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Phase diagram of disordered quantum antiferromagnets

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Phase diagram of disordered quantum antiferromagnets

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Content PHASE DIAGRAM OF DISORDERED QUANTUM ANTIFERROMAGNETS by Kien Trong Trinh 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) December 2014 Copyright 2014 Kien Trong Trinh Acknowledgments I would like to express my deep gratitude to Stephan Haas, my Ph.D. advisor, for being a great teacher and friend. Stephan patiently provided the vision, encouragement and advice necessary for me to proceed through the doctoral program. He introduced me to the physics of strongly correlated systems, and without his insightful advice several critical stages of this work would not have been passed. I am also grateful to Stephan for sharing his thoughts about career and life. A lot of work presented in this dissertation was the result of numerous long-distance collaborations with Tommaso Roscilde and Rong Yu. I would like to thank them for sharing their insights and stimulating ideas. I am indebted to Tommaso for his patience and guidance during our work together on the spin-spin correlation paper. I would like to thank Professors Nelson Bickers, Aiichiro Nakano, Susumu Taka- hashi, and Christoph Haselwandter for serving on my dissertation committee. I wish to thank my friends Ming Chak Ho, Benjamin Gross, Nicholas Guggemos, Zachary Levine, Anirban Das, Rodrigo Muniz and Mohammad Vedadi for memorable discussions and for the joyful time that we spent together. I have gone over many cultural differences and integrated quickly into American life with the help of these great friends. I also enjoyed many discussions with my colleagues Lorenzo Campos Venuti, Nick Chancellor, Silvano Garnerone and Siddhartha Santra. I would like to thank all members of the Department of Physics & Astronomy at USC for providing an excellent work environment. The numerical simulations for this dissertation were carried out on Linux clusters of the HPCC at USC, for which I really appreciate the access and support. I am deeply thankful for the love and support of my family, including my parents Tuyen Trinh and Chin Tang as well as my sister Giang Trinh. I appreciate very much their continuous encouragement in all years of study. To my parents, thank you. ii TableofContents Acknowledgments ii ListofFigures v Abstract ix Chapter1: Introduction 1 1.1 Quantum Spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Spin Dimers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Heisenberg Models . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Quantum Phases in Disordered Quantum Antiferromagnets . . . . . 7 1.5 Subjects of Investigation . . . . . . . . . . . . . . . . . . . . . . . 10 1.6 Organization of this Dissertation . . . . . . . . . . . . . . . . . . . 11 Chapter2: MonteCarloSimulation 13 2.1 Statistical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Classical Monte Carlo Simulation . . . . . . . . . . . . . . . . . . 15 2.3 Stochastic Series Expansion Quantum Monte Carlo Simulation . . . 17 2.3.1 Diagonal Updates . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.2 Loop Updates . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.3 Measuring Thermal Quantities . . . . . . . . . . . . . . . . 29 2.3.4 Measuring Off-Diagonal Quantities . . . . . . . . . . . . . 34 Chapter3: CorrelationsinQuantumSpinLadderswithSiteandBond Dilution 36 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Model and Quantities of Interest . . . . . . . . . . . . . . . . . . . 38 3.3 Correlation Length of Site-Diluted Ladders . . . . . . . . . . . . . 39 3.3.1 Effective Interactions among Localized Moments . . . . . . 39 3.3.2 Statistical Properties of the Couplings for 2-Leg Ladders . . 41 3.3.3 Correlations as a Function of Doping . . . . . . . . . . . . . 44 3.3.4 Doped 2-Leg Ladders and the Random-Exchange Heisen- berg Model . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3.5 Low-Temperature Scaling of the Correlation Length . . . . . 48 3.3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.4 Bond-Diluted Ladders . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.4.1 2-Leg Ladders . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4.2 4-Leg and 6-Leg Ladders . . . . . . . . . . . . . . . . . . . 64 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 iii Chapter4: Phase Diagram of Even-Leg Ladder with Bond Disorder 71 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.3 Bond-Doped Heisenberg Ladders . . . . . . . . . . . . . . . . . . . 74 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Chapter5: Conclusions 83 Bibliography 87 iv ListofFigures 1.1 (Color figure) Energy diagram of a pair of spin-1=2 as a function of applied magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 (Color figure) Scheme of different magnetic dimemsionalities. In the figure each dot represents a localized spin-1/2 degree of free- dom, and the lines represent the (nearest neighbor) interactions between these spins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 (Color figure) Sequences of ground-state phases in the Heisenberg bilayer in a uniform field h. Upper panel: clean case; lower panel: site-dilute case. For the phases indicated with an acronym: OBD=order-by-disorder, BG=Bose glass. The light shadow regions correspond to ordered (and gap- less) phases, the medium-shaded regions to gapped disordered phases, and the dark-shaded regions to gapless disordered phases. . . . . . . . . . . . 9 2.1 (Color figure) The six different vertices corresponding to the matrix ele- ments in Eqs. 2.19. The horizontal bar represents the full bond operator H b and the circles represent the spin state (solid and open are for spin-# and spin-" respectively) before and after operation with the diagonal and off-diagonal part ofH b . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 (Color figure) Example of transformation from one vertex to the other. In this example vertexW 4 is transformed to vertexW 6 . . . . . . . . . . . . 23 2.3 (Color figure) Transitions from one vertex to the others form directed equa- tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4 (Color figure) Schematic example of an operator string configuration on a 8-siteS = 1=2 chain with periodic boundary conditions. Only the off- diagonal terms cause a spin-flip. The truncationL is chosen to be larger than the sum of diagonal and off-diagonal termsn = 8. . . . . . . . . . . 