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Disordered quantum spin chains with long-range antiferromagnetic interactions
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Disordered quantum spin chains with long-range antiferromagnetic interactions
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DISORDERED QUANTUM SPIN CHAINS WITH LONG-RANGE ANTIFERROMAGNETIC INTERACTIONS by Nicolas Moure Gomez A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) May 2019 Copyright 2019 Nicolas Moure Gomez ii This work would not have been possible without Stephan. I will always be grateful to him for all his support, patience, and companionship. Special thanks to Prof. Kettemann for his guidance and for his key role in the project, and to Prof. Bhatt for all the useful insight and discussions. Y a todos mis amigos y familia por estar ahí siempre, gracias totales. Contents ii iii List of Figures v 1 Motivation, Theoretical Model, and Preliminary Concepts 1 1.1 Motivation and Model ......................... 1 1.2 Strong Disorder Renormalization Group ............... 5 1.3 Infinite Randomness Fixed Point and the Random Singlet Phase.. 10 1.4 Disordered Systems with long range Interactions: A short Overview 12 1.4.1 RTIM with long range Interactions .............. 13 2 Magnetic Properties and Localization 15 2.1 Magnetic Susceptibility......................... 15 2.1.1 Width of Couplings Distribution Function during the RG flow 20 2.1.2 Eect of the Exponential Cuto ................ 22 2.2 Distribution Function of Excitation Energies and the Delocalization Transition ................................ 24 2.2.1 Continuous Chain........................ 28 3 Entanglement entropy and spatial correlations in the SDRG con- text 32 3.1 Concurrence............................... 32 3.1.1 Mean and typical Concurrence at the IRFP ......... 34 3.1.2 Corrections to the Random Singlet State ........... 36 3.2 Entanglement Entropy ......................... 42 4 Conclusion 52 Reference List 54 iv List of Figures 1.1 Left: Sketch of electronic orbitals at low doping concentration n D . All states are localized and magnetic, as indicated by red arrows. Right: At larger n D states at Fermi energy are delocalized, coexist- ing with localized magnetic states.................... 2 1.2 Schematic representation of spins randomly placed in a diluted lat- tice with lattice spacinga. The empty circles represent empty sites, whereas full ones represent the sites that are occupied by spins. .. 4 1.3 Schematic representation of the SDRG procedure where the system size is reduced from N to N≠ 2 by eliminating spins l and m, the spin pair with the biggest couplings, while all remaining couplings are renormalized. Here only the coupling between sites i and j is explicitly shown, but in reality every spin is coupled to one another. 6 1.4 Graphic representation of the Random Singlet State of a chain with N=10 sites. The spins connected with the continuous lines form a singlet, e. g., spins 2 and 7 in this particular case. Note that bonds are not allowed to cross each other and that they can occur between arbitrarily distant spins. ........................ 10 v 1.5 When the biggest coupling occurs between sites 2 and 3 (double line), a new bond ˜ J 14 is created between spins 1 and 4 after the SDRG step is performed......................... 12 2.1 Magnetic susceptibility, normalized by ‰ 0 © ‰ (T = J 0 ),foran XX-chain with N =1280 randomly placed spins, interacting via antiferromagnetic long-range couplings given in Eq. (1.2), chain length L/a=100N, and – =0.6,...,2.0. The cuto length is set to › = Œ . Continuous lines: fits to ‰ ≥ T 1/z≠ 1 with finite z at low temperatures. Inset: susceptibility for – =10.0 along with the same power law fit with z=9.36 (green line) and the IRFP result (red line). ................................ 16 2.2 Dynamicalexponentz extractedbyfittingthelow-temperaturesus- ceptibility in Fig. 2.1 to Eq. (2.5) as a function of the power – (squares). These numerical results are then fitted to Eq. (2.6) (dashed line), which allows us to extract the crossover value – ú = 1.066±0.002. ............................. 19 2.3 In this particular realization of the random placement of the spins, spinicanbepracticallyisolatedfromtherestofthechainfor– ∫ 1 and still contribute to the low temperature magnetic susceptibility. 20 vi 2.4 WidthW 1 ofthedistributionfunctionofnearest-neighborcouplings as a function of the fraction of remaining spins. The negative log- arithmic in base two is used to have an equally spaced horizontal variablethatgrowsasthenumberofspinsdecreasesbyonehalf. All values have been normalized by the width W 0 1 (– ) of the initial dis- tribution and the parameters are kept as in Fig. 2.1. Inset: W 1 /W 0 1 for a simple nearest neighbor model with uniformly distributed cou- plings (crosses) known to flow to the IRFP. The numerical results for – =20.0 (black circles) are included for comparison purposes. . 21 2.5 Log-log plot of the magnetic susceptibility for – =d=1 and›/L = 1,1/2,...,1/64. The remaining parameters are kept as in Fig. 2.1, and the susceptibility is rescaled in the same manner. ........ 22 2.6 Logarithmic plot of the susceptibility for a fixed and finite cuto length › = L/32 and – =0.6,0.8,...,2.0. The remaining parame- ters are kept as in Fig. 2.1........................ 23 2.7 Distributionofthelowestexcitationgap‘ 1 scaledbyitsmeanvalue, s = ‘ 1 /È‘ 1 Í,for › = Œ and – =0.6,0.8,...,2.0. The remaining parameters are as in Fig. 2.1. ..................... 26 2.8 Wigner (black), Poisson (blue), and critical (red) distributions as a function of≠ log(s) for visualization purposes. ............ 28 2.9 Continuous XX-chain distribution of excitation gaps scaled by their mean value s = ‘ 1 /È‘ 1 Í,for › = Œ and – =0.4,0.8,...,2.2. The chain has a total of N =320 spins and length L =10N. The continuous lines represent the Wigner (black), critical (red), and Poisson (blue) distributions which fit the data best for – =0.8. – =1.6, and – =2.2, respectively.................... 30 vii 3.1 Diagram of the two possibilities occurring in Eq. (3.16). The top figure shows the case {pq} / œ RS, while the bottom one depicts the case {pq}œ RS. ............................ 39 3.2 Mean concurrence of a long range XX-chain with N =800 sites as a function of l = |p≠ q|, the dierence between spin sites numbers p and q. The couplings are true long range with › æŒ and – = 0.6,...,2.0. A clear distinction between odd values of l (top curves) and even ones (bottom curves) can be observed. ........... 40 3.3 Separate curves for the mean concurrence of even (left) and odd (right)valuesofl. Thecontinuousblacklinesrepresentlinearregres- sion fits to – =0.6 and – =2.0, while the dashed black line depicts c e,o l ≠ “ e,o , where c e,o and “ e,o have been obtained by averaging their corresponding values for – =0.6 and – =2.0. ............ 41 3.4 TypicalconcurrenceofalongrangeXX-chainwithN=800sitesas a function of l = |p≠ q|, the dierence between spin sites numbers p and q. The couplings are true long range with › æŒ and – = 0.6,...,2.0. A clear power law decay is observed for all values of – . . 42 3.5 Decay power of the typical concurrence plotted in Fig. 3.4 as a function of – . The dashed line represents the linear regression fit giving “ (– )=1.02– +2.02. ...................... 43 3.6 SinglerealizationoftheRandomSingletStateillustratingtheentan- glement entropy calculation for two dierent boundaries between subsystems. When the boundary is given by line a, the entropy is 4·ln(2). When its given by line b, a value of 2·ln(2) is obtained. . 45 viii 3.7 Average block entanglement entropy for the long ranged XX-chain with N =1280 spins, › æŒ , and – =0.6,2.0 (blue and orange circles, respectively). The black dashed line labeled "Theory 1" cor- responds to Eq. (3.22) with a central charge c = ln(2) and k Õ =0.4. Theblackdotted-dashedlinelabeled"Theory2"depictstheentropy given by Eq. (3.24) with the concurrence given by Eq. (3.6). .... 46 3.8 Entanglement entropy as a function of c 2 for the three instances that occur after approximating the corrected state to Eq. (3.25). As mentioned in the text, the entropy for Case 1 (dashed line) is alwaysbiggerthanthatforCase3(continuousline), exceptatc=1 where they coincide. In contrast, the entropy for Case 2 is always lower than the entropy at the IRFP, again, with the exception of c=1, where Eq. (3.29)goesto kln(2). The plot is done for the specific case k=3, but the conclusions drawn from it remain true for all values of k............................. 49 3.9 Average block entanglement entropy calculated with the approxi- mated state in Eq. (3.25) for a chain with N =800 spins. The power – is varied from 0.60 to 2.00 and Calabrese and Cardy’s for- mula(Eq. (3.22))isplottedalongforreferenceasablackcontinuous line. ................................... 50 ix Chapter 1 Motivation, Theoretical Model, and Preliminary Concepts In this chapter we aim to give a motivation for our work along with a clarifica- tion of the tools and preliminary concepts necessary to study random spin systems via renormalization group techniques. The organization of this chapter goes as follows. In Section 1.1 a motivation and a detailed description of the theoretical model used throughout this work is given. Section 1.2 is devoted to the intro- duction of the Strong Disordered Renormalization Group technique introduced by Dasgupta and Ma in Ref. [13], as well as its generalization to long range systems. In Section 1.3, the notion of Infinite Randomness Fixed Point, an important con- ceptinvolvingdisorderedspinchains,wouldbepresented. Finally,ashortoverview on the state of the art of long range disordered systems is given in Section 1.4. 1.1 Motivation and Model Roughlyspeaking,asthetitleofthisworksuggests,weareinterestedinrandom quantum spin chains with long range interactions. Short range, nearest neighbor chains as well as other spin systems in higher dimensions have been successfully studied by means of the Strong Disorder Renormalization Group techniques intro- duced below in Section 1.2 (see Ref. [25] for an extensive review). However, there are only a few studies regarding random spin systems with long range interactions 1 Figure 1.1: Left: Sketch of electronic orbitals at low doping concentration n D . All states are localized and magnetic, as indicated by red arrows. Right: At larger n D states at Fermi energy are delocalized, coexisting with localized magnetic states. [29, 32, 28, 16, 39], making it an exciting branch of Condensed Matter Physics to work in. Our particular model specified below is inspired in doped semiconductors such as P-doped Si where the doping atoms form randomly positioned moments that interact via exchange interactions. In particular, the magnetic susceptibility of these doped semiconductors is known to diverge at low temperatures with an anomalous power law [52] which is evidence of the aforementioned local mag- netic moments, formed in random localized states as illustrated in Fig. 1.1 [3, 4, 40, 1, 5, 8, 7, 37]. At low dopant density n D π 1/a 3 B , where a B is the Bohr radius of the dopants, these magnetic moments are coupled weakly by the antiferromagnetic exchange interaction between the hydrogen-like dopant levels [5, 10]. In this regime, the magnetic susceptibility is observed to follow Curie’s law ‰ ≥ n D /T of free magnetic moments [5, 50, 33]. However, as n D is increased, the magnetic susceptibility diverges like ‰ ≥ T ≠ – m with a decreasing anomalous power – m (n D ) < 1. This has been identified as being a consequence of a random distribution of exchange couplings due to the random positions of dopants [9, 47]. 