Jordanblocks The typical view of the Jordan canonical form today is that it is primarily a conceptual tool, and not useful for real computations [65]. Either there is a 1 in the superdiagonal of the Jordan canonical form or there is a 0, and this 1 can be destroyed by an infinitesimal perturbation of a matrix element. However, in this chapter I have built an approximation to the Jordan form that remains stable to small perturbations. This fact allows the tools of analysis—limits and continuity—to be brought into a problem that had previously beenthoughttobediscontinuouslyall-or-nothing. Fortherank-2case,thereisacuriousobservation. Let()becontinuouslyparametrizedby,and() and() the corresponding eigenvectors that lead to an emerging Jordan block in some limit. In each case wehaveseensofar,oneoftheeigenvectorsisindependentoftheparameter (moregenerally,ameaningful combination of parameters that tends to zero, as shown in the following examples). For the eigenvectors of the test matrix in Eq. (6.2), the eigenvector(1,0) is obviously independent of as it forms an emerging Jordanblockwith(1,). Moregenerally,for(,)inEq.(6.10), isindependentof−. Intheexample of Section 9.5.2 beginning with Eq. (9.55), eigenvector 5 of Eq. (9.60) does not depend on . In the test matrix of Eq. (6.21) we can see three examples of this pattern: if → there is an emerging Jordan block between and ,with beingindependentof − ,where1≤ < ≤ 3. SoIwonderifthisisageneral patternthatcanbeproved. Ihaveworkedouttherank-2andrank-3emergingJordanblocksinfulldetail. Whilethegeneralrank- case will likely be fairly involved and certainly less commonly used than the lowest rank cases, it would be desirabletocompletethisderivationandgiveasenseofcoherencetotheframeworkaswellaslinkstoother fields of mathematics where it might be profitably used. Pushing this to its limit, an emerging countably infinite Jordan tower may also find practical application and provide new perspectives, as I alluded to in Section1.3withtheharmonicoscillator. Finally, I have assumed that all vector spaces were real, which simplified many calculations as = . What might we find for complex inner product spaces, or inner product spaces over other complete topologicalfields? Moregenerally,suppose isacommutativeringwithidentity,andaproperideal. Then the concept of Cauchy sequence can be defined [18, p. 162], and the -adic completion of is (isomorphic to)theinverselimitlim ←−− / . Modulesoverthe-adiccompletioncouldpotentiallyfurnishanothersetting for emerging Jordan blocks, since there is now the concept of the limit of a sequence tending to the unit element. 6.3.2 PossibleapplicationsofemergingJordanblocks Thematrix 1 1 2 1 (6.33) has eigenvalues1+ and1−, with respective eigenvectors(1,),(1,−). You and I agree that this would be a fairly asinine basis to use for practical computations when is small, and especially when is not certain,butarandomvariablefollowingadistributionwithzeromeanandsomesmallvariance. Thislatter situation is common, such as that resulting from compounded numerical errors. In this case, the emerging Jordan block would be a more useful form of the matrix, and the coefficients of other vectors expanded in theorthogonalbasisobtainedfromtheGram–Schmidtprocedurewouldbestabletosmallperturbationsin . Awell-knownresultfromthetheoryofsecond-orderlineardifferentialequations(withgeneralizations tohigher-orderequations)involvesaso-called circuit matrix [66, Theorem 15.2.1]. Theorem 5. If () and () are analytic for 1 <|− 0 | < 2 , then the second-order linear differential equation ′′ +() ′ +() = 0 admits a basis of solutions{ 1 , 2 } in the neighborhood of the singular point 0 , where 53 either 1 ()=(− 0 ) () and (6.34a) 2 ()=(− 0 ) (), (6.34b) or,inexceptionalcases(whenthecircuitmatrix is not diagonalizable), 1 ()=(− 0 ) () and (6.35a) 2 ()= 1 ()[ 1 ()+log(− 0 )]. (6.35b) Thefunctions (),(),and 1 () areanalytic and single-valued in the annular region. InTheorem5,theexponents and aretheeigenvaluesoftherank-2circuitmatrix. If,however, and areveryneareachother,thesolutions 1 and 2 inthetheoremcouldlookverysimilartoeachother,and may not furnish as useful a basis. Theorem 5 could be reformulated in terms of an emerging Jordan block forthecircuitmatrix,andonewouldseehow the logarithm emerges as well. The fact that logarithms appear in the correlation functions of logarithmic conformal field theory is well-connectedtothefactthatthedilationoperator 0 isnotdiagonalizable. Theselogarithmictermswere initiallyfoundbysubtractingdivergencesandtakinglimits,aprocedurethatprobablyalsohasaparallelto subtractingcomponentsalongearliervectors in the Gram–Schmidt process. 54 Chapter7 BiorthogonalanddualJordanquantum physics One of the postulates of essentially all formulations of quantum physics involves the stipulation that observablesare representedbyhermitian operators. This assumption has the important consequences that their eigenvalues, representing the possible values of measurements, are real, and that the time evolution e −i associated with an hermitian Hamiltonian is unitary. Furthermore, the eigenstates are orthogonal (Theorem1). Non-hermitian Hamiltonians have appeared in the physics literature, and are usually dealt with on an ad hoc basis. For instance, Hamiltonians with eigenvalues that have a negative imaginary part have been used to describe decay processes, and a complex eigenvalue automatically renders the Hamiltonian to be non-hermitian. To illustrate how such a Hamiltonian implies decay, suppose=(−iΓ), withΨ > 0. Thenthesquarednormoftheinitialstate decays as † ()()=(e −i ) † e −i = e −2Γ † , (7.1) withΓ proportional to the decay rate, or inversely proportional to the lifetime. Nevertheless, the use of a non-hermitian Hamiltonian can be regarded as an approximation to a more fundamental theory, since decay processes should really be treated in the formalism of quantum field theory, which allows for the creationand annihilationofparticles. In another context, Bender [67] and collaborators [68] have developed what is now known as - symmetric quantum mechanics. These involve Hamiltonians with symmetry rather than hermiticity. Under certain restrictions, -symmetric Hamiltonians can be shown to have real eigenvalues, thus pre- serving the measurement interpretation. However, calculations in this formalism are done by deforming the geometry of the problem (wave functions in one dimension, for instance, are no longer defined on the real line but in “asymptotic wedges” in the complex plane) and using the inner product⟨|⟩ (Eq. (78) in Bender [67]), which renders the Hamiltonian to be hermitian again (recall that hermiticity is definedwithrespecttosomeinnerproduct[Section1.1],sochangingtheinnerproductchangesthenotion ofhermiticity). Other uses for non-hermitian Hamiltonians have been described within some of the references [67, 68, 69]. SingleJordanblocksofsmallrankcanbehandledastheyariseexceptionallyinsomephysicalproblems. However, there is a growing class of models where Jordan blocks in the Hamiltonian proliferate, and physically relevant operators couple different blocks. Thus they cannot be handled in isolation. The rich indecomposable structures of logarithmic conformal field theory, for instance, can be traced to the nondiagonalizabilityofthedilationoperator,whichplaystheroleofaHamiltonian. Likewise,nonunitarity isnaturalingeometricproblemswhereobservablesarenotlocalwithrespecttothebasicdegreesoffreedom, andinsystemswhereonehastoaverageover disorder. In the absence of an analytical framework, or when setting one up becomes intractable, numerical studies can provide a valuable pathway for progress, and inspire the correct analytical approach. Though 55 not conceptually difficult, carrying out accurate and efficient numerical studies requires the synergy of a number of computational techniques. Some of these techniques particularly suited to the problem at hand aredescribedinChapter5. Takenasawhole,thisworkshouldallowthereadertocarryouttheprocedures describedinthischapterbeginningwithan arbitrary operator. The spectral theorem for normal operators (Section 1.2) and the measurement postulate of quantum mechanics (Section 3.1) may be framed in terms of orthogonal projection operators. The purpose of this chapter is to find these orthogonal projection operators in a situation where the assumption of hermiticity hasbeenlifted. Inthischapterweassumethatvectorspaces are finite-dimensional. 7.1 Biorthogonalprojectionoperators The first step is to consider Hamiltonians that are non-hermitian but nevertheless diagonalizable with real eigenvalues. The exposition of this section follows Brody [69], where many additional elementary results are spelled out for the non-degenerate case, and I add in some of my own observations. Because of the primacyoftheHamiltonianinquantumphysics,throughoutthissection,Iuse“Hamiltonian”tomeanany linear operator which may be non-hermitian, but is diagonalizable with real eigenvalues. The eigenvalues mayhaveanypatternofdegeneracies. Tofixnotation,let beaHamiltonian. Then{ , } is a biorthogonal basis if = , (7.2a) = , (7.2b) and{ } and{ } are linearly independent sets. Note that in the matrix picture, is a column vector, and isarowvector. Moreabstractly, belongstotheunderlyingvectorspaceand isalinearfunctionalon thevectorspace. Iwillcompactlydenotethissituationwiththenotation = =Λ= diag{ }. Thatis, −1 = −1 =Λ,where isthebasismatrixwithcolumns and isthebasismatrixwithrows . When ishermitianthebasesareidentified as = † . I am intentionally avoiding the use of Dirac’s bra-ket notation and being careful with the meaning of the symbol†. In practice, whether numerically or algebraically,{ } is determined via ∗ ∗ = ∗ , with∗ denotingthe conjugate-transpose. Nevertheless, the relation = ⇐⇒ † † = † (7.3) should be true for any anti-involution†, and, in particular, it should cause no issue to use the conjugate- transpose of to compute{ ∗ }, and consequently{ }, regardless of the proper hermitian adjoint deter- minedbytheinnerproduct. Put another way, for a generic Hamiltonian, and ∗ are not simply related. The conjugate-transpose, expressed in the symbol∗, loses its significance as a physically meaningful operation, and becomes a com- putationaltooltodetermine{ },whichcontainstheinformationrelevanttothephysics. Oncedetermined, all physically meaningful quantities should be expressed in terms of,{ },{ }, and{ }, which are all non-starred. Thus particularly careful attention must be paid to the ordering of factors since is a row-column matrix product that evaluates to a scalar and is an outer product that results in a linear operator or a matrix. When isnon-degenerate, = and = dependingonwhether actstotheleft or to the right. Hence( − ) = 0 implies = , or that is diagonal. If is degenerate, = is block diagonal with block sizes corresponding to the pattern of degeneracies. Diagonalize = −1 . Then{ ′ , ′ } is a biorthogonal basis satisfying ′ ′ = ′ ′ with ′ = and ′ = −1 . Sending →( ∗ )() where is block unitary (with the block sizes in the same pattern as the degeneracies, and unitary with respect to the standard inner product) preserves the eigenspaces and the fact that( ∗ )() is diagonal. Additionally, the eigenvectors can be rescaled as → , → ( , ≠ 0)withoutaffectingthisdiagonalstructure. 56 Thecompletenessrelationisnow Õ = 1. (7.4) Theprojectors Π = (7.5) satisfyΠ Π = Π . Since isdiagonalandtheproductofinvertiblematrices, ≠ 0,sotheprojectors arewell-defined. Thus,foranyrowvector , = Õ Π = Õ ≡ Õ (7.6) isitsexpansioninthebasis{ }. Similarly, for a column vector, = Õ Π = Õ ≡ Õ (7.7) isitsexpansioninthebasis{ }. One may ask about the status of the operators na¨ ıvely constructed by applying the expression from hermitianquantummechanics, Φ = ∗ ∗ . (7.8) TheyarenolongerorthogonalprojectorsasΦ Φ ≠ Φ , but merely idempotent: Φ 2 =Φ . However, Π Φ =Φ , (7.9a) Φ Π =Π , (7.9b) which might be termed ”pairwise idempotent” (I think these expressions are correct and Eq. (11) in Brody [69]iserroneous). Thefirstoftheserelations can be strengthened to Π Φ = Φ . (7.10) 7.2 Anotherdiagnosticfor emergingJordanblocks One of the facts demonstrated in the preceding section was that if{ , } is a biorthogonal basis, then is diagonal, or could be diagonalized, and is also invertible. If we now continuously parametrize the Hamiltonian and the biorthogonal basis, we may consider the diagonal elements of as continuous functionsoftheparameter. Thelimit → 0forsome shouldthensignalanontrivialJordanblockatthe eigenvalue . Thisframeworkhasyettobecompleted,sincetheapproachinChapter6wasmuchsimpler, dealing only with right eigenvectors. But completion of this program may yield additional insights and computationaltools. 7.3 DualJordanprojectionoperators Now suppose furthermore that is non-diagonalizable, with a nontrivial Jordan structure. There is a columnbasis{ |1≤ ≤,1≤ ≤ } and a row basis{ |1≤ ≤,1≤ ≤ } such that 1 = 1 , (7.11a) = + ,−1 , ( > 1) (7.11b) 1 = 1 , (7.11c) = + ,−1 . ( > 1) (7.11d) 57 TheindexlabelstheJordanblock(ofrank )andthepositionofthevectorwithinthattower. Bydefining 0 = 0 we can also use just the second equation in each pair without any restriction on . The do not have to be distinct—there can be an eigenvalue with multiple Jordan blocks. However, the eigenvalues are assumed real. As before, I use the notation = + , = − , where + is upper Jordan and − is lower Jordan. As before, the row basis is usually determined via ∗ ∗ = ∗ + ∗ ,−1 . Call such a pair{ , } a dualJordanbasis. Iwillbuilduptothegeneralresultinthree stages. 7.3.1 SingleJordanblock Forthesimplestcase,let = + ,where + consistsofasinglerank- Jordanblockwitheigenvalue. Thus 1 = 1 , and = + −1 when2≤ ≤ . As above, we define 0 = 0, and the preceding equation thenholdswithoutrestrictionon. Whatdegreesoffreedomremaininalteringthe withoutaffectingtheJordanblock + ? First,itisclear that rescaling all of the by the same scalar constant will preserve + . However, they cannot be rescaled independently as in the biorthogonal case, as the relative normalizations are fixed by the superdiagonal couplingsin + . Second,chooseasetofscalars{ |1≤ ≤−1}. Then in the basis{ ′ }, where ′ = + −1 Õ =1 − , (7.12a) stillhastheform + ,whichiseasilyverifiedbydirectcalculation. Thesetwopropertiesmaybecondensed byextendingthelistofcoefficientsto { |0≤ ≤−1} and setting ′ = + Õ =1 − , (7.12b) which represents a rescaling of all the vectors by a factor 0 +1 on top of taking linear combinations with other vectors. The numbers{ } represent all of the degrees of freedom in linearly shifting the basis whilepreserving + . Theproofofthisassertionfollowsfromthe= 1caseinthemoregeneralresulttobe discussedinSection7.3.2. Wecouldusethese degreesoffreedomtoset ∗ 1 1 = 1,andtomakethehighestvector, ,orthogonal toalltherest: ∗ = ∗ . Generically, the matrix of inner products is still fairly dense: ∗ =( ∗ )= © « 1 ∗ ··· ∗ 0 ∗ ∗ ··· ∗ 0 . . . . . . . . . . . . 0 ∗ ∗ ··· ∗ 0 0 0 ··· 0 ∗ ª ® ® ® ® ® ¬ . (7.13) Thisisthenicestwecanmake ∗ look. Thesameremarksapplytotherowbasis{ },where = + −1 and 0 = 0. Butaswehaveseen, ∗ and ∗ arenotparticularlymeaningfulquantities. Wewillabandon thisconventionhenceforthandadoptanother one shortly. In contrast with the biorthogonal case, the matrix is not diagonal, but lower Hankel; the entries are constantacrosseachanti-diagonalandthe upper left half of the matrix is empty: =( )= © « 0 0 ··· 0 1 0 . . . . . . 1 2 . . . . . . . . . . . . . . . 0 1 . . . . . . . . . 1 2 ··· ··· ª ® ® ® ® ® ® ® ® ¬ (7.14) 58 Putdifferently, = + dependsonlyon+,where + ≡ asaboveinthematrixaboveand = 0for ≤. To prove this, simply note that = (−) 2−− , which manifestly depends only on the total +. If+≤ then = + 0 = 0 since 0 = 0, or note also that(−) 2−− = 0 for+≤ . This computation also implies that is lower Hankel: = + −1 , and both terms on the right are represented by lower Hankel matrices. If we reorder the basis backwards (and form the matrix ˜ ), then ˜ = − and ˜ islowerToeplitz. Similarly, ˜ = + and ˜ is upper Toeplitz. Now we are going to renormalize and shift the to obtain the completeness relation. Define ′ as in Eq.(7.12b)and asinEq.(7.14). Thenwe demand ′ = + Õ =1 − = 1 . (1≤ ≤) (7.15) Invert the above equations to find a set of coefficients satisfying them, and let ′ → , dropping the primes. This is possible as long as 1 ≠ 0: the = 1 equation reads 1 (1+ 0 ) = 1 =⇒ 0 = 1/ 1 − 1, andinductivelyhavingsolvedthethequation,substitutingthevaluesupto −1 intothe(+1)thequation yieldsalinearrelationfor oftheform 1 += 0. Furthermore, 1 = 0cannothappen: and areboth invertible, since their rows and columns, respectively, form bases, anddet =± 1 ≠ 0, as can be seen by rotating intoToeplitzform. Having done this, = +,+1 , which has the nice matrix form (remember that is still lower Hankelafterredefining inthisway) =( )= © « 0 ··· 0 1 . . . . . . . . . 0 0 . . . . . . . . . 1 0 ··· 0 ª ® ® ® ® ® ¬ . (7.16) Thereisstillaremainingcomplexdegreeoffreedom,whichmaybeusedtoscalejointly → , → −1 . Thecompletenessrelationis Õ =1 +1− = 1. (7.17) Theprecedingresultslooknicerifwefliponeofthebases,rememberingthatdoingthisalsotransposesthe Jordanforminthenewbasis: ˜ = ˜ = ⇐⇒ ˜ = ˜ = 1, (7.18a) Õ =1 ˜ = Õ =1 ˜ = 1. (7.18b) Theprojectionoperators Π ≡ +1− = ˜ =˜ +−1 +−1 (7.19) satisfytheusualΠ Π = Π andaretherefore orthogonal. 7.3.2 MultipleJordanblocksforasingleeigenvalue The next level of complexity, building on the previous section, is to consider a matrix with a unique eigenvalue. Let{ , |1≤ ≤,1≤ ≤ 1 } beadualJordanbasissuchthat = + and = − . The total dimension of is the algebraic multiplicity of,≡ Í =1 , and Eq. (7.11) holds with all = . As before,let 0 = 0 forconvenience. 59 Let us follow the same procedure as in the previous section and ask again: what degrees of freedom remaininalteringthe withoutaffectingthe Jordan block + ? Suppose ′ = ′ + ′ ,−1 , with ′ = + Õ =1 Õ =1 . (7.20) Explicitcalculationyields ′ = + Õ =1 Õ =1 + ,−1 + Õ =1 Õ =1 ,−1 = ′ + ,−1 + Õ =1 Õ =1 ,+1 . (7.21) Wemustidentifythelasttwotermsas ′ ,−1 : ′ ,−1 = ,−1 + Õ =1 Õ =1 ,+1 . (7.22) Thus Õ =1 Õ =1 −1, = Õ =1 −1 Õ =1 ,+1 (7.23) implies −1, = ,+1 asthe formabasis, or that ≡ − (7.24) dependsonlyonthedifference −. Next, considering ′ 1 = 1 + Õ =1 Õ =1 1− + Õ =1 Õ =1 1− ,−1 = ′ 1 + Õ =1 Õ =1 1− ,−1 (7.25) implies 1− = 0 for > 1,or = 0. ( < 0) (7.26) Thustherangesoftheindicesare{ |1≤ ≤,1≤ ≤,0≤ ≤ −1}. Thenumberoffreeparameters isthus. As before, + ≡ = (− ) + −− depends only on the sum + . Furthermore, + = 0 if +≤ ≡ max{ , }. Once again, in order to obtain the completeness relation, we shift → ′ = + Í − anddemand ′ = + Õ =1 Õ =1 − = + + Õ =1 Õ =1 − + = +, +1 . (7.27) For fixed values of and, the range of+ is +1≤ +≤ + . The number of equations is then + − = ≡ min{ , }. Ranging over1≤ ,≤, the total number of constraints is bounded by: Õ =1 Õ =1 ≤ Õ =1 Õ =1 = Õ =1 =. (7.28) 60 ItisthuspossibletosolvethelinearequationsEq.(7.27)fortheparameters{ },thoughnotnecessarily uniquely, so long as + is invertible when regarded as a matrix with rows labeled by the composite index =() and columns labeled by the composite index =(). That this is true follows from = being a matrix product of invertible matrices. When all Jordan blocks have the same size, = for all, thenthecoefficientsareuniquelydetermined as follows. Define and to be matrices with entries( ) = and( ) = . Eq. (7.27) for = in matrix formis + + Õ =1 − + = 1 . (7.29) The= 1 equationreads +1 + 0 +1 = =⇒ 0 =( +1 ) −1 − . (7.30) Iteratively,havingsolvedthethequationto obtain 0 ,..., −1 , the(+1)th equation has the form +1 + ++1 + +1 Õ =2 +1− + ≡ +1 += 0 =⇒ =−( +1 ) −1 , (7.31) where iscomposedofknownquantities. Thus, the coefficients { } are determined. Redefine → ′ using the coefficients { } and drop the primes (no longer assuming all = , so a set of coefficients must be obtained by other means if this doesn’t hold). Then as before, is block diagonalwithblocksofsize alloftheform given in Eq. (7.16). The completeness relation reads Õ =1 Õ =1 , +1− = 1. (7.32) Theprojectionoperators Π = , +1− (7.33) satisfyΠ Π = Π andarethusmutually orthogonal. 7.3.3 ThegeneralJordancanonicalform HavinghandledJordanformswithasinglerepeatedeigenvalue,resultsforthegeneralcasefallneatlyinto place. Let{ , |1≤ ≤,1≤ ≤ } be a dual Jordan basis. I will show first that = 0 whenever ≠ . Thus a semblance of orthogonality among different eigenspacesisretained. Theprooffollowsbydoubleinductionon and. First,Iwillshowthat 1 1 = 0. Next,Iwillshowthat ,−1 1 = 0implies 1 = 0and 1 ,−1 = 0implies 1 = 0. FinallyIwillshow thatif ,−1 = ,−1 = 0,thenitfollowsthat = 0. Theprincipleofdoubleinductionthenimplies that = 0 forall, where1≤ ≤ ,1≤ ≤ . For the base case, note that 1 1 = 1 1 = 1 1 depending on whether acts to the left or to theright. Hence( − ) 1 1 = 0 implies 1 1 = 0 as ≠ by assumption. Forthefirstinductivestep,assume ,−1 1 = 0. Then 1 = 1 + ,−1 1 = 1 (7.34a) when actstotheleftand 1 = 1 (7.34b) whenactstotheright. Thus,onceagain,( − ) 1 = 0anditfollowsthat 1 = 0. That 1 ,−1 = 0 implies 1 = 0 issimilar. 61 Forthesecondinductivestep,assume ,−1 = ,−1 = 0 for some and. Then = + ,−1 = (7.35a) when actstotheleftand = + ,−1 = (7.35b) when actstotheright. Yetagain,( − ) = 0 implies = 0, and the proof is complete. Since eigenspaces with distinct eigenvalues are orthogonal, the procedure of the preceding section can beappliedtoeacheigenvaluetoobtainthe orthogonality and completeness relations, = +, +1 , (7.36a) Õ =1 Õ =1 , 1 +1− = 1. (7.36b) For purposes of practical computation, it is useful to introduce another index to label the distinct eigenvalues. Following this idea, let{ , |1≤ ≤ ,1≤ ≤ ,1≤ ≤ } be a dual Jordan basis, where is the number of distinct eigenvalues{ 1 ,..., }, is the number of Jordan blocks for the eigenvalue , and is the rank of the th Jordan block for the eigenvalue . In terms of this set of indices,thepreviousresultsreadasfollows. Theorthogonalityofdifferenteigenspacesissimplyexpressed as = . Thislastquantityhasadependenceonlyon+,andnottheirvaluesseparately: = ; + . To obtain orthogonality and completeness, let ′ = + Í ; − and solve for coefficients { ; } suchthat ′ = +, +1 . (7.37) Thecompletenessrelationisthen Õ =1 Õ =1 Õ =1 ,, +1− = 1, (7.38) having dropped the primes on ′ . The projection operators, the summands in the completeness relation, aremutuallyorthogonal. 7.4 Nextsteps In this chapter I have described the basic task of finding the coefficients in the expansion of a general state as a sum of generalized eigenstates of a Hamiltonian, which suffices for the purposes of the rest of the work. There is much more to be done, of course. Conspicuously absent is a discussion of time evolution, which, even if we maintain the same postulate (number 4 in the three formulations of Section 3.1), takes on a profoundly different character—it is not unitary. Another natural question to ask concerns the nature of measurement. My intuition tells me that the right place to look for this is in terms of the positive operator-valuedmeasurement[31]. TheprocedureofSection7.3assumedanontrivialJordanformfortheHamiltonianunderstudy. Given that I wrote Chapter 6 on emerging Jordan forms, since Jordan forms may only appear in some limit, can the procedure be adapted to that situation? It would have to differ from the biorthogonal formalism since theendresultoftheemergingJordanform is a matrix form for the Hamiltonian that is not diagonal. Another possible extension of the theory here is a density matrix formulation. Since the left and right Jordan bases are treated asymmetrically (one is shifted while keeping the other fixed in order to find the projection operators) it may be the case that the projection operators rather than left or right states are the morefundamentalobject. Thisextensionwouldalsobringinthepossibilityofapplyingthedensitymatrix renormalization group (DMRG) technique to systems with nonunitary dynamics—I have been told that DMRG for the nonunitary statistical models discussed in Part III have failed because of the nature of the states[70]. 62 PartIII Applications 63 Chapter8 The actionoftheVirasoroalgebrainthe two-dimensionalPottsandloopmodels ThespectrumofconformalweightsfortheCFTdescribingthetwo-dimensionalcritical-statePottsmodel (or its close cousin, the dense loop model) has been known for more than 30 years [50]. However, the exact nature of the corresponding Vir⊕ Vir representations has remained unknown up to now. Here, we solve the problem for generic values of, whose relation to is described in Section 4.2.1. This is achieved by a mixture of different techniques: a careful study of the Koo–Saleur generators (Section 8.3), combined with measurements of four-point amplitudes, on the numerical side, and OPEs and the four-point amplitudes recently determined using the “interchiral conformal bootstrap” [71] on the analytical side. We find that null-descendants of diagonal fields having weights (ℎ 1 ,ℎ 1 ) (with ∈ ℕ ∗ ) are truly zero, so these fields come with simple Vir⊕ Vir (Kac) modules. Meanwhile, fields with weights (ℎ ,ℎ ,− ) and(ℎ ,− ,ℎ ) (with,∈ℕ ∗ ) come in indecomposable but not completely reducible representations mixing four simple Vir⊕Virmoduleswithafamiliardiamondshape. Thetopandbottomfieldsinthesediamondshaveweights (ℎ ,− ,ℎ ,− ),andformatwo-dimensionalJordanblockfor 0 and 0 . Thisestablishes,amongotherthings, that the Potts-model CFT is logarithmic for generic. Unlike the case of non-generic (root of unity) values of, these indecomposable structures are not present in finite size, but we can nevertheless show from the numericalstudyofthelatticemodelhowthe rank-two Jordan blocks build up in the infinite-size limit. 8.1 Overview The full solution of the CFT describing the critical -state Potts model for generic (or its cousins, the critical and dense() models) in two dimensions still eludes us, more than 30 years after the pioneering work of Dotsenko and Fateev [72]. While most critical exponents of interest were quickly determined (for some, even before the advent of CFT, using Coulomb-gas techniques) [73, 43, 74], the non-rationality of the theory(forgeneric)aswellasitsnonunitarity(inheritedfromthegeometricalnatureofthelatticemodel) madefurtherprogressusingtop-downapproaches (such as the one used for minimal unitary models[75]) considerably more difficult (see also Introduction). Several breakthroughs took place, however, in the last decade. First,manythree-pointfunctionsweredeterminedusingconnectionswithLiouvilletheoryat < 1 [76, 77, 78]. Second, a series of attempts using conformal bootstrap ideas [7, 71, 79, 80, 81, 82] led to the determinationofsomeofthemostfundamentalfour-pointfunctionsintheproblem(namely,thosedefined geometrically, and hence for generic), also shedding light on the OPE algebra and the relevance of the partitionfunctionsdeterminedbydiFrancesco,Saleur,andZuber[50]. Inparticular,thesetofoperators— the so-called spectrum—required to describe the partition function [50] and correlation functions [7] was settled. While the picture remains incomplete, a complete solution of the problem now appears within reach. AnintriguingaspectofthespectrumproposedbydiFrancesco,Saleur,andZuber[50]andJacobsenand Saleur[7]istheappearanceoffieldswithconformalweightsgivenbytheKacformula Δ= ℎ ,with,∈ℕ ∗ . 