27 2.5 (Color figure) Loop updates are constructed differently in order to measure the Green’s functions. . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1 (Color figure). Probability distribution (P ) and cumulative distribu- tion (E) of the effective couplings J (e) between nearest-neighbor localized moments in site-diluted 2-leg ladders, with parameters z = 2%, 0 = 7:5 (upper panels), andz = 2:5%, 0 = 15:3 (lower panels). The red dashed lines show the probability distributions in the continuum approximation according to Eq. (3.8). . . . . . . . . 42 v 3.2 (Color figure). Correlation length of site-diluted ladders with L = 128 and J l = J r = J. Upper panel: comparison between 2-leg, 4-leg and 6-leg ladders at inverse temperature J = 1024. Lower panel: correlation length of the 2-leg ladder at different tem- peratures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 (Color figure). Temperature dependence of the correlation length of site-diluted 2-leg ladders withJ l = 1, and power-law fits at low temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 (Color figure). Comparison of our data with those by Ref. [24]. Our data refer to ladders withL = 384, andJ l = 2J r = 1. . . . . . . . . 49 3.5 (Color figure). Values of the exponent extracted from the low- temperature behavior of site-diluted 2-leg ladders. . . . . . . . . . . 50 3.6 (Color figure). Correlation length of bond-diluted 2-, 4-, and 6- leg ladders with L = 128, J l = J r = J, and temperature T = J=1024 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.7 (Color figure). Sketch of the bond dilution effects on 2-leg ladders. a) pure ladder; b) ladder at low dilution: enhancement of rung sin- glets by leg-bond dilution, and low-energy singlets formed between LMs after rung-bond dilution; c) ladder at stronger dilution: cou- plings between LMs belonging to different diluted rungs. . . . . . . 55 3.8 (Color figure). Correlation length for 2-leg and 4-leg ladders with dilution of (center-)rung bonds. The average length of ladders seg- mentshli is obtained upon homogeneous dilution. . . . . . . . . . 55 3.9 (Color figure). (a) Correlation length of bond-diluted 2-leg ladders withL = 128 andJ l = J r = J at various temperatures. (b) Tem- perature dependence of the correlation length of above system. The curves with z > 22% are omitted due to is limited by the geo- metric length. (c) and (d) Fitting parameters of correlation length A(z) +B(z)j log(T=J)j, as a function of bond dilution. . . . . 57 3.10 (Color figure). (a) Temperature dependence of correlation lengths for four cases of rung-bond and random dilution. . . . . . . . . . . 57 3.11 (Color figure). (a) Low-temperature uniform susceptibility of bond- diluted ladders with L = 256 and J l = J r = 1. (b) The Curie coefficient increases as a power-law function of the bond dilution concentrationC(z)z 2:66 . . . . . . . . . . . . . . . . . . . . . . . 60 3.12 (Color figure). (a) Uniform susceptibility of even-numbered spin clusters of randomly bond-diluted 2-leg ladders. The solid lines are (a) logarithmic fits of the form u;c =C=(Tj logTj ) and (b) power- law fits u;c =C 0 T 0 1 . . . . . . . . . . . . . . . . . . . . . . . . . 61 vi 3.13 (Color figure). Fitting coefficients as a function of bond dilution. Left panels are fitting coefficients of the logarithmic function u;c = C=(Tj logTj ). Right panels are fitting coefficients of the power- law function u;c =C 0 T 0 1 . . . . . . . . . . . . . . . . . . . . . . 63 3.14 (Color figure). Bond-dilution effects on 4-leg ladders: (1) rung- singlet enhancement upon leg-bond dilution; (2) center-rung dilu- tion leading to the formation of two 2-spin rung singlets; (3) outer- rung dilution with formation of a low-energy singlets between a LMs and a 3-spin doublet. . . . . . . . . . . . . . . . . . . . . . . . 65 3.15 (Color figure). Correlation length with selective bond dilution. Bond dilution is distinguished as: (a) rung-bond and leg-bond dilution in 2-leg ladders; (b) center-rung bond, outer-rung bond and leg bond dilution in 4-leg ladders. The dashed line is the average correlation length from above selectively bond-diluted mechanisms . . . . . . . 67 4.1 (Color figure) (a) Distribution of local bond energies in pure (a) two- leg and (b) four-leg AF spin-1/2 Heisenberg ladders. For each case, there are different contributions due to rung bonds and leg bonds. Simulations are performed at temperatureT = J=2048, and lattice lengthL x = 256. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 (Color figure) Effects of local bond impurities, denoted by dashed lines, on the neighboring bond energies in two- and four-leg lad- ders. Simulations are performed at temperatureT =J=2048, lattice length L x = 256, with the coupling of the bond impurity J 0 = J=5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 (Color figure) (a) Distribution of local bond energies in bond-disordered (a) two-leg and (b) four-leg ladders. Simulations are performed at temperatureT =J=2048, and lattice lengthL x = 256. . . . . . . . 78 4.4 (Color figure) Staggered and uniform magnetization of pure and bond-disordered even-leg ladders. Bond couplings are randomly replaced by couplingsJ 0 = J=5. The values of the magnetization are averaged over at least 700 realizations for each lattice size. The uniform magnetization is measured at lattice lengthL x = 256. The thermodynamic limit of the staggered magnetization, i.e., infinite length, is extrapolated via finite-size scaling usingL x = 104; 128; 160; 200; 256; 320, and 400 for two-leg ladders and L x = 96; 128; 160; 192; 256; 320, and 400 for four-leg ladders. The error bars fall within the symbol size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 vii 4.5 (Color figure) Schematic illustration of the response of the uniform and staggered magnetizations to an applied magnetic field. In the pure case (a), the system remains in an RVB spin fluid state up to a lower critical field h c 1 . Then it undergoes a BEC transition into the regime h c 1 < h < h c2 . All spins are fully polarized beyond the saturation field h c2 . In the doped case (b), a field scan reveals the following sequence of phases: I: RVB spin liquid; II: Bose glass phase; III: BEC; IV: Bose glass phase; V: fully polarized phase. . . . 81 5.1 (Color figure) Effects of local site/bond impurities on the neighbor- ing local bond energies in three-leg ladders. Site impurities are denoted by blue dots with no connections. Bond impurities are denoted by dashed lines. Simulations are performed at tempera- tureT =J=2048, lattice lengthL x = 256, with the couplings of the impurity bondsJ 0 =J=10. . . . . . . . . . . . . . . . . . . . . . . 