2 With increasing doping concentration the density of magnetic momentsn FM , is observedtodecrease. Bothmagneticsusceptibilityandspecificheatmeasurements indicate a finite density n FM at the metal-insulator transition and deep into the metallic phase [52]. Onthemetallicsideofthetransition,theindirectexchangeinteractionbetween the magnetic moments mediated by the itinerant electrons becomes long-range [48, 31, 57]. The typical value of this Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling decays with a power law with exponent – = d, oscillating in sign with a period equal to the Fermi wavelength ⁄ F . Its amplitude is widely, log-normally distributed [35]. Thus, aiming to get a better understanding of the magnetic properties of doped semiconductors, we consider the Hamiltonian H = ÿ i<j J x ij 1 S x i S x j +S y i S y j 2 +J z ij S z i S z j , (1.1) describing N long-range interacting S =1/2 spins that are randomly placed on a periodic lattice of length L using a uniform distribution (see Fig. 1.2). The couplings between all pairs of sites i,j, are taken to be antiferromagnetic decaying as J x,z ij =J x,z 0 |(r i ≠ r j )/a| ≠ – exp(≠| r i ≠ r j |/› ), (1.2) where a is the lattice spacing and we include an exponential cut o quantified by › , that allows us to tune between the limit of short-ranged coupled spins for › æ L/N 1/d , and long-range couplings as › æŒ 1 . It is worth noting that the initial probability distribution of J x,z ij depends directly on – and › by virtue of Eq. (1.2) together with the fact that the spin positions r i are random. 1 Here we restrict ourselves and avoid the alternating sign. Inclusion of the sign alternation will likely lead to dierent fixed points of mixed character, such as the ones discussed in Ref. [54]. 3 Figure 1.2: Schematic representation of spins randomly placed in a diluted lattice with lattice spacing a. The empty circles represent empty sites, whereas full ones represent the sites that are occupied by spins. Now, the long-range interactions do not allow an analytical solution of the Hamiltonian in Eq. (1.1), and direct diagonalization requires large amounts of computationalresources, evenforrelativelysmallvaluesofN 2 . Therefore, weturn to the Strong Disorder Renormalization Group (SDRG) techniques developed first by Dasgupta and Ma [13], and later refined by Fisher [18, 19], which have made great contributions to the understanding of disordered systems [25]. In particular, SDRG techniques allow us to calculate the magnetic susceptibility eciently [ 8]. This work is specifically focused on the study of random spin chains with long range interactions modeled by Eq. (1.1), using the Strong Disorder Renormal- ization Group techniques to be presented in the next section. From a numerical perspective, one-dimensional models oer the possibility of exploring larger length scales, and therefore allow us to get more accurate asymptotic behavior. From a theoretical perspective, one-dimensional models with power-law hopping [38, 59], as well as power-law correlated disorder [14, 15, 27], are interesting since they 2 A large number of random realizations (≥ 20000) needs to be computed in order to obtain good statistics on the physical quantities desired. 4 exhibit an Anderson localization-delocalization transition for non-interacting elec- trons, which has critical properties quite similar to the ones observed in three- dimensional Anderson models [34, 17]. Thus, we can expect a delocalization tran- sition in random spin chains with long range interactions at a critical power – c [41]. Moreover, based on Eq. (1.1), the situation in three-dimensional doped semi- conductors for experimentally accessible temperatures has been at least semi- quantitatively explained [5, 8, 44, 49]. However, as will be explained in detail below, ferromagnetic bonds are created in three dimensions at intermediate renor- malization steps [8, 54], leading to the possibility of new fixed points, and making the asymptotic low-energy and low-temperature behavior a challenging open prob- lem. 1.2 Strong Disorder Renormalization Group Here we focus on the derivation of the SDRG decimation rules of random spin chains with long range interactions modeled by Eq. (1.1). This derivation is a simple generalization of the SDRG procedure developed by Dasgupta and Ma [13]. In a few words, the idea is to choose the spin pair with the largest couplings, say J x,z lm . Taking the expectation value of the Hamiltonian in the ground state of that pair and performing second-order perturbation theory [8, 13, 18, 19, 54, 25, 58], we can obtain a renormalized Hamiltonian with new couplings between all other spins ˜ J x,z ij andreducethesystemsizefromN toN≠ 2aspicturedinFig. 1.3. This process can be iterated numerically to obtain not only the magnetic susceptibility as a function of temperature [8], but by iterating until only two spins are left, it can give valuable information about the excitation energy of the system (also 5 Figure 1.3: Schematic representation of the SDRG procedure where the system size is reduced from N to N≠ 2 by eliminating spins l and m, the spin pair with the biggest couplings, while all remaining couplings are renormalized. Here only the coupling between sites i and j is explicitly shown, but in reality every spin is coupled to one another. referred as the ’exit gap’), as would become clearer in the following discussion (see Eq. (1.9)). To get into the details, let us assume as above that the spin pair (l,m) is connected by the largest couplings J x,z lm . Therefore, we can write the Hamiltonian in Eq. (1.1)as H N =J x lm (S x l S x m +S y l S y m )+J z lm S z l S z m ¸ ˚˙ ˝ H 0 +H Õ , (1.3) 6 where if the disorder is strong enough (J x,z lm ∫ J x,z ij ) we can consider H Õ as a perturbation to H 0 given by H Õ = ÿ i”=l,m J x im (S x i S x m +S y i S y m )+J z im S z i S z m + ÿ j”=l,m J x lj 1 S x l S x j +S y l S y j 2 +J z lj S z l S z j + ÿ i<j”=l,m J x ij 1 S x i S x j +S y i S y j 2 +J z ij S z i S z j . (1.4) Since we are interested in ground state properties, we can use second order pertur- bation theory in the ground state formed by the spin pair (l,m), |G lm Í, to define a new eective Hamiltonian with N≠ 2 spins H eff N≠ 2 =ÈG lm |H Õ |G lm Í+ ÿ {S,M}”=G ÈG lm |H Õ |(S,M) lm ÍÈ(S,M) lm |H Õ |G lm Í E G ≠ E SM . (1.5) Here, the states |(S,M) lm Í are the eigenstates of H 0 labeled by their total spin S and its projection M, while |G lm Í represents the ground state whose identity dependsonthesignofthecouplingsandtheratioJ z lm /J x lm ,becausetheeigenvalues of H 0 are given by E 00 =≠ 1 2 J x lm ≠ 1 4 J z lm , (1.6a) E 10 = 1 2 J x lm ≠ 1 4 J z lm , (1.6b) E 1,±1 = 1 4 J z lm . (1.6c) Since we are interested in purely antiferromagnetic couplings, the spin pair (i,j) alwaysformsasingletinitsgroundstate,ascanalsobeconjecturedbyexamination 7 of Eq. (1.6). Carrying the explicit calculation of the expressions in Eq. (1.5) with |G lm Í = |(0,0) lm Í, we obtain an eective Hamiltonian of the same form H eff N≠ 2 = ÿ (i<j)”=(l,m) ˜ J x ij 1 S x i S x j +S y i S y j 2 + ˜ J z ij S z i S z j , (1.7) where the new couplings are given by ˜ J x ij = J x ij ≠ (J x il ≠ J x im )(J x lj ≠ J x mj ) J x lm +J z lm , ˜ J z ij = J z ij ≠ (J z il ≠ J z im )(J z lj ≠ J z mj ) 2J x lm . (1.8) These rules determine the RG flow that can be numerically iterated until only two spins remain, say (p,q), and therefore obtain the exit gap energy ‘ 1 , which according to Eq. (1.6) would be given by ‘ 1 = Y ___] ___[ ˜ J x pq if ˜ J x pq < ˜ J z pq , 1 2 ( ˜ J x pq + ˜ J z pq ) if ˜ J x pq Ø ˜ J z pq . (1.9) It is important to note that, as shown throughout numerical realizations, the RG flow described by Eq. (1.8) holds two important properties when the initial chain has couplings given by Eq. (1.2). First, we have found that if we start with an antiferromagnetic chain, the chain remains antiferromagnetic during the process, i.e., ˜ J x,z i,j remains positive throughout the flow. This is not true when artificial cutos are introduced and only a finite number of neighbors for each spin is con- sidered. Inthiscase, ferromagneticcouplingswouldoccurandtheRGflowbreaks. The second property is that, as the RG flow advances, the energy scale at each step decreases with respect to the previous one, i.e., the maximum couplings J x,z lm 8 become smaller as the RG flows. This is important for the convergence of the flow and does not occur trivially. For example, when we have an XX-chain (J z ij =0) whose strongest coupled pair (l,m) has a ferromagnetic bond (J x lm < 0), the ground state ofH 0 is |(1,0) lm Í, one of the triplet states. In this case, Eq. (1.5) would lead us to the flow equation ˜ J x ij =J x ij + (J x il +J x im )(J x lj +J x mj ) J x lm , (1.10) which increases the energy scale at each step. A situation which is not physical and makes the numerical code break. This situation, as well as cases where the ground state of H 0 is degenerate and Eq. (1.5) does not hold 3 , would need a dierentapproach,perhapsresemblingthatofWesterberg et al.inRef. [54]. Their approach is valid for any value of the spin S in a short range (nearest neighbor) chain,soitcandealwiththeformationoftripletsthroughouttheflow. However,it does not allow a simple generalization to long range models like ours and therefore it will not be given further discussion. Now, as mentioned at the beginning of this section, the SDRG scheme can also be used to easily calculate the magnetic susceptibility which would be explained at the beginning of next chapter in Section 2.1. However, let us first discuss a very important concept in the framework of SDRG, the so called Infinite Randomness Fixed Point. 3 Take for example the Heisenberg ferromagnetic chain (J x lm = J z lm < 0), in which H 0 has a three-fold degenerate ground state. 