64 Wecalltheseweightsdegenerate. Itisknownthatforsomeofthesefields—suchastheenergyoperatorwith weights(ℎ 21 ,ℎ 21 )—the null-state descendants are truly zero, and the corresponding four-point functions obey the Belavin–Polyakov–Zamolodchikov (BPZ) differential equations [83]. It is also expected that this does not hold for all fields with degenerate weights. In fact, it was suggested by He, Jacobsen, and Saleur [71] and Jacobsen and Saleur [7] that, in the Potts-model case, only fields with weights (ℎ 1 ,ℎ 1 ) give rise to null descendants. Since the spectrum of the model is expected to contain non-diagonal fields with weights (ℎ ,ℎ ,− ) and(ℎ ,− ,ℎ ) for ,∈ ℕ ∗ , this means that the theory should contain fields with degenerate (left orright) weightswhose nulldescendants are nonzero, even though their two-point function vanishes. It has been well understood since the work of Gurarie [84] that in this case, “logarithmic partners” must be invoked to compensate for the corresponding divergences occurring in the OPEs. Such partners give rise toJordanblocksfor 0 or 0 ,andmakethetheorya logarithmicCFT—i.e.,a theorywheretheactionofthe productofleftandrightVirasoroalgebrasVir⊕Virisnotfullyreducible. This,inturn,ismadepossibleby thetheorynotbeingunitaryinthefirstplace [85]. A great deal of our understanding of the fields with degenerate weights in the Potts model comes from indirect arguments, such as the solution of the bootstrap equations for correlation functions and the presence of an underlying “interchiral” algebra, responsible for relations between some of the conformal- block amplitudes [71]. This chapter explores the issue much more directly using the lattice regularization ofVir⊕VirfirstintroducedbyKooandSaleur[3]. MycollaboratorsperformaparallelanalysisoftheXXZ spinchaininacompanionpaper[54],alsoforthenon-rationalcase. Otherapplicationshavedemonstrated theutilityofthelatticeapproach[5,86,87]. Thischapterrelatestotheremainderoftheworkasfollows. Basicfactsaboutthetwo-dimensionalPotts model and its CFT appeared in Section 4.2. The algebra of local energy and momentum densities turns out to be the affine Temperley–Lieb algebra. Various incarnations of algebra and its representation theory were discussed in Chapter 2. While somewhat technical, the results for the affine Temperley–Lieb algebra are crucial, since they will be used as a starting point to understand the corresponding representations of Vir⊕ Vir in the continuum limit. Returning to the CFT, general strategies to study the action of Vir⊕ Vir startingfromlatticemodelsappearedinChapter4. Puttingallthesepiecestogether,thefirstresultsappear inSection8.2.2wherewearguefortheexistenceofindecomposablemodulesofVir⊕Virinthecontinuum limit of the Potts model for generic. Our main results are given as Propositions 3 and 4. These results are to be contrasted with the non-degenerate case, a reminder of which is presented first in 8.2.1. Evidence fromthelatticethatsupportsourmainresults is discussed in Section 8.3. Theprimarythemeofthischapteristhepresenceofindecomposablestructuresinthecontinuumtheory. In the following chapter, we associate quantities to these indecomposable structures and describe methods ofmeasuringthemonthelattice,where,strictly speaking, they are absent. 8.2 Vir⊕VirmodulesinthePottsmodelCFT 8.2.1 Thenon-degeneratecase Werecalloncemorethatinthischapterisassumedtotakegenericvalues(notarootofunity). Whenever is such that the resonance criterion Eq. (2.11) is not met we say that is generic, and when Eq. (2.11) is satisfied isreferredtoas non-generic. Sinceisgenericthroughout,both anditsparametrization inEq.(4.8)takegeneric,irrationalvalues. The conformal weights may or may not be degenerate, depending on the lattice parameters. In the non- degenerate case, which corresponds to generic lattice parameters (the opposite does not always hold), it is natural to expect that the Temperley–lieb module decomposes accordingly into a direct sum of Verma modules: ,e i⇝ Ê ∈ℤ − ,− ⊗ − , . (8.1) The symbol⇝ means that the action of the lattice Virasoro generators restricted to scaling states on ,e i corresponds to the decomposition on the right hand side when →∞. This statement is discussed in considerabledetailbyGrans-Samuelsson, Jacobsen, and Saleur [54]. 65 RecallthataVermamoduleisahighest-weight representation of the Virasoro algebra, [ , ]=(−) + + 12 ( 2 −1) +,0 , (8.2) generated by a highest-weight vector|ℎ⟩ satisfying |ℎ⟩ = 0 for > 0, and for which all the descendants − 1 ··· − |ℎ⟩, with 0 < 1 ≤ 2 ≤···≤ and > 0, are considered as independent, subject only to the commutation relations of Eq. (8.2). In the non-degenerate case where the Verma module is irreducible, it is the only kind of module that can occur, motivating the identification in Eq. (8.1). We note that this identificationisindependentofwhetherwe consider the loop model or the XXZ spin chain. 8.2.2 Thedegeneratecase In the degenerate case the conformal weights may take degenerate values ℎ = ℎ with,∈ℕ ∗ , in which caseasingularvectorappearsintheVermamodule. Bydefinition,asingularvectorisavectorthatisbotha descendant and a highest-weight state. For instance, starting with|ℎ 11 ⟩=|0⟩ we see, by using the Virasoro commutationrelationsofEq.(8.2),that 1 ( −1 |0⟩)= 2 0 |0⟩= 0, (8.3) while ( −1 |0⟩) = 0 for > 1. Hence −1 |0⟩ is a singular vector. Under the action of the Virasoro algebra, this vector generates a submodule. For generic, this submodule is irreducible, and we have the decomposition (d) 11 : 11 1,−1 (8.4) where we have introduced the notation (d) to denote the degenerate Verma module, and we also denote by theirreducibleVirasoromodule(inthiscase,technicallya“Kacmodule”),withgeneratingfunction oflevels = ℎ −/24 1− () . (8.5) Thesubtractionofthesingularvectoratlevel gives rise to a quotient module. In cases of degenerate conformal weights, there is more than one possible module that could appear, and the identification in Eq. (8.1) may no longer hold. Furthermore, the identification now depends on the representation of () one considers. We restrict here to the loop–cluster representation, while corresponding results about the XXZ representation are discussed by Grans-Samuelsson, Jacobsen, and Saleur[54]. For the modules 0, ±2, without through-lines, the Verma structure is seen even at finite size—see Eq.(2.17). UsingthenumericalmethodsdescribedinSection8.3,wefindthatthecorrespondingloopstates areneverannihilatedbythe 1 or 1 combinations of Virasoro generators. We recall now from Section 2.3 that the module 0, ±2 appears in the loop model by keeping track of how points are connected across the periodic boundary. However, the Potts model where non-contractible loops have the same weight as contractible ones naturally involves the quotient 0, ±2 for which there arenodegeneratestatesonthelattice. Thespectrumgeneratingfunctionforthismoduleinthecontinuum limitisthen 0, ±2 = 0, ±2− 11 = ∞ Õ =1 1 1 , (8.6) which involves only Kac modules. It is thus natural to formulate the following conclusion for the scaling limit. 66 Conjecture3 (Quotientloop-modelmodule without through-lines). We have the scaling limit 0, ±2⇝ ∞ Ê =1 1 ⊗ 1 . (8.7) Notethatthisstructureimpliesthatthecorrespondinghighest-weightstates|ℎ,ℎ⟩ arenowannihilated: 1 |ℎ 1 ,ℎ 1 ⟩= 1 |ℎ 1 ,ℎ 1 ⟩= 0. (8.8) In particular, the ground state is indeed annihilated by −1 and −1 , a satisfactory situation physically. SupportingnumericalevidenceforConjecture 3 is found in Section 8.3.2. For the modules 1 with > 0, the numerical results in Section 8.3.3 indicate that the highest-weight stateswithconformalweightℎ and > 0areneverannihilatedbythecorrespondingoperators ,whether inthe holomorphicorantiholomorphic sector. It would be tempting to conclude that the modules are now systematicallyofVermatype,butthisnotpossible. Indeed,recallthatforgeneric,the ()modules 1 are irreducible and thus self-dual. The Virasoro generators, being obtained as continuum limits of () generators,shouldalsoobeythisself-duality(cf.Gainutdinovetal.[15,Section4.3]). Vermamodulesclearly do not, as their structure is not invariant under reversal of theVir⊕Vir action. To understand what might happen,letusdiscussinmoredetail,asanexample,thecase= 2. Thegeneratingfunctionoflevelsshows apairofprimaryfields Φ 12 ≡ 12 ⊗ 1,−2 , (8.9a) Φ 12 ≡ 1,−2 ⊗ 12 , (8.9b) withconformalweights(ℎ 12 ,ℎ 1,−2 )and(ℎ 1,−2 ,ℎ 12 ). Note herethat by we simplymean a chiralprimary field with conformal weight ℎ : the structure of the associated Virasoro module will be discussed below. Thismeansinparticularthat = −,− . By expanding the factor 1/()() in the spectrum generating functions, we see that the model also has four descendants at level two—i.e., with conformal weights(ℎ 1,−2 ,ℎ 1,−2 ), where we have used that ℎ 1,−2 = ℎ 12 +2. Now, if the modules generated byΦ 12 andΦ 12 in the continuum limit were a product of two Verma modules, these four descendants would be the two independent fields −2 Φ 12 and 2 −1 Φ 12 , as well as the two fields obtained by swapping chiral and antichiral components, −2 Φ 12 and 2 −1 Φ 12 . The chiral/antichiral symmetry corresponds to exchanging right and left (i.e., exchanging momentum for momentum−) and is present on the lattice as well, by reflecting the site index →+1−. This means one would expect to observe, in the finite-size transfer matrix, two eigenvalues, both converging (once properly scaled) to ℎ 1,−2 = ℎ 12 + 2, and corresponding to two linear combinations of −2 Φ 12 and 2 −1 Φ 12 andtheirconjugates—hencebothappearingintheformofdoublets. Thisisnot,however,whatisobserved numerically (see Section 8.4). Instead, we see one doublet and two singlets, which means that the module in the continuum limit and at level two does not have, as a basis, a pair of independent states and their chiral/antichiralconjugates. Introducing 12 = −2 − 3 2+4ℎ 12 2 −1 , (8.10a) 12 = −2 − 3 2+4ℎ 12 2 −1 , (8.10b) wenowclaimthat,inthecontinuumlimit, the identity 12 Φ 12 = 12 Φ 12 (8.11) is satisfied. Note that both sides of the equation are primary fields—i.e., they are annihilated by Vir⊕ Vir generators , with > 0. They are also of vanishing norm-square⟨·|·⟩ . Corresponding numerical resultsappearinSection8.3.3. 67 Wehavethereforeidentifiedpartofthemoduleasaquotientof ( (d) 12 ⊗ 1,−2 )⊕( 1,−2 ⊗ (d) 12 ),corresponding tothefollowingdiagramforthedegenerate fields: Φ 12 = 12 ⊗ 1,−2 Φ 12 = 1,−2 ⊗ 12 12 Φ 12 = 12 Φ 12 12 12 (8.12) Note we have the quotient modules (obtained by taking the quotient by the submodule generated by the bottom field) 12 ⊗ 1,−2 and 1,−2 ⊗ 12 and with generating functions( ℎ 1,−2 −/24 /())× 12 and 12 ×( ℎ 1,−2 −/24 /()). ThebottomfieldgeneratesaproductofVermamodules 1,−2 ⊗ 1,−2 withgenerating function( ℎ 1,−2 −/24 /())×( ℎ 1,−2 −/24 /()). However, this cannot be the end of the story, since the quotient identified so far is not self-dual— nor does it account for the proper multiplicity of fields. Invariance of the diagram under reversal of the arrows demands that there exists a field “on top,” with a quotient that is also a product of Verma modules 1,−2 ⊗ 1,−2 . Thisshouldgiverise,interms of fields, to the diagram ˜ Ψ 12 Φ 12 Φ 12 Ψ 12 ≡ 12 Φ 12 = 12 Φ 12 † 12 † 12 0 −ℎ −1,2 12 12 (8.13) with ˜ Ψ 12 afieldtobedetermined(Section9.3). The sameconstruction seems to apply to all cases in the characters 1 . The simplest example occurs, in fact,in 11 —eventhoughthismoduledoesnotappearinthePottsmodel,asdiscussedafterEq.(4.19)—with Φ 11 ≡ 11 ⊗ 1,−1 andΦ 11 ≡ 1,−1 ⊗ 11 . In this case, the quotient is simply given by −1 Φ 11 = −1 Φ 11 . The indecomposable structure for arbitrary positive integer values of and can then be conjectured to be ˜ Ψ Φ = ⊗ ,− Φ = ,− ⊗ Ψ ≡ Φ = Φ † † 0 −ℎ −1,2 (8.14) ThevalidityofEq.8.14ingeneralcomesfromstrongnumericalevidenceforsmallvaluesof and. Itisalso thesimpleststructurewecanimaginesolvingtheproblemsofpolesintheOPEs,basedonourindependent knowledge of the spectrum of the theory. More complete evidence should come from the construction of four-pointfunctionsusingthecorresponding regularized conformal blocks [88]. ThecorrespondingstructureofVirasoro modules defines the quotient modules ℒ : ,− ⊗ ,− ℒ ≡ [( (d) ⊗ ,− )⊕( ,− ⊗ (d) )] : ⊗ ,− ,− ⊗ ,− ⊗ ,− (8.15) 68 Accordingly, weareledtoournextresult. Conjecture 4 (Loop-model modules with through-lines). For > 0 and 2 through lines we have the scaling limit 1 ⇝( 0,− ⊗ 0 )⊕ ∞ Ê =1 ℒ . (8.16) As mentioned, an important piece of evidence for the correctness of the structure in Eq. (8.14) is based on the numerical observation of a pair of singlet states in the transfer matrix spectrum. In Section 8.4 we identifythispairofsingletspreciselyinthecases(,)=(1,1),(2,1),(1,2)and(1,3). Theseobservationsin turnlendcredencetothemoregeneralConjecture 4. 8.3 EvidencefromthelatticeviaKoo–Saleurgenerators Exceptwherenecessary,inthissectionwe suppress notating the explicit dependence on lattice size. Within this section we provide evidence for the main results given in Propositions 3 and 4, by acting directly with the Koo–Saleur generators of Eq. (4.28) on eigenstates of the lattice Hamiltonian in Eq. (4.24). In these numerical studies we partition our state space at each system size into eigenspaces of the translation operator, with eigenvalues{e 2i/ |0≤ ≤ −1}. As the Hamiltonian is manifestly invariant undertranslationwemaydiagonalizeitindependentlywithineachsuchsector. TheKoo–Saleurgenerators exactlyreproducethefactthattheactionof ( )onastateofmomentumproducesastateofmomentum − (+),atfinitesize. Forastatewithaneigenvalue oftheHamiltonianatagivensystemsize,we consideritseffectiveconformalweights,which we also denote (ℎ,ℎ), defined as the solutions to = 2 ℎ+ℎ− 12 , (8.17a) = ℎ−ℎ. (8.17b) By“following”astate(say,thelowest-energystatewithinagivensectoroflatticemomentum)as increases, and extrapolating the effective conformal weights ℎ and ℎ, we can identify the conformal weights in the continuum limit. This process is carried out in Section 8.4. We will omit the qualifier “effective” and simplyreferto“conformalweights”whenthecontextmakesitclearthatthetermisbeingappliedtolattice quantities. Similarly,wewillfrequentlyassignconformalweightsℎ givenbytheKacformulatofinite-size states—bythiswemeanthattheeffectiveconformalweightsofastateforincreasing convergetoℎ= ℎ . Inpractice,wewillonlybeabletoaccesssmallvaluesoftheKaclabels and,sincelargersystemsizesare neededtoaccommodatealargerlatticemomentum(whichgoverns)andalargernumberofthrough-lines (whichgoverns). Before discussing details of the numerics we must eliminate an ambiguity that may arise in the results duetophasedegreesoffreedom. Inthefollowingsectionswewilldiscussquantitiesoftheform∥−∥ 2 , where and are (descendants of) eigenstates of the Hamiltonian (e.g., = −1 Φ 11 and = −1 Φ 11 ), and∥·∥ 2 2 =(·|·)is the norm induced by the native positive-definite scalar product. and need not be simply related (aside from the fact that they are predicted to be equal in a certain limit), as the notation mightsuggest. Thediscussionappliestoanypairofexpressionsthatmaysufferfromthisphaseambiguity. Nevertheless, for the situations in this work where this discussion applies, and represent expressions thatarerelatedbytheexchangeofholomorphicandantiholomorphiccomponents. Inquantummechanics theoverallphaseofavectororwavefunctionhasnoobservableconsequencesande i foranyrealwould serve just as well in computations of observables. Typically one chooses the phase of a state such that its componentsinsomebasisareentirelyreal,wherepossible. Inthesituationathand,theeigenvectorsofthe Hamiltonian are generically complex—no choice of phase can make all of the components real—and there is no canonical way to fix the relative phase between eigenvectors. The measurement of ∥−e i ∥ 2 thus takes on a continuum of values. Where this ambiguity occurs, we fix the relative phase by choosing the 69 valueof thatminimizesthisquantity: ∥−∥ 2 ≡ inf ∥−e i ∥ 2 . (8.18) Thisoptimizationissuccinctlydenotedby the underlined 2 in the notation∥−∥ 2 . Our main goal is to establish certain identities by observing whether deviations from these identities at finite size decay to zero with increasing system size. Let us give two examples. In order to provide evidence for Conjecture 3 we will check that −1 → 0, with the identity state with conformal weights (0,0). Meanwhile,toprovideevidenceforConjecture4wewouldliketoestablishthat −1 Φ 11 → −1 Φ 11 ,or that −1 Φ 11 − −1 Φ 11 → 0 as→∞. Using the positive-definite norm, we examine equivalently whether ∥ −1 ∥ 2 → 0 and∥ −1 Φ 11 − −1 Φ 11 ∥ 2 → 0. Aswillbeseeninthetablesbelow,thissimplemeasurementfailstofurnishtheevidenceweseek. Indeed, as increases, the values observed actually grow in magnitude in most cases. An interpretation of this observationisthefactthat,sincethefinite-sizeKoo–Saleurgeneratorsdonotyetfurnisharepresentationof theVirasoroalgebra,theactionof −1 onΦ 11 ,forinstance,producesastatewithnonzerocomponentseven inhighlyexcitedeigenstatesoftheHamiltonian. Whileeachsuchcomponentwouldtendtozeroonitsown, the number of these so-called “parasitic couplings” grows rapidly, yielding a non-decaying contribution in total. Toavoidtheissueofthisrapidgrowthweprojecttothelowest-energystateswithintherelevantsector oflatticemomentum,keeping fixedas →∞. 8.3.1 Scaling-weakconvergence Forthefollowingdiscussionweconsideraconcreteexample,thefields −1 Φ 11 and −1 Φ 11 intheloopmodel. Inthecontinuumlimit,thesefieldshaveconformalweights (1,1). Theirlatticeanaloguesbothbelongtothe sector of lattice momentum=/2. By following the energies of states within this sector for increasing latticesizes,wefindthatthetwolowest-energy states correspond to these conformal weights. Letuswriteschematically −1 Φ 11 =+, (8.19) where is a linear combination of these two lowest states and represents all other states in the sector =/2. In order to exclude the effects of the parasitic couplings, we build a projection operator Π (2) such thatΠ (2) −1 Φ 11 =. TheoperatorΠ (2) weseekispreciselythesumofthebiorthogonalprojectionoperators for the two lowest states,Π (2) =Π 1 +Π 2 , described in Chapter 7. Since −1 Φ 11 has conformal weights(1,1) aswell,theprojectorΠ (2) alsotruncatesthe corresponding lattice quantity to the same two states. Itisnotnecessarytorestricttoonlythecomponentsin (givenbytheprojectiontothetwoloweststates intheexampleathand)—onecouldalsoincludehigherenergystates. Aslongastherankoftheprojection operatorisfixed,weexpecttheinfluenceofparasitic couplings within the image of the projection operator tovanishas→∞. Wecallconvergenceofvaluesinthecontextofthisprocedurescaling-weakconvergence. Todemonstratethistypeofconvergence,weapplyprojectorsofincreasingrank to −1 Φ 11 − −1 Φ 11 before measuring its norm. We expect that for any fixed projector rank (independent of ), so long as Π () eventuallyincludesallscalingstatesas→∞, ∀∈ℕ, lim →∞ ∥Π () ( −1 Φ 11 − −1 Φ 11 )∥ 2 = 0; (8.20) i.e.,scaling-weakconvergenceofthelattice quantities towards the continuum identity −1 Φ 11 = −1 Φ 11 . Thenotionofscaling-weakconvergenceisdefinedanddiscussedingreaterdetailbyGrans-Samuelsson, Jacobsen, and Saleur [54], where it is shown that a crucial difference compared to weak convergence is that limits of products of Koo–Saleur generators are in certain cases different compared to the cor- responding products of limits, necessitating the insertion of projectors. This difference is found to af- fect the products with dual operators that are induced by the positive-definite inner product, as in ∥ −1 Φ 11 ∥ 2 =(Φ 11 | ∗ −1 1 |Φ 11 ),butnottheproduct 2 −1 inside the singular vector operator 12 . 70 Ingeneral,foranyofthefields relevantbelow,wesaythatitslatticeanaloguescaling-weaklyconverges tozeroif ∀∈ℕ, lim →∞ ∥Π () ()∥= 0, (8.21) with∥·∥some positive-definite norm, and where () is the lattice analogue of at lattice size . The meaningofΠ () iscontext-dependent,butshouldbebuiltinsuchawaythatlim →∞ Π () effectivelyfunctions astheidentityoperator: ∀∈ 2ℕ, lim →∞ Π () ()=(). (8.22) Wesay“effectively,”since lim →∞ Π () doesnotnecessarilyhavetoequaltheidentityoperator. Forinstance, in the discussion of scaling-weak convergence of −1 Φ 11 − −1 Φ 11 to zero,Π () is built from states of lattice momentum = /2. Thus lim →∞ Π () is the identity operator in the subspace of momentum /2 and zeroelsewhere. However, −1 Φ 11 − −1 Φ 11 iszeroinallmomentumsectorssavefor/2. Thuslim →∞ Π () effectively functions as the identity in this measurement. It is also possible to construct Π () using the loweststatesoftheentireHamiltonian,regardlessofmomentum. ThisdoesnotaffectthelimitinEq.(8.22), butmerelytherateofconvergence. Inthis caselim →∞ Π () becomes the identity operator. Ananalogousdiscussionappliestothedemonstrationoftheidentity 12 Φ 12 = 12 Φ 12 ,mutatismutandis. Wepresentnumericalevidencethat 12 Φ 12 − 12 Φ 12 scaling-weakly converges to zero. We show in Figures 8.1 and 8.2 that when applying projectors of different rank , the numerical results extrapolate to almost the same values. We expect that the difference in the extrapolations can be made arbitrarily small by including data points for large enough system sizes, though they are not numerically accessibleatthetimeofwriting. 2 7 15 0.02 0.04 0.06 0.08 0.10 1/N 0.05 0.10 0.15 0.20 0.25 Figure 8.1: Comparison of lattice results using projectors of different of rank, illustrating the concept of scaling-weak convergence for = 1. The horizontal axis is 1/. The vertical axis is∥Π () ( −1 Φ 11 − −1 Φ 11 )∥ 2 /∥Π () −1 Φ 11 ∥ 2 . The tags on the graphs indicate the rank of the projectorΠ () . The dotted lines arefourth-orderpolynomialfits(in 1/)to the five leftmost data points. 8.3.2 Numericalresultsfor 0, ±2 Withinthemodule 0, ±2,thelinkstatescorrespondingtoprimaryfieldswithdegenerateconformalweights are never annihilated by the singular vector operators 1 or 1 . However, the module of interest for the 71 4 9 17 0.02 0.04 0.06 0.08 0.10 1/N 0.2 0.4 0.6 0.8 Figure 8.2: Comparison of lattice results using projectors of different of rank, illustrating the concept of scaling-weak convergence for = 1. The horizontal axis is 1/. The vertical axis is∥Π () ( 12 Φ 12 − 12 Φ 12 )∥ 2 /∥Π () 12 Φ 12 ∥ 2 . The tagson the graphs indicate the rank of the projectorΠ () . The dotted lines arethird-orderpolynomialfits(in 1/)to the four leftmost data points. study of the loop model is rather the quotient module 0, ±2. In this module we consider in particular the lowest-energy link state of lattice momentum = 0, which in the continuum limit will correspond to the identity state with conformal weights(ℎ,ℎ)=(0,0). We act on this state with the Koo–Saleur generator −1 . Thenormoftheresultingstate −1 isshowninTable8.1. (Thenormof −1 isthesamebysymmetry.) Inthesetables,thepeculiarvalue= /sec −1 (2 √ 2)−1correspondsto= 1/2,whichisfurtherstudiedin Section8.4. /3 /2 /sec −1 (2 √ 2)−1 e 8 0.00105459 0.0134764 0.0140696 0.0319876 0.0360179 10 0.00151863 0.0183461 0.0191326 0.0430288 0.0484952 12 0.00180350 0.0212243 0.0221219 0.0495035 0.0558428 14 0.00200139 0.0231978 0.0241704 0.0538990 0.0608373 16 0.00215397 0.0247167 0.0257464 0.0572352 0.0646218 18 0.00228117 0.0259884 0.0270657 0.0599884 0.0677341 20 0.00239306 0.0271153 0.0282345 0.0623992 0.0704482 22 0.00249505 0.0281508 0.0293087 0.0645961 0.0729122 24 0.00259016 0.0291244 0.0303188 0.0666511 0.0752099 Table 8.1: The values of∥ −1 ∥ 2 for various lengths and parameters . is the field in the = 0 sector withconformalweights(ℎ 11 ,ℎ 11 )=(0,0). Withinthismodulethereisnostatetoprojecttothatweexpecttogiveanonzerocontributioninthelimit →∞,theonlystatewiththeproperconformalweightshavingbeenexcludedbythequotient. Projecting on the lowest-energy state still remaining in the sector of the appropriate lattice momentum we therefore expecttheresulttoapproachzeroas→∞(Table 8.2). 72 /3 /2 /sec −1 (2 √ 2)−1 e 8 0.00105459 0.0134764 0.0140696 0.0319876 0.0360179 10 0.00154453 0.0201482 0.0209567 0.0434477 0.0485887 12 0.00140952 0.0207429 0.0216739 0.0447805 0.0501347 14 0.00121929 0.0192739 0.0202699 0.0427614 0.0480059 16 0.00103467 0.0170396 0.0180351 0.0394988 0.0445480 18 0.000875168 0.0147437 0.0156912 0.0359407 0.0407787 20 0.000742847 0.0126649 0.0135396 0.0325055 0.0371365 22 0.000634490 0.0108785 0.0116714 0.0293585 0.0337921 24 0.000545883 0.0093767 0.0100887 0.0265463 0.0307936 extrapolation 0.0000850643 0.00442133 0.00495163 0.000157526 −0.000258504 Table8.2: Thevaluesof∥Π (1) −1 ∥ 2 forvariouslengths andparameters. Π (1) isaprojectiontothestate oflowestenergywithinthe= 0,= 1sector. Thisisastatethathasconformalweights(ℎ 1,−1 ,ℎ 11 )=(1,0). Theextrapolationisobtainedbyfittingthelast five data points to a fourth-order polynomial in 1/. 