85 viii Abstract The presence of disorder can deeply affect the critical behavior of quantum spin sys- tems. Disorder can localize the long-wavelength modes developed at the quantum phase transition of the clean systems, and thus introduce glassy phases. Furthermore, it has been shown that many novel phases can be induced by the simultaneous presence of both magnetic field and geometric randomness. For instance, in doped spin-gap lad- ders one observes a very rich phase diagram, with a sequence of magnetic field con- trolled phases, including superfluidity, Bose-Einstein condensate, Bose glass and full polarization. Geometric randomness can also induce local magnetic moments giving rise to long-range order through an order-by-disorder mechanism in a gapped quantum- disordered ground state. In this dissertation, we investigate the physical properties and phase diagrams of doped even-leg ladders. Even-leg ladders are the minimal system in which we can study these effects: pruned spins, bond vs. rung vacancies, and order- by-disorder. Furthermore, such minimal even-leg ladders do not have any intrinsic low- energy degrees of freedom, because the pure system is gapped. Thus the effects of impurity vacancies, introducing new low-energy features, can be cleanly studied. In the first part, we uncover the effects of quenched disorder in the form of site and bond dilution on the physics of the S = 1=2 antiferromagnetic Heisenberg model in even-leg ladders. Site dilution is found to prune rung singlets and thus create local- ized moments which interact via a random, unfrustrated network of effective couplings, realizing a random-exchange Heisenberg model in one spatial dimension. This system exhibits a power-law diverging correlation length as the temperature decreases. Contrary to previous claims, we observe that the scaling exponent is non-universal, i.e., doping dependent. This finding can be explained by the discrete nature of the values taken by the effective exchange couplings in the doped ladder. Bond dilution of even-leg ladders ix leads to a more complex evolution with doping. Correlations are weakly enhanced in 2-leg ladders, and are even suppressed for low dilution in the case of 4-leg and 6-leg ladders. We clarify the different aspects of correlation enhancement and suppression due to bond dilution by isolating the contributions of rung-bond dilution and leg-bond dilution. In the second part, we investigate random bond disorder in antiferromagnetic spin- 1/2 Heisenberg ladders. We find that the effects of individual bond impurities vary strongly, depending on their positions on legs and rungs. We initially focus on how the distribution of local bond energies depends on the impurity concentration. Then we study how the phase diagram of even-leg ladders is affected by random bond doping. We observe Bose glass phases in two regimes (h 0 < h . h c1 andh 00 < h < h c2 ) and a Bose-Einstein Condensate inbetween. Their presence is discussed in relation to the local bond energies. To study the physical properties and phase diagram of antiferromagnetic spin-1/2 Heisenberg ladders, we apply the stochastic series expansion quantum Monte Carlo method. This algorithm has been proved to be a very powerful technique to investi- gate the low-temperature properties of quantum spin systems at large scales. Details of the method are discussed before we explore the physical results. x Chapter1 Introduction 1.1 QuantumSpins Quantum spins are encountered when studying magnetism. Quantum spin systems con- sist of two or more particles, each of which has an orbital angular momentum and intrin- sic magnetic momentum or spin. In atoms, electrons moving in their orbitals around the nuclei cause orbital angular momenta. These momenta are the quantum analogue of classical angular momentum. Spin appears such as a fundamental property of electrons. This is contrary to a common belief that spin angular momentum is due to the inter- nal spinning of a particle. Electrons are believed to be point particles with no internal structure. In magnetic materials the magnetic moment of atoms is due to a combination of spin and orbital angular momentum. The actual moment is a non-trivial function of the two types. It is described in terms of the ”Lande g-factor” of the atom or ion which lies between 1 and 2. We refer to the angular momentum of a magnetic atom as a spin and use the symbol ^ S for the momentum, even if it is the combination of both types. The physical properties of quantum spin systems are determined by their interac- tions. Their magnetic dipole interactions are assumed to be small compared to their exchange interaction. Indeed, experiments have shown that the interaction energy of two magnetic dipoles in magnetic materials is far below 1 K. In contrast, the ordering temperature reaches values of several 100 K for typical magnetic materials. Therefore the magnetic dipole interactions are typically too small to cause magnetism. 1 In fact, the exchange interactions are responsible for macroscopic scale magnetism. This is a direct consequence of the Pauli exclusion principle, and it is basically due to the electric forces between the electrons in the atoms. Since forces of electric origin are much stronger than those of magnetic origin at an atomic scale, they are power- ful enough to permit macroscopic scale magnetism, even at room temperature. The exchange interactions are very short ranged and dependent on the overlap of atomic wave functions. 