9 1.3 Infinite Randomness Fixed Point and the Random Singlet Phase After the SDRG process is completed, one is left out with an approximate ground state composed of the direct product of singlets formed between randomly positioned spins that can be arbitrarily separated, as illustrated in Fig. 1.4 for a chain with N =10 sites. Naturally, this was mentioned by Dasgupta and Ma when they developed the SDRG rules in Ref. [13] and was later given the name Random Singlet Phase [8], which will also be referred as the Random Singlet State (RSS). Later, Daniel Fisher showed that the method by Dasgupta and Ma was not only an approximation method, but that in certain short range models, the system flows to what he called an Infinite Random Fixed Point (IRFP) which makes the SDRG asymptotically exact and the RSS a proper description of the ground state of these systems [19]. Figure 1.4: Graphic representation of the Random Singlet State of a chain with N =10 sites. The spins connected with the continuous lines form a singlet, e. g., spins 2 and 7 in this particular case. Note that bonds are not allowed to cross each other and that they can occur between arbitrarily distant spins. Fisher’s derivation was based on the short range XX-model, but as he also mentions, the Heisenberg (XXX-model), which was studied in the original work 10 of Dasgupta and Ma, shows alike behavior. The SDRG rules for both short range modelscanbeobtainedasspecialcasesofEq. (1.8), themaindierencebeingthat now, since only nearest neighbors interact with each other, the RG flow creates bonds that were not existent before the decimation process (see Fig. 1.5). Assum- ing the biggest coupling occurs between spins 2 and 3, a new link ˜ J 14 is created between spins 1 and 4, which according to Eq. (1.8) is given by ˜ J 14 = Y _____] _____[ J 12 J 34 J 23 (XX-model,J z ij =0), J 12 J 34 2J 23 (Heisenberg model,J x ij =J z ij ). (1.11) Here, the superscript x has been eliminated due to its redundancy. We will not get into the details of Fisher’s derivation here (see Refs. [19, 25]), but it is worth noting that, due to their multiplicative nature, the SDRG rules in Eq. (1.11)can be turned into sum rules via logarithmic variables, which in turn allow to write an integro-dierential equation for the probability P(J,) of obtaining a coupling J between to spins at energy scale © max(J) 4 . The fixed point solution to this integro-dierential equation is the IRFP distribution P(J,) = 1 3 J 4 (1/ ≠ 1) ( ≠ J), (1.12) whosewidth ©≠ ln() growsmonotonicallywitheveryRGstep,sincetheenergy scale isalwaysdecreasinginaccordancewiththerulesgiveninEq. (1.11)(hence the "infinite randomness" adjective). In fact, for an infinite system æŒ at the fixed point and the distribution becomes singular. Moreover, this singularity is 4 To be more precise, the integro-dierential equation was written in terms of the logarithmic variables ’ © ln( /J) and= ≠ ln() . 11 responsible for the vanishing of the probability of obtaining a couplingJ>c withc< 1,ascanbeseenfrom lim æŒ ⁄ c P(J,) dJ = lim æŒ 1≠ c 1/ =0. (1.13) This important feature is what makes the transformation rules in Eq. (1.11)exact at the fixed point [25], i. e., the SDRG flow becomes asymptotically exact and the RSS describes the ground state of the system accurately. Figure 1.5: When the biggest coupling occurs between sites 2 and 3 (double line), anewbond ˜ J 14 iscreatedbetweenspins 1and 4aftertheSDRGstepisperformed. 1.4 DisorderedSystemswithlongrangeInterac- tions: A short Overview The number of published works about SDRG applied to disordered systems with longrange interactions is still very limited (seeSection III of thelatest review by F. Iglói and C. Monthus in Ref. [26]). Here we will discuss shortly the results byR.Juhász et al. inRef. [29]regardingtheRandomTransverse-fieldIsingModel (RTIM). Out of all current literature on long range systems, this is the one closest 12 to our model. Another interesting study by R. Juhász considers the RTIM with stretched exponential interactions of the form J ≥ exp(≠ cr a ) and finds that only for 0<a< 1/2, the system’s critical behavior is controlled by infinite-disordered fixed points dierent in nature from those of the know short range IRFP [ 28]. However, we will not discuss it further since it does not allow true long range couplings by its own nature. 1.4.1 RTIM with long range Interactions The Random Transverse-field Ising Model with long range ferromagnetic inter- actions described in terms of the Pauli matrices by the Hamiltonian H =≠ ÿ i”=j b ij r – ij ‡ x i ‡ x j ≠ ÿ i h i ‡ z i , (1.14) is studied via SDRG in Ref. [29]. Here, the disorder comes from the parameters b ij and the fields h i , which are positive, independent random variables drawn from some distributions p 0 (b) and q 0 (h). This way of introducing disorder is signifi- cantly dierent from the one in our model, where the spins themselves are placed randomly over an empty lattice and changing the power – aects the distribution of couplings, i. e., the amount of disorder introduced in the system. In contrary, varying the power – in Eq. (1.14) simply changes the interaction range, without modifying the probability distributions. The way the authors get around the issue of having long range couplings is by using a maximum rule in the following way: if at a given RG step, the energy scale is given by coupling J ij , the bond is eliminated and a new cluster with moment ˜ µ = µ i +µ j is formed and experiences an eective magnetic field ˜ h = h i h j /J ij . The coupling between this new cluster and spin (cluster) k is given by 13 ˜ J ij,k = max(J ik ,J jk ). If, on the other hand, the energy scale is controlled by a local field h i , spin i is decimated and all couplings are renormalized following the rule ˜ J jk = max(J ji J ik /h i ,J jk ). The authors also claim that this approximation becomes asymptotically exact at the IRFP. By using these rules they numerically found that for b ij and h i uniformly dis- tributed over the intervals (0,1] and (0,h], respectively, the distribution of exit gaps has a power law tale of the form g( ˜ h)≥ ˜ h 1/z≠ 1 , (1.15) where z is the dynamical exponent and the exit gap in a given realization is given by 2 ˜ h, two times the last decimated field. At criticality, ◊ © ln(h)¥ 1, they found a critical z c ¥ – , but they also found Eq. (1.15) to hold for the paramagnetic Griths phase ( ◊>◊ c ) with z(◊ )<z c . Given that the dynamical exponent remains finite, they conclude that the critical behavior is controlled by a finite disorder fixed point instead of an IRFP. The authors also present a simplified model that allows them to solve the SDRG flow equations exactly. In this model the couplings stop being random with b ij =1 for all spin pairs and the disorder comes uniquely from the transverse fields. This approximation makes sense, since, as the authors noted that almost always the transverse fields were being decimated out keeping the same couplings through the maximum rule, i. e., ˜ J jk =J jk . In addition, the extension of the non- decimated clusters were typically smaller than the distance between them. In fact, the analytic solution of this simplified model agrees with their numerical findings, justifying the simplified model as a good way to study the fully random system [29]. 14 Chapter 2 Magnetic Properties and Localization The main results discussed in this chapter are part of a collaborative process and have been published in Phys. Rev. B [42]. In Section 2.1, we consider the magnetic susceptibility of random spin chains in the context of SDRG. In particular, we calculate numerically the low temperature magnetic susceptibility of the XX-model with long-range, power law, interactions (› æŒ in Eq. (1.2)) and we study briefly the eect of including an exponential screening in the coupling strengths (finite › ). The RG evolution of the couplings distribution width is also computed as an eort to clarify the nature of the fixed point occurring at large values of – . Section 2.2 is dedicated to the exit gap distribution function of the XX-chain and its relation to the localization state of the system. 2.1 Magnetic Susceptibility The low temperature magnetic susceptibility ‰ (T) can be calculated via ‰ (T)à n FM (T) T , (2.1) where n FM (T) is the concentration of free paramagnetic moments at temperature T. In the context of SDRG, n FM (T) is the fraction of spins that have not formed 15 a singlet at RG energy scale , which in turn can be directly associated with the temperature, i. e., = T when setting the Boltzmann constant to unity [8]. Therefore, the numerical computation of ‰ (T) consists in the iteration of the SDRG rules in Eq. (1.8). At each RG step we record the energy scale= T and the number of remaining spins to get n FM (T) and calculate ‰ (T) by means of Eq. (2.1). This process is then repeated for all realizations of bare couplings (≥ 20000) to obtain the average susceptibility. Figure 2.1: Magnetic susceptibility, normalized by ‰ 0 © ‰ (T = J 0 ), for an XX- chain with N =1280 randomly placed spins, interacting via antiferromagnetic long-range couplings given in Eq. (1.2), chain length L/a =100N, and – = 0.6,...,2.0. Thecutolengthissetto › =Œ . Continuouslines: fitsto‰ ≥T 1/z≠ 1 with finite z at low temperatures. Inset: susceptibility for – =10.0 along with the same power law fit with z=9.36 (green line) and the IRFP result (red line). 16 In Fig. 2.1 we show numerical results for the susceptibility of the long-ranged, › =Œ ,XX-spinchain. Notethatthelowesttemperaturescalethatcanbereached for the finite system size L is of the order of T min =J min /k B =J 0 (L/2a) ≠ – , which explains why the data for dierent values of – terminates at dierent values of T/J 0 . At low temperatures, we can see a power law behavior, which appears linear on a double logarithmic scale, consistent with a finite dynamical exponent z as would become clear in the following discussion. In each RG step, a fraction dn FM /n FM () of the remaining spins at renormal- ization energy are taken away. Since this is due to the formation of a singlet with coupling J = , this fraction should equal 2P(J = ,) d , leading to the dierential equation dn FM d =2P(J= ,) n FM () , (2.2) where P(J,) is the probability distribution of couplings J at a given renormal- ization energy [19]. As discussed in Section 1.3, at the IRFP this distribution is given by Eq. (1.12) which in conjunction with Eq. (2.2) turns into the simple dierential equation dn FM d = 2 n FM , (2.3) whose solution is n FM () ≥ 1/ln 2 () , where the substitution= ≠ ln() has been made. This, in turn, yields the magnetic susceptibility at the IRFP via Eq. (2.1) ‰ IRFP (T)≥ 1 T ln 2 (T) , (2.4) as obtained in Ref. [19]. However, if we consider a finite disorder fixed point, i.e., we keep finiteandfixed, thesolutionofEq. (2.2)isn FM (T)≥T 2/ =T 1/z , with 17 the dynamical exponent z© /2 1 . In this case, Eq. (2.1) gives rise to a power law behavior for the low temperature susceptibility of the form ‰ (T)≥T 1/z≠ 1 , (2.5) consistent with our numerical results shown in Fig. 2.1 for z = z(– ), a mono- tonically increasing function of – that can be extracted by linear regression fits of the susceptibility in a logarithmic scale (continuous lines). Ifz> 1, the magnetic susceptibility diverges as T æ 0, with an anomalous power – m =1≠ 1/z < 1 that also grows with – . In the regionz< 1, this power becomes negative and we have a vanishing susceptibility at zero temperature, consistent with the formation of a pseudo-gap in the density of states fl (‘), since n FM (T)à s T 0 fl (‘)d‘. A similar behavior has been observed previously in Refs. [8] and [6]. The crossover value z=1, where the susceptibility saturates to a constant, occurs at a given – = – ú , which from Fig. 2.1 can be concluded to be somewhere between 1.0 and 1.2. Now, assuming a linear dependence on – of the form z = 1≠ b – ú – +b, (2.6) we find this crossover value to be – ú =1.066 ± 0.002 by a linear regression fit of z(– ) as shown in Fig. 2.2 (dashed black line), where the error only includes the fitting uncertainty. All values of z used for the fit are found by fitting the low temperature susceptibility curves to Eq. (2.5) as it is done in Fig. 2.1 for – =1.2,1.6, and 2.0. 