8.3.3 Numericalresultsfor 1 In this section we numerically illustrate our conjectured identities −1 Φ 11 = −1 Φ 11 and 12 Φ 12 = 12 Φ 12 , fromSection8.2.2,inthescaling-weaksense. For = 1, we find Φ 11 and Φ 11 in the momentum sectors = /2− 1 and /2+ 1, and their respective descendants −1 Φ 11 and −1 Φ 11 in the sector with = /2. We show the val- ues of∥ −1 Φ 11 − −1 Φ 11 ∥ 2 /∥ −1 Φ 11 ∥ 2 and see that they do not decay (Table 8.3), then the values of ∥Π (2) ( −1 Φ 11 − −1 Φ 11 )∥ 2 /∥Π (2) −1 Φ 11 ∥ 2 and see that they do (Table 8.4). Additional values of interest— ∥ −1 Φ 11 ∥ 2 and∥Π (2) −1 Φ 11 ∥ 2 —may be found in Appendix F.1. While we numerically do observe scaling- weak convergence of −1 Φ 11 − −1 Φ 11 to zero in the sense of Eq. (8.20), here we report the values of ∥ −1 Φ 11 − −1 Φ 11 ∥ 2 /∥ −1 Φ 11 ∥ 2 and∥Π (2) ( −1 Φ 11 − −1 Φ 11 )∥ 2 /∥Π (2) −1 Φ 11 ∥ 2 to give a measure of relative deviation from zero. The decay of the latter quantity to zero implies the scaling-weak convergence of −1 Φ 11 − −1 Φ 11 to zero so long as the norm∥Π (2) −1 Φ 11 ∥ 2 does not grow too quickly. That this is the case can be seen in Table F.2, in Appendix F.1. A similar discussion applies to the scaling-weak convergence of 12 Φ 12 − 12 Φ 12 tozero. /3 /2 /sec −1 (2 √ 2)−1 e 8 0.0106586 0.104747 0.108207 0.187030 0.199715 10 0.0114921 0.118893 0.123110 0.224566 0.241811 12 0.0117599 0.124101 0.128674 0.243359 0.263743 14 0.0119545 0.127043 0.131823 0.255388 0.278124 16 0.0121676 0.129476 0.134403 0.264557 0.289152 18 0.0124147 0.131931 0.136979 0.272425 0.298551 20 0.0126930 0.134557 0.139716 0.279679 0.307105 22 0.0129960 0.137370 0.142638 0.286636 0.315194 Table8.3: Thevaluesof∥ −1 Φ 11 − −1 Φ 11 ∥ 2 /∥ −1 Φ 11 ∥ 2 for various lengths and parameters. Similarly, for = 2 we find Φ 12 and Φ 12 in the momentum sectors = /2− 2 and /2+ 2, and their respective descendants 12 Φ 12 and 12 Φ 12 in the sector with=/2. As before, we show the values ∥ 12 Φ 12 − 12 Φ 12 ∥ 2 /∥ 12 Φ 12 ∥ 2 andseethattheydonotdecay(Table8.5),andthatactingwithaprojectorΠ (4) 73 /3 /2 /sec −1 (2 √ 2)−1 e 8 0.0000527492 0.00584360 0.00627048 0.0215309 0.0251946 10 0.0000191654 0.00297587 0.00322369 0.0133976 0.0161156 12 8.26633×10 −6 0.00167699 0.00183148 0.00901364 0.0111168 14 4.03988×10 −6 0.00102009 0.00112185 0.00641199 0.00809058 16 2.16718×10 −6 0.000658400 0.000728471 0.00475643 0.00612835 18 1.25094×10 −6 0.000445437 0.000495467 0.00364534 0.00478799 20 7.62288×10 −7 0.000313082 0.000349894 0.00286778 0.00383424 22 4.87306×10 −7 0.000227096 0.000254879 0.00230490 0.00313292 extrapolation 4.38043×10 −7 −0.0000700002 −0.0000675678 0.000161454 0.0000896441 Table8.4: Thevaluesof∥Π (2) ( −1 Φ 11 − −1 Φ 11 )∥ 2 /∥Π (2) −1 Φ 11 ∥ 2 for various lengths and parameters. does show a decay for∥Π (4) ( 12 Φ 12 − 12 Φ 12 )∥ 2 /∥Π (4) 12 Φ 12 ∥ 2 (Table 8.6). Additional data for∥ 12 Φ 12 ∥ 2 and∥Π (4) 12 Φ 12 ∥ 2 aresimilarlyfoundinAppendix F.1. /3 /2 /sec −1 (2 √ 2)−1 e 10 0.211762 0.451277 0.458346 0.621868 0.653152 12 0.151572 0.359414 0.365882 0.523169 0.555244 14 0.115725 0.304912 0.310899 0.459319 0.490658 16 0.0930500 0.275545 0.281251 0.420970 0.450592 18 0.0783010 0.264041 0.269673 0.402367 0.429735 20 0.0688143 0.265678 0.271421 0.399548 0.424477 22 0.0631192 0.277084 0.283096 0.409437 0.432035 Table8.5: Thevaluesof∥ 12 Φ 12 − 12 Φ 12 ∥ 2 /∥ 12 Φ 12 ∥ 2 for various lengths and parameters. Both for = 1 and = 2, we find that the results gathered in Tables 8.4 and 8.6 support the proposed identities −1 Φ 11 = −1 Φ 11 and 12 Φ 12 = 12 Φ 12 , when we use projection operators of ranks 2 and 4, respectively, before taking norms. In both cases we have used the lowest rank such that all states with the same total conformal dimension as the descendant fields in the proposed identities, and those with lower dimensions, have been included in the projectors. As in the discussion surrounding our definition of scaling-weak convergence, and as illustrated by Figures 8.1 and 8.2, we expect that the result in the limit →∞ will remain the same for higher rank projectors. However, we do not expect the result to remain the same when no projector is applied. Indeed, the values in Tables 8.3 and 8.5 do not tend to zero with increasing. The numerical proximity of∥ −1 Φ 11 ∥ 2 to∥Π (2) −1 Φ 11 ∥ 2 and of∥ 12 Φ 12 ∥ 2 to∥Π (4) 12 Φ 12 ∥ 2 indicatesthatthelackofconvergencecomesfromdeviationsthatcanbeattributedtoparasiticcouplingsto higherstates,howeversmallthesecouplings may be. 8.4 Observationofsingletstatesintheloopmodel Inthissectionwereporttransfermatrixcomputations that support some of our main results. Weconsiderthetransfermatrixoftheloopmodelon = 2sites,with2through-lines,inthegeometry thatcorrespondstoaPottsmodelonasquarelattice(seeSection4.2.1),withspinsineachrowandperiodic boundaryconditions. Thecorrespondingloopmodelthenlivesonatiltedsquarelattice(themediallattice oftheoriginal,axiallyorientedsquarelattice). Theparticularmethodofdiagonalizationusedherehasbeen explained in much detail by Jacobsen and Saleur [7, Appendix A], but it may also be carried out using the 74 /3 /2 /sec −1 (2 √ 2)−1 e 10 0.210763 0.439128 0.445749 0.600649 0.631230 12 0.148284 0.332095 0.337881 0.482952 0.514155 14 0.110152 0.257777 0.262750 0.395760 0.426568 16 0.0852125 0.204972 0.209224 0.329727 0.359515 18 0.0679876 0.166510 0.170153 0.278915 0.307392 20 0.0555742 0.137790 0.140930 0.239167 0.266240 22 0.0463202 0.115847 0.118574 0.207563 0.233243 extrapolation −0.0015914 −0.000137804 0.0000985782 −0.000155346 0.000403050 Table8.6: Thevaluesof∥Π (4) ( 12 Φ 12 − 12 Φ 12 )∥ 2 /∥Π (4) 12 Φ 12 ∥ 2 for various lengths and parameters. algorithmsinChapter5andAppendixA. We focus here on one well-chosen value, = 1/2, which can be considered representative for the case of generic values of. For each size= 5,6,...,13, we compute the first several hundred eigenvalues in each module , 2 with= 1,2,3 and 2 = 1, extracting the multiplicity, finite-size scaling dimension, and lattice momentum of each eigenvalue. The multiplicities are always found to be either 1 or 2, and we pay specialattentiontothesinglets. Thelatticemomentum canbeidentifiedonlyuptoasign,anditcoincides withtheconformalspinmodulo. Onemajordifficultyinthestudyisthatthe thlargesteigenvalueinthefinite-sizespectrumcorresponds tothethlowest-lyingscalingstateonlyforsufficientlylarge,andforallbutthesmallestfewvaluesof this simplesituationisreachedonlywhenismuchlargerthantheattainablesystemsize. Tostudythescaling states numerically nevertheless, one therefore has to identify the sequences( 5 , 6 ,..., 13 ) that correspond to any desired scaling field, using a general methodology that requires a lot of patience [ Id. at Appendix A.5]. Polynomialextrapolationsofthefinite-sizescalingdimensionsarethenpossible,mostoftenusingthe data for all sizes (and only occasionally excluding the first few sizes), leading to quite accurate estimates of the conformal scaling dimensionΔ= ℎ+ℎ. Moreover, comparing the values of for several different in thesequencewillpermitustoliftthe“modulo” qualifier and determine (again up to a sign). ThevaluesofΔand allowustoidentifythecorrespondingscalingfield,uptoafewambiguities. Tobe precise,weareabletoidentifythecorrespondingprimaryfieldandthedescendantlevelonboththechiral andantichiralsides,uptoapossibleoverallexchangeofchiralandantichiralcomponents(again,because isdeterminedonlyuptoasign). Ournotationbelowimplicitlyincorporatesthisambiguity. Forinstance,a fieldthatisdescendantatlevel (3,2)ofaprimaryfield Φwillbedenoted −3 −2 Φ,althoughitmightinfact be any linear combination of the form( −3 + −1 −2 + 3 −1 )( −2 + 2 −1 )Φ for some unknown coefficients ,,and —orindeedthesamefieldwithchiral and antichiral components being exchanged. 8.4.1 = 1 Resultsforthemodule 11 areshowninTable8.7. Wehave,infact,identifiedthescalingfieldsforalllines with 13 ≤ 60 in this case, but to keep the table concise we show only the first 10 fields, along with several other fields that are either a primary or a singlet (or both). The ranks (meaning their position within the ordered spectrum of the transfer matrix, sorted by decreasing eigenvalue without repetition) of singlet fieldsareshowninitalics,whilethoseofthedoubletsareinregulartype. Smallnumbersrefertofinite-size levelsforwhichthelatticemomentum differsfromtheconformalspin byanontrivialmultipleof (for moredetails,seeid.). Thetableshowsallsingletswith 12 ≤ 200(forthelastsevenlinesthediagonalization for= 13 was numerically too demanding). The extrapolation of the scaling dimension is shown to about thenumberofsignificantdigitstowhichit agrees with the exact result. The primariesΦ = ⊗ ,− andΦ = ,− ⊗ with = 1 can be seen in the table for = 0,1,2, corresponding to the lines with 13 = 1,2,22. The latter two are doublets, while the first one is a singlet, as Φ 01 = Φ 01 because of the identification = −,− . For the same reason, we find several of the spinless 75 || ( 5 , 6 ,..., 13 ) Δ identification of scaling field 5 6 7 8 9 10 11 12 13 numerics exact 0 1 1 1 1 1 1 1 1 1 0.187014 0.187027 Φ 01 1 2 2 2 2 2 2 2 2 2 1.000003 1 Φ 11 1 3 3 3 3 3 3 3 3 3 1.187040 1.187027 −1 Φ 01 2 4 4 4 4 4 4 4 4 4 1.9998 2 −1 Φ 11 0 6 6 6 5 5 5 5 5 5 2.0016 2 −1 Φ 11 0 9 8 7 7 7 7 7 7 6 1.9985 2 −1 Φ 11 2 5 5 5 6 6 6 6 6 7 2.1882 2.1870 −2 Φ 01 0 7 9 8 8 8 8 8 8 8 2.18708 2.18703 −1 −1 Φ 01 2 8 10 10 9 9 9 9 9 9 2.18710 2.18703 −2 Φ 01 3 5 7 9 10 10 10 10 10 10 2.992 3 −2 Φ 11 2 19 22 24 23 23 23 24 22 22 3.43895 3.43892 Φ 21 0 18 23 27 28 27 27 28 28 27 4.002 4 −2 −1 Φ 11 0 23 30 32 37 40 38 37 38 38 4.00016 4 −2 −1 Φ 11 0 20 24 29 34 36 34 35 37 39 4.1864 4.1870 −2 −2 Φ 01 0 33 45 47 47 49 52 49 46 46 4.18707 4.18703 −2 −2 Φ 11 0 25 43 56 71 82 88 99 103 6.03 6 −3 −2 Φ 11 0 51 78 95 111 114 119 118 115 5.423 5.439 −2 Φ 21 0 53 79 100 116 120 123 121 117 5.433 5.439 −2 Φ 21 0 — — 57 76 94 112 117 120 6.178 6.187 −3 −3 Φ 01 0 — — 65 88 103 116 129 132 5.989 6 −3 −2 Φ 11 0 55 86 115 133 144 145 151 155 5.994 6 −3 −2 Φ 11 0 52 84 119 144 166 174 175 182 6.32 6.19 −3 −3 Φ 01 Table8.7: Conformal spectrum in the sector 11 for= 1/2. descendants ofΦ 01 tobesinglets(e.g., 13 = 8,39and 12 = 120,182). Amoreremarkablefindingisthesingletnatureofthepairoflineswith 13 = 5,6. Ifwehadbeendealing with a product of Verma modules, these would have formed a degenerate doublet. Instead, we see here a manifestationoftheduality −1 Φ 11 = −1 Φ 11 . Much lower in the spectrum, we similarly draw attention to the singlet nature of the lines with 12 = 115,117. Theyshowtheduality 21 Φ 21 = 21 Φ 21 . 8.4.2 = 2 Inthesameway,weshowresultsforthemodules 2, 2 inTable8.8. Notethattheresultsforallpermissible cases, 2 = 1,areshowninthesametableandthecorrespondingscalinglevelsaremarkedbytheadditional label = 0,1 for 2 = e 2i/ =(−1) . Furthermore, primary fields with fractional Kac labels appear; they arestillgivenbyΦ = ⊗ ,− andΦ = ,− ⊗ ,where isnotrequiredtobeaninteger. Wedonot studythesefieldsfurther,thoughtheyappear in the table to show that they can be identified. The primaryΦ 02 on the line with 13 = 1, and its descendant at level(1,1) on the line with 13 = 12, are both singlets due to self-duality. But more importantly, we find a pair of singlets on the lines with 12 = 24,35. They show the duality 12 Φ 12 = 12 Φ 12 . By contrast, the remaining two states with conformal weights(ℎ 12 +2,ℎ 12 +2) can be identified as the doublet on the line with 13 = 25. They correspond to a descendant ofΦ 12 at level(2,0) and a descendant ofΦ 12 at level(0,2), with coefficients that are unknown butdifferentfromthoseoftheoperators 12 and 12 , respectively. 76 || ( 5 , 6 ,..., 13 ) Δ identification of scaling field 5 6 7 8 9 10 11 12 13 numerics exact 0 0 1 1 1 1 1 1 1 1 1 1.1095698 1.1095673 Φ 02 1 1 2 2 2 2 2 2 2 2 2 1.312824 1.312810 Φ 1/2,2 0 2 3 3 3 3 3 3 3 3 3 1.92264 1.92254 Φ 12 0 1 5 4 4 4 4 4 4 4 4 2.1099 2.1096 −1 Φ 02 1 2 4 5 5 5 5 5 5 5 5 2.31304 2.31281 −1 Φ 1/2,2 1 0 7 7 6 6 6 6 6 6 6 2.31297 2.31281 −1 Φ 1/2,2 0 3 6 8 8 7 7 7 7 7 7 2.9230 2.9225 −1 Φ 12 0 2 6 9 9 9 9 8 8 8 8 3.1075 3.1096 −2 Φ 02 1 3 8 10 11 11 10 10 10 9 9 2.9393 2.9388 Φ 3/2,2 0 1 13 13 13 13 11 11 11 10 10 2.9229 2.9225 −1 Φ 12 0 0 9 11 12 15 12 12 12 12 12 3.1101 3.1096 −1 −1 Φ 02 0 0 19 24 28 30 28 29 27 25 24 3.9244 3.9225 −2 Φ 12 0 0 21 27 31 32 32 33 31 30 25 3.9231 3.9225 −2 Φ 12 0 0 31 40 40 42 41 40 36 37 35 3.9202 3.9225 −2 Φ 12 0 4 23 34 37 43 47 48 45 46 4.356 4.361 Φ 22 Table8.8: Conformalspectruminthesector 2, 2 for= 1/2. The label corresponds to 2 =(−1) . 8.4.3 = 3 Finally, the results for the modules 3, 2 are given in Table 8.9. The label now takes on three possible values,0,1,and2,againsignifying 2 = e 2i/ = e 2i/3 . || ( 5 , 6 ,..., 12 ) Δ identification of scaling field 5 6 7 8 9 10 11 12 numerics exact 0 0 1 1 1 1 1 1 1 1 2.64732 2.64713 Φ 03 1 1 2 2 2 2 2 2 2 2 2.73773 2.73746 Φ 1/3,3 2 2 3 3 3 3 3 3 3 3 3.00867 3.00846 Φ 2/3,3 0 3 5 5 4 4 4 4 4 4 3.463 3.46011 Φ 13 0 0 10 13 16 17 19 17 17 17 4.637 4.647 −1 −1 Φ 03 2 0 — — 36 48 61 74 84 84 6.640 6.647 −2 −2 Φ 03 0 0 31 55 79 100 107 109 111 108 6.454 6.460 −3 Φ 13 0 0 39 66 98 122 140 149 144 6.6478 6.6471 −2 −2 Φ 03 0 0 43 85 126 155 163 175 169 6.468 6.460 −3 Φ 13 0 0 39 91 153 216 280 327 367 8.70 8.64 −3 −3 Φ 03 0 0 31 91 169 251 332 402 456 8.85 8.64 −3 −3 Φ 03 Table8.9: Conformalspectruminthesector 3, 2 for= 1/2. The label corresponds to 2 = e 2i/3 . Weobservehereapairofsingletsonthelineswith 11 = 111,169. Theyshowtheduality 13 Φ 13 = 13 Φ 13 . * * * Summarizing, through the particular cases(,)=(1,1),(2,1),(1,2) and(1,3), we have identified pairs ofsingletstogiveevidenceofthestructureproposedinConjecture4. Whilethemerepresenceofsingletsis insufficienttoclaimdefinitivelythatthestructuremustbeexactlyasdescribed,thepossibilityofaproduct 77 of Verma modules has been ruled out, and as discussed, the diamond structure is the simplest alternative consistentwithself-duality. 8.5 Parityandthestructureofmodules In Section 8.4, we showed that the presence of certain singlets ruled out the possibility of the structure of thecontinuumfieldtheorybeingaproductofVermamodules. Thoughthediamondstructureproposedis consistentwiththeseobservations,thenumberofsingletsordoubletsobservedcannotdirectlyyieldinsight intothestructureofthecontinuumlimitVirasoromodules. Thatthisissocanbeseenbycomparisonwith astudyoftheXXZspinchain[54],whereasimilarinvestigationisperformed. There,oneobtainsthesame spectrum of the Hamiltonian, and thus one must have the same numbers of singlets and doublets, while the numerical results from using the Koo–Saleur generators indicate that the structures of the continuum limitVirasoromodulesarenotthesame. To distinguish between the two types of modules that appear for loop models and the XXZ spin chain, we must instead think more carefully about symmetry under parity, under which chiral and antichiral sectors are mapped to each other. On the lattice this corresponds to a mapping of site to−, which by Eq. (4.28) maps the Koo–Saleur generators and to each other. The parity operation is an involution, so we can distinguish the states by its eigenvalues,=±1. Let us consider the two statesΦ 12 = 12 ⊗ 1,−2 andΦ 12 = 1,−2 ⊗ 12 ,whicharemappedtoeachotherunderparity. Differentsituationscanoccurfortheir descendants 2 −1 Φ 12 , −2 Φ 12 , 2 −1 Φ 12 ,and −2 Φ 12 . Onthelattice,thefourscalingstatesthathavethecorrect lattice momenta and energies to be identified with these descendants form two singlets and one doublet. The states 1 and 2 in the doublet are mapped to each other by parity and can thus be combined into one = 1 state 1 + 2 and one =−1 state 1 − 2 . We now distinguish between the types of modules using theparityof thesinglets. If the four descendants are independent, we can form four linear combinations that are eigenstates of the parity operator, 2 −1 Φ 12 ± 2 −1 Φ 12 and −2 Φ 12 ± −2 Φ 12 . Now consider instead the states depicted in the followingdiagram(thesameasthatinEq. (8.13)): ˜ Ψ 12 Φ 12 Φ 12 12 Φ 12 = 12 Φ 12 † 12 † 12 0 −ℎ −1,2 12 12 (8.23) The four descendants are no longer independent. The bottom field 12 Φ 12 = 12 Φ 12 is clearly invariant under parity. Meanwhile, the top field satisfies 12 † 12 ˜ Ψ 12 = 12 † 12 ˜ Ψ 12 and therefore also has = 1. These two fields should both appear as singlets, while the doublet would correspond to the two linear combinationsthatcanbeformedwithwhat remains (not shown in the diagram). Theexpositionoftheprecedingargumentreliesonthecontinuumformulation. Inordertovalidateour approachofinferringpropertiesofthecontinuumtheoryfromlatticediscretizations,asimilarscenariohad better hold on the lattice as well. We therefore return to the finite-size numerics to seek the verdict. When acting with : ↦→− on the two singlets, we find that the results depend on the representation. In the loop model, both singlets have= 1, corresponding to the situation in the diagram, while in the XXZ spin chainwefindthatonehas = 1andtheotherhas=−1,sothatweratherhavetheparitiesexpectedfrom four linearly independent descendants. The two lattice discretizations thus confirm the general argument, and we find that only the loop model has the Jordan-block structure of Eq. (8.13) with the dependence 12 Φ 12 = 12 Φ 12 ,whichisoneofthemain points of this study. 78 8.6 Summary OneofthelessonsofthischapteristhatJordanblocksfor 0 and 0 areexpectedtoappearinthecontinuum limit of the-state Potts model and the loop models, even though there are no such Jordan blocks in the finite-sizelatticemodel. ThispossibilitywasalreadymentionedbyGainutdinovetal.[15]intheparticular case = 0, but occurs quite generically, whenever fields with degenerate conformal weights ℎ (with ,∈ℕ ∗ )appearinthespectrum. Itis,infact,alogicalconsequenceoftheself-dualityofthemodules 1 , andthuscanbearguedonverygeneralgrounds. Atthetimethepaperonwhichthischapterisbasedwas written,westatedthat[13,footnote11] TheabsenceofJordan[blocks]onthelatticemakesmeasuringthelogarithmiccouplings [see Section 9.3] appearing in the indecomposable [structures of Eq. (8.14)] quite difficult, as there seemstobenosimplewayofnormalizing the lattice version of the field ˜ Ψ . We will see in Section 9.4 how the concept of emerging Jordan blocks, introduced in Chapter 6, surmounts thisdifficultyandallowsfordirectmeasurement of the logarithmic couplings. The CFT for the XXZ spin chain seems well described by the somewhat mundane Dotsenko–Fateev twisted boson theory [54]. By contrast, the -state Potts model or loop model CFTs appear to be new objects,relatedtobutnotidenticalwiththe < 1Liouvilletheory[76,77,78],andareslowlygettingunder control thanks to this and other recent work. A possible direction for future progress in understanding theseCFTsbetterwouldbetorevisitthebootstrapapproachofHe,Jacobsen,andSaleur[71]bytakinginto account properly regularized conformal blocks [88]. More pressing qualitative questions, perhaps, include abetterunderstandingoftheOPEs: inparticular,theOPEsforthehulloperators,whichshouldhavesome interesting geometrical [89] and algebraic [90] meanings, or the OPEs of the currents, where logarithmic features should explain why there are much fewer than 2 adj fields with weights (1,1)—or the behavior when approaches a root of unity, and more Jordan blocks appear, probably of rank higher than two. This lastquestion seemsparticularlysuitedfora treatment via the concept of emerging Jordan blocks. 79 Chapter9 NumericalstudyofJordanblocksinthe denseloopmodelCFT InChapter8,wearguedforthegeneralstructureoftheVirasoromodulesthatariseinthecontinuumlimitof thedenseloopmodelCFT,andgaveevidencethatessentiallyruledoutsimplerpossiblities. Inthischapter, weturntomoredirectmeasurableconsequences of such indecomposable structures. Thischapterisorganizedasfollows. Aftersomebriefremarkstoputourfindingsincontext,Section9.2 describesanindirectprobeoftheJordan-blockstructureviameasurementsoffour-pointfunctions,building upon some of the findings of the previous chapter. We turn back to the continuum field theory in Section 9.3toassignquantifiableparameterstothediamondstructuresof Conjecture4. Finally,in Sections9.4and 9.5, we bring in the new machinery of emerging Jordan blocks to observe directly the gradual formation of Jordanblocks,andalsotomeasuredirectly the quantities described in Section 9.3. 9.1 Overview The bootstrap approach and the detailed study of four-point correlation functions has helped clarify the logarithmicpropertiesofthe-statePotts-modeland()-modelCFTsforgenericvaluesoftheparameters and. Many of the results in this field, however, rely on self-consistency arguments, and have not been checkeddirectly,inparticularvianumericalcalculations. Anoticeableexceptionisthe= 0case,wherethe Jordan block mixing the stress–energy tensor and its logarithmic partner was first observed by Dubail, Jacobsen,andSaleur[57],leadingtomeasurementsofthefamous“-numbers”(thelogarithmiccouplings), andthesolutionofseveralparadoxesinvolvingthepolymer/percolationorthebulk/boundarydifferences [56, 58]. This progress at = 0 was rendered considerably easier by the presence, in finite size, of Jordan blocksfortheHamiltoniangoingovertheCFT Jordan blocks for 0 and 0 in the continuum limit. By contrast, despite the conjectured appearance of Jordan blocks in 0 and 0 for the Potts model in the continuum limit (Conjecture 4), their known lattice versions have Hamiltonians that, although non- hermitian, are fully diagonalizable for generic values of , making the confirmation of their logarithmic structure—letalonethemeasurementofthe corresponding—significantly more difficult. WewillseeinthischapterhowtheframeworkofemergingJordanblocksisnaturallysuitedtoovercoming these difficulties and thus strengthen, in particular, the evidence supporting Conjecture 4. While we focus hereonaspecificlatticemodel—thedenseloop model—we believe our technique is completely general. 9.2 NumericalamplitudesandJordanblocks In Section 8.4 we have identified some singlet levels in the transfer matrix of the loop model that support theexistenceoftheindecomposablestructureofConjecture4,inthesensethatthesimpleproductofVerma modules has been ruled out. To go further and find numerical evidence for the existence of the expected Jordanblocksfor 0 and 0 ismoresubtle,sinceitturnsoutthattheHamiltonianandtransfermatricesofthe 80 Pottsmodelforgeneric remain,forthelevelsweareinterestedin,completelydiagonalizableinfinitesize. Inotherwords,theJordanblocksappearonlyinthecontinuumlimit. Whilethispossibilitywasforeseenby Gainutdinovetal.[15],itmakestheproblemquitedifferentfromtheonestudiedbyVasseur,Jacobsen,and Saleur [56] and Dubail, Jacobsen, and Saleur [57], where Jordan blocks were present for finite systems as a resultofTemperley–Liebrepresentationtheory,withtheindecomposablestructuresinthecontinuumlimit beingidenticaltothoseobservedinthelatticemodel. Wewillseethat,evenbeforeinvokingtheformalism of emerging Jordan blocks, it is nonetheless possible for the case at hand to observe the build-up of Jordan blocksinthelatticemodelindirectly. Tothisend,wenowreturntothefour-pointfunctionsoftheorderoperatorinthePottsmodel. Inlattice terms,theyareoftheform 1 2 3 4 ,wherealabel isassociatedtoeachofthefourinsertionpoints (with = 1,2,3,4), the convention being that points are required to belong to the same FK cluster if and only if theircorrespondinglabelsareidentical. Forinstance, denotesthefour-pointfunctioninwhich 1 and 3 belongtothesamecluster,while 2 and 4 belongtoadifferentcluster(seeJacobsenandSaleur[7,Figure 2]). To study such correlation functions on the lattice by the transfer matrix technique, it is convenient to place points 1 and 2 on the same time slice (i.e., lattice row) and points 3 and 4 on a different, distant slice [Id., Figure 1]. This geometric arrangement amounts to performing the -channel expansion of the correlation function [71, 7, 82]. The simplest example of the structure conjectured in Eq. (8.14) involves the fields Φ andΦ fromthestandardmodule , 2 with= 1,butwehaveseeninEq.(4.19)andtheensuing discussion that these fields decouple from the Potts-model partition function, and the results of Jacobsen andSaleurshowthattheyalsodecouplefrom the correlation functions of the order parameter. Itisthereforenaturaltoturntothenextavailablecase,= 2,andthustherepresentation 2, 2. and bothhavethepropertyofcoupling 21 and 2,−1 intheir-channelexpansions,andtheyaretheonly four-point functions that contain these two representations as their leading contributions (other correlation functionscoupleto 0, ±2 and/or 0,−1 as well) [Id.]. Moreover, the symmetric combination S = + (9.1) decouplesfrom 2,−1 forsymmetryreasons,andsince 21 containsthefields Φ 2 andΦ 2 forintegers≥ 0, ittranspiresthat S isthemostconvenientcorrelationfunctiontoinvestigateinthepresentcontext. Finally, the lowest-lying levels that can give rise to the indecomposable structure correspond to the case = 1. For allthesereasonswehenceforthfocusonthe case(,)=(1,2). Denotingtheseparationbetweenthetwogroupsofpoints( 1 , 2 )and( 3 , 4 )alongtheimaginarytime directionby,thecorrelationfunctioninthe cylinder geometry generically takes the form S = Õ Λ Λ 0 , (9.2) where the sum is over the contributing eigenvaluesΛ (withΛ 0 referring to the ground state), and are the corresponding amplitudes. A shift between the two groups of points along the spacelike direction was shown to be irrelevant [Id.]. In the notations of Figure 1 therein, one can therefore consider the two groups to be aligned—i.e., with a shift= 0. A rank-2 Jordan block for the transfer matrix on the lattice manifests itself by a “generalized amplitude,” with of the form + . This structure can be observed in many cases when is a root of unity [55]. In our problem, however, the Jordan blocks are not present for finite , andonlyexpectedtoappearinthelimit→∞. Anaturalscenarioforhowthismighthappenisasfollows: we should have two eigenvalues which become close as→∞, with divergent and opposite amplitudes. Assuming thatΛ 1 = Λ(1+ ) andΛ 2 = Λ(1− ) appear with respective amplitudes 1 = +/ and 2 =−/,wherethesmallparameter → 0 as→∞, we have then 1 Λ 1 Λ 0 + 2 Λ 2 Λ 0 ≈ + Λ Λ 0 (1+)+ − Λ Λ 0 (1−) =(2+2) Λ Λ 0 , (9.3) reproducing as→∞ the behavior expected from the presence of a Jordan block for the continuum-limit Hamiltonian. 81 ThemethodbestadaptedtoidentifyingthescenarioofEq.(9.3)isbasedonscalarproducts,asdiscussed by Jacobsen and Saleur [7, Section 4.3.2]. Notice that although this method measures the amplitudes directly in the→∞ limit, the hypotheses leading to the scaling form can still be tested, and in particular thescalingoftheamplitudesundertheapproach to the thermodynamic limit→∞. WenowinvestigatethisissueinthecontextofthestructurecontainingΦ 12 andΦ 12 ,whichisnumerically themostaccessiblecaseforthereasonsgiven above. Thefinite-sizelevelcorrespondingtothepairoffields (Φ 12 ,Φ 12 )hasbeenidentifiedinSection8.4asthe line with 13 = 3 in Table 8.8. Note that this is a twice-degenerate level (a doublet) in the transfer matrix spectrum, because the fields Φ 12 andΦ 12 are related by the exchange of chiral and antichiral components. Thecorrespondingcombinedamplitude(i.e.,summedoverthedoublet)forthecontributionofthislevelto S isshowninthefirstlineofTable9.1. Theamplitudesarenormalizedbythatoftheleadingcontributionto S —namelytheamplitudeofthelinewith 13 = 1inTable8.8. Tobeprecise,thetableshowstheamplitudes forcylindersofcircumference= 5,6,...,10,andinallcasesthedistance betweenthetwopointsineach group( 1 , 2 ) and( 3 , 4 ) is taken the largest possible: = /2 for even , and =(− 1)/2 for odd . This choice (which was also used in similar numerical work [71, 7]) corresponds to a fixed, finite distance between the two points in the continuum limit. Unfortunately, it also leads to parity effects in , which are clearly visible in Table 9.1. It is nevertheless clear that the amplitude of the line with 13 = 3 converges to a finiteconstant,asexpectedforthisnon-logarithmicpairoffields,andthiscanbeconfirmedbyindependent fitsofevenandoddsizes. Regrettably,thesituationfortheremaininglinesofthetableislessclear. Naïvely, the amplitude for each one of the last three lines appears to grow with, but our attempts to quantify this havenotbeenverycompelling,duetothe fact that we only have three sizes of each parity at our disposal. 13 5 6 7 8 9 10 3 0.515846 0.537395 0.514353 0.534697 0.524267 0.539497 24 −0.00418366 −0.0124738 −0.0186079 −0.0326019 −0.0417739 −0.0596335 25 −0.0118071 −0.0252680 −0.0241138 −0.0342282 −0.0331532 −0.0404785 35 0.0230056 0.0532078 0.0610276 0.0930659 0.102977 0.134391 Table 9.1: Amplitudes of the correlation function S corresponding to selected fields within 21 , in finite size . The distance between the two points within each group is taken as =⌊/2⌋. The lines of the table are labeled, as in Table 8.8, by the index 13 . These values are given to 8 significant figures in Grans-Samuelssonetal.[13]. We therefore turn to another strategy, in which the same amplitudes are measured with the smallest possible distance = 1 between the two points in each group. This will eliminate the parity effects, so that more reliable fits can be studied. Note that the choice = 1 corresponds to a vanishing distance in the continuum limit, so one might expect the finite-size amplitudes to pick up an extra factor of 1/. In particular, the amplitude of a generic, non-logarithmic field contributing to S is the expected to vanish as −1 in the →∞ limit. Indeed, the amplitude of the line with 13 = 3 in Table 9.2 fits very nicely to 0 + 1 −1 + 2 −2 +··· , and the absolute value of the constant term 0 can be determined to be at least 80 timessmallerthanthedatapointwith= 10. We therefore conjecture that, in this case, 0 = 0 indeed. For the line with 13 = 24 (a singlet level) we attempt a fit of the form 0 + 1 − + 2 −2 + 3 −3 . This matches the data nicely with ≈ 1.005, indicating that = 1 might be the exact value of the exponent. But we find now that the absolute value of the constant term 0 is about 3 times larger than the data point with = 10, which is strongly indicative of 0 being nonzero in this case. We therefore conjecture that this line shouldbeidentifiedwithoneofthetoporbottom fields in the Jordan block of Eq. (8.13). The same type of fit for the line with 13 = 35 (the other singlet level) yields ≈ 2.05 and a constant term 0 which is about 4 times smaller than the= 10 data point. Finally, the line with 13 = 25 (a doublet) matchesthefitwith ≈ 1.3and 0 about3timessmallerthanthedatapointwith= 10. Seeninisolation, these fits do not permit us to convincingly conclude whether the value of 0 is finite or zero for those two lines. However, structural considerations provide more compelling evidence. According to the argument giveninEq.(9.3),thelogarithmicsingletwith 13 = 24needstobeaccompaniedbyanothersingletfieldwith 82 13 5 6 7 8 9 10 3 0.296317 0.233276 0.190424 0.159168 0.135394 0.116790 24 −0.0000022603 −0.0000057466 −0.0000088156 −0.0000107666 −0.0000116358 −0.0000117097 25 −0.00194026 −0.00264918 −0.00297698 −0.00305503 −0.00298975 −0.00285018 35 0.0000380855 0.0000508765 0.0000532218 0.0000497383 0.0000439510 0.0000377665 Table9.2: Amplitudes ofthecorrelationfunction S correspondingtoselectedfieldswithin 21 ,infinite size. The distance between the two points within each group is now chosen the smallest possible,= 1. Thesevaluesaregivento8significantfigures in Grans-Samuelsson et al. [13]. an opposite and diverging (for finite conformal distance) amplitude. Being a singlet, the line with 13 = 35 istheonlypossiblecandidateforsuchalogarithmic partner. Asadecisivetest,wethereforeplotinFigure9.1theratiobetweentheamplitudesofthetwosinglets. A second-order polynomial in 1/ fits the data nicely and gives an extrapolated value of the ratio of −0.985, very close to the exact ratio of−1 expected from Eq. (9.3). We believe that this shows that the two singlets correspond to the conformal fields 12 Φ 12 + 12 Φ 12 and ˜ Ψ 12 , and that the indecomposable structure of Eq. (8.14) builds up only in the →∞ limit (recall that 12 Φ 12 = 12 Φ 12 only holds in the limit; the symmetrized combination is the natural guess to consider here). On the other hand, Figure 9.1 vividly illustratesthatamaximumsizeof= 10isstillquitefarfromthethermodynamiclimit,andwithhindsight it is therefore hardly surprising that only a combination of arguments can reveal the detailed nature of the fourfieldsfromTable8.8havingconformal weights (ℎ 12 +2,ℎ 12 +2). 