1.2 SpinDimers In magnetic materials, the study of microscopic behavior focuses on how the angular momenta of the atoms interact rather than the magnetic moments. In fact, the exchange interaction of angular momenta can be expressed as E =S 1 :S 2 +S z 1 S z 2 ; (1.1) whereE is the interaction energy between two atoms with angular momentumS 1 and S 2 . And the special case =J; = 0 is called the Heisenberg interaction. The special case = 0; =J is called the Ising interaction. IfJ is positive then the lowest energy is obtained when the angular momentum vectors are anti-parallel. And the vectors are parallel if J is negative. For the Heisenberg interaction there is no favored axis of alignment, but for the Ising case the favored alignment is along thez-axis. Let us consider the example of two spin-1=2’s with the Heisenberg interactionJ, H = JS 1 :S 2 = JS z 1 S z 2 + J 2 S + 1 S 2 +S 1 S + 2 : (1.2) 2 We call the basic states of the spinj"i andj#i corresponding to the spin up and down. Then the basis for a pair of spins (dimer) isfj""i;j#"i;j"#i;j##ig. In this basis the Hamiltonian matrix has the form J 0 B B B B B B B @ 1 4 0 0 0 0 1 4 1 2 0 0 1 2 1 4 0 0 0 0 1 4 1 C C C C C C C A : The eigenenergies of this Hamiltonian matrix areE =f 3J 4 ; J 4 ; J 4 ; J 4 g. The lowest energy state is not degenerate, called a singlet, and can be expressed as jsi = 1 2 (j"#ij#"i): (1.3) The three remaining states, energetically degenerate, are called triplets, fjt + i =j""i;jt 0 i = 1 2 (j"#i +j"#i);jt i =j##ig: (1.4) While the second triplet has zero total spin, the other two triplets have non-zero total spin. This means these two states are separated in a uniform magnetic field pointing in thez direction. The energy of the statej##i increases proportional to the magnetic field, while the energy of the statej""i decreases with the same ratio. Figure 1.1 shows the energy diagram of a pair of spin-1=2 as a function of the applied magnetic field. A sufficiently strong magnetic field eventually closes the energy gap between the singlet and the triplets. In an ensemble of dimers, the system undergoes a phase tran- sition, in which the singlets in their ground state are gradually replaced by triplets with total spin 1 and a symmetric wave function. We will see later that the presence of these 3 ∣s〉=(∣↑ ↓ 〉−∣↓ ↑ 〉)/2 ∣t〉 ∣↑ ↑ 〉 ∣↓ ↓ 〉 ε J/4 −3J/4 0 (∣↑ ↓ 〉+∣↓ ↑ 〉)/2 J/gμ B h Figure 1.1: (Color figure) Energy diagram of a pair of spin-1=2 as a function of applied magnetic field. excited triplets (bosons) explains many novel phases in doped quantum antiferromag- nets. 1.3 HeisenbergModels We have seen that a system with only two electrons is relatively simple. In real mate- rials, however, magnetic systems are composed of large numbers of electrons. The Schrodinger equation of these many body systems cannot be solved without assump- tions. Here we assume that the most important part of such an interaction is the exchange interaction between neighboring atoms. This consideration leads to a Hamiltonian, H = X ij J i;j S i :S j ; (1.5) 4 withJ ij being the exchange integral between spinsi andj. Often a good approximation is given by nearest neighbor interations: J i;j = 8 > > < > > : J hi;ji for nearest neighbor spins; 0 otherwise: (1.6) The one-dimensional S = 1=2 antiferromagnetic Heisenberg model defined on a chain is one of the earliest studied models of interest. Its exact solution given by Bethe in 1931 showed that there is no long-range order for one dimensional chains. [7] Recently field theoretical methods, e.g. bosonization and conformal field theory, have been used to obtain the critical exponent = 1 of the algebraically decaying spin-spin correlation function,hS 0 S r i_ (1) r (logr) 1=2 =r , and the finite size corrections to the low energy spectrum. [23, 47] However long-range order in the ground state has been predicted for a S = 1=2 antiferromagnetic Heisenberg model on a square lattice. [13, 41] Hamiltonian Eq.1.5 is assumed to describe the antiferromagnetic undoped insulator La 2 CuO 4 and the oxygen- deficient YBa 2 CuO 6 and other undoped copper-oxide materials. Anderson (1973) con- jectured that the ground state of the two-dimensional (2D) spin-1=2 antiferromagnet might be disordered and postulated the resonating valence bond state as a possible lowest-energy state. So far, there is no exact theorem proof of this conjecture. Nev- ertheless, there is significant evidence which strongly supports that the ground state of theS = 1=2 antiferromagnetice Heisenberg model on the square lattice is characterized by antiferromagnetic long-range order. [13, 41] In the last two decades, a new class of antiferromagnetic materials, spin ladders, has attracted significantly both theory and experiment. [15, 16, 34, 80, 81] The spin lad- der is a crossover from the one-dimensional Heisenberg chain to the two-dimensional 5 J J J J J J 1D Spin ladders 2D Figure 1.2: (Color figure) Scheme of different magnetic dimemsionalities. In the figure each dot represents a localized spin-1/2 degree of freedom, and the lines represent the (nearest neighbor) interactions between these spins. square lattice via this novel lattice structure. A scheme displaying this crossover is shown in Figure1.2. Here each dot represents a local spin-1=2, and the lines repre- sent the nearest neighbor interactions J between these spins. Ladders made from an even number of legs have a spin-liquid ground state and display short-range order. An exponential decay of the spin-spin correlations is produced by their finite spin gaps, hS 0 :S r i_ r 1=2 exp(r=), which is the energy gap from the ground state to the low- est excited state. In a simple limit obtained by taking the exchange coupling along the rungs much larger than along the legs, the spin gap is the energy gap of a dimer in the previous part. Ladders with odd number of legs behave quite differently. [1, 9, 62, 63] They display properties similar to those of single chain at low energies, namely gapless spin excitations and power law decay of the spin-spin correlations. This dramatic differ- ence between even-leg and odd-leg ladders predicted by theory has also been confirmed experimentally in a variety of systems. [4, 32] 6 1.4 Quantum Phases in Disordered Quantum Antifer- romagnets Mathematical models give physicists powerful tools to investigate the physical proper- ties of condensed phases of matter and to understand the behavior of these phases. We are all familiar with the solid, liquid, or gas phases in daily life. Some of the more exotic condensed phases including the superconducting phase, spin liquid phase, Bose-Einstein condensate, are only observed in experimental laboratories at very low temperature. The study of these quantum phases and the transition between them is a very active field of contemporary physics. We would like to recall some basic properties of the few most common quantum phases which will be encountered in this dissertation. Spin liquid is an exotic state of matter which was first predicted by Haldane in the antiferromagnetic S = 1 Heisenberg chain. [49] Its ground state is disor- dered, and the spin-spin correlation decrease exponentially at large distance,hS 0 S r i_ (1) r exp(r=), where is a finite correlation length. The presence of a spin gap is a consequence of the exponential decay of the spin-spin correlation: the system has no low-lying excitations. The spin gap is the energy