1 Thefactoroftwodierencebetween andz comesfromthefactorinEq. (2.2)introducedto take into account the fact that at each RG step two sites are being decimated. In the Disordered Quantum Ising Model studied in Ref. [19] this factor does not appear since only one site is decimated and hence= z. 18 Figure 2.2: Dynamical exponent z extracted by fitting the low-temperature sus- ceptibility in Fig. 2.1 to Eq. (2.5) as a function of the power – (squares). These numerical results are then fitted to Eq. (2.6) (dashed line), which allows us to extract the crossover value – ú =1.066±0.002. Another important feature is found in the inset of Fig. 2.1. Here, the suscep- tibility for – =10.0 is displayed along with the curve given by Eq. (2.5) with the value z=9.36 (predicted by Eq. (2.6)), together with ‰ IRFP (T) (Eq. (2.4)). A better agreement to the numerical results is obtained with the finite z curve, indicating the flow to a finite z fixed point and not to the IRFP, as it would occur if we were dealing with nearest neighbor interactions. We note that at very large – ∫ 10 we find a finite number of free moments even at the smallest renormalization energies which are accessible in the finite spin chain. In our model, spins are randomly placed in a very diluted lattice, a situation in which nearest neighbor distances bigger than one lattice spacing a is 19 highlyprobable(seeFig. 2.3). Therefore,atverylargevaluesof– ,giventhepower law nature of the coupling strengths, one starts with an initial distribution P(J) heavily weighted near J =0, which might explain the above mentioned residual free moments and why a flow to the IRFP might not occur for large values of – , in contrary of what one might expect. Figure 2.3: In this particular realization of the random placement of the spins, spin i can be practically isolated from the rest of the chain for – ∫ 1 and still contribute to the low temperature magnetic susceptibility. 2.1.1 WidthofCouplingsDistributionFunctionduringthe RG flow Another way to investigate whether there is a strong disorder fixed point with a finite dynamical exponent z(– ) for finite – ∫ 1, or a transition to the IRFP at a specific finite power – IR , is to numerically inspect the evolution of the logarithmic width of the distribution function of couplings P(J) during the RG flow, W © 1 Èln 2 (J)Í≠È ln(J)Í 2 2 1/2 . (2.7) According to Eq. (1.12), when the system flows to the IRFP, this distribution gets wider at every RG step, since W == ≠ ln() increases monotonically as is always being lowered during the RG flow 2 . However, as shown in Fig. 2.4, our 2 In this context, the name Infinite Randomness Fixed Point becomes an obvious choice. 20 systemdoesnotfollowthistrendfor – ∫ 1(seeinset). Instead, thewidthisfound to saturate to a constant value after a non-monotonic transient behavior, which is a strong indication of a finite z fixed point. It is worth noting, that given the large number of couplings present in our system(N(N≠ 1)/2beforeanyrenormalizationisperformed), wehaveonlypicked the largest coupling to every spin in order to approximate P(J),. The width of this approximate distribution is denoted by W 1 . Figure 2.4: WidthW 1 of the distribution function of nearest-neighbor couplings as a function of the fraction of remaining spins. The negative logarithmic in base two is used to have an equally spaced horizontal variable that grows as the number of spins decreases by one half. All values have been normalized by the width W 0 1 (– ) oftheinitialdistributionandtheparametersarekeptasinFig. 2.1. Inset: W 1 /W 0 1 for a simple nearest neighbor model with uniformly distributed couplings (crosses) known to flow to the IRFP. The numerical results for – =20.0 (black circles) are included for comparison purposes. 21 2.1.2 Eect of the Exponential Cuto The eect of turning on a finite cuto › is briefly studied in this subsection. The appearance of a finite cuto can be thought to be caused by the finite electron localization length when the magnetic moments are surrounded by an electronic system. First, we keep the power – fixed to 1.0 and change › as shown in Fig. 2.5. There we see that the magnetic susceptibility diverges for›<L/ 4 and that at low temperatures we observe a power law behavior, which indicates a finite disorder (finitez)fixedpoint. Increasingthecutoto ›>L/ 4,resultsinalow-temperature suppression of the magnetic susceptibility, clearly demonstrating the opening of a pseudo gap as the range of the interaction increases. Figure 2.5: Log-log plot of the magnetic susceptibility for – = d =1 and ›/L =1,1/2,...,1/64. The remaining parameters are kept as in Fig. 2.1, and the susceptibility is rescaled in the same manner. 22 Figure2.6: Logarithmicplotofthesusceptibilityforafixedandfinitecutolength › = L/32 and – =0.6,0.8,...,2.0. The remaining parameters are kept as in Fig. 2.1. For fixed, small › = L/32 we observe in Fig. 2.6 that at suciently low temperatures, after some transient behavior, the magnetic susceptibility recovers a power law divergence consistent with finitez> 1. As expected, due to the presence of a finite › , this divergence is faster for every – when compared to the pure power-law couplings model, i.e., z(–,› = L/32)>z(–,› =Œ ).Infact,for – =0.6,0.8, and 1.0, we still observe a divergence in ‰ (T) at finite › , in contrast with the results shown in Fig. 2.1 for › =Œ , in which for these same values of – we found z(– ) to be smaller than unity, implying the existence of a pseudo gap. 23 2.2 Distribution Function of Excitation Energies and the Delocalization Transition Inthissectionweareinterestedindeterminingthelocalizationstateofthelong- ranged XX-model as a function of – ,at › =Œ , and whether or not it is correlated with the magnetic properties discussed above. In other words, we would like to know if the value of – ú that separates a vanishing susceptibility at T =0 from a diverging one, coincides with some – c that separates localized from delocalized chains. It is worth noting, that the existence of this delocalization transition is not trivial, and it might as well occur that the system does not goes through such transition. However, from the form of the interactions in Eq. (1.2), the system can be expected to be localized for large values of – , and delocalized when – is decreased. As discussed in Section 1.2, the Strong Disorder RG technique leads us directly to an approximation of the excitation gap of the system that becomes asymptot- ically exact for short range systems at the Infinite Randomness Fixed Point. It is also well known that the statistical properties of the spacing between energy levels gives information about the localization state of the system [55, 56, 51, 12]. Intuitively, this can be understood from the fact that delocalization causes level repulsion due to the correlations between all parts of the system, which means the probability to obtain a null gap should be zero in this case. On the other hand, when the system is completely localized, the energy levels cross each other and 24 this probability should be finite and maximal. Then, when put only in terms of the exit gap ‘ 1 , this translates to lim ‘ 1 æ 0 P(‘ 1 ) Y ___] ___[ =0 for delocalized systems (level repulsion), > 0 for localized systems (level crossing). (2.8) In Refs. [21, 30, 29, 32], the distribution of excitation gaps for the transverse Ising model was found to be P W (‘ 1 )= u 1/z 0 L z ‘ 1/z≠ 1 1 exp(≠ (u 0 ‘ 1 ) 1/z L), (2.9) where u 0 is a nonuniversal constant and the average excitation energy scales with system size L as È‘ 1 Í = (1+ z) u 0 L ≠ z3 . Here we claim that this distribution, also known as the Weibull distribution [45], should also be valid for the XX-model. This can be argued by noticing that the power law in Eq. (2.9) resembles that of P(J) in Eq. (1.12) for finitez. This comes from the fact that ‘ 1 ≥ , which means the probability of obtaining a gap ‘ 1 should be proportional to P(J æ ‘ 1 ). Now, taking into account that after the RG flow, the eective coupling that gives rise to the exit gap can be found in≥ L dierent positions of the chain, the exponential factor in Eq. (2.9) arises naturally, keeping in mind that the average value of ‘ 1 should scale like L ≠ z [30, 20]. We notice that the distribution in Eq. (2.9) exhibits level repulsion forz< 1 and level crossing forz> 1. Therefore it can properly describe localized and delo- calized systems depending on the value of the dynamical exponent z.Moreover, this transition occurs at z =1, the same value at which the pseudo gap opens. 3 In Ref. [20], D. Fisher showed that for the transverse Ising model,È‘ 1 Í≥L ≠ z in conjunction with a power law average susceptibility of the form presented in Eq. (2.5). 25 Thus, if the scaling scheme of the SDRG holds at the delocalization transition, z(– c )=1 and – c = – ú , signaling that the crossover in the magnetic susceptibility would coincide with the delocalization transition. Figure 2.7: Distribution of the lowest excitation gap ‘ 1 scaled by its mean value, s = ‘ 1 /È‘ 1 Í,for › =Œ and – =0.6,0.8,...,2.0. The remaining parameters are as in Fig. 2.1. In Fig. 2.7 we show the distribution function of the lowest excitation energy ‘ 1 using the logarithmic variable x = ≠ ln(s) (where s = ‘ 1 /È‘ 1 Í) in the limit of long-range interactions, i.e., › æŒ . The continuous black curves correspond to fits to the Weibull distribution in Eq. (2.9) multiplied by the cuto function introduced in Ref. [41], f c (x) = exp(c/(x≠ x max )), (2.10) 26 where c is a constant. This cuto function is included in order to account for the fact that at finite size L, when periodic boundary conditions are applied, we have a maximum value x max =≠ ln(‘ min /( – )) arising from the minimal energy scale ‘ min =(1/2)J 0 (L/6a) ≠ – . Here, the factor 6 is included since the numerical data is obtained at the third to last RG step as an eort to minimize the eect of the sharp cuto due to the finite size of the system. We found c =16 to work for all fits independent of the value of – , while u 0 was freely changed for each curve. The fact that the values of z(– ) obtained from the susceptibility data, used in conjunction with Eq. (2.9), can represent fairly well the numerical data for the excitation energy, should not be taken for granted. Eqs. (2.5) and (2.9) are derived using dierent arguments, and it could well occur that the values of z(– ) shown in Fig. 2.2 would not fit properly the numerical excitation gap distributions. Since they indeed do, as can be seen in Fig. (2.7), we can conclude that forz< 1 the existence of the pseudo-gap in the density of states and the level repulsion dictated by Eq. (2.9) occur simultaneously. Therefore, we can reuse the results obtained via the analysis of Fig. 2.2 and confidently claim that, indeed, the delocalization transition occurs at the same value at which the pseudo-gap appears, i.e., – c = – ú =1.066±0.002 and we see a clear correlation between the magnetic properties and the localization state of the system. At this point it is important to recall and compare the results by Juhász et al. on the ferromagnetic RTIM published in Ref. [29] and discussed previously in Section 1.4. Their main finding was that at criticality z RTIM = – , whereas away from criticality z RTIM <– . In contrast, we find z¥ – to hold both away and at criticality, where z c ¥ – c =1.066±0.002. On the other hand, as the power – = d=1 corresponds to the typical decay of the RKKY coupling in a 1D electron system in the metallic regime, we may 27 conclude from the results in Fig. 2.1 that the magnetic susceptibility due to the randomly coupled magnetic moments decays to zero in the metallic regime. 