0.05 0.10 0.15 0.20 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 Figure9.1: Ratio 1 / 2 betweentheamplitudesofthetwosingletfields(seeTable9.2),correspondingtothe lines with 13 = 24 and 13 = 35 (see Table 8.8), plotted against1/. The curve is a second-order polynomial fittothelastthreedatapoints. 83 9.3 Modules for the loop model in the degenerate case: the OPE point ofview As in the early works on logarithmic CFTs [56, 91], it is possible to understand the appearance of inde- composable modules in the continuum limit of 1 by carefully examining the OPEs and their potential divergenceswhenoneofthefieldsinthe -channel has a degenerate conformal weight. Tostart,imaginethatwehavesomeOPEofafieldofdimension Δwithitselfwhereafieldwithconformal weights(ℎ 12 ,ℎ 12 ) appears. In ordinary CFT, the descendants of this field at level two in the chiral and in the antichiral sector would not be independent: this fact is crucial to cancel the divergence arising in the OPE coefficients from the fact that ℎ 12 is in the Kac table, resulting in a finite OPE such as the ones arising in the minimal-model CFTs [83]. Let us now see what happens if the null descendants are not zero, and the divergences potentially remain. To proceed, we factor out the() −2Δ term, withΔ = Δ denoting the conformal weight of the fields being fused, and analyze the potential divergences by slightly shifting the conformalweightsofthefieldontherighthand side of the OPE: Φ ℎℎ ()Φ ℎℎ (0)∼ () () 2Δ n () ℎ 1+,2 h + 2 + (−2) (Δ,ℎ 1+,2 ) 2 −2 + (−1,−1) (Δ,ℎ 1+,2 ) 2 2 −1 ⊗ h. c. i +··· o , (9.4) where() is a number to be determined, the dots stand for other fields, and we have used the shorthand notations = 1+,2 and (9.5a) = 1+,2 . (9.5b) Fields in OPEs without position arguments are understood to be evaluated at= 0 (= 0 for antiholomor- phicfields). Thecoefficients inEq.(9.4) are fully determined by conformal invariance: (−2) (Δ,ℎ)= (ℎ−1)ℎ+2Δ(1+2ℎ) 16(ℎ−ℎ 12 )(ℎ−ℎ 21 ) , (9.6a) (−1,−1) (Δ,ℎ)= (1+ℎ)(+8ℎ)−12(Δ+ℎ) 64(ℎ−ℎ 12 )(ℎ−ℎ 21 ) , (9.6b) andnotethatwehave (−1,−1) (Δ,ℎ) 2 −1 + (−2) (Δ,ℎ) −2 = (−2) (Δ,ℎ)(ℎ)+ (−1,−1) 0 (ℎ) 2 −1 , (9.7) where (ℎ)≡ −2 − 3 2+4ℎ 2 −1 , and (9.8a) (−1,−1) 0 (ℎ)= 1+ℎ 4(1+2ℎ) . (9.8b) It is important to notice that in writing Eq. (9.7), the dependence on the external field Δ only appears in the coefficient (−2) —i.e., the operator that will turn out to give rise to the Jordan-block structure is independentoftheexternalfield. Thispoint will become more clear in the following discussion. With → 0, we maintain all calculations only to the lowest necessary orders in . Going back to ℎ= ℎ 1+,2 ,andwriting ≡(ℎ 1+,2 ),itis convenient to define ()≡⟨ | † | ⟩= 8(ℎ−ℎ 12 )(ℎ−ℎ 21 ) 1+2ℎ = , (9.9) with =− 2(1−2 2 − 4 +2 6 ) 6 , (9.10) 84 where we have used the parametrization 2 = /(+ 1). On the other hand, notice that as → 0, the coefficient (−2) (Δ,ℎ 1+,2 ) has a simple pole, since the denominator is proportional to the Kac determinant, as is clear from Eq. (9.6a). This means that the OPE potentially presents singularities, which must be properlycanceledbythecontributionofotherfieldswiththeproperdimensions—apointwellunderstood since earlier studies on indecomposability in logarithmic CFT [56, 57, 58, 91]. The leading singularity in the OPE is a second-order pole coming from the descendants at level 2 of ⊗ . Keeping in mind that ℎ 12 +2= ℎ −1,2 ,andofcourseℎ = ℎ −,− , we therefore introduce the other fields = −1+,2 and (9.11a) = −1+,2 (9.11b) inordertocancelsuchsingularities,andwe complete the OPE as follows: Φ ℎℎ ()Φ ℎℎ (0)∼ () () 2Δ n () ℎ 1+,2 h + 2 + (−1,−1) 0 () 2 2 −1 + (−2) () 2 ⊗ h. c. i +() ℎ −1+,2 () ⊗ o , (9.12) wherewehaveadoptedtheshorthandnotations (−2) ()≡ (−2) (Δ,ℎ 1+,2 ), (−1,−1) 0 ()≡ (−1,−1) 0 (ℎ 1+,2 ),and thenewcoefficient () isyettobedetermined. Tostudythenecessarycancelationofsingularities, we focus on the most divergent term at level 2: () ℎ 1+,2 +2 [ (−2) ()] 2 ⊗ +()() ℎ −1+,2 +2 ⊗ = log()() ℎ −1,2 [ (−2) ()] 2 ⊗ + 1 √ () ℎ −1+,2 Φ , (9.13) wherewehavedefined ≡ ℎ 1+,2 +2−ℎ −1+,2 = 1 2 (9.14) andintroducedthenewfield Φ ≡ √ {[ (−2) ()] 2 ⊗ +() ⊗ }. (9.15) Thetwo-pointfunctionofthisfieldisgiven by ⟨Φ (,)Φ (0,0)⟩= {[ (−2) ()] 4 2 ()() −2ℎ 1+,2 −4 + 2 ()() −2ℎ −1+,2 }. (9.16) RecallEq.(9.9)andthat (−2) () hasasimple pole in . One can write [ (−2) (Δ,ℎ 1+,2 )] 2 ()≡ ++(). (9.17) It is then clear that the coefficient of the first term in Eq. (9.16) has a double pole that must be canceled by thedivergencefromthesecondterm. This requires 2 () to be of the form 2 ()= 2 + +(1). (9.18) Such behavior can in fact be established using that 21 is degenerate in the theory [13, Section 7]. The singularitycancelationconditionthenreads =− 2 , (9.19) andthetwo-pointfunctionofEq.(9.16)becomes as → 0 ⟨Φ(,)Φ(0,0)⟩= −2 2 log()+2+ () 2ℎ −1,2 . (9.20) 85 Takingintoaccountthe1/ √ factorinEq.(9.13),wemustthereforetake()= √ ,sothatthecontribution ofΦ intheOPEisoforderunity. Atthispoint,itisnaturaltointroducethe normalized field ˆ ≡ 1 √ , (9.21) and identify it as another copy of in the limit → 0, since both then have dimension ℎ −1,2 and are annihilatedby 1 and 2 . ThefirsttermontherighthandsideofEq.(9.13),uponmultiplicationby (),is thengivenby 2 √ () ℎ −1,2 log()(⊗+⊗). (9.22) Combiningwiththeremainingtermsinthe OPE of Eq. (9.12), √ () ℎ 1+,2 h + 2 + (−1,−1) 0 () 2 2 −1 ⊗ (−2) () √ 2 ˆ +h. c. i , (9.23) andrecallingEq.(9.17),wethenhavethefull OPE as → 0, after factoring out a global factor of √ : h ℎ 12 ℎ −1,2 + 2 + (−1,−1) 0 2 2 −1 ⊗+h. c. i + () ℎ −1,2 2 √ log()(⊗+⊗)+ 2 √ Φ . (9.24) Setting ⊗= √ ˆ ⊗= √ ⊗ ˆ =⊗, (9.25) usingtheidentificationsof ˆ and ˆ with and in the → 0 limit, the OPE becomes h ℎ 12 ℎ −1,2 + 2 + (−1,−1) 0 2 2 −1 ⊗+h. c. i +() ℎ −1,2 √ log()⊗+ √ Φ . (9.26) Aswillbecomeobviousbelow,thishasthe interpretation that 0 − 0 is diagonalizable. We are interested in the logarithmic mixing at level 2—i.e., the second term of the OPE. Inspecting the terms,itisnaturaltoredefinethefield ˜ Ψ≡ √ Φ, (9.27) which, as we will see, becomes the logarithmic partner of ⊗ = ⊗. It is a simple exercise to calculatetheirtwo-pointfunctionsandone arrives at ⟨(⊗)(,)(⊗)(0,0)⟩= 0, (9.28a) ⟨ ˜ Ψ(,)(⊗)(0,0)⟩= −1 () 2ℎ −1,2 , (9.28b) ⟨ ˜ Ψ(,) ˜ Ψ(0,0)⟩= −2 −1 log()+() −2 (2+) () 2ℎ −1,2 . (9.28c) Werecognizetheusuallogarithmicstructure of a rank-2 Jordan block [84]. Asafinalstep,wecomputetheactionof the Virasoro generator 0 on the pair(⊗, ˜ Ψ): 0 ⊗= ℎ −1,2 ⊗, (9.29a) 0 ˜ Ψ= ℎ −1,2 ˜ Ψ+ √ [ (−2) ()] 2 (ℎ 1+,2 +2−ℎ −1+,2 ) ⊗ = ℎ −1,2 ˜ Ψ+⊗, (9.29b) 86 andsimilarlyfor 0 . Thereforeweseethat in the basis(⊗, ˜ Ψ)=(⊗, ˜ Ψ), we have 0 = ℎ −1,2 1 0 ℎ −1,2 = 0 , (9.30) formingarank-2Jordanblock. Inaddition, we find † ˜ Ψ= √ [ (−2) ()] 2 ⊗ = −1 ⊗, (9.31) wherewehaveusedEqs.(9.9),(9.17),and(9.21). Notealsothat 1 ˜ Ψ= 0. Hence,themoduleisdepictedas ˜ Ψ ⊗ ⊗ ⊗=⊗ † / −1 † / −1 0 −ℎ −1,2 (9.32) ThisstructurecoincideswiththatofEq.(8.13). As we have briefly commented, the logarithmic coupling −1 in Eq. (9.28c), which characterizes the Jordan-block structure, does not depend on the dimensionΔ of the external fields. More explicitly, from Eqs.(9.10)and(9.14),wehave −1 =− 2(1−2 2 − 4 +2 6 ) 4 , (9.33) which is entirely determined by the Kac formula and the Kac determinant. By contrast, the coefficient √ / √ in the OPE of Eq. (9.26) does depend onΔ through, due to Eq. (9.17). Similarly, the constant in thetwo-pointfunctionofEq.(9.28c)alsodependson . Thisis,however,compatiblewiththeJordan-block structure,sincethefield ˜ Ψalwaysadmits a shift by a multiple of the null field [84], ˜ Ψ→ ˜ Ψ+⊗= ˜ Ψ+⊗, (∈ℂ) (9.34) whichdoesnotchangeEq.(9.30). The construction also generalizes to the case of operators and ,− . In general, the module has the structure in Eq. (9.32) with → ,→ ,− , and replaced by the proper combination of Virasoro generators . Setting ⟨ + | † | + ⟩= , (9.35) andobservingthat ℎ +, +−ℎ −+, = , with (9.36) = 2 , (9.37) we find that the free parameter of the module (the so-called logarithmic coupling, or indecomposability parameter)is = −1 , (9.38) sothat ( 0 −ℎ −, ) ˜ Ψ =( 0 −ℎ ,− ) ˜ Ψ = ⊗ ,− = ,− ⊗ , (9.39a) † ˜ Ψ = ⊗ ,− , (9.39b) † ˜ Ψ = ,− ⊗ (9.39c) 87 withthestructure ˜ Ψ ⊗ ,− ,− ⊗ ⊗ ,− = ,− ⊗ † / † / 0 −ℎ −, (9.40) Thisstructureagaincoincideswiththatof Eq. (8.14). Forthespecialcase== 1,forinstance, we find that 11 =−1+1/ 2 and therefore 11 = 1− 2 . (9.41) Similarly,thecalculationofEq.(9.33)yields, using these notations, 12 = −1 12 12 =− 2(1−2 2 − 4 +2 6 ) 4 . (9.42) 9.4 Measurement of indecomposability parameters via emerging Jor- danblocks As a result of indecomposable operator product expansions in a logarithmic CFT, a parameter called was firstintroducedbyGurarieandLudwig[91]toquantifythisindecomposability,intermsofthestress–energy tensor and its logarithmic partner. The indecomposability manifests as non-diagonalizability of 0 and 0 : 0 = 2, (9.43a) 0 = 2+, (9.43b) 0 = 0, and (9.43c) 0 =. (9.43d) Forthetwo-pointfunctionsinvolving and, we have (at= 0) ⟨()(0)⟩= 0, (9.44a) ⟨()(0)⟩= 4 , and (9.44b) ⟨()(0)⟩= −2log+ 4 , (9.44c) where is an arbitrary constant that can be adjusted using the freedom→ +, with an arbitrary scalar. RecallingtheVirasoroinnerproduct (for holomorphic fields) [33] ⟨Φ|Ψ⟩= 2ℎ Φ lim →∞ ⟨Φ()Ψ(0)⟩, (9.45) wehave,intheCFT,⟨|⟩= 0,⟨|⟩=∞,and⟨|⟩=. The measurement of on the lattice would thus seem straightforward—identify and on the lattice and evaluate their conformal scalar product. The difficulty with this approach is the otherwise-mundane matter of normalization. Because⟨|⟩= 0 in the CFT, also observed in finite size on the lattice, the value of⟨|⟩ on the lattice is completely dependent on the way one normalizes. The solution is to write an 88 expression for that is independent of the normalization of. In the CFT, = −2 , with the identity field. Beingprimary,wemaynormalizeits lattice analogue to unity: ⟨|⟩= 1. Then = 2 = |⟨| −2 |⟩| 2 ⟨|⟩ (9.46) isindependentofthenormalizationof,sincethenormalizationof islinkedtothatof viathecoefficients in(9.43). Themethodofmeasuring between and generalizestoanylogarithmicpair,especiallyregardingthe factthat isanullstate. Ingeneral,wheneveraJordanblockforms,thereisatopandabottomstate. Ifthis JordanblockcomesfromaHamiltoniandescribingalogarithmicCFT,fromgeneralargumentsinvolvingthe actionoftheVirasoroalgebra,itisexpectedthatthebottomstateinsuchaJordanblockhaszeroconformal normsquare. IntheCFT,calling ˜ Ψ,Ψthetopandbottomstate,theresultfollowsfromthefactthatwehave 0 ˜ Ψ= ℎ ˜ Ψ+Ψ (9.47a) 0 Ψ= ℎΨ (9.47b) Hencewehave ⟨Ψ|Ψ⟩=⟨Ψ|( 0 −ℎ) ˜ Ψ⟩=⟨( 0 −ℎ)Ψ| ˜ Ψ⟩= 0 (9.48) where in the last equation we used the fact that † = − (i.e. and − are conjugate for the Virasoro norm). On the lattice, recall that we observe pairs of singlets (Section 8.4) that have close eigenvalues at finite size that approach each other in the limit→∞. Nevertheless, both singlets are still proper eigenvectors, andweexpectthattheybothconvergetothebottomstateΨofanemergingJordanblock. Theyshouldthus have a loop norm going to zero in the limit→∞. This will be checked in cases where we find emerging Jordanblocksusingthe measure(Chapter 6). Themethodforcalculating issummarized as follows: 1. Identify an emerging Jordan block by finding pairs of vectors Ψ,Ψ ′ on the lattice corresponding to givenconformalfieldssuchthat lim(Ψ,Ψ ′ )= 1. 2. ConstructtheemergingJordanvector ˜ ΨbyorthogonalizingΨ ′ againstΨwithrespecttothestandard innerproduct(·|·) . Normalize ˜ Ψsothat(Ψ| 0 | ˜ Ψ)= 2 (recall generally that = + − ). 3. Using the CFT, expressΨ=Φ as a descendant of a primary field Φ, with the corresponding null vectoroperator. Identifythelatticeanalogue ofΦ, and normalize it to⟨Φ|Φ⟩= 1. 4. Finally, = |⟨ ˜ Ψ||Φ⟩| 2 ⟨ ˜ Ψ|Ψ⟩ , (9.49) where,optionally, − → − and − → in the expression for. There are many possibilities for the unspecified limit in step 1. In this work we focus on the Jordan blocks thatformas→∞and→ 0,bystudyingtheemergingJordanblocksatfinite andgeneric,respectively. Additionally, the procedure gives two possible measurements of. SinceΨ andΨ ′ both converge to the samevector,theyshouldbetreatedonanequalfooting,andbyswitchingtherolesofΨandΨ ′ weobtaina second value for. Generically, this value will be different from the first one, but should become the same whenthelimitistaken. Inpractice,wemay have reason to favor one measurement over the other. 89 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.2 0.4 0.6 0.8 1.0 J α, β 6 8 10 12 14 16 18 20 Figure 9.2: The formation of a Jordan block between two singlets in 11 . The legend indicates the value of = 2. 9.4.1 EmergingJordanblocksin 11 atgeneric In 11 therearetwoeigenstatesatzeromomentumwhoseconformalweightsgotoℎ+ℎ= 2as→∞(Table 8.7, 13 = 5 and 6). For generic and finite , these eigenstates are non-degenerate, and we refer to them by the labels and . In practice, is the first excitation (above the ground state) in the zero-momentum sectorof 11 . isthesecondorthirdexcitation,dependingonthevaluesof and. Atvaluesof closeto 1 and small values of, it is the third excitation, but crosses over to the second excitation at large values of . The emerging Jordan block involving and , based on the measure(,), can be seen in Figure 9.2. TheplotsuggeststhatthereisaJordanblock at finite size for =−2, and this is indeed the case [92]. Following our procedure, we calculate the parameter 11 that describes this Jordan block at finite size. Both and converge to the same vector as→∞, the primary field Ψ 11 = −1 Φ 11 = −1 Φ 11 of Eq. (8.14) (again, the equality holds in the continuum limit). Take, for the lattice approximation ofΦ 11 , the lowest field of momentum 1 ( 13 = 2 in Table 8.7) and normalize it to⟨Φ 11 |Φ 11 ⟩ = 1. Then, orthogonalize and by subtracting from its component along , resulting in ˜ , and normalize it such that(| 0 | ˜ )= 2. Then, weobtainalatticemeasurementofthelogarithmic coupling by calculating (1) 11 (,)= |⟨ ˜ | −1 |Φ 11 ⟩| 2 ⟨ ˜ |⟩ . (9.50) As and should be treated on an equal footing, we similarly define (2) 11 by exchanging the roles of and inthisprocedure. (2) 11 willgenericallybedifferentfrom (1) 11 ,butwhenagenuineJordanblockforms,they should be equal, as we see in the analytic examples. These two measurements of 11 are given graphically inFigures9.3and9.4. WepredictedinEq.(9.41)that 11 = 1 +1 , (9.51) nowwrittenintermsof. Thecomparison of this value with the extrapolations is given in Figure 9.5. 90 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.1 0.2 0.3 0.4 0.5 0.6 b 11 1 8 10 12 14 16 18 extrapolation Figure9.3: The measurement of (1) 11 between and. -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.2 0.4 0.6 0.8 b 11 2 8 10 12 14 16 18 extrapolation Figure9.4: The measurement of (2) 11 between and. 91 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.2 0.4 0.6 0.8 b W 11 b 11 (1) b 11 (2) b 11 Figure9.5: Thetwomeasurements of 11 compared with the expected value. 9.4.2 EmergingJordanblocksin 21 atgeneric In 21 therearefourfields,whichhave ℎ+ℎ= 2ℎ 1,−2 . Forgeneric andfinite ,thesefieldscorrespondto two non-degenerate eigenvalues (singlets; lines 13 = 24 and 35 of Table 8.8), which we denote by and , andonedoublydegenerateeigenvalue(line 13 = 25ofTable8.8). Atgeneric,thetwosingletsareexpected to form a Jordan block in the→∞ limit. The doublet occurs because of a left-right symmetry (Section 8.5),andisnotexpectedtobeinvolvedinthe Jordan-block structure. Using the measure, we can directly observe the emerging Jordan block involving and (Figure 9.6). As with the fields and in 11 , the plot correctly suggests that there is a Jordan block at finite size for =−2. We again obtain estimates for the parameter 12 that characterizes the Jordan block involving and . Both and converge to the same vector as→∞, the field Ψ 12 in Eq. (8.13). The field Φ 12 is realized on thelatticeasthelowestfieldofmomentum 2(line 13 = 3inTable8.8),whichwenormalizeto⟨Φ 12 |Φ 12 ⟩= 1. Orthogonalizeand bysubtractingfrom itscomponentalong,resultingin˜ ,andnormalizeitsothat (| 0 |˜ )= 2. Thenwehave (1) 12 (,)= |⟨˜ ||Φ 12 ⟩| 2 ⟨˜ |⟩ , (9.52) where = −2 − 3 2(2ℎ 12 +1) 2 −1 . (9.53) Wesimilarlydefine (2) 12 byexchangingtherolesofand inthisprocedure. Thetwomeasurementsof 12 aregivengraphicallyinFigures9.7and9.8. FromEq.(9.42)wehave 12 = 4 +1 − 2 2 . (9.54) Thecomparisonofthisvaluewiththetwo extrapolated measurements is shown in Figure 9.9. 92 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.2 0.4 0.6 0.8 1.0 J(μ, ν) 10 12 14 16 18 20 Figure 9.6: The formation of a Jordan block between two singlets in 21 . The legend indicates the value of = 2. -2.0 -1.5 -1.0 -0.5 0.5 1.0 c -1 1 2 3 b 12 1 10 12 14 16 18 extrapolation Figure9.7: The measurement of (1) 12 between and. 93 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.5 1.0 1.5 2.0 2.5 b 12 2 10 12 14 16 18 extrapolation Figure9.8: The measurement of (2) 12 between and. -2.0 -1.5 -1.0 -0.5 0.5 1.0 c -0.5 0.5 b W 21 b 12 (1) b 12 (2) b 12 Figure9.9: Thetwomeasurements of 12 compared with the expected value. 94 * * * Particularly with the measured values of (1) 11 and (1) 12 , we see excellent agreement with the theoretical values in Eqs. (9.41) and (9.42). As these values of were derived from the diamond structure of Eq. (8.14) in Section 9.3, these measurements almost certainly establish the correctness of Conjecture 4. Admittedly, themeasurementisnotascompellingnear= 1,andthevaluesof (1) 11 and (1) 12 differsignificantlyfrom (2) 11 and (2) 12 , even though they should give the same values in the limit. On the former point, the emergence of the Jordan block is much slower near = 1, as seen in Figures 9.2 and 9.6, and this may be remedied once wehaveaccesstolargerlatticesizes. Onthelatterpoint,theequivalenceofthetwomeasurementsfollowed from the fact that in the limit→∞, the two eigenvectors eventually became equal. But at finite size, one ofthetwomaybemoreequalthantheother, as we now describe. InSection6.3.1,itwasremarkedthatineachexamplewithacontinuousparameter,oneofthetwovectors that formed an emerging Jordan block was already equal to the proper eigenvector, even before the limit was taken. In the case at hand, we studied the limit→∞ and there is no sense in which an eigenvector can be considered independent of, as the dimension of the vector strongly depends on. But if there is a sense in which a similar idea can be made precise—that and are less strongly dependent on than and —it would favor the former fields over the latter and break the symmetry of treating each pair on an equal footing. The “strength” of the dependence on could very well vary with, a concept we explore in another context in Chapter 11. This could explain why (1) 11 and (1) 12 are more successful measurements of 11 and 12 than (2) 11 and (2) 12 for a wide range of values, with their degrees of success trading places near = 1. 9.5 Thelimit→ 0: Jordanblocksbetweendifferentmodules 9.5.1 Gluedcellrepresentations We previously described cell representations of the Jones–Temperley–Lieb algebra, the standard modules , 2 and 0, ±2. The salient feature of the modules , 2 was that the number of through-lines was kept fixed: whenever the action of a generator resulted in a state with fewer through-lines, the result was annihilated. Onemayequallywellconsiderarepresentationconsistingofacollectionofstandardmodules with a range of consecutive values of . The underlying vector space is simply the vector-space direct sum of the underlying vector spaces of the involved modules, and the basis is simply the union of their bases. The generators act naturally on the basis states, and the number of through-lines is allowed to decrease (except possibly for the module with the lowest value of , which is usually 0). The new representation is thus not a module direct sum, so it is indecomposable. But it is not irreducible, since the standard modulewiththelowestvalueof isaninvariantsubmodule. Inthissectionweconsidertherepresentation 0, ±2+ 11 + 21 —theprimary“glued”module or representation of this section. Recalltherulespreviouslysetoutfordetermining the loop scalar product between link states: First, unless all through-lines connect from top to bottom the result is zero. We also take into accounttheweightofstraighteningtheconnectedthrough-lines,usingtheideaofthegraduated phase: a through-line that has moved to the right (left) is assigned the weight e i/2 (e −i/2 ) foreachstep. Eachcontractibleloopcarriestheweight=+ −1 ,whileeachnon-contractible loopcarriestheweighte i/2 +e −i/2 . In the glued representations, we define an inner product that is the same as in the standard-module case, except we drop the requirement that through-lines connect. Importantly, the ex ante inner product of anytwostates,eventhosewithdifferentnumbersofthrough-lines,isnonzero—ifthediagramproducedby gluingtwolinkstatesproducesnoloopsandmovementofthrough-lines,itsinnerproductis 0 (e i/2 ) 0 = 1. Undertheserulestheinnerproductofthefirstexample ⟨(12)(3)(4)|(1)(2)(34)⟩ inEq.(4.35)likewisebecomes 1 insteadof0. TheJTLgenerators remain self-adjoint with respect to this new inner product. 95 9.5.2 AbsenceorpresenceofJordanblocksatfinitesize Strictly at = 0, it so happens that there are no Jordan blocks on the lattice for the dense loop model Hamiltonian at = 1, even in the glued module 0, ±2+ 11 + 21 . This is a somewhat surprising fact since we know from representation theory that the JTL algebra is not semisimple at that point, and that the modules 0, ±2 are glued by the action of generic elements of the algebra (Section 2.5). Furthermore, other representations, such as the ℓ(2|1) spin chain, do exhibit the expected Jordan blocks. Nonetheless, the fact remains that the Hamiltonian itself is fully diagonalizable, and that, for instance, the eigenvalues corresponding to the conformal states and its logarithmic partner are degenerate without being part of aJordanblock. ThisfactwasactuallyobservedpreviouslybyDubail,Jacobsen,andSaleur[57]forthecase of open boundary conditions, though it was not clear at the time whether this was a bizarre effect due to thesmallsizesstudied. However,thesituationdoesnotseemtochangewhenexploringlargersizes,andit seemstobeadefinitefeatureofthemodel. In the preceding work as well as in subsequent studies, this problem was circumvented by introducing aparameter thatdidnotchangethe(generalized)eigenvaluesoftheHamiltonianbutallowedtheJordan blocks at= 0 to “reappear” somewhat miraculously (Appendix E.2). The meaning of this parameter was notveryclear,exceptforthefactthatitbrokethesymmetryoftranslationbyonesite—asomewhatpleasant feature, since the dense loop model is naturally described using an underlying oriented lattice that is only compatible with translation invariance by two sites. With ≠ 1 for open boundary conditions ( ≠±1 for periodic boundary conditions) the measured values of the logarithmic couplings known at the time of those studies for = 0 were found to be independent of and in excellent agreement with theoretical expectations. Some illustrative calculations are carried out analytically, in Appendix E.3, to show that the absenceofaJordanblockfor =±1 isno accident. The parameter only affects matrix elements between different modules. As long as one restricts to a singlestandardmodule(e.g., 11 or 21 ),thevaluesoftheoverlaps ortheparameters 11 and 12 studied inthischapterforgeneric areunaffectedbytheintroductionof —andthisistrueevenat= 0. However, it turns out that, if we measure the same quantities by embedding these in the glued module, the values of, 11 , and 12 depend on in a nontrivial, most unsatisfactory way: they encounter seemingly-random divergencesandsignchangeswhenconsideredasfunctionsof,andthesedivergencesandsignflipschange positions as varies. This occurs despite the fact that the JTL algebra is simple for ≠ 0, so a change of basisprovidesmodulesisomorphicto 11 or 21 . Theonlystrategytorecoverresultswhicharecompatible with conformal invariance seems to be to consider the limit → 1 at finite , and then extrapolate to the thermodynamiclimit,ifitcanbedone. Itisfairtoaskwhyonewouldhavetogothroughthesegymnastics, andwhattheymean. We now observe that the JTL algebra has a symmetry under →− and→−. We can therefore study the loop model with opposite signs for the Hamiltonian and loop fugacity (as well as the ground- state energy density ∞ ). While it is widely expected that these two descriptions are equivalent, they do differ right at = 1 ( = 0). To see this, consider the simple example of = 2 ( = 4). Using the basis {(12)(34),(23)(14),(2)(3)(41),(3)(4)(12),(1)(4)(23),(1)(2)(34),(1)(2)(3)(4)} (the notation is defined in Section 2.3),wehave,excludingtheprefactor F in Eq. (4.24), (, ∞ )= © « 4 ∞ −2 −2 0 −1 0 −1 0 −2 4 ∞ −2 −1 0 −1 0 0 0 0 4 ∞ − −1 0 −1 −1 0 0 −1 4 ∞ − −1 0 −1 0 0 0 −1 4 ∞ − −1 −1 0 0 −1 0 −1 4 ∞ − −1 0 0 0 0 0 0 4 ∞ ª ® ® ® ® ® ® ® ® ¬ . (9.55) 96 Both (1,1)= © « 2 −2 0 −1 0 −1 0 −2 2 −1 0 −1 0 0 0 0 3 −1 0 −1 −1 0 0 −1 3 −1 0 −1 0 0 0 −1 3 −1 −1 0 0 −1 0 −1 3 −1 0 0 0 0 0 0 4 ª ® ® ® ® ® ® ® ® ¬ (9.56) and −(−1,−1)= © « 2 2 0 1 0 1 0 2 2 1 0 1 0 0 0 0 3 1 0 1 1 0 0 1 3 1 0 1 0 0 0 1 3 1 1 0 0 1 0 1 3 1 0 0 0 0 0 0 4 ª ® ® ® ® ® ® ® ® ¬ (9.57) have eigenvalues{ 1 ,..., 7 } ={0,1,3,3,4,4,5}. It turns out however that(1,1) is diagonalizable with the7correspondinggeometriceigenvectors 1 =(1,1,0,0,0,0,0), (9.58a) 2 =(2,2,−1,−1,−1,−1,0), (9.58b) 3 =(0,0,1,0,−1,0,0), (9.58c) 4 =(0,0,0,1,0,−1,0), (9.58d) 5 =(1,−1,0,0,0,0,0), (9.58e) 6 =(1,1,−2,−2,−2,−2,6), (9.58f) 7 =(2,−2,1,−1,1,−1,0), (9.58g) but−(−1,−1) is not. It has an identical spectrum, but its generalized eigenbasis in Jordan canonical form is 1 =(1,−1,0,0,0,0,0), (9.59a) 2 =(2,−2,−1,1,−1,1,0), (9.59b) 3 =(0,0,0,1,0,−1,0), (9.59c) 4 =(0,0,1,0,−1,0,0), (9.59d) 5 =(1,1,0,0,0,0,0), (9.59e) ˜ 6 = 0,0, 1 2 , 1 2 , 1 2 , 1 2 ,− 1 2 , (9.59f) 7 =(2,2,1,1,1,1,0), (9.59g) wherethetildeon˜ 6 indicatesitisnotaproper eigenvector. The reader may be unwilling to believe the astonishing fact that a symmetry of the algebra as a whole leads to such profoundly different behavior. My collaborators and I are also displeased with this state of affairs,althoughrepeatedcalculationwithvariedparameterscontinuestoyieldthisconclusion. Ontheplus side, we have now recovered nontrivial Jordan blocks without invoking the parameter . Let us reinforce this observation using the formalism of emerging Jordan blocks. With left unspecified, we have more generallythecorrespondingeigenvectors(of(, ∞ ) and−(−,− ∞ ), where > 0) 5 =(1,−1,0,0,0,0,0), (9.60a) 6 =(1,1,−(+1),−(+1),−(+1),−(+1),(+1)(+2)), (9.60b) 5 =(1,1,0,0,0,0,0), (9.60c) 6 =(1,1,−1,−1,−1,−1,(−1)(−2)). (9.60d) 97 Onecanimmediatelyseethat( 5 , 6 )= 0, whilelim →1 ( 5 , 6 )= 1. More generally, ( 5 , 6 )= s 2 2+4(−1) 2 +(−1) 2 (−2) 2 . (9.61) Furthermore,˜ 6 is obtained by orthogonalizing 6 against 5 , then rescaling the result in the limit→ 1. Finally,weillustratetheconceptoftheconnectionbetweentherateofconvergenceandtheJordancoupling. For generic, the eigenvalues are{ 5 , 6 }={4 ∞ ,4 ∞ +2−2}. Their difference is 6 − 5 = 2−2. If weorthonormalize 6 against 5 instead,we obtain ˆ 5 = 1 √ 2 (1,1,0,0,0,0,0), (9.62a) ′ 6 = sgn(−1) √ 2 −4+8 (0,0,1,1,1,1,−2). (9.62b) Wethenhave lim →1 ′ 6 =± 1 √ 5 (0,0,1,1,1,1,−1)=± 1 √ 2 × r 8 5 ˜ 6 . (9.63) The first prefactor 1/ √ 2 comes from the normalization of 5 . Otherwise, ′ 6 differs from ˜ 6 by a constant, whichmustbetheJordancoupling. Asin Eq. (6.20), one may compute lim →1 ( 6 − 5 )( 5 , 6 ) p 1−( 5 , 6 ) 2 = r 8 5 (9.64) (theabsolutevalueistakentoremovethesignambiguitypresentalsoinSection6.2). Therefore,ifwescale ′ 6 downbythisfactor,adjustitssignifnecessary,andthenrescalebothˆ 5 and ′ 6 by1/ √ 2,weobtain 5 and ˜ 6 —thebasisoftheJordancanonicalform. 