difference between the singlet ground state and the first excited triplet. Magnetic systems with a finite correlation length at zero temperature and a spin gap are called spin liquids, by analogy with standard liq- uids, which, in contrast to solids, have only short-range order. Haldane’s prediction has been verified in many organic and inorganic compounds. [35, 60] Since the first obser- vations inS = 1 chains, spin liquid states were also discovered in several systems such as spin ladders and frustrated magnets. [6] Some magnetic materials also display Bose-Einstein condensation (BEC), which is a state of matter of dilute gases of bosons. In 1999, Bose condensation of bosons was demonstrated in the antiferromagnet TlCuCl 3 . [54, 66] This condensation was observed 7 at temperatures below 14 K. In this material, two Cu 2+ ions are antiferromagnetically coupled to form dimers embedded in a crystalline network: the dimer ground state is a singlet, separated by an energy gap from the excited triplet state. A triplet excited on one dimer can hop to a neighboring dimer; as a result, the triplets delocalize in a way similar to electrons that become delocalized in crystals. These triplets can be regarded as bosons with hard core repulsion, which prevents more than one boson from being present on a single dimer. In an external magnetic field, the field does not alter the singlet ground states but lowers linearly the energy of the triplet components with S z = +1. At a critical magnetic field, the energy of the triplet intersects the ground-state singlet, resulting in long-range magnetic order. This transition represents a quantum critical point at which Bose-Einstein condensate occurs. [21, 61, 67] Disorder can have a very strong effect on quantum fluids. Owing to their wave-like nature, boson particles are subject to interference when scattering against disordered potentials. Such random potentials suppress or may even completely destroy the long- range order related to Bose-Einstein condensates. Fisher et al. [18] have suggested that sufficiently strong disorder in a lattice leads to the appearance of a new phase, called Bose glass. It is characterized by a finite compressibility, the absence of a gap in the single particle spectrum, and a nonvanishing density of states at zero energy. [22, 102, 103] Bose glasses can be made by adding bond disorder or site dilution. [18, 55, 64, 65, 100] The transitions between quantum phases, namely quantum phase transitions, in low- dimensional quantum Heisenberg antiferromagnets have been the subject of extensive investigations during the last two decades. [67] Quantum phase transitions can be con- trolled by various parameters, including lattice dimerization, frustration, and a magnetic field. Examples of field-induced quantum phase transitions can be found in Haldane chains, [26, 106] unfrustratedS = 1=2 weakly-coupled dimer systems arranged in spin 8 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 h 0.01 disordered free moments 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 h 0.01 B D O plateau dimer singlet polarized BG BG polarized XY ordered XY ordered Figure 1.3: (Color figure) Sequences of ground-state phases in the Heisenberg bilayer in a uni- form fieldh. Upper panel: clean case; lower panel: site-dilute case. For the phases indicated with an acronym: OBD=order-by-disorder, BG=Bose glass. The light shadow regions corre- spond to ordered (and gapless) phases, the medium-shaded regions to gapped disordered phases, and the dark-shaded regions to gapless disordered phases. ladders, [12, 88] in coupled bilayers, [21, 31, 73] and in more complex 3D geome- tries. [54, 66] The application of a uniform field overcoming the spin gap brings these systems from a gappedS = 0 state to a state with finite magnetization parallel to the field and (ind> 1) spontaneous finite staggered magnetization transverse to the field. More recently, special interest has focused on phase transitions driven by geometric randomness of the lattice, including site and bond disorder. Strong geometric disorder not only breaks translational invariance and perturbs the ground state of the pure system, but it can also destabilize renormalized classical phases with long-range order and drive the system to novel disordered phases, (see Figure 1.3). [87, 98] Site disorder can be realized by replacing magnetic ions Cu 2+ by non-magnetic impurities. [3, 43, 57, 58, 86] The effect of non-magnetic impurities is the formation of local free S = 1=2 moments exponentially localized around the impurities. The overlap between two exponentially localized moments produces an effective coupling between them which decays exponentially with the impurity-impurity distance. [75] Despite the fact that the impurities are randomly located, such couplings are perfectly 9 unfrustrated and have signs so that they induce spontaneous long-range N´ eel order in the free moments, giving rise to a paradigmatic order-by-disorder phenomenon. [74] Bond disorder occurs instead when the dopant ions replace the ions which act as bridges between the magnetic ions. For instance, bond disorder can be intro- duced in IPA-CuCl 3 by a partial substitution of non-magnetic Br for the likewise non- magnetic Cl, affecting the bond angles in the Cu-halogen-halogen-Cu superexchange pathways. [38, 40, 44] The reduced strength of magnetic interactions on the affected bonds leads to Bose glass when a strong magnetic field is applied. [27] Many novel phases can be induced by the simultaneous presence of both magnetic field and disorder. [58, 65] For example, in doped spin gap ladders one observes a very rich phase diagram, with a sequence of magnetic field controlled phases, including superfluidity, BEC, Bose glass and full polarization. [38, 58, 97] Using quantum Monte Carlo simulations on site diluted bilayer system, Roscilde et al. have found a sequence of three distinct quantum-disordered phases induced by the field. The three quantum disordered phases are a gapless disordered-free-moment phase, a gapped plateau phase, and two gapless Bose glass phases, (see Figure 1.3). 