2.2.1 Continuous Chain Here we discuss the main results obtained in a previous collaborative work publishedinEurophys. Lett. [41],whereasimilarstudyoftheexitgapdistribution was made for a continuous chain, i. e., in this continuous model, the interactions were still given by Eq. 1.2 with › æŒ , but the spins were placed uniformly distributed in a continuous chain of length L=10N, allowing the position to be any real number between 0 and L, in contrast to the lattice model that has been discussed throughout this work. Figure 2.8: Wigner (black), Poisson (blue), and critical (red) distributions as a function of≠ log(s) for visualization purposes. 28 The localization state of the system and the critical point are determined via dierent distribution functions resembling the approach of Cuevas in Ref. [ 12]. Since our system is invariant under time reversal, delocalization and its character- istic level repulsion can be described by the Wigner surmise [55, 56] P WS (‘ 1 )= fi 2 (‘ 1 /)exp 3 ≠ fi 4 (‘ 1 /) 2 4 , (2.11) where ©È ‘ 1 Í. The level crossing found in localized systems can be modeled via a Poisson level distribution P P (‘ 1 ) = exp(≠ ‘ 1 /) . (2.12) As the value of the power – increases, we then expect the distribution to move from the former to the latter going through a critical distribution of the form P C (‘ 1 )=4(‘ 1 /)exp( ≠ 2‘ 1 /) , (2.13) which determines the critical point of the delocalization transition [12]. This pro- cedure to determine the localization state can be schematically visualized in Fig. 2.8, where these three distributions are plotted as a function of the random vari- able≠ log(s), withs = ‘ 1 / being the exit gap in units of its mean value . Here, we can see how the distribution of a delocalized system plotted as a function of ≠ log(s) is sharper and narrower than that of a localized system which would have a distribution closer to Poisson (blue line). In Ref. [41] we used SDRG to obtain the exit gap distributions of a continuous XX chain with N =320 spins for various values of – , resorting to ≥ 150000 realizations of the initial bare couplings. The results are plotted in Fig. 2.9 along 29 with the Wigner (continuous black curve), Poisson (continuous blue curve), and critical (continuous red curve) distributions, multiplied by the same cuto given in Eq. (2.10) with c=8 4 . The critical distribution fits the data for – =1.6 best, which allows us to conclude that in the continuous XX-chain the delocalization transition occurs at – c =1.6. Additionally, the Wigner surmise fits best the data for – =0.8 and the Poisson level distribution the data for – =2.2. Figure 2.9: Continuous XX-chain distribution of excitation gaps scaled by their mean value s = ‘ 1 /È‘ 1 Í,for › =Œ and – =0.4,0.8,...,2.2. The chain has a total ofN=320 spins and lengthL=10N. The continuous lines represent the Wigner (black), critical (red), and Poisson (blue) distributions which fit the data best for – =0.8. – =1.6, and – =2.2, respectively. It is worth mentioning that after these results were obtained, a transition to the lattice model was made. The lattice model is preferred since it brings the study closer to the physical reality of doped semiconductors, were dopant atoms 4 The analogous of Fig. 2.9 found in Ref. [41] (Fig. 4 there) has a dierent aesthetic style, so we have replotted the same data to fit the current style of this work. 30 are positioned randomly in the underlying lattice structure of the semiconductor substrate. Therefore, the continuous model will not be discussed further and the restofthisworkwillbeentirelydedicatedtothelatticemodeldescribedinSection 1.1. 31 Chapter 3 Entanglement entropy and spatial correlations in the SDRG context The present chapter is dedicated to the study of the entanglement of random spin chains in the context of Strong Disorder RG. In particular, we will focus on the concurrence between two spins of the chain (Section 3.1) and the entanglement entropy (Section 3.2) of the XX model. Previous studies on the entanglement entropy of short range models, which will also be discussed here, are all based on the assumption that the Random Singlet State (RSS) is a correct description of the ground state of the system [46, 24]. However, as it would become clear below, the RSS is insucient to calculate both concurrence and entanglement entropy in a reliable way when long ranged couplings are taken into account. 3.1 Concurrence Weareinterestedintheentanglementbetweenanytwospinsofthechainwhich can be quantified by the concurrence between them [22]. For a pure state of the chain |Â Í , the concurrence between spins m and n is given by C mn = |È |‡ y m ‡ y n | Í|, (3.1) the absolute value of the overlap between the original state and the state obtained after spins m and n have been flipped. For the XX-chain, since the Hamiltonian is 32 isotropic, the concurrence and the spin-spin spatial correlationÈS m ·S n Í hold the relationship C mn = 1 3 |ȇ m ·‡ n Í| = 4 3 |ÈS m ·S n Í|. (3.2) Therefore, we will not make any distinctions between the two when referring to their dependence on the distance l between spins. Instead, we will limit ourselves to talk about concurrence even though the majority of the papers cited in this section were concerned in calculating the spatial spin-spin correlation itself. As discussed in detail in Section 1.3, after the SDRG procedure, even when long range couplings are present, we obtain the Random Singlet State (RSS) as an approximation of the ground state of the system, an approximation that becomes asymptotically exact for short range models. This RSS can be written in the form | RS Í = p {i,j}œ RS |0 ij Í, (3.3) where |0 ij Í=(|ø i ¿ j Í≠|¿ i ø j Í)/ Ô 2 is the singlet state between spins i and j, and the direct product goes over all singlets forming the Random Singlet State. From Eq. (3.3)itbecomesapparentthatwhenthesystemisintheRSS,theconcurrence between two spins is given by C mn = Y ___] ___[ 1 if n and m form a singlet in the RSS, 0 otherwise. (3.4) This underlying feature of the RSS clearly represents an obstacle in properly describing the more subtle properties arising in the ground state of the system when long range interactions are present. In such case, due to their existing weak couplings, a finite amount of entanglement prevails even between spins that do 33 not form a singlet in the RSS. Moreover, the spins that do form a singlet after the SDRGprocedure,shouldnotbemaximallyentangledbetweeneachother,sincethe weaker interactions with the rest of the chain should reduce their mutual quantum correlations. Before proceeding further into a solution to this issue, it is worth making a short review of the known results for the scaling behavior of the mean and typ- ical concurrence at the IRFP, which were calculated by D. Fisher in the same paper where he presented the IRFP as an asymptotically exact fixed point of the Dasgupta-Ma SDRG procedure [19]. 3.1.1 Mean and typical Concurrence at the IRFP AsD.FishercleverlynoticedinRef. [19], themeancorrelationatlongdistance l isdominatedbyrareevents. Typically,twodistantspinsmandn =m+l willnot form a singlet and will be very weakly correlated. However, in the rare event that they do form a singlet in the RSS, they will be strongly correlated and therefore will dominate the mean concurrence behavior at large distances. As a result, the mean correlation function must be proportional to the total number of singlets formed at length scale l. Now, since the probability of m and n = m+l forming a singlet is proportional to the probability that they have not been decimated yet at energy scale l , n FM ( l ) 2 , we obtain C(l)© C mn ≥n FM ( l ) 2 ≥ 1 l 2 . (3.5) Here, the fact that the average density of free spins is inversely proportional to the average length scale at l , i. e., n FM ( l )≥ l ≠ 1 , has been used. As D. Fisher also noted, the mean correlations decay faster when disorder is introduced, since 34 in the clean case they decay asl ≠ 1/2 . Later, Hoyos et. al., found the prefactor that should be present in Eq. (3.5)tobe 1/3 for the XX-chain and explicitly noted that C(l)=0 for even l/a, which allows an equality of the form C(l)= 1 3 3 a l 4 2 ◊ Y ___] ___[ 1 if l/a is odd, 0 if l/a is even, (3.6) to be written [24] 1 . Duetothedominanceofrareeventsinthecalculationofthemeanconcurrence, we expect a dierent behavior at large length scales for the typical value. The typical concurrence is then proportional to the typical value of J/ l which can be calculated via the IRFP distribution in Eq. (1.12)[19, 24], i. e., C typ (l)≥ exp A ⁄ l 0 ln 3 J l 4 P(J, l )dJ B , ≥ exp A 1 l 1/ l l ⁄ l 0 J 1/ l ≠ 1 ln(J)dJ B ≠ 1 l . (3.7) Since the integral gives exactly≠ l 1/ l l ( l ≠ ln( l )) =≠ 2 2 l 1/ l l , we obtain C typ (l)≥ l , (3.8) were we used the definition l ©≠ ln( l ). Now, to find the relationship between length and energy scales, we note that l ≠ 1 ≥ n FM ( l )≥ 1/ln 2 ( l ) (see Eq. (2.3) 1 Since in Ref. [24] they calculate the correlation function instead of the concurrence, the prefactor appearing here has been adjusted in accordance with Eq. (3.2). 35 and its following discussion). Solving for l , we obtain the scaling behavior of the typical concurrence as an extended exponential C typ (l)≥e ≠ k Ô l , (3.9) where k is a nonuniversal constant of order unity [19]. It is then clear that the typical value decays significantly faster than the mean as one would expect, since, as we know from the discussion above, the later is dominated by the rarely found singlets at large distances. 