9.5.3 Measurementof Although we now have real, rather than emerging, Jordan blocks, it is useful to consider the measurement of using emerging Jordan blocks anyway, because we already have measurements of itself using the Jordanblocks[58],andwewouldliketoseeagreementbetweenthetwomethods. TheprocedureofSection 9.4 reads as follows. For fixed , identify an eigenvector of in the representation 0, ±2+ 11 + 21 correspondingtothestress–energytensor. (Actually,itisknownthat livesin 0, ±2,andsincetheaction ofJTLcanonlydecreasethefirstindex ,itsufficestofind in 0, ±2 andembeditinthefullgluedmodule.) Findaneigenvector ′ thatbecomesdegenerateandparallelwith at= 0,butisotherwisedistinct—ithas nonzerocomponentsin 21 . Define viathe Gram–Schmidt process and normalize it as specified. Then ()= lim →0 |⟨| −2 |⟩| 2 ⟨|⟩⟨|⟩ , (9.65) where isthegroundstate(fullycontained in 0, ±2) and= −2 as= −2 . The last expression requires some explanation. The modifications →− ,→−, and ∞ →− ∞ mustbepropagatedthroughouttheentireprocess. ThisincludesnotonlytheHamiltonianbutalsotheinner productandKoo–Saleurgenerators. Whenwedothis,itisfoundthat⟨|⟩=(−1) uponnormalization—i.e., itisnegativeforodd. Thesignofthenormsquareisinvariantandcannotbechangedbyrescaling,evenby acomplexconstant. However,becauseCFTdemandsthatprimaryfieldshaveunitnormsquare,particularly the ground state, in these cases we must enforce this by manually adding in the sign, and declaring this to bethecorrectinnerproduct. Thisisthemeaning of the appearance of⟨|⟩ in the denominator. Theresultsofthisprocessareshownin Table 9.3. 98 = 2 4 −1.96028 6 −3.24046 8 −3.94952 10 −4.33295 12 −4.55079 14 −4.68234 16 −4.76633 18 −4.82253 conjectured −5 Table9.3: Differentmeasurements for =⟨|⟩ in 0, ±2+ 11 + 21 . In this case, when the roles of and ′ are exchanged, the same value of() obtains, unlike what is observed in Section 9.4. The two measurements match here because the limit → 0 is actually reached, rather than extrapolated as must be the case for the limit→∞. Furthermore, the measurements match those reported in Vasseur et al. [58]. The crucial difference is that we were able to compute the same quantities without ever calculating a Jordan form, a demanding task (Chapter 5). We needed only to diagonalizetheHamiltonianandtakelimits; the emerging Jordan block found itself for us. Because exact diagonalization is also algebraically much less daunting a task than finding a Jordan canonical form, we are able to calculate closed form expressions for() for sufficiently small . For= 2, and ′ are 5 and 6 of Eq. (9.60), and is 1 of Eq. (9.59), but normalized to⟨|⟩ = 1. This leads to two measurementsof: (1) =− 8(+1) F , and (9.66a) (2) = 4(+1)( 4 +2 3 + 2 −12+10) 2 F ( 3 +7 2 +6−16) . (9.66b) When → 1, F → 3 √ 3/2, and (1) , (2) →−32 √ 3/9≈−1.96028, precisely the measured value. An analogouscalculationispossiblefor= 3. One value of is (1) =− 3 F 4 2 − 2(2 2 −9) √ 2 +48 −6 −1 . (9.67) Theexpressionfor (2) ismuchtoocomplicatedtobeworthshowinghere,andisnotparticularlyilluminating. (A related convoluted expression does appear in Appendix E.3.) However, for → 1 both tend to =−288 √ 3/49≈−3.24026, again the measured value. (Despite the denominator in the expression for (1) ,itisaremovablesingularity,andyields a finite value as → 1.) It seems plausible that the finite-size numbers have the form() =−() √ 3/, where ()∈ ℚ. If this sequence could be found, one could rigorously extract the conjectured limit =−5. This would also furthervalidatethelatticeapproach. 99 Chapter10 The periodicalternatingℓ(2|1) superspinchainanditsscaling-weak convergenceto alogarithmicconformal field theory Theℓ(2|1) superspin chain (Section 4.3) is only defined for the JTL parameter = 1. There is no known modificationthatallowsforrepresentationsforothervaluesof . Inparticular,wecannotusetheframework of emerging Jordan blocks to study the Jordan blocks observed at finite size, since there is no way to take the limit→ 1. However, precisely because Jordan blocks proliferate in this model even at finite size, this modelwillservewellasatestinggroundfor the dual Jordan projection operators of Chapter 7. 10.1 IdentificationoffieldsandJordanstructureatfinitesize The identification of fields proceeds as described in Section 4.1.2. This task may seem to be much more difficult, because the Hilbert space of the ℓ(2|1) chain at any given size is much larger than that of the corresponding loop model. However, since the ℓ(2|1) representation and the loop model representation share the JTL standard modules ,e i (for = 1,= e i/3 ), we may transfer the field identifications from the loop model to ℓ(2|1) by matching the spectrum. However, we must use the ℓ(2|1) model itself to determine the Jordan block structure. Examples of this field and structure identification for = 10 sites is shown in Table 10.1. (The identification of the continuum scaling fields was performed by my collaborator Jesper Jacobsen, following Jacobsen and Saleur [7, Appendix A.5].) The Jordan form of the Hamiltonian wasobtainedusingtheKågström–Ruhealgorithm (Section 5.3). The analysis of the = 0 logarithmic CFT makes a clear prediction that the stress–energy tensor and its logarithmic partner are found in a diamond structure much like Eq. (9.40), with an associated =−5. Using the field–vector correspondence, we are able to identify the lattice analogue of the and. We may obtain a measurement of following the procedure of Section 9.4, except we can start from step 2 as there isalreadyaJordanblock. TheresultsareagainidenticaltothoseinTable9.3. Wethenrenormalize and by jointly rescaling them by the same constant—→ and→—so that⟨|⟩ matches the finite-size measuredvalueof. Fromthediamondstructure, one then has [93] † =, (10.1) 100 Jordan structure module(s) ℎ+ℎ scaling field(s) 1 0 1 1 0, ±2 0 2 0.248568 2 3 11 1/4,5 Φ 01 ,(0,4)Φ 11 ,(4,0)Φ −1,1 3 1.18586 4 7 2 0, ±2, 21 5/4,5 21 ⊗ 21 ,(5,0),(0,5), Φ 02 ,(0,3)Φ 12 ,(3,0)Φ −1,2 4 1.69315 2 9 11 2,21/4 (1,0)Φ 11 ,(5,0)Φ 01 ,(0,5)Φ 01 5 2.00969 2 11 11 2 (0,1)Φ −1,1 6 2.01198 2 13 11 9/4,5 (1,1)Φ 01 ,(0,4)Φ 11 ,(4,0)Φ −1,1 7 2.16868 4 17 21 39/16,79/16 (1,0)Φ 1/2,2 ,(0,1)Φ −1/2,2 , (0,2)Φ 3/2,2 ,(2,0)Φ −3/2,2 8 2.58777 8 25 3, ±2 35/12 Φ 03 9 2.63744 4 29 2 0, ±2, 21 13/4,5 (1,1) 21 ⊗ 21 , (5,0),(0,5),(1,1)Φ 02 10 2.91833 2 31 11 4 11 3.05347 2 33 11 17/4 (2,2)Φ 01 12 3.17526 4 37 2 0, ±2, 21 4,21/4 31 ⊗ 31 ,(2,2) 21 ⊗ 21 , (2,0)Φ 12 ,(0,2)Φ −1,2 13 3.19435 4 41 21 71/16 (2,1)Φ 1/2,2 ,(1,2)Φ −1/2,2 14 3.23433 2 43 11 4 15 3.24694 24 67 8×2 11 , 21 , 3, ±2 4,21/4 (2,1)Φ 11 ,(1,2)Φ −1,1 , (2,0)Φ 12 ,(0,2)Φ −1,2 , (1,0)Φ 1/3,3 ,(0,1)Φ −1/3,3 , (0,1)Φ 4/3,3 ,(1,0)Φ −4/3,3 16 3.29693 4 71 11 17/4 (2,2)Φ 01 17 3.50028 2 73 11 17/4 (2,2)Φ 01 18 3.52212 4 77 21 71/16 (2,1)Φ 1/2,2 ,(1,2)Φ −1/2,2 19 3.59093 8 85 3, ±2 59/12 (1,1)Φ 03 20 3.67553 4 89 2 0, ±2, 21 4 (2,2),(2,0)Φ 12 ,(0,2)Φ −1,2 21 3.74941 24 113 8×2 11 , 21 , 3, ±2 5,21/4 Φ −5/3,3 ,Φ 5/3,3 22 3.90951 4 117 11 23 4.00967 4 121 2 0, ±2, 21 6 (3,3) 24 4.05177 4 125 21 Table 10.1: Jordan structure of the lowest 125 of 711 eigenvalues of 0 on = 10 sites, in the vacuum sector at momentum 0. is the algebraic multiplicity of the eigenvalue on line . = Í =1 is the runningdimension. Inthe“Jordanstructure”column,× means rank- Jordanblocksappearforthat eigenvalue,and≡ 1×.(,)Φ meansa level-(,) descendant ofΦ. 101 (⟨| † |⟩) 1/2 ⟨| † |⟩ 10 3.94536 −0.00856632 12 4.24481 −0.0237170 14 4.43279 −0.0275028 16 4.55716 −0.0262987 conjectured 5 0 Table10.2: Tests of⟨| † |⟩= 0 and⟨| † |⟩= 2 . ⟨| 1 −1 |⟩ ⟨| 1 −1 |⟩ 10 −6.34905 0.00287950 12 −7.25285 0.00285605 14 −7.88091 0.00261639 16 −8.32548 0.00233339 conjectured −10 0 Table10.3: Testsof⟨| 1 −1 |⟩= 0 and⟨| 1 −1 |⟩= 2. with = −2 − 3 2 2 −1 , (10.2a) † = 2 − 3 2 2 1 . (10.2b) Itfollowsthat ⟨| † |⟩= 0, (10.3a) ⟨| † |⟩= 2 . (10.3b) Thesevaluesonsomelatticesizesarereported in Table 10.2. AnothercheckcomesfromRidout[94]. Since 1 −1 = 2 0 = 2, (10.4) itfollowsthat ⟨| 1 −1 |⟩= 0, (10.5a) ⟨| 1 −1 |⟩= 2. (10.5b) Thevalidationoftheseidentitiesonthelattice are reported in Table 10.3. One reasonable objection to the preceding tests is that measurements involving the supersymmetric inner product don’t necessarily imply convergence or equalities of the underlying fields; since the inner product is indefinite, lots of independent random fluctuations could be hidden by the fact that they tend to cancel out. So, for instance, to test † = , we should see if∥ † −∥ = 0, with∥·∥ some positive definite norm. It will become apparent that the measurements using positive definite norms don’t converge as nicely as those using the supersymmetric inner product, if at all. A possible solution to this is to find restricted subspaces of interest of some fixed dimension (for instance, by working in a subspace with certain conformal dimensions) while increasing the lattice size and measuring quantities there and seeing if there is convergence for every. This would be a notion of convergence even weaker than “weak convergence”—itispreciselythenotionof scaling-weak convergence introduced in Section 8.3.1. 102 10.2 Scaling-weakconvergenceofconformalidentities Adetailedanalysisofthescalinglimitoftiltingmodules,composedofthestandardmodules,predictsthat in the full logarithmic CFT, = and = , where and are the antiholomorphic counterparts to and [15, Figure 2 and Section 7]. Because we are able to identify these four fields on the lattice and constructthesingularvectoroperators,westudytheseidentitiesinthecontextofscaling-weakconvergence, nowusingthedualJordanprojectionoperators of Section 7.3. These identities involve descendant fields with zero spin (i.e., ℎ = ℎ). We identify and ( and ) as eigenvectors of the Hamiltonian with momentum 2 and conformal weights(2,0) (momentum−2 and conformal weights(0,2)), and bring it to the momentum 0 sector by applying (). To begin applying the projectors, we first find a reduced Hamiltonian in the sector of momentum 0 at the lowest energy levels, using the implicitly restarted Arnoldi method (Section 5.2) with dynamic multiplicity adjustment (Appendix A), which greatly improves convergence. Next, we find a dual Jordan basis of the reduced Hamiltonian, using the Kågström–Ruhe algorithm. We partition the two bases by eigenvalue, and follow theprocedureofSection7.3toconstructtheresolutionoftheidentityinthisreducedsubspace. Inorderto apply these projection operators as constructed, one must express the descendant field in the basis of the reduced Hamiltonian, using the partial change of basis found by the implicitly restarted Arnoldi method. We subtract the two and take the 2-norm, with the phase optimization described in Section 8.3. Finally, with an exact Jordan block, another complication is that is not uniquely defined either: there is a degree offreedom→+ for∈ℂ(andsimilarly→+ ∗ ). We must therefore measure instead ∥−∥ 2 ≡ inf , ∥(+ ∗ )−e i (+)∥ 2 (10.6) toobtainawell-definedmeasurement(there are probably others but this seems to be the simplest). Numerical results from this procedure are found in Tables 10.4–10.7. The second of each pair of tables differs in that the normed difference is divided by the norm of one of the two vectors, to give a relative measureofthedeviationfromzero. Since∥Π () ∥ 2 =∥Π () ∥ 2 and∥Π () ∥ 2 =∥Π () ∥ 2 ,thechoiceis immaterial. Forafixedvalueof ,thevaluesinTables10.4and10.5appeartodecaytozerowithincreasing quiteconvincingly. Wethussaytheidentity= holds in the scaling-weak sense. Thesituationwith= isnotsoclear,andevenabitdeceptive. TherelevantresultsareinTables10.6 and10.7. InTable10.6Iseenoclearpattern,exceptforthe“noprojector”rowincreasingmonotonically. The = 73rowinTable10.7ismonotonicallydecreasingforthosethreevalues. Nearthe= 77rowthevalues drop suddenly for each column (although the row is different for different columns). I believe this drop is deceptive. RecallfromEq. (10.6)that∥−∥ 2 isreallyshorthandforinf , ∥(+ ∗ )−e i (+)∥ 2 . Thus∥Π () (−)∥ 2 /∥Π () ∥ 2 isshorthand for ∥Π () [(+ ∗ )−e i (+)]∥ 2 ∥Π () (+ ∗ )∥ 2 , , (10.7) where the entire expression is evaluated with the parameter values and that minimize the numerator alone. The minimal values of the denominators are reported in Table 10.6. However, the value of the denominator can be unbounded if the value of that happens to minimize the numerator is very large. I believe this is responsible for the sudden drop that I mentioned. For instance, in the= 16, = 73 entry the parameter that minimizesthe difference is =−22.3744+14.6383i. But the value that minimizes the = 16,= 77 entry is = 27.5284+154.983i. For= 16,∥∥ 2 ≈ 8.54 and∥∥ 2 ≈ 29.4 as they come out of thediagonalizationprocess( willvarybecauseithasthe degreeoffreedom,butthesenumbersallrefer tothesameevaluation). Furthermore∥∥ 2 ≈ 2.08and∥∥ 2 ≈ 52.4. HencethesmallvaluesinTable10.7 are probably controlled by a large∥ ∗ ∥ in the denominator. Thus I cannot say how meaningful Table 10.7reallyis since canvarysomuch. Note that the ranks of the projectors should be chosen so that they completely include all of the eigenspaces they encompass, otherwise they are ill-defined. As increases, level crossings cause these rankstochange,buttheyshouldstabilize for large enough. 103 10 12 14 1 0* 0* 0* 3 0* 0* 0* 7 0* 0* 0* 9 0* 0* 0* 11 0* 0* 0* 13 0* 0* 0* 17 0* 0* 0* 25 0* 0* 0* 29 0* 0* 0* 31 0* 0* 0* 33 0* — — 35 — 0* 0* 37 0* 0* — 41 0* — — 43 0* — — 59 — — 0* 61 — 0* 0* 63 — 0* 0* 67 0* 0* 0* 71 0* 0* 0* 73 0* 0* 0* 77 0* 0* 0.129267 81 — 0* 0.129267 85 0* 0.140133 0.129267 87 — 0.140133 — 89 0.153817 — — 93 — — 0.129267 95 — 0.140133 0.129267 noprojector 0.153611 0.181995 0.320142 Table 10.4: The norm∥Π () (−)∥ 2 for various projector ranks and system lengths. In this table “0*” means a number that is less than about2×10 −6 . I estimate the uncertainty in the nonzero numbers to beabout10 −6 . “—”meansthattheprojectorofagivenrank atlength isill-definedduetodegeneracies. 104 10 12 14 1 * * * 3 0* 0* * 7 0* 0* * 9 0* 0* * 11 0* 0* * 13 * * * 17 * * * 25 * * * 29 * * * 31 * * * 33 * — — 35 — 0* 0* 37 0* 0* — 41 0* — — 43 0* — — 59 — — 0* 61 — 0* 0* 63 — 0* 0* 67 0* 0* 0* 71 0* 0* 0* 73 0* 0* 0* 77 0* 0* 0.015756 81 — 0* 0.015756 85 0* 0.0220225 0.015756 87 — 0.0220225 — 89 0.0315451 — — 93 — — 0.015756 95 — 0.0220225 0.015756 noprojector 0.0317082 0.0288243 0.0393106 Table 10.5: The norm∥Π () (−)∥ 2 /∥Π () ∥ 2 for various projector ranks and system lengths . In this table “*” means a highly variable number of order 1. They are likely the result of a 0/0 since the corresponding values in Table 10.4 are so small. “0*” means a number that is less than10 −6 . I estimate the uncertainty in the nonzero numbers to be about 10 −6 . “—” means that the projector of a given rank at length isill-definedduetodegeneracies. 105 10 12 14 1 0* 0* 0* 3 0* 0* 0* 7 0* 0* 0* 9 0* 0* 0* 11 0* 0* 0* 13 0* 0* 0* 17 0* 0* 0* 25 0* 0* 0* 29 0* 0* 0* 31 0* 0* 0* 33 0* — — 35 — 11.052 12.3243 37 9.97002 10.6092 — 41 9.97002 — — 43 9.97002 — — 59 — — 12.8422 61 — 11.136 12.8422 63 — 11.0192 12.8422 67 10.8326 11.4142 12.8422 71 10.8326 11.2607 12.8422 73 10.8326 11.2429 12.8422 77 10.8326 11.4929 12.5025 81 — 11.4057 12.5025 85 10.8326 11.5618 12.5025 87 — 11.5618 — 89 10.8411 — — 93 — — 12.5025 95 — 11.5618 12.5025 noprojector 11.0886 12.0692 13.2406 Table10.6: Thenorm∥Π () (−)∥ 2 forvariousprojectorranks andsystemlengths. Inthistable“0*” means a number that is less than10 −5 . I estimate the uncertainty in the nonzero numbers to be about10 −4 . “—”meansthattheprojectorofagivenrank at length is ill-defined due to degeneracies. 106 10 12 14 1 0* 0* 0* 3 0* 0* 0* 7 0* 0* 0* 9 0* 0* 0* 11 0* 0* 0* 13 0* 0* 0* 17 0* 0* 0* 25 0* 0* 0* 29 0* 0* 0* 31 0* 0* 0* 33 0* — — 35 — 0.163094 0.120853 37 0.230968 0* — 41 0.230968 — — 43 0.230968 — — 59 — — 0.123166 61 — 0* 0.123166 63 — 0* 0.123166 67 0.24383 0* 0.123166 71 0.24383 0* 0.123166 73 0.24383 0* 0.123166 77 0.24383 0* 0.0452794 81 — 0* 0.0452796 85 0.24383 0.0674564 0.0452796 87 — 0.0674562 — 89 0.0998496 — — 93 — — 0.0452792 95 — 0.0674561 0.0452794 noprojector 0.147268 0.318627 0.299784 Table 10.7: The norm∥Π () (−)∥ 2 /∥Π () ∥ 2 for various projector ranks and system lengths . In this table “0*” means a number that is less than10 −6 . I estimate the uncertainty in the nonzero numbers to beabout10 −5 . “—”meansthattheprojectorofagivenrank atlength isill-definedduetodegeneracies. 107 The full interpretation of these results in terms of conformal fields requires a further set of tables. One ofthemisTable10.1,andtheothersarein Appendix F.2, for which the analysis is analogous. Considerthe= 10columnofTable10.4. Thistablemeasuresthedeviationfromtheidentity= by projecting the difference − to an increasing chain of subspaces of given ranks and taking the 2-norm. Accordingtothetablethismeasurementisessentiallyzerountiltherankoftheprojectorsurpasses 85 and becomes 89. Thus the first substantial deviation of this identity is associated to the eigenspace whoseeigenvaluesareassociatedwithpositions86–89whensortedinincreasingorder. AccordingtoTable 10.1, this eigenvalue is 3.67553, in row = 20. The eigenvalues of row 20 extrapolate to scaling fields with (ℎ,ℎ) =(2,2), and the four possibilities are given in the scaling fields column there. (The smaller values of order 10 −6 in these tables are not very large, but they are not negligibly small either. They change every timeIrunthecode,butalwaysremainataboutthesameorderofmagnitude. Itispossiblethattheyshould reallybenumericallyzero,andarejustthe result of compounded numerical error.) Fromthisobservationwecouldmakethedeductionthattheidentity= holdsatfinitesizeinthe subspace of states that extrapolate to a total conformal dimension of ℎ+ℎ < 4. In fact it holds for almost all of the states that do extrapolate to ℎ+ℎ = 4 as well, except for row 12. There are a few good reasons for the deviation here. At finite size the Koo–Saleur operators we use for have not yet converged well to the Virasoro operators. Similarly, the states also have not yet converged well to their continuum limits. By way of example, the stress-energy tensor is found with ℎ = 1.79319 at = 10, when it should be ℎ = 2 in the continuum limit. Nevertheless, the very small measurements found in the rows with= 1 to 85 are a very good indication of the internal consistency of the algebraic analysis of the ℓ(2|1) chain, the extrapolationofthefinite-sizedata,andtheconstructionoftheprojectionoperatorswhentheHamiltonian is not diagonalizable, all put together. This result furthermore suggests that and should be linear combinations of −2 −2 =, −2 ′ 1 , and −2 Φ −1,2 in the continuum limit. The prediction is indeed that ==. The analysis proceeds similarly for the = 12 column, referencing the appropriate table in Appendix F.2. For Table 10.6,= 10, we find jumps in the measurement at = 37,67, and89. Turning again to Table 10.1,thesearerows= 12,15,and20,whichagaincorrespondtofieldsthatextrapolateto ℎ+ℎ= 4,another goodsign. OnemightnoticeinTables10.4–10.7thatforfixed ,thenumbersdonotalwaysbehavemonotonically withincreasing. Recallthatif{ } dim() =1 isa(generalized)righteigenbasisofthehamiltonian,sortedin anyorder(thoughpracticallybyincreasing eigenvalue), then for an arbitrary vector = dim() Õ =1 , (10.8) Π () = Õ =1 (10.9) isitsrestrictiontothefirst states. The(squared) 2-norm of this vector is given by ∥Π () ∥ 2 2 = Õ =1 Õ =1 ∗ ∗ . (10.10) In hermitian quantum mechanics we are guaranteed orthonormality of the basis (or it can be made so by Gram–Schmidt) and ∗ = . The above sum collapses to Í =1 | | 2 and is monotonically increasing in. In dual Jordan quantum mechanics we do not have such a nice property of the basis, and, as far as I know currently, the double sum cannot be simplified. As a consequence, non-monotonic behavior is not ruled out. Wecouldconsiderdefininganewnorm ∥·∥that would be monotonic in: ∥∥ 2 ≡ dim() Õ =1 ∥ ∥ 2 2 , (10.11) 108 where{ } are the coefficients of in the basis{ }. On the one hand, this reduces to the standard 2-norm when applied to hermitian quantum mechanics, so it is a reasonable guess. On the other hand, it is not obvious to me yet that this is a norm, and it is not obvious that it is induced by an inner product, which seemstobeanimportantpropertyforquantum mechanics. One might guess the inner product (,)≡ dim() Õ =1 ∗ , , ∥ ∥ 2 2 (10.12) (withobviousnotation)inducesthenormbutitisnotobviouswhetherthisisaninnerproduct. Thisnorm might not even be meaningful here since it uses the 2-norm, which is induced by the conjugate-transpose inner product, and that is obviously meaningful in hermitian quantum mechanics, thus giving the 2-norm meaning in that context. We do have the notion of inner product and hermitian conjugation from the CFT, but it is generally not positive-definite and not as convincing to say “ = 0 because∥∥ = 0” when the normisindefinite. We may be able to get better results by carefully choosing our projection steps. By carefully following states as the lattice size increases and tracking various properties, one can associate a pair of conformal weights(ℎ,ℎ) to each state on the lattice. The Koo–Saleur generators generally do a good job of coupling vectors to those expected in the continuum theory. For example, one expects to have(ℎ,ℎ) =(4,4) in the limit, and on the lattice we generally find the strongest components of in this sector. Nevertheless we occasionally find some unexpected contributions. For instance, on the lattice I observe that has a significant component along the ground state. Fortunately, this ground state component tends to decrease with. With this in mind, there are a few options we could try. First is to pick our lattice projection operator Π to restrict− (as well as the identity) to states whose conformal weights have been identified as (2,2). ThisprojectorΠisdistinctfromΠ () asΠ () projectstothelowest statesbyenergy,whichisclosely related to (but not quite the same as, due to finite-size effects) the conformal weights. Another possibility is to apply a projection operator after every action of a Koo–Saleur operator. Thus in a term like 2 −1 we considerΠ −1 Π −1 where (abusing notation) eachΠ is the correct projection operator for the context. I willusesquarebracketsΠ[·]todenotetheabovedescribedprocedureofapplyingprojectionoperatorsafter eachKoo–Saleuroperator. Thus we will be interested in∥Π(−)∥ 2 and∥Π[−]∥ 2 , as well as the versions and the normed versions. In each of these measurements the and optimizations remain implicit. Thus the notation∥Π[−]∥ 2 ishidingthecomplexity of a rather long expression: ∥Π[−]∥ 2 = inf , Π (ℎ+ℎ=4) −2 (+ ∗ )− 3 2 Π (ℎ+ℎ=4) −1 Π (ℎ+ℎ=3) −1 (+ ∗ ) −e i Π (ℎ+ℎ=4) −2 (+)− 3 2 Π (ℎ+ℎ=4) −1 Π (ℎ+ℎ=3) −1 (+) 2 . (10.13) So far, I have performed these measurements for = 10,12, and 14, and they do not shed much light nor do they show much structure. It is plausible that larger lattice sizes will reveal same patterns. But for now, thedatahasbeenomitted. Inotethatthesamemeasurementscanbeperformedusingoneofthemodifiedloopmodels,described inSection9.5.2. Thoughthenumbersdiffer,theconclusionsareverysimilar. Tablesforthesemeasurements canbefoundinAppendixF.2. 109 10.3 Actionoftheℓ(2|1)superalgebraonsingletstates The ℓ(2|1) algebra is spanned by 8 generators whose commutation relations are standard [95]. For the fundamentalrepresentationonevensites I use = † + 1 2 ( † 1 1 + † 2 2 )= † + (−1) 2 ( † 1 1 + † 2 2 ), (10.14a) = 1 2 ( † 1 1 − † 2 2 )= (−1) 2 ( † 1 1 − † 2 2 ), (10.14b) + = † 1 2 =(−1) † 1 2 , (10.14c) − = † 2 1 =(−1) † 2 1 , (10.14d) + = † 2 =(−1) † 2 , (10.14e) − = † 1 =(−1) † 1 , (10.14f) + = † 1 =(−1) † 1 , (10.14g) − = † 2 =(−1) † 2 , (10.14h) andonoddsites Iuse = † − 1 2 ( † 1 1 + † 2 2 )= † + (−1) 2 ( † 1 1 + † 2 2 ), (10.15a) =− 1 2 ( † 1 1 − † 2 2 )= (−1) 2 ( † 1 1 − † 2 2 ), (10.15b) + =− † 2 1 =(−1) † 2 1 , (10.15c) − =− † 1 2 =(−1) † 1 2 , (10.15d) + =− † 2 =(−1) † 2 , (10.15e) − =− † 1 =(−1) † 1 , (10.15f) + =− † 1 =(−1) † 1 , (10.15g) − =− † 2 =(−1) † 2 . (10.15h) The signs(−1) are added to emphasize where relative signs are important. Note that I have used different formsofthegeneratorscomparedtoGainutdinovetal.[15,AppendixA]. ButIhavecheckedcarefullythat my generators do produce the ℓ(2|1) relations, and deviations here represent corrections to errors in the reference. Also note the proliferation of signs one must keep track of carefully, in addition to the signs in the odd generators. First, when applying an operator to a state, the super product of elements in a graded tensor productisused. Foranoperator= 1 ⊗···⊗ with builtupfrom{ 1 , 2 , , † 1 , † 2 , † } andabasis state|⟩=| 1 ··· ⟩=| 1 ⟩⊗···⊗| ⟩∈ Ë =1 { † 1 |0⟩, † 2 |0⟩, † |0⟩}, |⟩=( 1 ⊗···⊗ )(| 1 ⟩⊗···⊗| ⟩)=(−1) 1 | 1 ⟩⊗···⊗ | ⟩, (10.16) where is the number of fermionic exchanges needed to bring the operators to the corresponding states. Second, |2⟩ = † |0⟩ =((−1) − † )|0⟩ =(−1) |0⟩, not just|0⟩. This consideration is important for operators such as ± for odd and ± for even . These signs were not an issue previously since (and consequently the Koo–Saleur operators) is built of pairs of neighboring fermionic operators and has a nonzeroactiononlyonstatescontainingadjacentpairsoffermions. Similarly,the= = 0Hilbertspace was built by iterating the procedure of replacing pairs with pairs (see the discussion at the end of Section 4.3). From theℓ(2|1) commutation relations (or just by looking at the operators) it is clear that ± , ± , and ± will take us out of the= = 0 sector. However, since we are most interested in the action ofℓ(2|1) generatorsonstatesfromthatsector,suchasthegroundstate,wecanfirstidentifyallthestatesofinterestby 110 workinginthe= = 0sectorandthenembeddingtheminthewhole3 -dimensionalspacetocompute theactionof thegenerators. Asafirstcheck,thegroundstate,whichshouldbelongtoan ℓ(2|1)singletrepresentation,isannihilated byallgenerators whereasusual = Õ =1 , = Õ =1 3 ↑ 1 ⊗···⊗ 3 ⊗ , ↑ ⊗ 3 ⊗···⊗ 3 ↑ . (10.17) This is identically true in finite size. (It is known for this Hamiltonian that the vector components of the ground state are all integers before normalization. Therefore after numerically finding the ground state I canrationalizeitbydividingthroughbythesmallestnonzeroelementandroundingalltheresults,setting them to integers. Then I divide through by the ℓ(2|1) norm, which ends up being an integer. Results of thistypearegenerallyknownasRazumov–Stroganovconjecturesandhavebeenproveninsomecases. The takeawayisthatthegroundstatecanbetreated exactly, to infinite precision.) Similarly,thestress-energytensor shouldalsobelongtoasinglet. Whilethisdoesnothappenexactly, for= 10, typical values are∥ ± ∥ 2 ≃ 3×10 −12 and∥ ± ∥ 2 ≃∥ ± ∥ 2 ≃ 3×10 −7 , and for= 12, typical values are∥ ± ∥ 2 ≃ 7× 10 −12 and∥ ± ∥ 2 ≃∥ ± ∥ 2 ≃ 3× 10 −6 . The value of∥ ± ∥ 2 is of the same order as 1/2 , where is the dimension of the vacuum sector and is the precision of the eigenvalues. So thesenumberscanprobablybeimprovedbydetermining tohighernumericalprecision. Neverthelessthe evidenceisinfavorof transformingtrivially under the action of the generators. 