1.5 SubjectsofInvestigation In this dissertation we investigate site and bond disordered antiferromagneticn-leg lad- ders with n = 2; 4, and 6. These even-leg ladders are minimal systems in which we can study these effects: pruned spins, bond vs. rung vacancies, order-by-disorder. Fur- thermore, even-leg ladders do not have any intrinsic low-energy degrees of freedom, because the pure system is gapped. So the effects of impurity vacancies, introducing new low-energy features, can be cleanly studied. We focus on the effects of site and bond disorder on the spin-spin correlations. For site disorder, we study the low-temperature 10 scaling of correlation lengths. For bond disorder, we clarify different aspects of corre- lation enhancement and suppression due to bond dilution by isolating the contributions of rung-bond dilution and leg-bond dilution. Finally, we study the phase diagram of even-leg ladders in the presence of both a magnetic field and bond disorder. These disordered-induced phases are related to recent experiments. In this dissertation we apply the stochastic series expansion (SSE) quantum Monte Carlo (QMC) method to study quantum phases in disordered antiferromagnetic even- leg ladders. [68, 72, 79] The SSE algorithm has been proved to be a very powerful QMC technique to investigate the low-temperature properties of quantum spin systems at large scales. The non-local update in the algorithm also helps to avoid critical slowing down. [71] 1.6 OrganizationofthisDissertation Chapter 2: We summarize the basic ideas of Monte Carlo simulations. As an exam- ple, we briefly describe a classical MC simulation of the Ising model. We emphasize on building the transition probabilities, transitions between configurations, and impor- tance sampling. We then build the configuration space of the SSE QMC simulation for a general Heisenberg model in an applied magnetic field. We discuss analogies between classical and quantum Monte Carlo simulations. The detailed balance condition is dis- cussed. The probabilities for changing between configurations are derived. We then introduce an implementation of how to perform a SSE QMC simulation of Heisenberg models in practice, which is the object of this dissertation. Finally, methods how to measure physical qualities are presented. Chapter3: We investigate the effects of quenched disorder, in the form of site and bond dilution, on the physics of the S = 1=2 antiferromagnetic Heisenberg model in 11 even-leg ladders. Site dilution is found to prune rung singlets and thus create localized moments which interact via a random, unfrustrated network of effective couplings, real- izing an effective random-exchange Heisenberg model in one spatial dimension. This system exhibits a power-law diverging correlation length as the temperature decreases. Contrary to previous claims, we observe that the scaling exponent is non-universal, i.e., doping dependent. Bond dilution on even-leg ladders leads to a more complex evolu- tion with doping of correlations, which are weakly enhanced in 2-leg ladders, and are even suppressed for low dilution in the case of 4-leg and 6-leg ladders. We clarify the different aspects of correlation enhancement and suppression due to bond dilution by isolating the contributions of rung-bond dilution and leg-bond dilution. [84] Chapter 4: In this chapter, random bond disorder in antiferromagnetic S 1=2 Heisenberg ladders is investigated using the SSE QMC method. We find that the effects of individual bond impurities vary strongly, depending on their position on legs and rungs. We initially focus on how the distribution of local bond energies depends on the impurity concentration. Then we study how the phase diagram of even-leg ladders is affected by random bond doping. We observe Bose glass phases in two regimes (h 0 < h . h c1 andh" < h < h c2 ) and a Bose-Einstein Condensate inbetween. Their presence are discussed in the relation to the local bond energies. [83] Chapter 5: We summarize the results obtained in this dissertation. We leave open questions on the physics of disordered odd-leg ladders, where there is a zero-energy mode even in the clean system. We also propose future research on the disordered odd- leg ladders. 12 Chapter2 MonteCarloSimulation 2.1 StatisticalPhysics At any given time a physical system can be said to occupy one of many possible states. The set of all possible states of the system constitutes its phase space, in which each pos- sible state corresponds to one unique point. As the system evolves, its states follow one of the lines (trajectories) in phase space. The expectation value of a physical observable A is found by integrating over a certain trajectory in phase space along the equation of motion. If the number of degrees of freedom of a system is very large or even infinity, the expectation value is approximated by averaging over a set of sample states, hAi = R A(x)p(x)dx R p(x)dx ! P i A(x i )p(x i ) P i p(x i ) ; (2.1) where x i is the configuration of state i and p(x i ) is the probability of the system to appear in that configuration. Physically, we can imagine that under thermal fluctuations, the system can jump from one state to another. If we let the system travels long enough in its phase space the system can go back and forth a certain state many times. The ratio of the times the system appears in one state over the total number of jumps is proportional to the probabilityp(x i ). In general problems with arbitrary distributions, p(x i )-distributed configuration is not created from scratch. Instead, it is built by using a Markov chain. A Markov chain of states is generated starting from an initial pointx 0 : x 0 !x 1 !x 2 !:::!x n1 !x n !:::: (2.2) 13 The probability for the system to move from one configurationx to anothery is given by a transition matrixW xy . One of the many applications of Markov chain is in finding long-range predictions. Although it is not possible to make long-range predictions with all transition matrices, it is always possible for a large set of transition matrices. The first property of transition matrixW is derived from the fact that the system can stay in the same state or jump to any other states with the probability 1. Mathematically, it means: X y W xy = 1: (2.3) In other words, the column vectors of matrixW are normalized. Thus the largest possi- ble entry of transition matrixW is 1. We find a transition matrix so that the probability distributionp(x n ) changes at each step of the Markov process: X x p(x n )W xy =p(y n+1 ); (2.4) but it asymptotically approaches to the desired probabilityp(x). The transition matrix need to satisfy a set of following conditions: [25, 85] Ergodicity: It has to be possible to reach any configuration x from any other configurationy in a finite number of Markov steps. This means that there exists a finite numbern such that (W n ) xy 6= 0 for allx andy. Such a transition matrix is called a regular matrix. And its corresponding Markov chain is a regular Markov chain. 