3.1.2 Corrections to the Random Singlet State Let us now move forward into calculating the concurrence when long range interactions are included and the RSS, as shown in Eq. (3.4), does not give an appropriate answer. As mentioned above, the weaker long range couplings occur- ring between spins that do not form a singlet are responsible for a non-zero con- currence in such cases. Therefore, to find corrections to the RSS, perturbation theory on these weaker couplings comes naturally as a solution using an eective Hamiltonian ˜ H = ÿ {ij}œ RS ˜ J ij 1 S x i S x j +S y i S y j 2 ¸ ˚˙ ˝ ˜ H 0 + ÿ {ij}/ œ RS ˜ J ij 1 S x i S x j +S y i S y j 2 ¸ ˚˙ ˝ ˜ H Õ , (3.10) that can be written after the RSS has been obtained and whose couplings ˜ J ij are taken when one of the spins i,j has been decimated during the RG procedure. The”unperturbed"Hamiltonian ˜ H 0 containstheinteractiontermsofthespinpairs that form a singlet in the RSS, while ˜ H Õ consist of all other possible interactions between all spin pairs that did not form a singlet. Consequently, up to first order 36 in the couplings included in ˜ H Õ , we can write the corrected ground state of the XX-chain as |Â Í = | RS Í+ ÿ — È — | ˜ H Õ | RS Í E RS ≠ E — | — Í, (3.11) where the sum runs over all excited states of ˜ H 0 denoted with the index — , which correspond to all combinations of direct products of triplet and singlet states of spins pairs formed in the RSS. We note that ˜ H Õ | RS Í = 1 4 ÿ {nl}”={mk} (J nm +J lk ≠ J nk ≠ J ml )(|+ nl Í|≠ mk Í+|≠ nl Í|+ mk Í) p {ij}”={nl} ”={mk} |0 ij Í, (3.12) where |± nl Í©| (S=1,M = ±1)Í are two of the triplet states and the notation {ln} implies that l and n form a singlet in the RSS, i. e., the double sum and the direct product run over all singlet pairs in the RSS with the exceptions specified under the summation and direct product signs 2 . From this result, it becomes apparent that the only excited states that contribute to the sum in Eq. (3.11)are of the form | — Í = |± nl Í|û mk Í p {ij}”={nl} ”={mk} |0 ij Í, (3.13) whose energy dierence with respect to the RSS (see Eq. ( 1.6)) is given by E RS ≠ E — =≠ (J nl +J mk )/2. With this in mind, Eq. (3.11) transforms into the final form of the corrected ground state |Â Í =c| RS Í ≠ c 2 ÿ {nl}”={mk} J nm +J lk ≠ J nk ≠ J ml J nl +J mk (|+ nl Í|≠ mk Í+|≠ nl Í|+ mk Í) p {ij}”={nl} ”={mk} |0 ij Í, (3.14) 2 This notation will be used throughout this section to simplify the long expressions involved. 37 with a normalization constant c given by c = Q a 1+ 1 2 ÿ {nl}”={mk} 3 J nm +J lk ≠ J nk ≠ J ml J nl +J mk 4 2 R b ≠ 1/2 . (3.15) It is worth noting that all states included in |Â Í are orthogonal to each other. Afterasomewhatlengthybutstraightforwardcalculation,wefindaconditional expression for the concurrence calculated with the corrected ground state in Eq. (3.14) C pq = Y ____] ____[ c 2 - - - - - J pq +J rs ≠ J pr ≠ J qs J ps +J qr - - - - - if {pq} / œ RS, 1≠ c 2 2 q {mk}”={pq} A J pm +J qk ≠ J pk ≠ J mq J pq +J mk B 2 if {pq}œ RS, (3.16) where the indexes r and s in the first line of the equation correspond to the spins that form a singlet in the RSS with spins q and p, respectively, as pictured in Fig. 3.1 . Eq. (3.16) has all the properties expected for the concurrence between two spins in a chain with long range couplings. It does not only give a non-zero value for pairs that do not form a singlet in the RSS, but it also gives a concurrence smaller than 1 for spins that do 3 . Numerical results for the mean concurrence of the long range XX-chain with N =800 sites, obtained via Eq. (3.16), are shown in Fig. 3.2. First, we note that, in contrast with the short range case, there is a finite concurrence for even values of l. This is expected by looking at the form of Eq. (3.16) and recalling that C(l)=0 for even l was due to the impossibility of crossing singlets in the 3 Notethatwiththegiventhedefinitionofc, theconcurrencefor {pq}œ RS isalwayspositive. 38 Figure3.1: DiagramofthetwopossibilitiesoccurringinEq. (3.16). Thetopfigure shows the case {pq} / œ RS, while the bottom one depicts the case {pq}œ RS. RSS. However, there is still a clear dierence between even values of l (bottom) and odd ones (top), as indicated by the formation of dierent sets of curves. Both sets of curves have a low dependence on – and a power law regime in which C≥l ≠ “ e,o , (3.17) where “ e,o are the decay powers for even and odd values of l, respectively. In fact, by using linear regression fits in the logarithmic scale for the – =0.6 and – =2.0 curves (see continuous black curves in Fig. 3.3), we find an average decay represented by “ e =1.75±0.04 and “ o =1.95±0.04, whose central values are used in conjunction with Eq. (3.17) to create the dashed black lines in Fig. 3.3. The most remarkable thing to notice is that, even with the corrections to the RSS, the concurrence for odd values of l is still dominated by rare events (singlets formed at long distances), as can be concluded from its roughly≥l ≠ 2 decay for all values of – . On the other hand, for even values of l, the decay is slower, the power law 39 Figure 3.2: Mean concurrence of a long range XX-chain with N =800 sites as a function of l = |p≠ q|, the dierence between spin sites numbers p and q. The couplings are true long range with › æŒ and – =0.6,...,2.0. A clear distinction between odd values of l (top curves) and even ones (bottom curves) can be observed. regimeissmaller,andtheoverallvalueshaveahigherdependenceon– asexpected from Eq. (3.16). For both sets of curves, a saturation at large values of l can be seen, but this is simply due to finite size eects. Another indication that the mean concurrence is dominated by rare events is that the typical value diers significantly from the mean as can be seen in Fig. 3.4. A clear power law behavior of the form C typ (l)≥l ≠ “ (– ) , (3.18) 40 Figure3.3: Separatecurvesforthemeanconcurrenceofeven(left)andodd(right) values of l. The continuous black lines represent linear regression fits to – =0.6 and – =2.0, while the dashed black line depicts c e,o l ≠ “ e,o , where c e,o and “ e,o have been obtained by averaging their corresponding values for – =0.6 and – =2.0. is found. Here, the power “ (– ) has a strong dependence on – , unlike the decay powers of the mean value. In fact, as can be seen in Fig. 3.5 , “ (– ) is linear in – . A simple linear regression fit gives “ (– )=1.02– +2.02, (3.19) which indicates that the typical concurrence decays faster than the mean for all values of – . It is also worth noting, that since the typical concurrence decays as a power law, it decays slower than in the IRFP case, where it has the extended exponential behavior stated in Eq. (3.9). This should not come as a surprise, since given the long range interactions, we expect spatial correlations to survive at longer distances. 41 Figure 3.4: Typical concurrence of a long range XX-chain with N =800 sites as a function of l = |p≠ q|, the dierence between spin sites numbers p and q. The couplings are true long range with › æŒ and – =0.6,...,2.0. A clear power law decay is observed for all values of – . 3.2 Entanglement Entropy The entanglement between two subschains A and B can be quantified by the von Neumann entropy of the reduced density matrix S =≠ Tr(fl A lnfl A )=≠ Tr(fl B lnfl B ), (3.20) where fl A = Tr B (| ÍÈ |) (fl B = Tr A (| ÍÈ |)) is obtained by partially tracing the complete density matrix of the system over all degrees of freedom of subchain B (A). 42 Figure3.5: DecaypowerofthetypicalconcurrenceplottedinFig. 3.4asafunction of– . Thedashedlinerepresentsthelinearregressionfitgiving“ (– )=1.02– +2.02. This entanglement entropy constitutes a very important quantity in the con- text of quantum phase transitions. For clean chains, it has been shown that at criticality, the entropy of a subchain A of length l scales as [53] S(l)= c+c 6 ln(l/a)+k, (3.21) where c and c correspond to the central charges of the corresponding 1+1 Con- formal Field Theory, and k is a nonuniversal constant [23]. This scaling behavior violates the area law, which predicts no dependence on the length of the subchain fortheone-dimensionalcase. Thearealawisrecoveredawayfromcriticality,where it has been found that the entropy saturates at large l [53, 23]. 43 InRef. [46],itwasshownbyRefaelandMoorethatEq. (3.21)alsoholdsforthe average entanglement entropy of disordered systems. In particular, using SDRG, they found that in the disordered Ising Model, the eective central charges where given by c = c = ln(2)/2, while in the Heisenberg and XX-chain c = c = ln(2). Both cases correspond to a factor of ln(2) dierence with the central charge of their analogous pure system found earlier by Vidal et al. in Ref. [53]. Refael and Moore’s method to calculate the average entanglement entropy in the presence of disorder is based on the assumption that the system has been drawn to the IRFP and the Random Singlet State is a correct representation of its ground state, an assumption that can be safely done when only nearest neighbor interactions are present. Since the RSS corresponds to a product state of maximallyentangledspinpairs, theynotethatthetotalentanglemententropycan be calculated by counting the number of singlets that cross the boundary between subsystemsAandB andthenmultiplyingthisnumberwiththeentropyofasinglet S 0 = ln(2) [46, 24]. A schematic representation of this method is found in Fig. 3.6 for a specific RSS. Here, when the boundary between A and B is defined by line a, we obtain an entanglement entropy of 4 ·S 0 , since four singlets cross over the boundary line. But if the boundary is defined by line b, the entanglement entropy is reduced to 2·S 0 , because in this case only two singlets have spins in opposites sides of the boundary between subsystems. The formula in Eq. (3.21) is specific for infinite systems. In Ref. [11], Cal- abrese and Cardy found analogous formulas for finite systems with, and without, periodic boundary conditions. When periodic boundary conditions are applied, the entanglement entropy of a chain with total length L has a scaling behavior of the form S pbc (l)= c 3 ln 3 L fia sin(fil/L ) 4 +k Õ , (3.22) 44 Figure3.6: SinglerealizationoftheRandomSingletStateillustratingtheentangle- ment entropy calculation for two dierent boundaries between subsystems. When the boundary is given by line a, the entropy is 4·ln(2). When its given by line b, a value of 2·ln(2) is obtained. where k Õ is again a nonuniversal constant and the assumption c =c has been done [11]. If the boundaries are kept open, and additional term related to the boundary entropy of Aeck and Ludwig ( g) appears [2], and the prefactor changes in such a way that the entanglement entropy is given by S obc (l)= c 6 ln 3 L fia sin(fil/L ) 4 +2g +k Õ . (3.23) Additionally, Hoyos et. al., by using Refael and Moore’s method described above, were able to relate the entanglement entropy at the IRFP with the average con- currence via S(l)=2S 0 Y ] [ l ÿ ls=0 l s C(l s )+l N/2 ÿ ls=l+1 C(l s ) Z ^ \ , (3.24) where S 0 = ln(2) and, in the specific case of the XX-chain, the concurrence C(l s ) is given by Eq. (3.6)[24]. Moreover, by taking the limit N ∫ 1 of Eq. (3.24), they were able to arrive at Eq. (3.21) with c = c = ln(2), recovering Refael and Moore’s results for the infinite XX-chain [46, 24]. 