111 Chapter11 Mixingofconformalfieldsatfinitesize Muchofthesuccessofthelatticeapproachtoconformalfieldtheorycamefromtheidentificationoflattice eigenvectors as analogues of continuum fields. Indeed, we have made the innocuous assumption, justified bymostofthedata,thateachvectoratfinitesizecouldcleanlybeassociatedtoasingleconformalfield,even ifwedidnotknowpreciselywhatthatfieldwas. However,twolatticeexcitationsin 0, ±2 poseachallenge tothisassumption,anditappearsthatthey each represent mixtures of the same two conformal fields. N.B.: InthischapterΦ referstothe diagonalfield ⊗ , not ⊗ ,− as in Eq.(8.14). 11.1 Mixingof andΦ 31 Begin with the loop model in the standard module 0, ±2, with the lattice size denoted by. Consider the lowest three right eigenstates of the Hamiltonian in the sector of momentum zero, labeled|0⟩,|1⟩, and|2⟩. |0⟩ is the ground state. Various calculations show that for > 0,|1⟩ has the expected properties of|Φ 31 ⟩ and,likewise,|2⟩ tendstobehavelike|⟩. Thisbehavioris“sharper”thecloseronegetsto= 1. Itisalso observed that for < 0, the roles are exchanged;|1⟩ has the properties of|⟩ and|2⟩ has the properties of|Φ 31 ⟩, with such behavior most clear approaching =−2. Close to = 0, one observes that both states |1⟩ and|2⟩ “contain”afairlyevenmixtureofthetwo. Theidentificationof |1⟩ or|2⟩ with|⟩ or|Φ 31 ⟩ also sharpenswithincreasing,exceptat= 0. Anansatzthatwouldaccountforthese observations is |2⟩= 2 (,) |⟩+ 2 () 2 () |Φ 31 ⟩ , (11.1a) |1⟩= 1 (,) |Φ 31 ⟩+ 1 () 1 () |⟩ , (11.1b) where 1 and 2 are overall normalization constants, and 1 and 2 represent a relative normalization betweenthetwoterms. Thismaybewritten more symmetrically as |2⟩= ′ 2 (,) 2 ()/2 |⟩+ 2 () 2 ()/2 |Φ 31 ⟩ , (11.2a) |1⟩= ′ 1 (,) 1 ()/2 |Φ 31 ⟩+ 1 () 1 ()/2 |⟩ , (11.2b) with ′ (,)= (,) − ()/2 . Let us suppose that the exponents () have the property that their sign is that of: sgn ()= sgn. This needs to be demonstrated, either numerically or analytically. However, assumingthistobethecase,theansatzdisplays exactly the exchange of states observed numerically. 112 We turn now to numerical calculation of the exponents . Hereafter, as much as possible, we will suppressnotationoffunctionaldependence on and. Write |2⟩= |⟩+|Φ 31 ⟩, (11.3a) |1⟩= |⟩+|Φ 31 ⟩. (11.3b) Wealsodefineanormalizedversion, | ˜ 2⟩=˜ |⟩+ ˜ |Φ 31 ⟩, (11.4a) | ˜ 1⟩=˜ |⟩+ ˜ |Φ 31 ⟩ (11.4b) such that⟨ ˜ 2| ˜ 2⟩ =⟨ ˜ 1| ˜ 1⟩ =±1. This is not possible exactly at = 0 since⟨1|1⟩ =⟨2|2⟩ = 0 exactly. Up to phases we must have| ˜ 1⟩ =|1⟩/ p ⟨1|1⟩ and| ˜ 2⟩ =|2⟩/ p ⟨2|2⟩. Furthermore,| ˜ 1⟩ and| ˜ 2⟩ are orthogonal with respecttotheloopscalarproduct. It turns out we must partition the range of,(−2,1), into three regions. These will be 0 =(−2,−3/5)∪ (1/2,1), 2 =(−3/5,0), and 1 =(0,1/2). The boundaries at =−2,−3/5,0,1/2,1 correspond to = 1,3/2,2,3,∞. The regions are characterized by the loop norms of the states above. Let =⟨ ˜ | ˜ ⟩. In 0 we have 1 = 2 = 1. In 2 we have 1 = 1, 2 =−1. In 1 we have 1 =−1, 2 = 1. The labeling is such that =−1 in . Wehavethefollowingexpansionsofageneric field |⟩: |⟩= 0 |0⟩+ 1 |1⟩+ 2 |2⟩+··· (11.5a) =˜ 0 | ˜ 0⟩+˜ 1 | ˜ 1⟩+˜ 2 | ˜ 2⟩+···. (11.5b) Combiningthelastequationabovewith(11.4) we have |⟩= 0 |0⟩+(˜ 1 ˜ +˜ 2 ˜ )|⟩+(˜ 1 ˜ +˜ 2 ˜ )|Φ 31 ⟩+···. (11.6) 0 =˜ 0 and|0⟩=| ˜ 0⟩ since⟨0|0⟩= 1 already. The question then turns to the extraction of the coefficients ˜ and the parameters˜ , ˜ ,˜ , and ˜ . To be clear, the˜ are properties of the expansion, and so depend on the field |⟩. ˜ , ˜ ,˜ , and ˜ are properties of the guess (11.4) and are fixed once determined. All of ˜ and˜ , ˜ , ˜ ,and ˜ dependon and. It is rather straightforward to get˜ . The Hamiltonian is self-adjoint with respect to the loop inner product,andthus⟨ ˜ | ˜ ⟩= . Thus˜ =⟨ ˜ |⟩. Let us concretely try to choose|⟩=|⟩. On the lattice, we do not have direct access to this quantity. Theclosestwecangetistochooseadefinitionfor |⟩ thatishopefullyclosetoits“correct”identification. We thus let|⟩ = −2 −2 |0⟩, or, on the lattice, a symmetrized combination( −2 −2 + −2 −2 )|0⟩/2, which I will leave implicit. We should thus have a unit coefficient for |⟩ and zero for|Φ 31 ⟩. Let =⟨|⟩, which is numerically calculated and thus given. Finally, let =⟨|Φ 31 ⟩, which is left as an unknown, as itmaybenonzeroonthelattice. The expected coefficients of the expansion, along with orthonormality of the states | ˜ 1⟩ and| ˜ 2⟩, gives us 5equationsin5unknowns: ˜ 1 ˜ +˜ 2 ˜ = 1, (11.7a) ˜ 1 ˜ +˜ 2 ˜ = 0, (11.7b) ˜ ∗ ˜ +˜ ∗ ˜ + ∗ ˜ ∗ ˜ + ˜ ∗ ˜ = 2 , (11.7c) ˜ ∗ ˜ +˜ ∗ ˜ + ∗ ˜ ∗ ˜ + ˜ ∗ ˜ = 1 , (11.7d) ˜ ∗ ˜ +˜ ∗ ˜ + ∗ ˜ ∗ ˜ + ˜ ∗ ˜ = 0. (11.7e) Theinputstothisequationare 1 , 2 ∈{1,−1},˜ 1 ,˜ 2 ∈ C, and∈ R. The unknowns are,˜ , ˜ ,˜ , and ˜ . Eq.(11.7b)gives ˜ =− ˜ 1 ˜ ˜ 2 . (11.8) 113 Eq.(11.7e)gives ˜ =− ˜ (˜ ∗ + ˜ ∗ ) ˜ ∗ + ∗˜ ∗ = ˜ 1 ˜ (˜ ∗ + ˜ ∗ ) ˜ 2 (˜ ∗ + ∗˜ ∗ ) . (11.9) PuttingthisintoEq.(11.7a)andrearranging gives ˜ 1 (˜ ∗ ˜ +˜ ∗ ˜ + ∗ ˜ ∗ ˜ + ˜ ∗ ˜ )= ˜ ∗ + ∗ ˜ ∗ . (11.10) BecauseofEq.(11.7d),thelefthandsideis 1 ˜ 1 . Thus = 1 ˜ ∗ 1 −˜ ˜ , (11.11) and,inturn, ˜ = 1 ˜ ∗ 1 ˜ ∗ + ˜ ∗ ˜ −˜ ∗ ˜ 1 ˜ 2 . (11.12) PuttingthisbackintoEq.(11.7a),onefindsthat ˜ ∗ ˜ = ˜ ∗ ˜ + 1 (1−˜ 1 ˜ −˜ ∗ 1 ˜ ∗ ). (11.13) UsingthispreviousequalityandusingEqs.(11.7a)and(11.7b)towrite˜ and ˜ intermsof˜ and ˜ ,Eq.(11.7c) givesus 1 ˜ ∗ 1 ˜ 1 + 2 ˜ ∗ 2 ˜ 2 = . (11.14) Sincethisisarelationamonggivenparametersanddoesnotcontaintheunknowns,itwouldappearthatone of the equations is redundant. At the same time, we are short one to determine a value for˜ . Numerically, Eq. (11.14) is not satisfied exactly, though it is close. However, the discrepancy seems to grow with . The appearance of this relation can likely be traced back to the assumptions encoded in Eqs. (11.7a) and (11.7b)that −2 −2 canbeidentifiedexactlywith withnocomponentalongΦ 31 oranyotherfield(other calculations indicate that there might be a component along the ground state). Another way to put it is that, with|⟩ =|⟩, the|⟩ on the left hand side of Eq. (11.6) is not the same as the|⟩ on the right handside,becausethelefthandsideisdefinedandingeneralthisdefinitionisinconsistentwithitscorrect identification. Despite the preceding, numerical evidence suggests that the equality appears to be satisfied exactly for =−3/5, = 0, and = 1/2. The ratio of the left hand side to the right hand side approaches unity for =−3/5 and = 1/2 and all values of , though it diverges at = 0 (where both quantities approach0,thoughclearlyatdifferentrates). As a first approximation, let’s suppose = 0 (or that it is the least significant component to the above system). Thenwehaveasolutionwhere ˜ ∗ ˜ = 1 2 ˜ ∗ 2 ˜ 2 , (11.15a) ˜ = 2 ˜ 2 , (11.15b) ˜ =− ˜ 1 ˜ ˜ 2 , (11.15c) ˜ = 1 ˜ 1 1− 2 ˜ ∗ 2 2 . (11.15d) Again, these are based on the approximate equality above. We may simply choose ˜ to be real, which amounts to a choice of phase forΦ 31 . Since ˜ is proportional to ˜ , this does not lead to an inconsistency. Thus ˜ = r 1 2 ˜ ∗ 2 ˜ 2 . (11.16) 114 LetusnowreconcileEqs.(11.1),(11.3), and (11.4). We may trivially pull out various factors: |2⟩= |⟩+ |Φ 31 ⟩ , (11.17a) |1⟩= |Φ 31 ⟩+ |⟩ , (11.17b) | ˜ 2⟩=˜ |⟩+ ˜ ˜ |Φ 31 ⟩ , (11.17c) | ˜ 1⟩= ˜ |Φ 31 ⟩+ ˜ ˜ |⟩ . (11.17d) We thus have = 2 , = 1 . However, the overall normalization is not particularly meaningful at the moment, and besides, we do not have and . What is consistent, though, are the relative normalizations betweenthetwostates,regardlessoftheoverall normalizations. Thus we should have 2 2 = = ˜ ˜ , (11.18a) 1 1 = = ˜ ˜ . (11.18b) For a fixed value of , we can thus measure˜ 1 and˜ 2 for various values of , and compute the values of the ratios ˜ /˜ and˜ / ˜ using the relations derived above. We can then fit the data to a curve of the form ˜ /˜ = − andlettheparametervalues which give the best fit define 2 = , and similarly for 1 . We can perform the preceding procedure for both definitions of |⟩. To distinguish between the cases where|⟩= −2 −2 |0⟩ and( −2 −2 + −2 −2 )|0⟩/2, we will append a subscript “s” to the exponents correspondingtothesymmetriccombination. (Whenthesymmetryofthedefinitionisnotimportanttothe discussion I will simply refer to the exponents as with no subscript. The subscripts appear primarily in theplots.) Plotsoftheexponents ()aregiveninFigures11.1and11.2. (Forallplots,thenonsymmetrized versionsappearinAppendixF.3.) Another quantity that is consistent regardless of the normalization of the states is˜ 1 /˜ 2 . Let us also guessthatforfixed ,thisquantityalsoobeysapowerlaw(tobejustifiedlater)sothat ˜ 1 /˜ 2 =/ () . We may perform the procedure described above in order to get a numerical measurement for. The results of this calculation are given in Figures 11.3 and F.3. Finally, all three exponents are shown together in Figures 11.4andF.4. From Figures 11.4 and F.4, the relation 1 ()= 2 ()= () obviously suggests itself. Aside from some artifacts near = 0, it seems quite plausible that the three exponents should, in fact, be equal. Assuming this to be the case, the deviations from equality would come from finite-size effects and the curve-fitting method, which currently is a least-squares optimization—it minimizes the sum of squares of the absolute deviations. Sincethedatavaluescanbecomequitelargeandindicatedivergences,largervaluesofthedata have greater weight in the fit. Thus using a fitting algorithm that minimizes the sum of squares of the relativedeviationsmayyieldcloserconvergence of the curves. I now give an analytical argument that implies the equality of all three, at least for the region 0 . Take large enough such that on the lattice,⟨|⟩ deviates negligibly from 2 /4 and⟨|Φ 31 ⟩ is essentially zero. Theloop-orthonormalityof| ˜ 1⟩ and| ˜ 2⟩ implies that = ˜ /2 ˜ ˜ /2 ˜ (11.19) isaunitarymatrix. Fixthephasesof|⟩ and|Φ 31 ⟩ suchthatthediagonalelementsof arereal. Thenthe mostgeneralformof is = cos e i sin −e −i sin cos , (11.20) 115 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c -4 -2 2 4 ϵ 1,s (c) Figure11.1: Plot of the exponent 1,s as a function of. 116 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c -4 -3 -2 -1 1 ϵ 2,s (c) Figure11.2: Plot of the exponent 2,s as a function of. 117 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c -4 -3 -2 -1 1 η s (c) Figure11.3: Plot of the exponent s as a function of. -2.0 -1.5 -1.0 -0.5 0.5 1.0 c -4 -2 2 4 ϵ 1,s (c) ϵ 2,s (c) η s (c) Figure11.4: Plotsofthe exponents 1,s , 2,s , and s as functions of. 118 region ˜ 1 ˜ 2 ˜ ˜ ˜ ˜ ˜ 1,s ˜ 2,s ˜ s ˜ s ˜ s ˜ s 0,− ℂ ℝ ∗ ℂ ℝ ℂ ℝ ℂ ℝ ℂ ℝ ∗ ℝ ℝ ℝ ℝ ℝ ℂ ℝ ∗ ℝ 2 ℂ ℝ ℂ iℝ ℂ iℝ ℂ iℝ ℂ ℝ ℝ ℝ iℝ iℝ iℝ ℂ ℝ ∗ ℝ 1 ℂ iℝ ℂ ℝ ℂ ℝ ℂ iℝ ℂ iℝ ℝ iℝ ℝ ℝ iℝ ℂ iℝ ℝ 0,+ ℂ ℝ ℂ ℝ ℂ ℝ ℂ ℝ ℂ ℝ ℝ ℝ ℝ ℝ ℝ ℂ ℝ ℝ Table 11.1: The domains in which the values of the given parameters lie. iR means purely imaginary. ℂ ℝ means that 10 < Re/Im < 1000,ℂ ℝ ∗ means Re/Im≥ 1000,ℂ iℝ means 10 < Im/Re < 1000, and ℂ iℝ ∗ meansIm/Re≥ 1000. InnocasedidIobservetherealandimaginarypartstobethesameorderof magnitude. withrealparameters and ,bothpossibly dependent on and. We thus have = ˜ ˜ = 2 e i tan, (11.21a) = ˜ ˜ =− 2 e −i tan. (11.21b) Sincee i isanumberofunitmodulus,asidefromanormalizingfactor(/2) ±1 ,theexponentialdependence on ofbothratiosisentirelycontainedintan,whichiscommontoboth. Thus 1 = 2 . Fromthiswealso seethat 1 () 2 ()=−1. Similarly, ˜ 1 ˜ 2 =− ˜ ˜ =−e i tan, (11.22) and thus= 1 = 2 . The plots indicate that the preceding equality is also true for parts of 2 and 1 , so it remains to be seen how 1 and 2 figure into this argument. (It seems to me that instead of ∈ (2), we wouldhave∈(1|1) instead. Ihavenot worked out the form of in this case.) Finally, there seem to be significant deviations from = 1 = 2 near = 0. A closer look at the data explains why these deviations occur, and how the data might be refined to yield closer agreement. Figures 11.5–11.13showthevaluesoftheindividualparametersandsomeoftheirratios. Theymayhelptoexplain some features that we see in the plots of the exponents. For each of these I have only given the absolute value of each, rather than the real and imaginary parts separately. Not much information is lost here since foragivenvalueof and (morebroadly,foragivenregion ofthe axis),allthevaluesplottedtendto beclosetopurelyrealorpurelyimaginary. ThisinformationissummarizedinTable11.1. Forcertainratios such as ˜ /˜ , I have instead plotted( ˜ ()/˜ ()) sgn =( ˜ ()/˜ ()) ±1 so that it is easier to view on the same plot—for < 0 onethussees˜ ()/ ˜ (). One noticeable feature is that for any given, there is a value < 0,∈ 2 where˜ () passes through zero. This value of increases (i.e., moves closer to 0) with. This means that there will be zeros in the ratio /, and these zeros are located in different places with different . Clearly such a behavior cannot bemodeledwithapurepowerlaw. Thewayoutofthismaybetowaitforlargeenough suchthatweare past the zero and fit only those points to the power law, instead of including all data points for small and large. Of course, this would mean the validity of our data and fits are reduced from all of 2 =(−3/5,0) to approximately(−3/5,()), where() is such that˜ (()) = 0. This would explain the departure of 1 ()inFiguresF.4and11.4fromasmooth curve at small negative values of. It appears to me that the locations of the zeros may correspond to the same places where the measured of =⟨|⟩ is zero, but 1 ˜ ∗ 1 ˜ 1 + 2 ˜ ∗ 2 ˜ 2 is close to 2 /4. Thus the equality (11.14) fails badly and the expressions for the coefficients are not valid. (The positions of the zeros are not quite the same [in fact for ≥ 20they movetopositivevalues]butI am still inclined to think that this is related.) Alongthesamelines,weseeadivergencein ˜ forsome∈ 2 thatismovingtowards= 0(equivalently, the discussion for˜ applies to 1/ ˜ with its moving zeros). Thus the result is the deviation of 2 () from a smoothcurvecloseto= 0. Itseemsthatthedefectsinthesegraphsdonotapplyto(),sincetheplotsof ˜ 1 and˜ 2 donotshowzerosexceptpossiblyattheboundariesoftheregions . Alongtheselinesthedirect measurement of˜ 1 and˜ 2 are not subject to approximations such as those used in the computation of˜ , ˜ , 119 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 5 10 15 α s (c) 10 12 14 16 18 20 22 Figure 11.5 ˜ ,and ˜ . Wemightbetemptedtoconcludethat isthe“true”valueoftheexponent,andmostaccurateof thethreeatfinitesize. 11.2 Theregion 1 Focusing on 1 , particularly close to= 0, we seek more detailed information on the behavior of the ratios /,/,and˜ 1 /˜ 2 ,astheydependon ratherthan. Fromtheoreticalargumentsgivenabove,weexpect that =− 2 ˜ 1 ˜ 2 , (11.23a) = 2 ˜ ∗ 1 ˜ ∗ 2 , (11.23b) with the caveat that these relations were derived for 0 . Plots of˜ 1 /˜ 2 indicate that it approaches a finite valueas→ 0,anddecreasesthereafter. For any given value of, we thus use the ansatz ˜ 1 ˜ 2 = 0 − 0 , (11.24) valid close to = 0. The results are given in Table 11.2 by fitting the first 10 data points to the given curve. AcomparisonisgiveninFigure11.14. Plotsof/= ˜ /˜ suggestazeroat= 0. As before, we fit the first 10 data points to = 2 2 . (11.25) 120 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.5 1.0 1.5 2.0 β s (c) 10 12 14 16 18 20 22 Figure 11.6 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 2 4 6 8 10 12 γ s (c) 10 12 14 16 18 20 22 Figure 11.7 121 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.5 1.0 1.5 2.0 2.5 δ s (c) 10 12 14 16 18 20 22 Figure 11.8 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.1 0.2 0.3 0.4 0.5 0.6 β s (c) α s (c) sgn(c) 10 12 14 16 18 20 22 Figure 11.9 122 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.2 0.4 0.6 0.8 1.0 1.2 γ s (c) δ s (c) sgn(c) 10 12 14 16 18 20 22 Figure 11.10 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.2 0.4 0.6 0.8 1.0 c 1,s (c) 10 12 14 16 18 20 22 Figure 11.11 123 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.1 0.2 0.3 0.4 c 2,s (c) 10 12 14 16 18 20 22 Figure 11.12 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.1 0.2 0.3 0.4 0.5 0.6 c 1,s (c) c 2,s (c) sgn(c) 10 12 14 16 18 20 22 Figure 11.13 124 0 0 10 0.828808 2.21892 0.947905 12 0.808445 2.43479 0.936954 14 0.818670 2.70472 0.926392 16 0.836131 2.97936 0.916099 18 0.853852 3.24119 0.906020 20 0.869871 3.48412 0.896121 22 0.883846 3.70654 0.886379 (a)|˜ 1 /˜ 2 | 0 0 10 0.828551 2.21891 0.947901 12 0.808445 2.43479 0.936954 14 0.818627 2.70474 0.926391 16 0.836056 2.97942 0.916098 18 0.853772 3.24127 0.906018 20 0.869801 3.48421 0.896119 22 0.883788 3.70662 0.886377 (b)|˜ 1,s /˜ 2,s | Table11.2: Parameters in the guess|˜ 1 /˜ 2 |= 0 − 0 . 0.005 0.010 0.015 0.020 0.025 0.030 c 0.75 0.80 0.85 c 1 (c) c 2 (c) 10 12 14 16 18 20 22 0.005 0.010 0.015 0.020 0.025 0.030 c 0.75 0.80 0.85 c 1,s (c) c 2,s (c) 10 12 14 16 18 20 22 Figure11.14: Theratio|˜ 1 /˜ 2 | close to= 0, along with curves from Table 11.2. TheseareinTable11.3andFigure11.15. Based on theoretical arguments, we expect = 2 ˜ 1 ˜ 2 = 0 2 + 0 +1 2 . (11.26) As → 0 this would suggest 2 = 1 if > 0. However, what is observed is that 2 < 1. While the convergenceto 2 = 1seemsplausible,theconsequenceofthisisaninfiniteslopeapproaching = 0rather thanlinearbehaviorforfinite . Similarly, duetoanapparentdivergence, we guess = 1 1 . (11.27) 2 2 10 0.105648 0.553664 12 0.104027 0.547802 14 0.104449 0.556083 16 0.105804 0.569144 18 0.107676 0.583945 20 0.109775 0.598920 22 0.111860 0.613150 (a)| ˜ /˜ | 2 2 10 0.105610 0.553536 12 0.104027 0.547802 14 0.104440 0.556060 16 0.105785 0.569095 18 0.107652 0.583882 20 0.109750 0.598855 22 0.111836 0.613089 (b)| ˜ s /˜ s | Table11.3: Parameters in the guess|/|= 2 2 . 125 0.005 0.010 0.015 0.020 0.025 0.030 c 0.005 0.010 0.015 β (c) α (c) 10 12 14 16 18 20 22 0.005 0.010 0.015 0.020 0.025 0.030 c 0.005 0.010 0.015 β s (c) α s (c) 10 12 14 16 18 20 22 Figure11.15: Theratio|/|closeto= 0,alongwithcurvesfromTable11.3. Forcomparisonisthestraight line/2. 1 1 10 4.72444 −0.602725 12 4.21977 −0.610617 14 4.06902 −0.627908 16 3.98118 −0.648606 18 3.87961 −0.670884 20 3.74892 −0.693805 22 3.59302 −0.716786 (a)|˜ / ˜ | 1 1 10 4.72675 −0.602800 12 4.21977 −0.610617 14 4.06924 −0.627923 16 3.98144 −0.648639 18 3.87975 −0.670928 20 3.74892 −0.693853 22 3.59292 −0.716833 (b)|˜ s / ˜ s | Table11.4: Parameters in the guess|/|= 1 1 . TheseareinTable11.4andFigure11.16. Based on theoretical arguments, we expect = 2 ˜ 1 ˜ 2 = 2 0 −1 +2 0 −1 . (11.28) If > 0,wethusexpect 1 =−1,butthisisnotobservedeither. However,theconvergenceseemsplausible. 11.3 Thefield Φ 21 To test some of our conjectures, we look at a situation where there is presumably no mixing—the field Φ 21 inthemomentum/2sector. Whennormalizedto⟨Φ 21 |Φ 21 ⟩= 1,itsstandard2-normbehavesasshownin 0.005 0.010 0.015 0.020 0.025 0.030 c 50 100 150 200 γ (c) δ (c) 10 12 14 16 18 20 22 0.005 0.010 0.015 0.020 0.025 0.030 c 50 100 150 200 γ s (c) δ s (c) 10 12 14 16 18 20 22 Figure11.16: Theratio|/| close to= 0, along with curves from Table 11.4. 126 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.5 1.0 1.5 2.0 2.5 3.0 Φ 21 10 12 14 16 18 20 Figure 11.17 Figure11.17. Interestingly,Φ 21 doesnotseemto“know”thatotherquantitiesbehaveunusuallyfor=−3/5 or = 1/2. Close to = 0, our analysis leads us to guess that∥Φ 21 ∥ 2 ∼ 1/ p ||. By fitting the points closest to= 0(|| < 0.025)toapowerlaw,wecanseeifthisholds. TheresultsaregiveninTable11.5. Thequality ofthefitcanbeinspectedinFigure11.18. The eigenvalue of the diagonal field Φ 21 = 21 ⊗ 21 is expected to approach2ℎ 21 . However, inspection of Figure 11.19 indicates that in some regions the data is repelled from the curve 2ℎ 21 . This might signal thatatfinitesize,thelowesteigenstateisstill a mixture of Φ 21 with some higher excitation. + 10 0.508331 12 0.509521 14 0.510640 16 0.511718 18 0.512768 20 0.513801 (a) > 0 − 10 0.491626 12 0.490427 14 0.489299 16 0.488212 18 0.487150 20 0.486105 (b) < 0 Table11.5: Best fit parameters for ∥Φ 21 ∥ 2 =|| − . 127 -0.02 -0.01 0.01 0.02 c 1 2 3 4 5 6 7 Φ 21 Ac -ϵ 10 12 14 16 18 20 Figure 11.18 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.5 1.0 1.5 2.0 2h 21 10 12 14 16 18 20 Figure 11.19: Measurements of the eigenvalue ofΦ 21 , giving the total conformal dimension. The black line istheexactvalue. 128 Conclusionandoutlook Muchofthestoryofstudyinglogarithmicconformalfieldtheoryhasinvolvedtheuseofstandardmethodsin studying its problems, only for those methods to be thwarted by null results, followed by the development of increasingly elaborate methods. For instance, progress in the numerical observation of a number of theoreticalpredictionsstalledforanumberofyearsduetothetypically-uninterestingissueofnormalization: fieldsofprimaryimportance,suchasthestress–energytensor,hadzeroconformalnorms,andstudiesusing the standard norm were initially not very promising. The resolution of these—the quotient expression of Eq. (9.49) that circumvents the issue of absolute normalization and the introduction of the concept of scaling-weakconvergence—restartedprogress,thougheachfixfeltlikeanadhocapproachthatwaslimited inapplicabilitytotheproblemathand. Againstthisbackground—thesearchforasystematicunderstanding oflogarithmicCFTs—theideasofemerging Jordan blocks began to, well, emerge. An immediate continuation of the present work is to search for an emerging rank-3 Jordan block in the continuum limits of ourlattice models, whose existence has been argued for by Gainutdinov et al. [15] and He and Saleur [96]. At present, the primary obstacle seems to be that the bottom field, , is difficult to identify—it is buried in a mixture withΦ 31 (Chapter 11). Disentangling this mixture would lead to further understanding of the representations of the various Temperley–Lieb algebras, why this mixture occurs in 0, ±2,butnotintheotherstandardmodules (as far as we have observed). ForthefieldoflogarithmicCFTsasawhole,Vasseuretal.[58]suggestthatit“maybesolvedbyacareful exercise in the representation theory of the (associative) algebras satisfied by the local energy terms (such as the Temperley–Lieb algebra).” This exercise could likely involve the use of emerging Jordan blocks to get a gradual handle on emerging indecomposability. The fact that logarithms appear in the correlation functions of logarithmic conformal field theory is well-connected to the fact that the dilation operator 0 is not diagonalizable. These logarithmic terms were initially found by subtracting divergences and taking limits, a procedure that probably also has a parallel to subtracting components along earlier vectors in the Gram–Schmidtprocess. Furthermore,thereexistformulasfortheindecomposabilityparameters interms of how quickly the eigenvalues converge—a fact almost certainly related to the connection between the Jordancouplingandtheratesofconvergence. I have mentioned that using the dual Jordan quantum formalism for a density matrix formulation of non-hermitian quantum spin chains could lead to the adaptation of the density matrix renormalization grouptechniquetotheseproblems. Thecompletionofthisundertakingwouldyieldawiderangeofuseful numericalresultsforlongspinchains,whoseabsenceispreventingusfromgettingclosertothecontinuum limit. The new frameworks of Jordan forms and dual Jordan quantum physics seem to be very general, and this is a very exciting prospect. 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Suppose one wants eigenvalues of an× matrix. For concreteness, assume that the desired eigenvalues are the lowest ones in magnitude—many criteria are possible, of course. In order to determine these lowest eigenvalues, the algorithm first determines + approximate eigenvalues, then throws out the largest values determined in this iteration, while projecting the information about the eigenvectors associatedtothese(approximate)eigenvaluesoutoftheArnoldibasis (tousethenotationofAlgorithm 2). Thealgorithmthencalculates moreapproximateeigenvalues,aggregatesthemwiththe eigenvalues kept from the first step (which become refined), then throws out the largest ones from this set, and this processisrepeateduntilconvergence. While performing some of the numerical calculations in Part III, the starting point was naturally the (partial)diagonalizationofsomeHamiltonianortransfermatrix. Ibeganbyusingsomeblack-boxsoftware, thoughtheresultswereinconsistent,andonoccasion,clearlyincorrect. Ithenwrotemyownimplementation of the implicitly restarted Arnoldi method. On some occasions, this algorithm would work very well, and on some occasions, it would reach the maximal number of iterations without converging. Paradoxically, I found in these latter cases that increasing to a larger value would sometimes lead to convergence, even thoughthiswouldseemtoimplyhighercomputational demands. Experimenting further, I found that asking for some values of would lead to rapid convergence, and othersthatwouldleadtonon-convergence. Itturnedoutthatwhenever = +1 ,thealgorithmwouldnot converge. Turning backto theintuitivedescription, this makes sense—whenever information about +1 is thrown out upon application of the filter, information about is thrown out with it. Since is one of the desired eigenvalues, it must eventually end up in the kept eigenvalues. At the same time, +1 is almost suretoendupinthespectrumof ,and it will get thrown out repeatedly, taking along with it. Most matrices naturally occurring in applications are completely non-degenerate. Indeed, the set of squarematricesofagivendimensionthathavearepeatedeigenvalueareasubsetofmeasurezerooftheset of all square matrices. Thus the implicitly restarted Arnoldi method works very well in a large majority of cases. IntheproblemsofPartIII,however,symmetriesofthelatticemodelsleadtopatternsofdegeneracies. Itissimilartohowrotationalinvarianceinthreedimensionsleadstoindependenceofthemagneticquantum number,witha(2+1)-folddegeneracywhenthetotalangularmomentumis,andfurthermoreforthe basic hydrogen atom with a Coulomb potential, conservation of the Runge–Lenz vector implies that all 2 orbitalstateswithprincipalquantumnumber are degenerate. 137 It would seem that in order to get the algorithm to converge for highly-degenerate matrices, one would have to specify a proper value of that does not “split” the eigenspace associated to a particular repeated eigenvalue. However, knowing exactly the pattern of degeneracies absent other input would essentially require the spectrum to be known, which makes the question moot. The solution is to have the algorithm dynamically adjust ifitencountersasituation of multiplicity. Asmentioned,thissituationcanbeencounteredwhenanycriterionisusedtodecidewhicheigenvalues tokeep. Thediscussionassumedthatonewaslookingtocomputethelowesteigenvalues,buttogeneralize thediscussion,Iusealinearordering⪰todenotethecriteriondeterminingthatthe(approximate)eigenvalue 1 ismoreimportantthan 2 ifandonlyif 1 ⪰ 2 . The improved algorithm is given in Algorithm 3. Algorithm3:ImplicitlyrestartedArnoldi method with dynamic multiplicity adjustment Data: × matrix,-dimensionalcolumn vector, positive integers, such that+≤ Result: , , asinEq.