14 Detailedbalance: There exists a single vectorp(x) that does not depend on initial p(x i ), such thatp(x i ):W n gets closer top(x) asn gets larger and larger. Vector p(x) is called equilibrium vector. It is an eigenvector with left eigenvalue 1, X x p x W xy =p y : (2.5) It easy to see the detailed balance condition satisfies when W xy W yx = p y p x : (2.6) Readers can find detail mathematical proofs of these conditions in a book by Grinstead.[25] Now let’s see how do we apply these conditions in practice. 2.2 ClassicalMonteCarloSimulation Let consider a simple Ising model system including a square lattice of classical spins =1, that can point either up = 1 or down =1. The Hamiltonian of this system is H =J X i;j i j ; (2.7) where the sum is taken over all nearest neighbor spin pairs. A pair of two antiparallel spins contribute an energy +J, while two parallel spins contributeJ. The system is ferromagnetic ifJ > 0 or antiferromagnetic ifJ < 0. Its ground state with lowest energy is the fully polarized state where all spins are aligned, either pointing up or down. Since each spin can either point up or down the system has 2 N configurations. In other words, the configuration space of this model is 2 N . 15 The thermal average of a quantityA at finite temperatureT is given by summarizing over all configurations hAi = 1 Z X i A i exp(E i ); (2.8) where = 1=k B T is the inverse temperature. A i is the value of the quantityA in the configurationi.E i is the energy of that configuration. AndZ is the partition function of the system. For small systems, it is possible to evaluate the above sum completely. For larger systems, Monte Carlo simulation is the method of choice. In MC simulations, instead of taking the sum over all states, we actually build a representative sample of configurations and evaluate the quantity A basing on these configurations. The more number configurations we choose, the closerA approaches its actual value. The simplest way to ”jump” from one configuration to another is the single-flip Metropolis algorithm: Starting from a configurationc i choose to flip a random single spin, leading to a new configurationc j . Calculating the energy difference E = E j E i between two configurationsc i andc j . If E < 0 the new configuration is acceptedc i+1 =c j . If E > 0 the new configuration is accepted only if the probability exp(E) < r, r is a randomly distributed number in the interval [0; 1]. If this condition is not fulfilled, old configuration is untouched and we repeat above procedure with a different random single spin. Measure all the quantities of interest in the new configuration. 16 This algorithm is ergodic since it allows to move to any configuration in a finite number of spin flips. It also satisfies the detailed balance condition. Note that the new configuration is not far from the old one. Therefore at low temperatures, the probability exp(E) is very small, and it takes an extremely long time for all spins to be flipped by the single spin flip algorithm. This problem arises as soon as we get close to the critical temperature. It is called the critical slowing down problem. A solution to this problem was found in 1987 by Swendsen and Wang [78] and in 1989 by Wolff [94]. Instead of flipping a single spin they proposed to flip a cluster of spins and choose them in a clever way so that the probability to flip them is large. The principles for a quantum Monte Carlo simulation are similar to the classical case. Now let’s see how we bring the ideas of: generating configurations, calculating the probabilities, and flipping spin clusters to simulate quantum systems. 2.3 StochasticSeriesExpansionQuantumMonteCarlo Simulation The central problem in quantum Monte Carlo simulations is to calculate the thermal expectation value of a quantum-mechanical operatorA, hAi = 1 Z TrfAe H g; (2.9) where is the inverse temperature,Z is the partition function, andH is the Hamiltonian of the system. The dimension of the Hilbert space of a quantum system is not linear in its degrees of freedom, but exponential. Therefore we cannot visit all states of the system. Instead we build an algorithm to evaluate the observables through a number of sample states of the system. The stochastic series expansion algorithm is one such 17 algorithm which has been proved to be a powerful technique for importance sampling. This algorithm was recently developed by Sandvik.[68, 69, 70, 71, 72, 79] Let’s consider a general XXZ spin- 1 2 antiferromagnetic Heisenberg model in a uni- form magnetic fieldh with site impurities. Its Hamiltonian can be written as H = X hiji i j J ij S i :S j X i i hS z i ; (2.10) wherehi;ji represents the nearest neighbor couplings, J ij is their coupling strength, i =f0; 1g is the occupation number at sitei. The Hamiltonian can then be written in a bond representation as H = X <ij> i j J ij S z i S z j + 1 2 (S + i S j +S + i S j ) X i i hS z i = Nb X b=1 H b = Nb X b=1 (H 1;b +H 2;b ); (2.11) whereN b is the number of bonds. Here we define two types of operator: the diagonal operatorH 1;b and the off-diagonal operatorH 2;b H 1;b =C b J b S z i(b) S z j(b) +h S z i(b) z i(b) + S z j(b) z j(b) ; (2.12a) H 2;b = 1 2 J b (S + i(b) S j(b) +S i(b) S + i(b) ); (2.12b) whereC b is a constant so that the diagonal operatorH 1;b is positive,J b =J ij is the bond coupling, andz i is the number of neighbors at sitei. Sandvik then proposed to expand the partition function in a Taylor series and truncated the series at a certain lengthL: Z = Trfe H g = X L X n=0 () n n! hjH n ji; (2.13) 18 by adding new unit operators H 0;0 = 1, all terms in the series now can be written as sequences of operators (or operator sequences) of equal length. Z = X X S L n S L (Ln S L )! L! hj L Y i=1 H a(i);b(i) ji; (2.14) S L is a series of operatorsH 1;b ;H 2;b ;H 0;0 andn S L is the number of non-unit operators of the sequences. Here we define the basis asfjig =fjS z 1 ;S z 2 ;:::;S z N ig with spins pointing either up or down. (As we see later, the length of the operator sequence is finite. It is proportional to the inverse temperature and the internal energy of the system). Finally, we define W (;S L ) = n S L (Ln S L )! L! hj L Y p=1 H a(p);b(p) ji = n S L (Ln S L )! L! L Y p=1 W (p); (2.15) as the weight or the probability for a configurationfji;S L g. The set of all configu- rationsfji;S L g creates our configuration space. Similarly, the observableA can be expanded and written in the language of new configuration