45 Figure3.7: AverageblockentanglemententropyforthelongrangedXX-chainwith N =1280 spins, › æŒ , and – =0.6,2.0 (blue and orange circles, respectively). The black dashed line labeled "Theory 1" corresponds to Eq. (3.22) with a central charge c = ln(2) and k Õ =0.4. The black dotted-dashed line labeled "Theory 2" depicts the entropy given by Eq. (3.24) with the concurrence given by Eq. (3.6). Figure 3.7 contains numerical calculations of the mean block entanglement entropy using Refael and Moore’s prescription illustrated in Fig. 3.6. The XX- chain has N =1280 spins with long range power law interactions (see Eq. (1.2) in the limit › æŒ ). As mentioned above, such prescription implies that the RSS is a good approximation of the actual ground state of the chain. From Chapter 2 we know that the long ranged XX-chain is not critical for either – =0.6 (blue circles)or– =2.0(orangecircles)andthereforewewouldexpecttheentanglement entropy to saturate to a constant in both cases. Instead, they behave in a similar fashion to Calabrese and Cardy’s formula in Eq. (3.22) with the known XX-chain central charge value c = ln(2) (and constant k Õ =0.4), as well as to Eq. (3.24) 46 which intrinsically assumes the same value of the central charge by setting the concurrence to its XX-chain value in Eq. (3.6). Therefore, the numerical results show that, even though we see a small variation with – , using the RSS as an approximation to calculate the entropy forces the results to be of a critical nature irrespective of the power – . As a consequence, the phase transition observed in Sections 2.1 and 2.2 can not be detected using the entanglement entropy without making corrections to the RSS. Naively, one would think of using the corrected state |Â Í in Eq. (3.14)to calculate the entanglement entropy beyond the RSS. However, given that |Â Í is not a product state, geometrical calculations similar to the counting of crossing singlets are not possible, since the entropy of a superposition state is not the sum of the individual entropies. Moreover, a closed formula using the definition in Eq. (3.20) is not feasible due to the dependence of the sums on the specific realization oftheRSS,whichmakestakingthepartialtraceinconceivablewithoutconsidering everysinglepossiblescenario, i.e., thereareasmanyoutcomesforthepartialtrace as there are possible Random Singlet States on the chain (≥N). Apossiblesolutiontothisproblemistomakeasomehowsevereapproximation to the corrected state in Eq. (3.14) and only take into account the term with the greatest coecient in the sum. We then write | ͥ c Q c c c a | RS Í+”J (|+ nl Í|≠ mk Í+|≠ nl Í|+ mk Í) p {ij}”={nl} ”={mk} |0 ij Í R d d d b , (3.25) where ”J = J nm +J lk ≠ J nk ≠ J ml J nl +J mk , (3.26) 47 is the maximum coecient appearing in Eq. ( 3.14), and the constant c needs to be redefined to c = 1 Ô 1+2”J 2 , (3.27) in order to keep the approximated state normalized. Within this approximation, after a lengthy but straightforward calculation, it is possible to arrive to a conditional closed form for the entanglement entropy that depends on the relationship between the boundary and the two converted singlet pairs {nl}and {mk}. Therearethreedistinguishingcasesthatgiverisetodierent expressions for the entropy as a function of the number of crossing singlets and triplets k: • Case 1: Each of the two converted singlets {nl} and {mk} are at opposite sides of the cut and none of them cross the boundary: S(k)=≠ c 2 ln A c 2 2 k B ≠ (1≠ c 2 )ln A 1≠ c 2 2 k+1 B . (3.28) • Case 2: Both converted singlets cross the boundary between subsystems (kØ 2): S(k)=≠ c 2 2 ln A c 2 2 k B ≠ c 2 4 (2”J +1) 2 ln A c 2 2 k (2”J +1) 2 B ≠ c 2 4 (2”J ≠ 1) 2 ln A c 2 2 k (2”J ≠ 1) 2 B . (3.29) • Case 3: Any other relationship between the converted pairs and the bound- ary, e.g., both pairs are part of the same subsystem or only one of them 48 crosses the boundary. In this case, the approximated state brings no correc- tion to the entropy, giving the same value obtained at the IRFP [46] S(k)=kln(2). (3.30) Figure 3.8: Entanglement entropy as a function of c 2 for the three instances that occur after approximating the corrected state to Eq. (3.25). As mentioned in the text, the entropy for Case 1 (dashed line) is always bigger than that for Case 3 (continuous line), except at c=1 where they coincide. In contrast, the entropy for Case 2 is always lower than the entropy at the IRFP, again, with the exception of c=1, where Eq. (3.29)goesto kln(2). The plot is done for the specific case k=3, but the conclusions drawn from it remain true for all values of k. First, we note that in the limit of no corrections (”J æ 0,cæ 1), we recover Eq. (3.30)frombothEq. (3.28) and Eq. (3.29), as one can anticipate. Moreover, as seen in Fig. 3.8 for the specific instance k =3,Case1(Eq. (3.28), dashed line) gives a higher entropy than the one at the IRFP (Eq. (3.30), continuous line). This is expected since the corrected state is a superposition of states that 49 dier only on spin pairs {nl} and {mk}, which live at opposite sides of the subsys- tem boundary, and therefore results in an increment of the quantum correlations between subchains. On the other hand, Case 2 (Eq. (3.29), dashed-dotted line) gives a lower entropy than that of Eq. (3.30), also for all values of c”=1. Again, this is expected due to the fact that the extra correction terms are destroying the RSS which in this case is the maximally entangled sate, given that both pairs cross the boundary. It is worth noting, that in order to plot the entropy in Eq. (3.29), Eq. (3.27)wasinvertedinordertoobtain”J (c 2 ), andthepositiverootwaschosen. However, since Eq. (3.29) is an even function of ”J , this choice becomes trivial. We observe that, even though the plot is only done for the a specific value of k =3, the above statements remain true for all values of k as can be easily inspected via Eqs. (3.28) and (3.29). Figure 3.9: Average block entanglement entropy calculated with the approximated state in Eq. (3.25) for a chain with N =800 spins. The power – is varied from 0.60 to 2.00 and Calabrese and Cardy’s formula (Eq. (3.22)) is plotted along for reference as a black continuous line. 50 Unfortunately, as seen in Fig. 3.9 for a chain of length N =800, the approx- imation done in Eq. (3.25) that gives rise to Eqs. (3.28-3.30) is not sucient to produce a significant dependence on the power – , and the entropy remains more or less critical. The numerical results (dashed lines), follow the same trend as the Calabrese and Cardy’s formula in Eq. (3.22) with identical parameters c = ln(2) and k=0.4 (continuous black line). Therefore, we can conclude that Cases 1 and 2, the two cases in which corrections appear, are not frequent enough throughout realizations to notably aect the average entropy. This means that our eorts to approximate the corrected state in Eq. (3.14) to obtain an – -dependent entangle- ment entropy were not sucient. In conclusion, even though the corrected state in Eq. (3.14), a product of including the weaker couplings by means of perturbation theory around the RSS, isusefultocalculatetheconcurrence,ithasproventobeunsuccessfulingettingthe numerical results of the average entanglement entropy out of criticality (behavior at the RSS) and has not showed a saturating entropy for non critical values of – . Therefore, a completely new approach is needed. Inthatdirection,acollaborativeeorthasfoundsomepreliminaryresultsusing Density Matrix Renormalization Group (DMRG) techniques that seem to detect the localization transition using the average block entanglement entropy, i. e., the – -dependenceandexpectedfeaturesdiscussedthroughoutthissectionareobserved [43]. 51 Chapter 4 Conclusion The main finding of the present work and the collaborative eort published in Ref. [42] is that the primary eect that long range interactions have on the antiferromagnetic XX-chain is to send the system into a finite disordered fixed point (z< Œ ), instead of the IRFP. This resembles the findings of Ref. [29]on the RTIM with ferromagnetic interactions. Additionally, we have found that, for z(– ) < 1, the system is delocalized and presents a pseudo-gap in the density of states that causes the magnetic susceptibility to vanish as T æ 0.Incontrast, in the regime z(– ) > 1, the system is localized and some magnetic moments remain free even at low temperatures causing an anomalous power divergence of the magnetic susceptibility, i. e., ‰ (T)≥T ≠ – m , with – m =1≠ 1/z < 1, just as it happensindopedsemiconductorsathighenoughdopantdensities[5,50,33,9,47]. The transition between this two phases occurs at z =1, which was found to correspond with – = – c =1.066±0.002. It is also worth mentioning that since in onedimensionalelectronsystemstheRKKYinteractiondecayswithapowerof– = 1, we may extrapolate our results and conclude that the magnetic susceptibility of such systems decays to zero when they are in the metallic regime. On the other hand, in Chapter 3, we made corrections to the Random Singlet State that allowed us to calculate the concurrence between two spins of the chain when long range interactions are present. It was found that the mean concur- rence will still decay approximately as l ≠ 2 for spins separated by an odd value of l/a, regardless of the interaction decay power – , just as in the short range 52 model. The only mayor dierence being that due to the long range interactions, the concurrence now has a finite value for even values of l/a and, in this case, it approximately decays as l ≠ 1.75 with a bigger dependence on – . An even more interesting fact is that the typical concurrence decays significantly slower than at the IRFP (power law vs. stretched exponential) with a highly – -dependent power “ (– )=1.02– +2.02. Additionally, we noted that using the Random Singlet State as an approximation to the ground state of the system, always leads to a critical blockentanglemententropyand,therefore,the– -inducedtransitionbetweenlocal- ized and delocalized states can not be detected in this way. An attempt to use the corrections to the RSS that were successful in computing the concurrence (with an extra approximation) was made. However, this attempt was not enough to show a significant dependence on – and draw the results out of criticality. There- fore, other approaches such as Density Matrix Renormalization Group need to be implemented in order to calculate the block entanglement entropy in a reliable way. 53 Reference List [1] A. M. Finkel’shtein, A. M. 46:513, 1987. [2] Ian Aeck and Andreas W. W. Ludwig. Universal noninteger “ground-state degeneracy” in critical quantum systems. Phys. Rev. Lett., 67:161–164, Jul 1991. [3] P.W.Anderson. Absenceofdiusionincertainrandomlattices. Phys. Rev., 109:1492–1505, Mar 1958. [4] P. W. Anderson. In S. Lundqvist, editor, Nobel Lectures in Physics 1971â1980 , page 376. World Scientific, Singapore, 1980. [5] K. Andres, R. N. Bhatt, P. Goalwin, T. M. Rice, and R. E. Walstedt. Low-temperature magnetic susceptibility of si: P in the nonmetallic region. Phys. Rev. B, 24:244–260, Jul 1981. [6] R. N. Bhatt. unpublished, 1982. [7] R. N. Bhatt. Magnetic properties of doped semiconductors. Physica Scripta, 1986(T14):7, 1986. [8] R. N. Bhatt and P. A. Lee. Scaling studies of highly disordered spin- antiferromagnetic systems. Phys. Rev. Lett., 48:344–347, Feb 1982. [9] R. N. Bhatt and T. M. Rice. Clustering in the approach to the metal-insulator transition. Philosophical Magazine B, 42(6):859–872, 1980. [10] Bhatt, R. N., Paalanen, M. A., and Sachdev, S. Magnetic properties of disordered systems near a metal-insulator transition. J. Phys. Colloques, 49(C8):C8–1179–C8–1184, 1988. [11] Pasquale Calabrese and John Cardy. Entanglement entropy and quantum field theory. Journal of Statistical Mechanics: Theory and Experiment, 2004(06):P06002, 2004. [12] Cuevas, E. Criticallevelspacingdistributioninlong-rangehoppinghamiltonians. Europhys. Lett., 67(1):84–89, 2004. [13] Chandan Dasgupta and Shang-keng Ma. Low-temperature properties of the random heisen- berg antiferromagnetic chain. Phys. Rev. B, 22:1305–1319, Aug 1980. [14] Francisco A. B. F. de Moura and Marcelo L. Lyra. Delocalization in the 1d anderson model with long-range correlated disorder. Phys. Rev. Lett., 81:3735–3738, Oct 1998. [15] Francisco A.B.F. de Moura and Marcelo L. Lyra. Correlation-induced metal-insulator tran- sition in the one-dimensional anderson model. Physica A (Amsterdam), 266(1):465 – 470, 1999. [16] AmitDuttaandR.Loganayagam.Eectoflong-rangeconnectionsonaninfiniterandomness fixed point associated with the quantum phase transitions in a transverse ising model. Phys. Rev. B, 75:052405, Feb 2007. 