(5.1) 1 =+; 2 computea-stepArnoldifactorization using Algorithm 1; 3 repeat 4 beginningwiththe-stepArnoldifactorization( , , ), apply additional steps of the Arnoldiprocesstoobtainan-step Arnoldi factorization( , , )[i.e., run Algorithm 1, lines2–9,withline2replacedby“for= to+−1 do”]; 5 compute ( ) andsortusing⪰,obtaining{ 1 ,..., }; 6 while = +1 do 7 ←+1;←−1; 8 end 9 select asetof shifts 1 , 2 ,..., based on ( ) or other information; 10 = ; 11 for= 1 to do 12 ( , )= QR( − ); 13 ← ∗ ;← ; 14 end 15 ← ; = +1 ; 16 ←( +1 )+ ( + ); ← (1 :,1 :); ← (1 :,1 :); 17 if wasmodifiedinline7itmayoptionally be reset to its original value; 18 until convergence; 138 AppendixB EfficientMathematicaimplementationof latticefieldtheories InthisappendixIdescribehowIhaveimplementedtheloopmodelandtheℓ(2|1)superspinchaintogive efficient computations, with an eye towards generating minimal overhead. Each description is followed by samplecodethatcanberuninMathematica,althoughthedescriptionsshouldsufficetoallowthereaderto createanalogouscodeinothersuitablecomputinglanguages. Thebeginningofmyimplementationofthese latticemodelsarerepresentationsofthebasisinaparticularsectorunderstudy. Nextisarepresentationof the translation operator, and the construction of its eigenstates. This is followed by the construction of the Temperley–Liebgenerators ,fromwhichtheHamiltonianfollows. ThediagonalizationofthisHamiltonian is accomplished using the methods of Chapter 5 and Appendix A. Finally, I describe considerations for the efficientcomputationofconformalinnerproducts, which are not needed in every calculation. Note: in the remainder of the text the total number of sites is usually denoted by = 2, so that is even. However, since N is a protected symbol in Mathematica, for this appendix I replace by, so that isanevennumberandthetotallength. Theloopmodelisassumedtobeperiodic—itshouldnotbeterribly difficulttoadaptthefollowingconsiderations to the open case. B.1 Commondefinitions Theuserfirstsetsvaluesfor xandL,correspondingtotheparametersthatdeterminethecentralchargeand thetotallengthofthespinchain(seepreceding note). c=1-6/(x(x+1)); \[Gamma]=\[Pi]/(x+1); M=-2Cos[\[Gamma]]; e\[Infinity]= -Sin[\[Gamma]]NIntegrate[Sinh[(\[Pi]-\[Gamma])t]/(Sinh[\[Pi] t]Cosh[\[Gamma] t]), {t,-\[Infinity],\[Infinity]}]; vF=\[Pi] Sin[\[Gamma]]/\[Gamma]; h[r_,s_]:=((r(x+1)-s x)^2-1)/(4x(x+1)); (The appearance of many of the code listings here should simplify substantially when copied and pasted intoMathematica,sincesymbolsdenoted in text by escaped brackets will condense to a single character.) B.2 Loopmodel B.2.1 Constructionofthebasis There are two representations I use for the link states. First is the pairing notation described in Section 2.3: ()()···( ). Second is the identification of each of these with a binary string. The correspondence is 139 as follows: in a length binary string, whenever an index appears as the first of a pair () or indicates a singleton(), then the binary string has a 1 in the corresponding position. In terms of diagrams, the ones representthrough-linesortheleftsideofahalf-loopwhereit“opens,”andzerosrepresenttherightsideof ahalf-loopwhereit“closes.” Tousetheexamples of Section 2.3, we have ↔(12)↔ 1011 (B.1a) ↔(34)↔ 1110 (B.1b) ↔(23)(41)↔ 0101 (B.1c) ↔(14)(23)↔ 1100 (B.1d) ↔(14)(23)↔ 110011 (B.1e) ↔(23)(45)↔ 110101 (B.1f) The binary string can be more compactly represented using its base-10 equivalent (offset by 1; thus note that00··· 00correspondsto1,achoicethatcaneasilybereversedbythereader’sownimplementation),an integerbetween1 and2 ,andIdosointerchangeably. Toconvertfromthebinaryrepresentationtothepairingrepresentation,Iindexthesites,anditeratively replace an appearance of “10” in the binary string with its corresponding pair using the labels, until only onesareleft. Thisisbestillustratedbyexample: 11001101→ 11001101 12345678 →(23)(67) 1011 1458 →(23)(67)(14) 11 58 →(14)(23)(67)(5)(8) (B.2) Ifahalf-loopcrossestheperiodicboundary,Itranslatethestatetotheleftuntilthediagramisplanaragain (to ensure that the binary string will never start with a zero), determine the pairings, then add the number of sites back to the indices in the pairings. This entire procedure is accomplished by the following code in Mathematica: pairings[n_]:=pairings[n]=Block[{bin=IntegerDigits[n-1,2,L],rot=0,l,m,r,s}, If[Total[bin]<L/2,Abort[]]; While[Min[Accumulate[2bin-1]]<0,rot++; bin=RotateLeft[bin]]; bin=MapIndexed[{#2,#1}&,bin]//.{l___,{{r_},1},{{s_},0},m___}->{l,m,{r,s}}/.{{l_},1}->{l}; Mod[bin+rot,L,1]]; This function takes the base-10 representation of a binary string and outputs the pairing representation of that binary string. It exits if the sum of digits of the binary string is less than/2 since such a binary string cannotdescribeavalidlinkstate. Toconstructthebasisinthesectorwith2 through lines, one simply runs the following. jbasis[j_]:=Select[Range[2^L],Total[IntegerDigits[#-1,2]]==j+L/2&]; The output of this function on a value of is a set of base-10 numbers that can be converted into the binary representations—validlinkstateswith2 more ones than zeros. Thecaseof 0, ±2 versus 0, ±2 mustbehandledseparately. Onesetsabooleanvariableidentifyequal to True if one only wants to consider states in 0, ±2, and False to construct the entire sector 0, ±2. If identify isTrue,thenwemusteliminate the extra states: If[identify,jbasis[0]=Select[jbasis[0],Min[Accumulate[2IntegerDigits[#-1,2,L]-1]]==0&]]; Thisisequivalenttokeepingstateswhosepairingrepresentationshaveonlyincreasingpairs()with < , butthisimplementationissomewhatfaster since it avoids callingpairings. Next,itisuseful,givenanintegerrepresentingastate,todeterminehowmanythrough-linesithas. This ismosteasilyachievedbytakingthesum of digits of the binary string and subtracting/2: 140 jval[n_]:=Total[IntegerDigits[n-1,2,L]]-L/2; This function takes an integer and outputs , where the associated link state has 2 through-lines. This function only makes sense when the integer represents a valid binary string, and we will see that it is only calledinthoseinstanceswherewearesure that we have one. The user must indicate which modules are of interest, in terms of the values of . This is stored in a user-specifiedvariable jvals,whichisasetofnonnegativeintegers. Tostudythemodule 0, ±2+ 11 + 21 , for instance, the value of jvals is set to {0,1,2} (and identify to True). If one is interested in a single module 1 ,thenjvalsisjust{j},withanumericalvalueforj. Thenthebasisconsistsofallintegersthat arethebase-10representationsoftheappropriate binary strings: basis=Join@@jbasis/@jvals; dim=Length[basis]; dim0=If[MemberQ[0,jvals],Length[jbasis[0]],0]; Do[pos[basis[[n]]] = n, {n, dim}]; I have additionally defined dim and dim0, the total dimension and the dimension of the = 0 sector, the latter defined to be 0 if = 0 is not part of jvals. Finally, the basis integers are given a position index from 1todim. Finally,twomorefunctionsareuseful: toBinary[diagram_]:=Block[{blank=ConstantArray[0,L]},blank[[First/@diagram]]=1;blank]; projPairs[n_]:=projPairs[n]=FromDigits[toBinary[Sort/@pairings[n]],2]+1; Thefirst, toBinary,issimplytheconversionofapairingdiagram()()···( )toitsbinaryrepresentation, which starts with an array of zeros and replaces the zeros corresponding to the first indices of pairings and singletons with ones. The second, projPairs, is simply the map 0, ±2 → 0, ±2 in the base-10 representation. Usingthefirstofthese,itiseasy to construct the action of the parity operator : P[n_?IntegerQ]:=pos[FromDigits[toBinary[L+1-Reverse/@pairings[basis[[n]]]],2]+1]; parity=P/@Range[dim]; P[vec_?ListQ]:=vec[[parity]]; B.2.2 Translationoperatoranddiagonalization The translation operator by one site simply increases all of the indices in the pairing representation by 1, modulo . In the binary representation, the entire string is simply shifted periodically by 1 as well. If identify is True and we are in the module with = 0, then the result must be projected back to 0, ±2. Using the base-10 representation, I have implemented this in the following, as a translation Tnum and its inverseTinvnum. Tnum[n_]:=Tnum[n]=Block[{val=FromDigits[RotateRight[IntegerDigits[n-1,2,L]],2]+1}, If[identify&&jval[n]==0,projPairs[val],val]]; Do[Tinvnum[Tnum[n]]=n,{n,basis}]; Itisalso usefultoconsiderthe actionof the translation operator in terms of the positions that index the basisvectors: shuffle=pos/@Tinvnum/@basis; translate=pos/@Tnum/@basis; T[i_]:=translate[[i]]; Tinv[i_]:=shuffle[[i]]; T[i_,n_]:=Which[n>0,Nest[T,i,n],n<0,Nest[Tinv,i,-n],True,i]; If translation takes the basis vector with index to the basis vector with index , then translate is the ordered image of the translation operator on the ordered range from 1 to dim and shuffle is its inverse. These are useful to apply the translation operator on vectors and matrices without having to construct its 141 matrix explicitly. The function T is the function↦→ as above, and Tinv its inverse. Finally, T[i,n] is the -foldapplicationofTtoi. Because of the translation invariance of the loop model it is useful to use translation to accomplish calculations using only a subset of the basis. I first define another set dist, consisting of the “distinct” link stateswhosetranslatesgeneratethewholebasis. Thissetisgeneratedbywritingtranslate,apermutation, as a product of cycles, and taking an element corresponding to one index of each cycle. For each distinct state, its order ord is also recorded, defined as the smallest positive integer such that translating this state thismanytimesgivesbacktheoriginalstate. Clearly this number divides. With[{cycles=PermutationCycles[translate,Identity]},dist=First/@cycles; Evaluate[orbit/@dist]=cycles; Evaluate[ord/@dist]=Length/@cycles]; For each link basis state it is also useful to have at hand the number of steps away that a given site is “linked” by the loops, keeping in mind the periodic boundary condition. For through-lines this number is zero. Forapairsuchas(16),thenumberof steps at site 1 is5, and the number of steps6 is−5. Block[{blank,a,b},Do[blank=ConstantArray[0,L]; Do[If[Length[link]==2,{a,b}=link; If[link==Sort[link], blank[[a]]=b-a;blank[[b]]=-blank[[a]], blank[[a]]=b-a+L;blank[[b]]=-blank[[a]]], blank[[link]]={0}];steps[i]=blank,{link,pairings[basis[[i]]]}],{i,dim}]]; Since we know the distinct basis elements and the order of each, it is easy to construct the basis that diagonalizesthetranslationoperator. Theeigenvaluesofare{e 2i/ },where isanyconsecutiverange of integers of length. For convenience, each of these eigenvalues will be referenced just using the value of in the exponent. To construct the basis diagonalizing , we begin by taking the distinct vectors and taking linear combinations of their translates, multiplied by phases. For example, to construct|,⟩, the eigenvectorofmomentum builtuponbasis vector, |,⟩∼ −1 Õ =0 e −2i/ |⟩. (B.3) However,itmustbecheckedthatthisvectoractuallybelongstothespaceassociatedtotheeigenvalue. It turnsoutthatthisisthecaseifandonlyiford()/∈ℤ. Whenthisisthecase,thenormalizedeigenvector isgivenby |,⟩= 1 p ord() ord(i)−1 Õ =0 e −2i/ |⟩. (B.4) Thisbasisisaccomplishedwiththefollowingcode. kbasisdetermineswhichelementsofdistgiveriseto a nonzero vector given a value of , and kvec constructs the vector above. Finally, kvecs creates a matrix whose rowsaretheconstructedeigenvectors. kbasis[k_]:={#,k}&/@Select[dist,IntegerQ[ord[#]k/L]&]; kvec[{n_,k_}]:=kvec[{n,k}]=Block[{rules}, rules=Table[{orbit[n][[m+1]]}->E^(-2\[Pi] I k m/L),{m,0,ord[n]-1}]; 1/Sqrt[ord[n]]SparseArray[rules,dim]]; kvecs[k_]:=SparseArray[kvec/@kbasis[k]]; B.2.3 ConstructionoftheTemperley–Liebgenerators Theconstructionoftheoperators ismosteasilydoneusingthepairingrepresentation. Theactionof on abasisstateyieldsthepair(,+1)intheresult. Thenanexaminationofcasesinwhichloopscanpossibly 142 connect to sites and+1 gives the rest of the state. The user must decide whether to allow contraction of through-lines, depending on the module under study. If it is one of the standard modules, then a boolean variable named irreducible should be set to True, and if it is a glued module, then it should be False. Thereisaclosedloopformedupontheactionof wheneither(,+1)or(+1,)ispresentinthepairing representation of the basis state. The preceding considerations suffices to construct all of the generators . Tosavecomputationtime,however,Iconstructonly 1 andusetranslationtogeneratetheothergenerators. eTL[1,m_]:=eTL[1,m]=Block[{step1=steps[m][[1]],step2=steps[m][[2]],val, return,n=basis[[m]],bin}, bin=IntegerDigits[basis[[m]]-1,2,L]; bin[[{1,2}]]={1,0}; return=Mod[Which[step1==0&&step2==0,{},step1==0,{2+step2}, step2==0,{1+step1},step2-step1+1>0,{1+step1,2+step2}, True,{2+step2,1+step1}],L,1]; bin[[return]]=Take[{1,0},Length[return]]; val=FromDigits[bin,2]+1; If[irreducible&&jval[val]!=jval[n],0, If[identify&&jval[val]==0,pos[projPairs[val]],pos[val]]]]; loopfactor[j_,m_]:=loopfactor[j,m]=Boole[steps[m][[j]]==1||steps[m][[j]]==1-L]; e[1]=SparseArray[Table[If[eTL[1,i]!=0,{eTL[1,i],i}->M^loopfactor[1,i], Nothing],{i,dim}],{dim,dim}]; e[j_]:=e[j]=e[j-1][[shuffle,shuffle]]; From these the Hamiltonian, the Hamiltonian when restricted to the sector of momentum, and Koo– Saleurgeneratorsfollow: Htl=(Sum[e[i],{i,L}]-L e\[Infinity] IdentityMatrix[dim,SparseArray])L/(2\[Pi] vF); Hk[k_]:=Hk[k]=Simplify[Conjugate[kvecs[k]].Htl.Transpose[kvecs[k]]]; Hv[n_Integer,vec_List]:= -L/(2\[Pi] vF)Sum[(e[j].vec-e\[Infinity] vec)E^(2\[Pi] I n j/L),{j,L}] +c/12 vec KroneckerDelta[n,0]; B.2.4 Innerproducts I begin with some practical aspects to consider when computing the loop inner product. As mentioned in Section 9.5, almost all inner products are nonzero, in contrast with, say, the standard inner product in an orthonormalbasis. Thismakesitverytime-consuming to compute inner products between all pairs. The standard inner product is easy to calculate on a computer, since it involves only multiplication and addition of complex numbers, with total terms, being the dimension of the space (dim in the code listings). Theloopinnerproductismuchmorecomputationallyexpensive,sincenotonlyarethere 2 terms, but for each pair of basis vectors one must do a “geometric” computation to count the number of loops (and also the number of steps the defect lines move in the case where 2 is different from 1). It is therefore desirabletocutdownonasmanyofthese geometric computations as possible. Hereafter, I will abbreviate = to = ; this means that if 1 ,..., is an enumeration of the basis vectors, then translating vector results in vector . This is helpful because on linear combinations, () = −1 . One helpful observation is that the inner product between basis vectors is obviously symmetric:⟨|⟩= ⟨|⟩. Thiscutsdownthenumberofcomputations by about a factor of 2. But we can do better. Theinnerproductisinvariantundertranslation:⟨|⟩=⟨|⟩. Wehave,ageneratingsetforthebasis under (thisisnotquitethesameasdistin the code listings, but ratherpos/@dist). That is, −1 Ø =0 ={1,2,...,}. (B.5) 143 For each ∈ , we have already computed ord(), the order of under translation. (Here I am abusing notationbyidentifyingthepositionindex of a basis vector with its base-10 representation.) We thenhave Õ =1 ()= Õ ∈ ord()−1 Õ =0 ( ). (B.6) Applyingthistotheloopinnerproductof vectors and, ⟨|⟩= Õ =1 Õ =1 ∗ ⟨|⟩= Õ ∈ ord()−1 Õ =0 Õ =1 ∗ ⟨ |⟩ = Õ ∈ ord()−1 Õ =0 Õ =1 ∗ ⟨| − ⟩= Õ ∈ ord()−1 Õ =0 Õ =1 ∗ ⟨|⟩. (B.7) It is therefore only necessary to compute those basis inner products in which the first state belongs to . In practice,|| is slightly larger than/ (which is not necessarily an integer), reflecting the fact that most statesin haveorder. Thuswehavereducedthenumberofbasisinnerproductcomputationsbyafactor ofabout. Thefollowingfunctioncountsthenumber of loops when two link states are sandwiched together. ipLoops[i_,j_]:=Block[{f,sign,start,startsign,a,b,k,loop=0,points=Range[L]}, f[-1]=steps[i]; f[1]=steps[j]; While[points!={},a=First[points]; sign=1; start=a; points=Rest[points]; Do[b=Mod[a+f[sign][[a]],L,1]; points=Complement[points,{b}]; If[b==a,Break[]]; If[b==start&&sign==-1,loop++;Break[]]; a=b; sign=-sign,L]];loop]; Then,thecomputationoftheinnerproduct,usingthereductioninthenumberofcalculationsabove,is givenbythe following,whencontractionof through-lines is allowed and 2 = 1. distLoopAssociation=Association[ParallelTable[i->(ipLoops[i,#]&/@Range[dim]), {i,Union[dist,Range[dim0]]}]]; ip[u_,v_]:=Block[{tot=0,u2=Conjugate[u]}, Do[Block[{vec=M^distLoopAssociation[i],v2}, v2=v; Do[v2=v2[[translate]]; tot+=u2[[T[i,m]]]vec.v2,{m,ord[i]}]],{i,dist}]; tot]; In the glued modules, when = 0 and 2 = 1 things would seem to be easy. = 1 so that⟨|⟩ = 1 for all and. Thus ⟨|⟩= Õ =1 Õ =1 ∗ ⟨|⟩= Õ =1 ∗ Õ =1 . (B.8) Hereonesimplyaddsupthecomponents of each vector and multiplies the two sums together. This simple picture changes once one introduces a parameter (usually when = 0, as that is the only case where the introduction of controls the appearance of Jordan blocks) that arises each time two 144 stringsarecontracted,withthefirstonelivingonanevensiteandthesecondoneonanoddsite(Appendix E.2). This breaks the symmetry of the original problem in two ways: first is that the system is no longer translationallyinvariant,andsecondisthatachiralityisintroducedintothesystembygivingitapreferred direction. B.2.5 Thecontractionparameter Thefollowingalgorithmsallowonetostudy the glued modules with a non-unity value of. Importantly, since the system is no longer invariant under translations by one site, but only translations bytwosites,thetranslationoperatormustbemodifiedappropriately. Iretain Ttoeffecttranslationsbyone site,anduseUasthetranslationoperatorby two sites. Uis most easily defined by the action of Ttwice. Unum[n_]:=Tnum[Tnum[n]]; Uinvnum[n_]:=Tinvnum[Tinvnum[n]]; shuffle=pos/@Uinvnum/@basis; translate=pos/@Unum/@basis; U[i_]:=translate[[i]]; Uinv[i_]:=shuffle[[i]]; U[i_,n_]:=Which[n>0,Nest[U,i,n],n<0,Nest[Uinv,i,-n],True,i]; Therearesomechangesinconstructing the basis that diagonalizesU. kbasis[k_]:={#,k}&/@Select[dist,IntegerQ[2ord[#]k/L]&]; kvec[{n_,k_}]:=kvec[{n,k}]=Block[{rules}, rules=Table[{orbit[n][[m+1]]}->E^(-4\[Pi] I k m/L),{m,0,ord[n]-1}]; 1/Sqrt[ord[n]]SparseArray[rules,dim]]; The introduction of also implies a modification to the even generators . It becomes necessary to constructboth 1 and 2 ,andtherestcanbe generated again by translation. yfactor[j_/;OddQ[j],m_]:=0; yfactor[j_,m_]:=yfactor[j,m]=Boole[steps[m][[j]]==0&&steps[m][[Mod[j+1,L,1]]]==0]; e[1]=SparseArray[Table[If[eTL[1,i]!=0,{eTL[1,i],i}->M^loopfactor[1,i], Nothing],{i,dim}],{dim,dim}]; e[2]=SparseArray[Table[If[eTL[1,Tinv[i]]!=0, {T[eTL[1,Tinv[i]]],i}->M^loopfactor[1,Tinv[i]] y^yfactor[2,i], Nothing],{i,dim}],{dim,dim}]; e[j_]:=e[j]=e[j-2][[shuffle,shuffle]]; Finally,Ireturntosomeconsiderationsforcomputingtheinnerproduct,nowwithanon-unityvalueof . In 0, ±2 therearenodefectlinessothat the calculation of⟨|⟩ remains the same. At= 0,⟨|⟩= 1. Let = 2 denote translations to the right by two sites. It is easily seen that⟨|⟩=⟨|⟩ when both and are in 0, ±2, or when both and are in 1 (henceforth collectively abbreviated to = Ð 1 , where the union is over all values of interest [typically = 1 and 2]). Simple counterexamples show that the factor and the projection 0, ±2→ 0, ±2 are responsible for the fact that⟨|⟩ need not equal⟨|⟩ when∈ 0, ±2 and∈ . Foraninnerproduct⟨|⟩,breakupthe sum into four pieces: ⟨|⟩= Õ =1 Õ =1 ∗ ⟨|⟩= Õ ∈ 0, ±2 + Õ ∈ Õ ∈ 0, ±2 + Õ ∈ ∗ ⟨|⟩ = Õ ∈ 0, ±2 Õ ∈ 0, ±2 ∗ ⟨|⟩+ Õ ∈ 0, ±2 Õ ∈ ∗ ⟨|⟩+ Õ ∈ Õ ∈ 0, ±2 ∗ ⟨|⟩+ Õ ∈ Õ ∈ ∗ ⟨|⟩ = Õ ∈ 0, ±2 ∗ Õ ∈ 0, ±2 + Õ ∈ 0, ±2 Õ ∈ ( ∗ + ∗ )⟨|⟩+ Õ ∈ Õ ∈ ∗ ⟨|⟩. (B.9) 145 Thelastequalityholdsonlyat= 0: thefirsttermfactorsintotwopiecesif ⟨|⟩= 1. Thetwo“off-diagonal” terms are reindexed and combined into one sum, using the symmetry⟨|⟩ =⟨|⟩. Finally, we can handle the last piece as before, with some modifications. Let be a generating set for , except now under insteadof. Similarly,ord() for∈ isthe smallest integer for which =. Thus ⟨|⟩= Õ ∈ 0, ±2 ∗ Õ ∈ 0, ±2 + Õ ∈ 0, ±2 Õ ∈ ( ∗ + ∗ )⟨|⟩+ Õ ∈ ord()−1 Õ =0 Õ ∈ ∗ ⟨|⟩. (B.10) Thustheonlyinnerproductsthatneedtobecomputedarethosebetween 0, ±2 and ,andthosebetween and . The reduction is not as great as in the = 1 case since has smaller orbits than—generically theyarehalfthesize. Thefollowingfunctioncountsthenumberoftimesafactorof mustbeintroducedwhentwolinkstates aresandwichedtogethertocomputetheirinner product. evens=Range[2,L,2]; evenLines[j_]:=evenLines[j]=Pick[evens,steps[j][[evens]],0]; evenLines/@Range[dim]; ipyfactor[i_,j_]:=Block[{f,sign,shift,start,startsign,a,b,m,n,count=0}, {n,m}=Sort[{i,j}]; f[-1]=steps[m]; f[1]=steps[n]; Do[b=start=k; shift=0; sign=startsign=1; Do[a=f[sign][[b]]; If[a==0,If[sign==-startsign,If[shift>0,count+=1]];Break[]]; b=Mod[b+a,L,1]; shift+=a; sign=-sign,L],{k,Complement[evenLines[m],evenLines[n]]}]; Do[b=start=k; shift=0; sign=startsign=-1; Do[a=f[sign][[b]]; If[a==0,If[sign==-startsign,If[shift>0,count+=1]];Break[]]; b=Mod[b+a,L,1]; shift+=a; sign=-sign,L],{k,Complement[evenLines[n],evenLines[m]]}]; count]; Finally, the computation of the inner product, following the considerations above, is as follows. Note thatanadditionaloptimizationhasbeenmadetoputthevectorwithfewernonzerocomponentsontheleft, sothatfewercomputationsareneeded. distyAssociation=Association[ParallelTable[i->(ipyfactor[i,#]&/@Range[dim]), {i,Union[dist,Range[dim0]]}]]; dist0=Intersection[dist,Range[dim0]]; dist12=Complement[dist,Range[dim0]]; ip[u_,v_]:=Block[{upos=Intersection[Union[Flatten[NestList[U,Flatten[ SparseArray[u]["NonzeroPositions"]],L/2-1]]], Union[Range[dim0],dist12]],vpos=Intersection[Union[Flatten[NestList[U,Flatten[ SparseArray[v]["NonzeroPositions"]],L/2-1]]], Union[Range[dim0],dist12]],short,long,shortpos,longpos,u0,u1,v0,v1,tot=0, translate0=Take[translate,dim0],translate1=Drop[translate,dim0]-dim0,v2,a}, If[Length[upos]<Length[vpos],{short,shortpos,long,longpos}={u,upos,v,vpos}, 146 {short,shortpos,long,longpos}={Conjugate[v],vpos,Conjugate[u],upos}]; u0=Conjugate[Take[short,dim0]];u1=Conjugate[Drop[short,dim0]]; v0=Take[long,dim0];v1=Drop[long,dim0]; Do[Block[{vec}, vec=M^Take[distLoopAssociation[i],dim0] y^Take[distyAssociation[i],dim0]; v2=v0; Do[v2=v2[[translate0]]; tot+=u0[[U[i,m]]]vec.v2,{m,ord[i]}]],{i,dist0}]; tot+=Sum[u0[[i]]((M^Drop[distLoopAssociation[i],dim0] y^Drop[distyAssociation[i],dim0]).v1),{i,Intersection[shortpos,Range[dim0]]}]; tot+=Sum[v0[[i]]((M^Drop[distLoopAssociation[i],dim0] y^Drop[distyAssociation[i],dim0]).u1),{i,Intersection[longpos,Range[dim0]]}]; Block[{vec}, Do[vec=M^Drop[distLoopAssociation[i],dim0] y^Drop[distyAssociation[i],dim0]; v2=v1; Do[v2=v2[[translate1]]; tot+=u1[[U[i,m]-dim0]]vec.v2,{m,ord[i]}],{i,Intersection[shortpos,dist12]}]]; tot]; B.3 ℓ(2|1) superspinchain Theℓ(2|1)superspinchainismuchsimplertoimplementcomputationallythantheloopmodel. However, thedrawbackisthatitsdimensiongrowsmuch more quickly with. B.3.1 Constructionofthebasis The computational basis can easily be identified with ternary strings of length . As with the loop model, these ternary strings can be identified with their base-10 equivalents, again with an offset of 1. If the user specifies ,,and ,thenonemayconstruct the basis of that sector as follows. Bnum[n_]:=Block[{str=IntegerDigits[n-1,3,L],k}, Sum[If[str[[k]]==2,(-1)^k,(-1)^k/2],{k,1,L}]]; Qznum[n_]:=Block[{str=IntegerDigits[n-1,3,L],k}, Sum[If[str[[k]]==2,0,(1/2-str[[k]])(-1)^k],{k,1,L}]]; sector[B_,Qz_]:=Select[Range[3^L],Bnum[#]==B&&Qznum[#]==Qz&]; basis=sector[B,Qz]; Do[pos[j basis[[i]]]=j i,{i,1,Length[basis]},{j,{-1,1}}]; dim=Length[basis]; With an eye on what follows, the basis has been stored as a positively indexed version and a negatively indexed version, with a corresponding sign for the basis element. For instance, if pos[m] gives n then pos[-m] gives-n . One may also construct the basis corresponding to an invariant submodule of the= = 0 sector, as describedattheendofSection4.3. Block[{old={},new=FromDigits[#,3]&/@Table[ConstantArray[i,L],{i,0,2}]+1,mid,str,vec}, While[old!=new, mid=Complement[new,old]; old=new; new=Reap[Do[str=IntegerDigits[j-1,3,L]; Do[If[str[[i]]==str[[i+1]], vec=str; Do[vec[[i]]=b; vec[[i+1]]=b; 147 Sow[FromDigits[vec,3]+1];,{b,Complement[{0,1,2},{str[[i]]}]}];],{i,1,L-1}]; If[str[[L]]==str[[1]],vec=str; Do[vec[[L]]=b; vec[[1]]=b; Sow[FromDigits[vec,3]+1];,{b,Complement[{0,1,2},{str[[L]]}]}];],{j,mid}];][[2,1]]; new=Union[new,old]]; basis=new]; B.3.2 Translationoperatoranddiagonalization Because of the alternating representations, the fundamental translation operator for this system is the translationbytwosites, = 2 . If † isany of the bosonic or fermionic creation operators for site,then † 1 1 † 2 2 ··· † |0⟩= † 1 1 † 2 2 ··· † −1 |0⟩= † 3 1 † 4 2 ··· † −2 † 1 −1 † 2 |0⟩ (B.11) defines the action of on the computational basis. That is, shifts all of the site indices on the creation operators by 2, but does not change the type of particle created. Note, however, that the expression on the right side of the equation isn’t necessarily equal to an element of the computational basis. To make this expression comparable to the computational basis, one must use the commutation relations to move † 1 −1 † 2 allthewaytotheleft,sothatthesitesareinorderagain. Ifeitherofthesearefermionicoperators, thentherearepotentialsignsthatarise. Thus,theactionof onacomputationalbasisstateisagainanother computationalbasisstate,uptoasign. Thisisrepresentedschematicallyas=±,wherethesignistobe determined. InMathematica,thisisaccomplished with the following function. Unum[0]=0; Unum[n_]:=Unum[n]=Block[{bas=IntegerDigits[basis[[Abs[n]]]-1,3,L], fcount1,fcount2,rot}, fcount1=Count[Drop[bas,-2],2]; fcount2=Count[Take[bas,-2],2]; rot=FromDigits[RotateRight[bas,2],3]+1; Sign[n](-1)^(fcount1 fcount2) pos[rot]]; Do[Uinvnum[Unum[n]]=n,{n,-dim,dim}]; U[vec_]:=Block[{new=ConstantArray[0,dim]}, Do[new[[Abs[Unum[n]]]]=Sign[Unum[n]]vec[[n]],{n,1,dim}]; new]; Onemaycheckthat applied/2 timesgives the identity operator. As before, it is useful to find the distinct elements that generate the whole basis under translation, and tousethemtoconstructthebasisthatdiagonalizes. split=GatherBy[(IntegerDigits[#-1,3,L]&/@basis),Sort[Tally[#]]&]; split=Flatten[GatherBy[#,Sort[Tally[ListConvolve[{1,-1},#,1]]]&]&/@split,1]; split=Flatten[GatherBy[#,Sort[Tally[ListConvolve[{1,1},#,1]]]&]&/@split,1]; Do[splitBasis[i]=1+FromDigits[#,3]&/@split[[i]],{i,1,Length[split]}]; dist=Reap[Do[Block[{track=splitBasis[i]}, While[Length[track]>0, Block[{n,abs,signs}, n=pos[track[[1]]]; Sow[n]; orbit[n]=NestList[Unum,n,L/2-1]; track=Complement[track,basis[[Abs[orbit[n]]]]]; orbit[n]=DeleteDuplicatesBy[orbit[n],Abs]; ord[n]=Length[orbit[n]]; torsion[n]=Unum[orbit[n][[ord[n]]]]/n;]]],{i,1,Length[split]}]][[2,1]]; dist=Sort[dist]; 148 kbasis[k_]:=kbasis[k]=Reap[Do[ If[(IntegerQ[2ord[n]k/L]&&torsion[n]==1)||((OddQ[L/(2ord[n])]||OddQ[4ord[n]k/L]) &&torsion[n]==-1),Sow[{n,k}]],{n,dist}]][[2,1]]; kvec[{n_,k_}]:=kvec[{n,k}]=Block[{rules}, rules=Table[{Abs[orbit[n][[m+1]]]}->Sign[orbit[n][[m+1]]]E^(-4\[Pi] I k m/L), {m,0,ord[n]-1}]; 1/Sqrt[ord[n]]SparseArray[rules,dim]]; kvecs[k_]:=kvecs[k]=SparseArray[kvec/@kbasis[k]]; B.3.3 ConstructionoftheTemperley–Liebgenerators Following the explicit expressions of the generators , the fact that each replaces pairs of identical neighboring particles with linear combinations of the same indicates that it is useful to work in the ternary representation. The following code creates Temperley–Lieb generators 1 , 2 , and (denoted as h in the following),andusestranslationbytwosites to generate the remainder. Do[h[i]=Block[{rules,fac}, rules={}; rules=Reap[Do[str=IntegerDigits[basis[[j]]-1,3,L]; If[str[[i]]==str[[i+1]], bos1=str; bos1[[i]]=0; bos1[[i+1]]=0; bos2=str; bos2[[i]]=1; bos2[[i+1]]=1; fer=str; fer[[i]]=2; fer[[i+1]]=2; bos1=FromDigits[bos1,3]+1; bos2=FromDigits[bos2,3]+1; fer=FromDigits[fer,3]+1; bos1=pos[bos1]; bos2=pos[bos2]; fer=pos[fer]; If[str[[i]]==2,fac=(-1)^(i+1),fac=1]; Sow[{{bos1,j}->fac,{bos2,j}->fac,{fer,j}->(-1)^i fac}];];,{j,1,dim}]][[2]]; rules=Flatten[rules]; SparseArray[rules,dim]];,{i,{1,2}}]; h[L]=Block[{rules,fac,f}, rules={}; rules=Reap[Do[str=IntegerDigits[basis[[j]]-1,3,L]; If[str[[L]]==str[[1]], bos1=str; bos1[[L]]=0; bos1[[1]]=0; bos2=str; bos2[[L]]=1; bos2[[1]]=1; fer=str; fer[[L]]=2; fer[[1]]=2; bos1=FromDigits[bos1,3]+1; bos2=FromDigits[bos2,3]+1; 149 fer=FromDigits[fer,3]+1; bos1=pos[bos1]; bos2=pos[bos2]; fer=pos[fer]; f=Count[Rest[Most[str]],2]; If[str[[L]]==2,fac=(-1)^f,fac=1]; Sow[{{bos1,j}->fac,{bos2,j}->fac,{fer,j}->(-1)^(f+1) fac}];];,{j,1,dim}]][[2]]; rules=Flatten[rules]; SparseArray[rules,dim]]; shuffle=Abs[Uinvnum/@Range[dim]]; Do[h[i]=h[i-2][[shuffle,shuffle]],{i,3,L-1}]; h[i_]:=h[Mod[i,L,1]]; Htl=-Sum[h[j],{j,1,L}]+L IdentityMatrix[dim,SparseArray]; Hv[n_,vec_]:=-L/(2\[Pi] vF)Sum[(h[j].vec-vec)E^(2\[Pi] I n j/L),{j,1,L}]; B.3.4 Innerproducts Theℓ(2|1)innerproductiscomputationallyverysimple,sincedifferentbasisstatesareorthogonalandthe inner product of a basis state with itself is±1. Its calculation is described in Section 4.3, and implemented asfollows. ipbas[i_]:=ipbas[i]=(-1)^Total[Flatten[Position[IntegerDigits[basis[[i]]-1,3,L],2]]]; ips=ipbas/@Range[dim]; ip[v_,w_]:=Conjugate[v].