54 [17] Ferdinand Evers and Alexander D. Mirlin. Anderson transitions. Rev. Mod. Phys., 80:1355– 1417, Oct 2008. [18] Daniel S. Fisher. Random transverse field ising spin chains. Phys. Rev. Lett., 69:534–537, Jul 1992. [19] Daniel S. Fisher. Random antiferromagnetic quantum spin chains. Phys. Rev. B, 50:3799– 3821, Aug 1994. [20] Daniel S. Fisher. Critical behavior of random transverse-field ising spin chains. Phys. Rev. B, 51:6411–6461, Mar 1995. [21] Daniel S. Fisher and A. P. Young. Distributions of gaps and end-to-end correlations in random transverse-field ising spin chains. Phys. Rev. B, 58:9131–9141, Oct 1998. [22] Scott Hill and William K. Wootters. Entanglement of a pair of quantum bits. Phys. Rev. Lett., 78:5022–5025, Jun 1997. [23] Christoph Holzhey, Finn Larsen, and Frank Wilczek. Geometric and renormalized entropy in conformal field theory. Nuclear Physics B, 424(3):443 – 467, 1994. [24] José A. Hoyos, André P. Vieira, N. Laflorencie, and E. Miranda. Correlation amplitude and entanglement entropy in random spin chains. Phys. Rev. B, 76:174425, Nov 2007. [25] F. Iglói and C. Monthus. Phys. Rep., 412:277, 2005. [26] Ferenc Iglói and Cécile Monthus. Strong disorder rg approach – a short review of recent developments. The European Physical Journal B, 91(11):290, Nov 2018. [27] F. M. Izrailev and A. A. Krokhin. Localization and the mobility edge in one-dimensional potentials with correlated disorder. Phys. Rev. Lett., 82:4062–4065, May 1999. [28] R.Juhász. Infinite-disordercriticalpointsofmodelswithstretchedexponentialinteractions. Journal of Statistical Mechanics: Theory and Experiment, 2014(9):P09027, 2014. [29] R. Juhász, I. A. Kovács, and F. Iglói. Random transverse-field ising chain with long-range interactions. EPL (Europhysics Letters), 107(4):47008, 2014. [30] Róbert Juhász, Yu-Cheng Lin, and Ferenc Iglói. Strong griths singularities in random systems and their relation to extreme value statistics. Phys. Rev. B, 73:224206, Jun 2006. [31] Tadao Kasuya. A theory of metallic ferro- and antiferromagnetism on zener’s model. Prog. Theor. Phys., 16(1):45–57, 1956. [32] IstvánA.Kovács, RóbertJuhász, andFerencIglói. Long-rangerandomtransverse-fieldising model in three dimensions. Phys. Rev. B, 93:184203, May 2016. [33] M. Lakner and H. v. Löhneysen. Thermoelectric power of a disordered metal near the metal-insulator transition. Phys. Rev. Lett., 70:3475–3478, May 1993. [34] Patrick A. Lee and T. V. Ramakrishnan. Disordered electronic systems. Rev. Mod. Phys., 57:287–337, Apr 1985. [35] Igor V. Lerner. Dependence of the ruderman-kittel-kasuya-yosida interaction on nonmag- netic disorder. Phys. Rev. B, 48:9462–9477, Oct 1993. [36] L.S. Levitov. Critical hamiltonians with long range hopping. Annalen der Physik, 8(7- 9):697–706, 1999. [37] Milica MilovanoviÊ, Subir Sachdev, and R. N. Bhatt. Eective-field theory of local-moment formation in disordered metals. Phys. Rev. Lett., 63:82–85, Jul 1989. 55 [38] Alexander D. Mirlin, Yan V. Fyodorov, Frank-Michael Dittes, Javier Quezada, and Thomas H. Seligman. Transition from localized to extended eigenstates in the ensemble of power-law random banded matrices. Phys. Rev. E, 54:3221–3230, Oct 1996. [39] C. Monthus. Dyson hierarchical long-ranged quantum spin-glass via real-space renormaliza- tion. Journal of Statistical Mechanics: Theory and Experiment, 2015(10):P10024, 2015. [40] Mott, N.F. Impuritybandconduction.experimentandtheorythemetal-insulatortransition in an impurity band. J. Phys. Colloques, 37(C4):C4–301–C4–306, 1976. [41] N. Moure, S. Haas, and S. Kettemann. Many-body localization transition in random quan- tum spin chains with long-range interactions. EPL (Europhysics Letters), 111(2):27003, 2015. [42] N. Moure, Hyun-Yong Lee, S. Haas, R. N. Bhatt, and S. Kettemann. Disordered quantum spin chains with long-range antiferromagnetic interactions. Phys. Rev. B, 97:014206, Jan 2018. [43] N. Moure, Hyun-Yong Lee, S. Haas, R. N. Bhatt, and S Kettemann. Work in progress (unpublished), 2019. [44] M. A. Paalanen, J. E. Graebner, R. N. Bhatt, and S. Sachdev. Thermodynamic behavior near a metal-insulator transition. Phys. Rev. Lett., 61:597–600, Aug 1988. [45] A. Papoulis and S. U. Pillai. McGraw-Hill, Boston, 4 edition, 2002. [46] G. Refael and J. E. Moore. Entanglement entropy of random quantum critical points in one dimension. Phys. Rev. Lett., 93:260602, Dec 2004. [47] Michel Rosso. Hierarchy of exchange interactions in a disordered magnetic system. Phys. Rev. Lett., 44:1541–1544, Jun 1980. [48] M. A. Ruderman and C. Kittel. Indirect exchange coupling of nuclear magnetic moments by conduction electrons. Phys. Rev., 96:99–102, Oct 1954. [49] Subir Sachdev. Local moments near the metal-insulator transition. Phys. Rev. B, 39:5297– 5310, Mar 1989. [50] M. P. Sarachik, A. Roy, M. Turner, M. Levy, D. He, L. L. Isaacs, and R. N. Bhatt. Scaling behavior of the magnetization of insulating si:p. Phys. Rev. B, 34:387–390, Jul 1986. [51] B. I. Shklovskii, B. Shapiro, B. R. Sears, P. Lambrianides, and H. B. Shore. Statistics of spectra of disordered systems near the metal-insulator transition. Phys. Rev. B, 47:11487– 11490, May 1993. [52] Hilbert v. Löhneysen. Disorder, electron-electron interactions and the metal-insulator tran- sition in heavily doped si:p. In Bernhard Kramer, editor, Advances in Solid State Physics 40, pages 143–167, Berlin, Heidelberg, 2000. Springer Berlin Heidelberg. [53] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev. Entanglement in quantum critical phenom- ena. Phys. Rev. Lett., 90:227902, Jun 2003. [54] E. Westerberg, A. Furusaki, M. Sigrist, and P. A. Lee. Low-energy fixed points of random quantum spin chains. Phys. Rev. B, 55:12578–12593, May 1997. [55] Eugene P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Annals of Mathematics, 62(3):548–564, 1955. [56] Eugene P. Wigner. Characteristics vectors of bordered matrices with infinite dimensions ii. Annals of Mathematics, 65(2):203–207, 1957. 56 [57] Kei Yosida. Magnetic properties of cu-mn alloys. Phys. Rev., 106:893–898, Jun 1957. [58] Eddy Yusuf and Kun Yang. Random antiferromagnetic spin- 1 2 chains with competing inter- actions. Phys. Rev. B, 68:024425, Jul 2003. [59] ChenggangZhouandR.N.Bhatt. One-dimensionalchainwithrandomlong-rangehopping. Phys. Rev. B, 68:045101, Jul 2003. 57
Abstract (if available)
Abstract
Disordered quantum spin chains with long-range antiferromagnetic interactions are studied via the Strong Disorder Renormalization Group (SDRG) technique. This technique allows a computationally efficient calculation of the low temperature magnetic susceptibility and excitation energy distribution, two quantities that contain information about the localization state of the chain and allow the detection of a localization-delocalization phase transition induced by the change of the decay power of the long-range couplings. The critical decay power is found to be approximately unity. Additionally, a brief review of the SDRG technique is given, including the main results for short range models obtained by D. Fisher on his seminal 1994 paper, as well as the modifications that need to be introduced in order to study long-range interactions. Furthermore, the two-site concurrence and the block entanglement entropy are computed. Corrections beyond SDRG are proposed in order to overcome the limitations of the random singlet state, the ground state approximation obtained by SDRG that has been proven to successfully describe short-range models, but fails when long-range interactions are present. These corrections manage to improve the two-site concurrence calculation, but fall short when it comes to the block entanglement entropy, which instead of working as another indicator of the phase transition, has a critical behavior for all decay powers of the interaction strengths.
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Creator
Moure Gomez, Nicolas
(author)
Core Title
Disordered quantum spin chains with long-range antiferromagnetic interactions
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Publication Date
04/29/2019
Defense Date
03/20/2019
Publisher
University of Southern California
(original),
University of Southern California. Libraries
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Tag
Anderson localization,concurrence,disordered systems,entanglement entropy,magnetic susceptibility,OAI-PMH Harvest,quantum phase transitions,quantum spin chains,strong disorder renormalization group
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English
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Haas, Stephan (
committee chair
), Boedicker, James (
committee member
), Nakano, Aiichiro (
committee member
), Saleur, Hubert (
committee member
), Venuti, Lorenzo Campos (
committee member
)
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mouregom@usc.edu,nmouregr@gmail.com
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https://doi.org/10.25549/usctheses-c89-159208
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Moure Gomez, Nicolas
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University of Southern California Dissertations and Theses
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The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
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Tags
Anderson localization
concurrence
disordered systems
entanglement entropy
magnetic susceptibility
quantum phase transitions
quantum spin chains
strong disorder renormalization group