(ips w); 150 AppendixC Directcalculationofthesubquotient structureofTemperley–Liebalgebra representations In Gainutdinov et al. [15] the subquotient structure of(1) modules for various numbers of sites was presentedbasedonalgebraicarguments. InthisappendixIdescribeamethodtocomputethissubquotient structuredirectlyandthuscorroboratetheir conclusions from a different approach. The identity module containing the ground state can be constructed following an algorithm presented in Section 4.3, where starting from a computational basis state along which the ground state is expected to have a nonzero component, act with the(1) generators repeatedly until no new states are generated. Thenthegroundstatecanbefoundbyconstructing the Hamiltonian within this subspace. Wemaygeneralizethisideatodeterminethe(1)structureofvariousindecomposablemodules. Start with the Hamiltonian restricted to a particular module. Its generalized right eigenstates are denoted by column vectors . For each, find the (1) submodule generated by . Explicitly, start with the vector , and apply to it all the operators and 2 , yielding vectors , 1 , 2 ,..., , 2 . Gather them all into a matrix[ ; 1 ; 2 ;···; ; 2 ] and column-reduce it. Keep the nonzero columns and repeat the procedure, applying and 2 to each column, then gathering them all and column-reducing, until the number of vectors stabilizes. This is a basis of the(1) submodule generated by . Thus to each we associate a set of vectors{ } =1 (or its span ), which may or may not contain itself (but ∈ either way). Ofcoursemanyofthesemoduleswill coincide for different indices. Thenwiththesesubmodulesgeneratedbyeigenvectorsinhandwecanobservethesubquotientstructure. Say ≤ . Take the vectors{ } and{ } and form one big matrix =[ 1 ;···; ; 1 ;···; ] and column-reduce it. If rank = then that implies ⊂ . We then have the subquotient structure ( − )→ , where labels the dimension of a factor. By doing this for every pair of submodules one shouldbeabletoinferthefullsubquotientstructure,particularlyforsmalllatticesizeswhichcanbehandled exactly. 151 AppendixD CorrectionstosometablesinKooand Saleur(1994) N. B.: Because it pertains only to the paper under discussion, this appendix will use notation consistent with that of Koo and Saleur [3]. In particular, the Koo–Saleur generators as elements of a lattice Virasoro algebra will be denoted by and . There exist several processes that serve as guardrails to maintain the integrity of published scientific research. One of these is peer review, in which experts in the same or a related field of study determine whetherapaperunderconsiderationmeritspublication. Becauseoftimelineconstraints,thereviewprocess typicallyaddressesmethodologyandlogic rather than the finer details of the study. Equallyimportantormoreispostreview,duringwhichattemptsaremadetoreplicatepublishedresults, sometimes using the same methodology described in an original publication. Rarely is this done to “find the fraud” in the literature. I would venture to guess that most of the time replicating established results serves as a prequel to follow-up studies that extend the methodology or apply it to different situations. It is in this context that I find myself proposing these corrections, thus engaging with the literature and participatinginthescientificprocess. Correcteddata Tables D.1 and D.2 contain my own measurements of the quantities given in the respective captions, that correctTables1and2inKooandSaleur[3]. Theextrapolationsareobtainedusingathird-orderpolynomial in1/ fitted to the four last data points. Comparing with the published tables, the obvious trend to notice is that the data values are quite different but the extrapolations are very close (and by transitivity, close to theconjecturedvalues). Re-generatingTables2,3,and4fromKooandSaleur[3]arequitesimilar. Oneonlyneedstochangethe parametervaluesof and andruntherestofthecomputationsasusual. Occasionally,thephase Iuse togeneratedataisoftheoppositesignasthatquotedinthepaper;whetherthesignerrorsareintheprogram orthepaperIdonotknow. Forexample,Table2issupposedtobegeneratedwith=−4 0 − = 2/(+1) (by Eq. (4.13)), but this results in a table that is identically zero (within about 10 −15 ), so I have used the oppositesign. Tosummarize,theevidenceremainsfirmthattheKoo–Saleurgeneratorsproducemanyoftheexpected results from conformal field theory. The extrapolated data remain close to the conjectured values, and only the finite-size data differ significantly. This pattern is consistent for the other tables, and so it is only necessary to show corrections to the first two. The reader capable of replicating the data in Tables D.1 and D.2likelyhassetuptheircomputationscorrectly. 152 3 4 5 6 7 4 0.172749 0.134770 0.110859 0.0942739 0.0820654 6 0.190565 0.148760 0.122293 0.103957 0.0904698 8 0.196899 0.153548 0.126146 0.107184 0.0932494 10 0.199752 0.155643 0.127795 0.108544 0.0944074 12 0.201253 0.156709 0.128611 0.109202 0.0949585 14 0.202128 0.157308 0.129054 0.109549 0.0952415 16 0.202679 0.157669 0.129308 0.109740 0.0953921 18 0.203045 0.157897 0.129461 0.109847 0.0954721 20 0.203298 0.158047 0.129552 0.109907 0.0955121 extrapolation 0.204065 0.158103 0.129210 0.109357 0.0948614 TableD.1: Numericalvalues of|⟨| 1 ⟩| in the = 0 sector with = 0. 3 4 5 6 7 4 0.331475 0.262672 0.217753 0.186055 0.162462 6 0.373264 0.293407 0.242177 0.206384 0.179910 8 0.388620 0.304308 0.250630 0.213303 0.185777 10 0.395791 0.309229 0.254344 0.216281 0.188264 12 0.399682 0.311811 0.256233 0.217757 0.189473 14 0.402018 0.313308 0.257290 0.218557 0.190110 16 0.403527 0.314241 0.257922 0.219016 0.190462 18 0.404554 0.314854 0.258317 0.219289 0.190660 20 0.405285 0.315273 0.258572 0.219453 0.190770 extrapolation 0.408197 0.316209 0.258395 0.218681 0.189692 TableD.2: Numericalvaluesof|⟨| 1 ⟩| in the = 0 sector with =−2/(+1). 153 AppendixE Additionalcalculations E.1 Anotherloopmodelscalarproduct Wehaveseenthatthemodule 0, ±2+ 11 + 21 ofthestandardloopmodelismissingtheexpectedJordan blocksat= 0. Thisproblemwasremediedintwodifferentways: eitherintroduceacontractionparameter ,orinvokeasymmetryoftheTLalgebrathatsimultaneouslynegatesthegenerators andtheloopweights (Section9.5.2). Anotherinnerproduct,obtainedviainspiredguesswork,isasfollows. Imaginethatanalternatingsign livesbetweenthesites: 1+2−3+4−···+−1 (E.1) where I have used the fact that is even and the chain is periodic. Now, whenever in⟨|⟩ there are two defectlines()and()onthesamesidethatareeventuallycontractedtogether,startingfrom()andending at(), multiply all the signs together between and . If the result is positive, do nothing. If the result is negative,callthisa signedcontraction,forlack of a better term. The inner product is then ⟨|⟩= ˜ (,) [ (,) ], (E.2) where˜ (,)countsthenumberofsignedcontractions. Thefactor (,) isoptional,as= 0andthus= 1, soitisplacedinbracketsasareminderthatitshouldbethereotherwise. Herearetwoexampleson6sites: ⟨(14)(23)(56)|(23)(56)⟩=,⟨(41)(23)(56)|(23)(56)⟩= 1. (E.3) In the first example, defect lines 1 and 4 in the right state are contracted from 1→ 4. The overall enclosed signis+−+=−,sothisisasignedcontraction. Therearenoothers. Inthesecondexample,thesamelines are contracted but from 4→ 1 (draw this out and you will see what I mean). The overall enclosed sign is −+−=+,sothisisnotasignedcontraction. It turns out that this inner product gives-independent measurements for (except = 1) when taken with the Hamiltonian deformation. The two deformations described also help with the comparison to ℓ(2|1) since now the deformed loop model is only invariant under translations by even numbers of sites as is the ℓ(2|1) chain. It turns out that the measurements of are quite off using this inner product, but the fact that the results are still -independent made this inner product worth recording, because it may indicateasymmetryoftheloopmodelthat has yet to be explored. E.2 Loopmodelcontractionparameter Thecorrectdeformedinnerproductforthegluedmodulesoftheloopmodel,infact,ismuchsimplerthan the alternating sign one. Whenever two defect lines() and() on the same side are contracted together, startingfrom()andendingat(),and is even, introduce a factor. For example, on = 6 sites ⟨(23)(14)(56)|(41)(56)⟩=⟨(23)|(25)(34)⟩=. (E.4) 154 In the first case (23) in the left state contracts(2) with(3) in the right state, and since the first of these, (2), is even, there results a factor (and there is one closed loop). In the second case,(4) and(5) are contracted thoughviaamorecomplicatedpath,and this again gives a factor. But note that ⟨(32)(45)(61)|(45)(61)⟩= 1, (E.5) sinceinthiscasethepaireddefectintherightstatestartsat(3)andendsat(2),andthedeformationrequires startingonanevensite. Then,thenewinner product is ⟨|⟩= (,) [ (,) ], (E.6) where(,) counts the number of paired defects as described above. Measurements of using this inner product give -independent outcomes, which match Table 9.3. Because of the smaller dimension, larger latticesizesarepossible. E.3 Illustration of the absence of Jordan blocks for =±1, and the -independenceofthemeasurementof at= 0 Intheperiodicloopmodelwithdeformation parameter, on four sites, in the basis {(12)(34),(23)(14),(2)(3)(41),(3)(4)(12),(1)(4)(23),(1)(2)(34),(1)(2)(3)(4)}, (E.7) theHamiltonianmatrixis 0 = 2 F © « 4 ∞ −2 −2 0 −1 0 −1 0 −2 4 ∞ −2 − 0 − 0 0 0 0 4 ∞ − −1 0 −1 − 0 0 −1 4 ∞ − −1 0 −1 0 0 0 −1 4 ∞ − −1 − 0 0 −1 0 −1 4 ∞ − −1 0 0 0 0 0 0 4 ∞ ª ® ® ® ® ® ® ® ® ¬ , (E.8) where,intermsof, = /(+1), F = sin/,= 2cos, and ∞ = sin ∫ ∞ −∞ sinh[(−)] sinhcosh d. (E.9) Excludingthe/2 F factor,theeigenvalues of this matrix are 1 = 4 ∞ −2−2, (E.10a) 2 = 4 ∞ −−2, (E.10b) 3 = 4 ∞ −, (E.10c) 4 = 4 ∞ −, (E.10d) 5 = 4 ∞ , (E.10e) 6 = 4 ∞ −2+2, (E.10f) 7 = 4 ∞ −+2. (E.10g) 155 Theyareorderedsothatfor= 2, ∞ = 1,= /3,{ 1 , 2 , 3 , 4 , 5 , 6 , 7 }={0,1,3,3,4,4,5}. For= 2, ≠ 1,itisknownthatthereisaJordanblock at 5 = 6 = 4. Their corresponding eigenvectors are 1 =(1,1,0,0,0,0,0) (E.11a) 2 = −2+4+4 (−4) , 4−2(−2) (−4) ,1,1,1,1,0 (E.11b) 3 =(0,0,0,−1,0,1,0) (E.11c) 4 =(0,0,−1,0,1,0,0) (E.11d) 5 = 2 − 2 +2(1−) 4 −5 2 +4 , 2 2 −+2(1−) 4 −5 2 +4 , 2− 2 −4 , −2 4− 2 , 2− 2 −4 , −2 4− 2 ,1 (E.11e) 6 =(−1,1,0,0,0,0,0) (E.11f) 7 = − 2(+2+2) (+4) , 2(+2)+4 (+4) ,−1,1,−1,1,0 (E.11g) Theinnerproductexpressionsignifyingthe appearance of Jordan blocks is then ( 5 | 6 ) p ( 5 | 5 )( 6 | 6 ) = (+1)( 2 −1) p 2(,) (E.12a) where (,)= 8 + 6 (2 2 −8)−16 5 + 4 ( 4 +4 2 +38)+ 3 (−4 3 −4 2 +28) + 2 ( 4 +8 3 −6 2 +8−53)+(−4 3 −16 2 −20)+16 2 +24. (E.12b) Importantly,thisexpressioncontainsafactor 2 −1inthenumerator. ThusJordanblockscannotbeobserved with = 1,asthen 5 and 6 arealwaysorthogonal. Thisisobservedtobethecasewhendirectlytryingto findtheJordancanonicalformof 0 at =±1. Wemaytakeaseriesexpansionaround= 1. Theresultis lim →1 ( 5 | 6 ) p ( 5 | 5 )( 6 | 6 ) = 1− 2 |1− 2 | − (−1) 2 ( 4 −8 3 +98 2 −136+153) 8(1− 2 )|1− 2 | +((−1) 3 ). (E.13) As long as ≠±1, the leading term goes to±1, as expected. In Dubail, Jacobsen, and Saleur [57], it was found that only = 1 led to diagonalizability. Here, this was due to expressions that contained −1 in the denominator. However, things seem to differ with the inclusion of =−1. Note that in that work open boundary conditions were used, whereas here they are periodic, which may be responsible for the additional special value =−1. The(1− 2 )|1− 2 | factor in the denominator of the quadratic term is also ratherbothersome. Clearly in the( 5 , 6 ) pair, 6 is the bottom state as 6 is contained in 0, ±2 while 5 is in the full 0, ±2+ 1,1 + 2,1 (although as→ 1, they converge to the same vector). The conformal norm square of 6 is ⟨ 6 | 6 ⟩= 2(−1), (E.14) withthecaveatthatitisnotnormalized. But the preceding expression vanishes at= 1 (= 0). Wehave lim →1 5 = 2 −1 6(−1) + 7 2 +12−13 36 , 1− 2 6(−1) + −13 2 +12+7 36 , −2 3 , 1−2 3 , −2 3 , 1−2 3 ,1 (E.15) The two diverging components proportional to(− 1) −1 are clearly parallel to 6 . Subtracting off this component (and a little more, in order to make the result orthogonal to 6 ), we must have the Jordan partner, ˜ 5 = − 2 +4−1 12 , − 2 +4−1 12 , −2 3 , 1−2 3 , −2 3 , 1−2 3 ,1 . (E.16) 156 Irescale˜ 5 sothat(ˆ 6 | 0 ˜ 5 )= 2. Thenormalization of˜ 5 is thus tied to that of 6 , and we should get = |⟨˜ 5 | −2 |˜ 1 ⟩| 2 ⟨˜ 5 |ˆ 6 ⟩ , (E.17) where⟨˜ 1 ,˜ 1 ⟩ = 1. For = 4, the result is =−32/3 √ 3≈−1.96028, which is -independent. Note that remained generic in this calculation, so that all terms ended up disappearing. This value is identical to theoneobtainedinthediscussionsurrounding Table 9.3. The preceding procedure makes sense for all values of, not just = 0. If we follow it and define in thesameway,Ifindthat =−8(+1)/ F , which is still independent of. Wealsoknowthat 5 and 6 becomeparallelas→ 1. Theyshouldthusbetreatedonanequalfooting, and I should be able to follow the same process but with the roles of 5 and 6 reversed. If I do this, I get a horrendousexpressionfor: = 16 8 +2 6 ( 2 −4)−16 5 + 4 ( 4 +4 2 +38)−4 3 ( 2 +−7) + 2 ( 4 +8 3 −6 2 +8−53)−4( 2 +4+5)+8(2 2 +3) 2 (+1) 2 ( 2 −1) 2 F 6 −2 5 ( 2 +1)+ 4 ( 4 +8−8)−2 3 (2 3 −6 2 +2−5) −2 2 ( 4 +2 3 −4 2 +22−11)+8( 3 +−1)−16(−1) 2 . (E.18) But, when I set= 1 in this expression, it reduces to=−32 √ 3/9, again without specifying (although thereisa( 2 −1) 2 inthedenominatorofthe full expression). To see that this is no accident, we can do the same thing for = 6. In the basis of momentum-2 states, whichIwillnotwriteout,onehastwoeigenvectors that degenerate at= 1: 1 =(1,0,0,0,0,0,0,0), (E.19) 2 = ( 2 −1) 3 − 2 −4+4 , ( √ 3−1)(−2) 2( 2 −4) , 2− 2 −4 , (1− √ 3)(−1) 2(−2) , (1+ √ 3)(−2) 2( 2 −4) ,− (1+ √ 3)(−2) 2( 2 −4) , 1 2 (1− √ 3),1 . (E.20) Ifind ( 1 , 2 )= ( 2 −1) s 2 6 −4 5 + 4 (3 2 −2−11)−2 3 ( 2 +10−15) + 2 ( 4 +5 2 +38+24)−4(5 2 +2+21)+12 2 −8+44 . (E.21) Aseriesexpansionaround= 1 gives lim →1 ( 1 , 2 )= 1− 2 |1− 2 | − (−1) 2 (19 2 −34+37) 2(1− 2 )|1− 2 | +((−1) 3 ). (E.22) 1 and 2 thusbecomeparallel,andareevidently can be orthogonalized to obtain and. A value of is = 3 F 4 2 + 2(9−2 2 ) √ 2 +48 −6 1− . (E.23) Despite the denominator, the function is perfectly analytic at = 1. We then have =−288 √ 3/49≈ −3.24046. Exchangingtherolesof 1 and 2 leads to a number of = 6(+1) 2 (− 2 + √ 2 +48+12) 2 F ( 2 −1) 2 × 2 6 −4 5 + 4 (3 2 −2−11)−2 3 ( 2 +10−15) + 2 ( 4 +5 2 +38+24)−4(5 2 +2+21)+12 2 −8+44 2 4 +50 2 +22 √ 2 +48 − √ 2 +48 3 +96 6 −2 5 ( 2 +1)+ 4 ( 4 +8−8)−2 3 (2 3 −6 2 +2−5) −2 2 ( 4 +2 3 −4 2 +22−11)+8( 3 +−1)−16(−1) 2 . 157 (E.24) The( 2 −1) 2 in the denominator says that there is no when =±1, but when is generic and→ 1, we get the same value as before, =−288 √ 3/49. This value is also identical to the one obtained in the discussionsurroundingTable9.3. Theconformalnormof 1 is ⟨ 1 | 1 ⟩=( 2 −1). (E.25) 158 AppendixF Additionaltablesandfigures F.1 SupplementarydataforSection8.3.1 /3 /2 /sec −1 (2 √ 2)−1 e 8 0.325797 0.350787 0.351607 0.372103 0.376185 10 0.329276 0.352661 0.353516 0.376516 0.381388 12 0.328903 0.351523 0.352390 0.376800 0.382216 14 0.327108 0.349302 0.350173 0.375408 0.381203 16 0.324801 0.346738 0.347609 0.373350 0.379417 18 0.322360 0.344140 0.345010 0.371078 0.377346 20 0.319948 0.341637 0.342507 0.368803 0.375224 22 0.317637 0.339281 0.340151 0.366621 0.373165 TableF.1: Thevaluesof∥ −1 Φ 11 ∥ 2 for various lengths and parameters. /3 /2 /sec −1 (2 √ 2)−1 e 8 0.325786 0.349822 0.350572 0.368581 0.372080 10 0.329244 0.350622 0.351340 0.369610 0.373372 12 0.328845 0.348905 0.349594 0.367684 0.371562 14 0.327027 0.346360 0.347031 0.364836 0.368753 16 0.324704 0.343605 0.344262 0.361773 0.365685 18 0.322251 0.340885 0.341533 0.358769 0.362653 20 0.319832 0.338301 0.338943 0.355932 0.359775 22 0.317515 0.335886 0.336524 0.353297 0.357092 TableF.2: Thevaluesof∥Π (2) −1 Φ 11 ∥ 2 for various lengths and parameters. 159 /3 /2 /sec −1 (2 √ 2)−1 e 10 0.724502 0.855299 0.859681 0.965547 0.985531 12 0.769370 0.879924 0.883733 0.977938 0.996234 14 0.794399 0.894860 0.898393 0.987034 1.00452 16 0.808303 0.903779 0.907193 0.993487 1.01062 18 0.815690 0.908984 0.912369 0.998190 1.01523 20 0.819132 0.912010 0.915425 1.00203 1.01916 22 0.820131 0.913885 0.917377 1.00577 1.02311 TableF.3: Thevaluesof∥ 12 Φ 12 ∥ 2 for various lengths and parameters. /3 /2 /sec −1 (2 √ 2)−1 e 10 0.724473 0.853535 0.857809 0.959148 0.977917 12 0.769136 0.874667 0.878230 0.964825 0.981331 14 0.793913 0.886208 0.889353 0.967146 0.982324 16 0.807582 0.891969 0.894866 0.967190 0.981519 18 0.814765 0.894091 0.896830 0.965628 0.979399 20 0.818030 0.893910 0.896546 0.962969 0.976361 22 0.818871 0.892283 0.894845 0.959593 0.972719 TableF.4: Thevaluesof∥Π (4) 12 Φ 12 ∥ 2 for various lengths and parameters. 160 F.2 SupplementarydataforSection10.2 Jordan structure module(s) ℎ+ℎ scaling field(s) 1 0 1 1 0, ±2 0 2 0.248922 2 3 11 1/4,6 Φ 01 ,(0,5)Φ 11 ,(5,0)Φ −1,1 3 1.20295 4 7 2 0, ±2, 21 5/4,6 21 ⊗ 21 ,(6,0),(0,6), Φ 02 ,(0,4)Φ 12 ,(4,0)Φ −1,2 4 1.75032 2 9 11 2 (1,0)Φ 11 5 2.03178 2 11 11 2 (0,1)Φ −1,1 6 2.07518 2 13 11 9/4 (1,1)Φ 01 7 2.23935 4 17 21 39/16 (1,0)Φ 1/2,2 ,(0,1)Φ −1/2,2 8 2.6721 8 25 3, ±2 35/12 Φ 03 9 2.79865 4 29 2 0, ±2, 21 13/4,6 (1,1) 21 ⊗ 21 ,(6,0),(0,6),(1,1)Φ 02 10 3.14032 2 31 11 4 11 3.33436 4 35 2 0, ±2, 21 4 31 ⊗ 31 ,(2,0)Φ 12 ,(0,2)Φ −1,2 12 3.36307 2 37 11 17/4 (2,2)Φ 01 13 3.43849 24 61 8×2 11 , 21 , 3, ±2 4 (2,1)Φ 11 ,(1,2)Φ −1,1 , (2,0)Φ 12 ,(0,2)Φ −1,2 , (1,0)Φ 1/3,3 ,(0,1)Φ −1/3,3 14 3.50199 2 63 11 4 15 3.51279 4 67 21 71/16 (2,1)Φ 1/2,2 ,(1,2)Φ −1/2,2 16 3.52945 4 71 11 17/4 (2,2)Φ 01 17 3.67553 2 73 11 17/4 (2,2)Φ 01 18 3.69711 4 77 2 0, ±2, 21 21/4 (2,2) 21 ⊗ 21 19 3.74793 4 81 21 71/16 (2,1)Φ 1/2,2 ,(1,2)Φ −1/2,2 20 3.81032 4 85 2 0, ±2, 21 4 (2,2),(2,0)Φ 12 ,(0,2)Φ −1,2 21 3.91012 2 87 11 22 3.92424 8 95 3, ±2 59/12 (1,1)Φ 03 23 4.11904 24 119 8×2 11 , 21 , 3, ±2 21/4 24 4.30698 24 143 8×2 11 , 21 , 3, ±2 25 4.32986 4 147 11 26 4.33042 4 151 11 27 4.41063 4 155 2 0, ±2, 21 6 (3,3) Table F.5: Jordan structure of the lowest 155 of 3991 eigenvalues of 0 on = 12 sites, in the vacuum sector at momentum 0. is the algebraic multiplicity of the eigenvalue on line . = Í =1 is the runningdimension. Inthe“Jordanstructure”column,× means rank- Jordanblocksappearforthat eigenvalue,and≡ 1×.(,)Φ meansa level-(,) descendant ofΦ. 161 Jordan structure module(s) ℎ+ℎ scalingfield(s) 1 0 1 1 0, ±2 0 2 0.249157 2 3 11 1/4 Φ 01 3 1.21393 4 7 2 0, ±2, 21 5/4,7 21 ⊗ 21 ,(7,0),(0,7),Φ 02 4 1.7908 2 9 11 2 (1,0)Φ 11 5 2.04203 2 11 11 2 (0,1)Φ −1,1 6 2.11598 2 13 11 9/4 (1,1)Φ 01 7 2.28522 4 17 21 39/16 (1,0)Φ 1/2,2 ,(0,1)Φ −1/2,2 8 2.72747 8 25 3, ±2 35/12 Φ 03 9 2.90368 4 29 2 0, ±2, 21 13/4 (1,1) 21 ⊗ 21 ,(1,1)Φ 02 10 3.2963 2 31 11 4 11 3.44746 4 35 2 0, ±2, 21 4 31 ⊗ 31 ,(2,0)Φ 12 ,(0,2)Φ −1,2 12 3.56541 24 59 8×2 11 , 21 , 3, ±2 4 (2,1)Φ 11 ,(1,2)Φ −1,1 , (2,0)Φ 12 ,(0,2)Φ −1,2 , (1,0)Φ 1/3,3 ,(0,1)Φ −1/3,3 13 3.56731 2 61 11 17/4 (2,2)Φ 01 14 3.6637 2 63 11 4 15 3.68765 4 67 11 17/4 (2,2)Φ 01 16 3.72401 4 71 21 71/16 (2,1)Φ 1/2,2 ,(1,2)Φ −1/2,2 17 3.79679 2 73 11 17/4 (2,2)Φ 01 18 3.89009 4 77 2 0, ±2, 21 4 (2,2),(2,0)Φ 12 ,(0,2)Φ −1,2 19 3.90027 4 81 21 71/16 (2,1)Φ 1/2,2 ,(1,2)Φ −1/2,2 20 4.04741 4 85 2 0, ±2, 21 21/4 (2,2) 21 ⊗ 21 21 4.14726 8 93 3, ±2 59/12 (1,1)Φ 03 22 4.30023 2 95 11 23 4.36054 2 97 11 24 4.3695 24 121 8×2 11 , 21 , 3, ±2 21/4 25 4.51031 4 125 21 26 4.56181 2 127 11 27 4.63272 4 131 11 Table F.6: Jordan structure of the lowest 131 of 23391 eigenvalues of 0 on = 14 sites, in the vacuum sector at momentum 0. is the algebraic multiplicity of the eigenvalue on line . = Í =1 is the runningdimension. Inthe“Jordanstructure”column,× means rank- Jordanblocksappearforthat eigenvalue,and≡ 1×.(,)Φ meansa level-(,) descendant ofΦ. 162 Jordan structure module(s) ℎ+ℎ scalingfield(s) 1 0 1 1 0, ±2 0 2 0.249321 2 3 11 1/4 Φ 01 3 1.22141 4 7 2 0, ±2, 21 5/4,8 21 ⊗ 21 ,(8,0),(0,8),Φ 02 4 1.82075 2 9 11 2 (1,0)Φ 11 5 2.04663 2 11 11 2 (0,1)Φ −1,1 6 2.14386 2 13 11 9/4 (1,1)Φ 01 7 2.31668 4 17 21 39/16 (1,0)Φ 1/2,2 ,(0,1)Φ −1/2,2 8 2.76578 8 25 3, ±2 35/12 Φ 03 9 2.97579 4 29 2 0, ±2, 21 13/4 (1,1) 21 ⊗ 21 ,(1,1)Φ 02 10 3.41025 2 31 11 4 11 3.53106 4 35 2 0, ±2, 21 4 31 ⊗ 31 ,(2,0)Φ 12 ,(0,2)Φ −1,2 12 3.65361 24 59 8×2 11 , 21 , 3, ±2 4 (2,1)Φ 11 ,(1,2)Φ −1,1 ,(2,0)Φ 12 , (0,2)Φ −1,2 ,(1,0)Φ 1/3,3 ,(0,1)Φ −1/3,3 13 3.70857 2 61 11 17/4 (2,2)Φ 01 14 3.76720 2 63 11 4 15 3.79947 4 67 11 17/4 (2,2)Φ 01 16 3.87063 4 71 21 71/16 (2,1)Φ 1/2,2 ,(1,2)Φ −1/2,2 17 3.88376 2 73 11 17/4 (2,2)Φ 01 18 3.93996 4 77 2 0, ±2, 21 4 (2,2),(2,0)Φ 12 ,(0,2)Φ −1,2 19 4.00740 4 81 21 71/16 (2,1)Φ 1/2,2 ,(1,2)Φ −1/2,2 20 4.29274 4 85 2 0, ±2, 21 21/4 (2,2) 21 ⊗ 21 21 4.30309 8 93 3, ±2 59/12 (1,1)Φ 03 22 4.54588 24 117 8×2 11 , 21 , 3, ±2 21/4 23 4.58776 2 119 11 Table F.7: Jordan structure of the lowest 119 of 143073 eigenvalues of 0 on = 16 sites, in the vacuum sector at momentum 0. is the algebraic multiplicity of the eigenvalue on line . = Í =1 is the runningdimension. Inthe“Jordanstructure”column,× means rank- Jordanblocksappearforthat eigenvalue,and≡ 1×.(,)Φ meansa level-(,) descendant ofΦ. 163 10 12 14 1 0* 0* 0* 2 0* 0* 0* 4 0* 0* 0* 5 0* 0* 0* 6 0* 0* 0* 7 0* 0* 0* 9 0* 0* 0* 10 0* 0* 0* 11 0* — — 12 — 0* 0* 13 0* 0* — 14 0* — — 16 — — 0* 17 — 0* 0* 18 0* 0* 0* 20 0* 0* 0* 21 0* 0* 0* 23 0.011488 0* 0.00296685 25 — 0.00585179 0.00296685 26 — 0.00585179 0.00296685 27 0.011488 — 0.00296685 29 0.011488 — — 30 — 0.00585179 — 31 0.0115711 — 0.00296685 32 — — 0.00296685 34 0.0115711 0.00585179 0.00296685 35 0.0115711 — — 36 0.0115711 0.00585179 — 37 — — 0.00296685 38 — 0.00585179 — 39 — 0.00617649 — 40 0.0115711 0.00627736 0.00296685 41 — — 0.00317961 noprojector 0.0115289 0.0076238 0.00712264 TableF.8: Thenorm∥Π () (−)∥ 2 forvariousprojectorranksandsystemlengths,intheloopmodel representation. Inthistable“0*”meansanumberthatislessthanabout2×10 −8 . Iestimatetheuncertainty inthenonzeronumberstobeabout10 −7 . 164 10 12 14 1 * * * 2 * * * 4 * * * 5 * * * 6 * * * 7 * * * 9 * * * 10 * * * 11 * — — 12 — 0* 0* 13 0* 0* — 14 0* — — 16 — — 0* 17 — 0* 0* 18 0* 0* 0* 20 0* 0* 0* 21 0* 0* 0* 23 0.035351 0* 0.0185908 25 — 0.0253789 0.0185908 26 — 0.0253789 0.0185908 27 0.035351 — 0.0185908 29 0.035351 — — 30 — 0.0253789 — 31 0.035679 — 0.0185908 32 — — 0.0185908 34 0.035679 0.0253789 0.0185908 35 0.035679 — — 36 0.035679 0.0253789 — 37 — — 0.0185908 38 — 0.0253789 — 39 — 0.0268784 — 40 0.035679 0.0272667 0.0185908 41 — — 0.0199863 noprojector 0.0355239 0.0330649 0.044643 Table F.9: The norm∥Π () (−)∥ 2 /∥Π () ∥ 2 for various projector ranks and system lengths , in theloop modelrepresentation. Inthistable “*” means a highly variable number of order 1. They are likely theresultofa0/0sincethecorrespondingvaluesinTableF.8aresosmall. “0*”meansanumberthatisless than10 −6 . Iestimatetheuncertaintyinthe nonzero numbers to be about10 −6 . 165 10 12 14 1 0* 0* 0* 2 0* 0* 0* 4 0* 0* 0* 5 0* 0* 0* 6 0* 0* 0* 7 0* 0* 0* 9 0* 0* 0* 10 0* 0* 0* 11 0* — — 12 — 0.236909 0.134669 13 0.424521 0.236909 — 14 0.424521 — — 16 — — 1.05087 17 — 1.99303 1.05087 18 3.79519 1.99303 1.05087 20 3.79519 1.99303 1.05087 21 3.79519 1.99303 1.05087 23 3.80601 1.99303 1.05369 25 — 1.99874 1.05369 26 — 1.99874 1.05369 27 3.80601 — 1.05369 29 3.80601 — — 30 — 1.99874 — 31 3.80617 — 1.05369 32 — — 1.05369 34 3.80617 1.9989 1.05369 35 3.80617 — — 36 3.80617 1.9989 — 37 — — 1.16385 38 — 1.9989 — 39 — 2.4453 — 40 3.86286 2.44529 1.16385 41 — — 1.99782 noprojector 3.92271 2.17113 1.2513 TableF.10: Thenorm∥Π () (−)∥ 2 forvariousprojectorranksandsystemlengths,intheloopmodel representation. In this table “0*” means a number that is less than 10 −5 . I estimate the uncertainty in the nonzeronumberstobeabout10 −4 . 166 10 12 14 1 0* 0* 0* 2 0* 0* 0* 4 0* 0* 0* 5 0* 0* 0* 6 0* 0* 0* 7 0* 0* 0* 9 0* 0* 0* 10 0* 0* 0* 11 0* — — 12 — 0.0572678 0.0451944 13 0.0739766 0.0572678 — 14 0.0739766 — — 16 — — 0.356867 17 — 0.485697 0.356867 18 0.650097 0.485697 0.356867 20 0.650097 0.485697 0.356867 21 0.650097 0.485697 0.356867 23 0.374967 0.485697 0.213 25 — 0.278863 0.213 26 — 0.278863 0.213 27 0.374968 — 0.213 29 0.374968 — — 30 — 0.278863 — 31 0.368193 — 0.213 32 — — 0.213 34 0.368193 0.278954 0.213 35 0.368194 — — 36 0.368194 0.278954 — 37 — — 0.240011 38 — 0.278954 — 39 — 0.00225377 — 40 0.373997 0.00225377 0.240011 noprojector 0.380182 0.302237 0.268341 Table F.11: The norm∥Π () (−)∥ 2 /∥Π () ∥ 2 for various projector ranks and system lengths , in the loop model representation. In this table “0*” means a number that is less than 10 −6 . I estimate the uncertaintyinthenonzeronumberstobe about10 −5 . 167 F.3 SupplementalfiguresforChapter11 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c -4 -2 2 4 ϵ 1 (c) FigureF.1: Plot of the exponent 1 as a function of. 168 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c -4 -3 -2 -1 1 ϵ 2 (c) FigureF.2: Plot of the exponent 2 as a function of. 169 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c -4 -3 -2 -1 1 η(c) FigureF.3: Plot of the exponent as a function of. -2.0 -1.5 -1.0 -0.5 0.5 1.0 c -4 -2 2 4 ϵ 1 (c) ϵ 2 (c) η(c) FigureF.4: Plotsof the exponents 1 , 2 , as functions of. 170 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 5 10 15 α (c) 10 12 14 16 18 20 22 Figure F.5 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.5 1.0 1.5 2.0 β (c) 10 12 14 16 18 20 22 Figure F.6 171 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 2 4 6 8 10 12 γ (c) 10 12 14 16 18 20 22 Figure F.7 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.5 1.0 1.5 2.0 2.5 δ (c) 10 12 14 16 18 20 22 Figure F.8 172 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.1 0.2 0.3 0.4 0.5 0.6 β (c) α (c) sgn(c) 10 12 14 16 18 20 22 Figure F.9 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.2 0.4 0.6 0.8 1.0 1.2 γ (c) δ (c) sgn(c) 10 12 14 16 18 20 22 Figure F.10 173 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.2 0.4 0.6 0.8 1.0 c 1 (c) 10 12 14 16 18 20 22 Figure F.11 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.1 0.2 0.3 0.4 c 2 (c) 10 12 14 16 18 20 22 Figure F.12 174 -2.0 -1.5 -1.0 -0.5 0.5 1.0 c 0.1 0.2 0.3 0.4 0.5 0.6 c 1 (c) c 2 (c) sgn(c) 10 12 14 16 18 20 22 Figure F.13 175
Abstract (if available)
Abstract
Two novel frameworks for handling mathematical and physical problems are introduced. The first, emerging Jordan forms, generalizes the concept of the Jordan canonical form, a well-established tool of linear algebra. The second, dual Jordan quantum physics, generalizes the framework of quantum physics to one in which the hermiticity postulate is considerably relaxed. These frameworks are then used to resolve some long-outstanding problems in theoretical physics, coming from critical statistical models and conformal field theory. I describe these problems and the difficulties involved in finding satisfactory solutions, then show how the concepts of emerging Jordan forms and dual Jordan quantum physics are naturally suited to overcoming these difficulties. Although the applications of these frameworks in this work are limited in scope to rather specific problems, the frameworks themselves are completely general, and I describe ways in which they may be used in other areas of mathematics and physics. Several appendices close the work, which include improvements to a widely used computational algorithm and corrections to some published data.
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Liu, Lawrence
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Emerging Jordan forms, with applications to critical statistical models and conformal field theory
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2023-05
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Arnoldi method
biorthogonal quantum physics
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