hAi = 1 Z TrfAe H g = X X S L n S L (Ln S L )! L! hjA L Y i=1 H a(i);b(i) ji = X X S L A (;S L ) W (;S L ): (2.16) The above formula means that the expectation value of operator A is the average of all of its values A (;S L ) in the configuration spacefji;S L g. Similarly, in classical Monte Carlo simulation, in order to measure the expectation value, we need to find a 19 way walking in the configuration space. This will allow us to build the configuration samples. If the simulation time is long enough, with numerous steps, the system passes by the configurationfji;S L g with a probability equal to its weightW (;S L ). We can now see how the system walks in its configuration space. But first, we have to look at the weightW (;S L ) in more detail, Eq.2.15, to see how the system can change from one configuration to another. We defineW (p) as a bare vertex weight. It is defined as a matrix element of the bond operatorH b =H 1;b +H 2;b at positionp in the operator sequence, W (p) =hS z i(b) (p)S z j(b) (p)jH b(p) jS z i(b) (p 1)S z j(b) (p 1)i: (2.17) These bond operators are described by Eq.2.12. Each bond operator acts only on two connected spins. We choose the constant C b = 1 4 J b + 1 2 (h=z i +h=z j ) +; (2.18) such that all the diagonal matrix elements are larger than (or equal to) zero. For our model, there are total six types of verticesW (p) =W i ;i = 1;::; 6 (see Fig. 2.1): W 1 =h##jH b j##i = (2.19a) W 2 =h#"jH b j#"i =J b =2 +h=z j + (2.19b) W 3 =h"#jH b j"#i =J b =2 +h=z i + (2.19c) W 4 =h""jH b j""i =h=z i +h=z j + (2.19d) W 5 =h#"jH b j"#i =J b =2 (2.19e) W 6 =h"#jH b j#"i =J b =2 (2.19f) 20 Figure 2.1: (Color figure) The six different vertices corresponding to the matrix elements in Eqs. 2.19. The horizontal bar represents the full bond operatorH b and the circles represent the spin state (solid and open are for spin-# and spin-" respectively) before and after operation with the diagonal and off-diagonal part ofH b . Monte Carlo is conducted by building the importance sampling of the basis fji;S L g. The system stateji is initially taken by generating random spins either in up or down directions. The operator sequenceS L at first is empty. Then the operators in the sequenceS L are updated via two steps. First, diagonal updates are implemented by exchanging operators H 0;0 $ H 1;b . This process can be carried out sequentially at every propagation level p = 0;:::;L. Second, loop updates are built to exchange between diagonal and off-diagonal operatorsH 1;b $ H 2;b . The diagonal update does not change direction of spins at all. However, the loop updateH 1;b $H 2;b flips direc- tion of every pair of spins whenever an operator change occurs. In both two steps, spins are flipped corresponding to vertex types, and importance samples are built. In fact, the loop update presented here is similar to the cluster update in the classical Monte Carlo simulation. We would like to present these updates so that it can be applied directly to our later research. The initial algorithm with single updates is presented somewhere else. [72] In the following parts, we would like to go into further details about diagonal update, loop update, as well as measuring the physical quantities. 2.3.1 DiagonalUpdates Let’s recall some definitions of Monte Carlo importance sampling. The system initially stays in an arbitrarily allowed configuration. From it a Markov chain of configurations is 21 generated by making changes. These configurations are accepted or removed according to probabilities chosen so that detailed balance is satisfied. According to the Metropolis method, the probability to accept a change of the old configuration into a new one is P accept = min W new W old ; 1 ; (2.20) P acc (X!Y ) = min W (Y )P select (Y!X) W (X)P select (X!Y ) ; 1 : (2.21) In the diagonal update, it is a change H 0;0 ! H 1;b , and the number of operators increments (n 1)! n. The Metropolis acceptance probabilities for such additions are: P acc (H p 0;0 !H p 1;b ) = min N b h p jH 1;b j p i Ln ; 1 : (2.22) This is the probability of choosing one bond arbitrarily fromN b bonds to fulfill one of theLn available positions. Inversely, the probability to remove one bond is P rem (H p 1;b !H p 0;0 ) = min Ln + 1 N b h p jH 1;b j p i ; 1 : (2.23) 2.3.2 LoopUpdates In order to understand how loop updates work, we investigate detailed balance processes from configurations!s 0 , which require: P (s!s 0 )W (s) =P (s 0 !s)W (s 0 ); (2.24) 22 P (s! s 0 ) is the probability to transform from configurations to configurations 0 and W (s) is the weight at configuration s, Eq. 2.15. If the weight is given by the Hamil- tonian, then the probability depends on the actual algorithm. According to the SSE algorithm introduced by Sandvik [70], after the diagonal update, the operator sequence S L contains a series of six types of vertices, Fig. 2.1. The configuration is mapped upon a linked-vertex list which includesn vertices and a total of 4n legs. Fig. 2.4 shows an example of configuration of lengthL = 11 with 8 vertices and 32 legs. Loop updates start by first picking up an initial entrance vertex leg at random among all 4n legs. Then an exit leg on the same vertex is chosen in a probabilistic way. Two spins on the entrance and exit legs are flipped with unit probability and the vertex is transformed into a new vertex relatively, Fig. 2.2. This exit leg is linked to an entrance leg of another vertex along the direction of time propagation. This process continues until the initial leg is reached and the loop closes. Figure 2.2: (Color figure) Example of transformation from one vertex to the other. In this example vertexW 4 is transformed to vertexW 6 . The probability for arriving at a new configurations 0 can therefore be written as P (s!s 0 ) = X P (e 0 )P (s;e 0 !s 1 ;e 1 )P (s 1 ;e 1 !s 2 ;e 2 ):::P (s n1 ;e n1 !s 0 ;e 0 ); whereP (e 0 ) is the probability for choosing the vertex lege 0 as the starting point and P (s i ;e i !s i+1 ;e i+1 ) is the probability given a configurations i and the entrance lege i to exit the vertex at x i , which is connected to the next entrance leg e i+1 , resulting in a new configurations i+1 . The exit legsx i do not explicitly appear in the probabilities 23 since they are uniquely linked to the following entrance legse i+1 . One can easily write the probability for reverse process as P (s 0 !s) = X P (e 0 )P (s 0 ;e 0 !s n1 ;e n1 ):::P (s 2 ;e 2 !s 1 ;e 1 )P (s 1 ;e 1 !s;e 0 ): The detailed balance condition, Eq. 2.24, therefore, is satisfied if, P (s i ;e i !s i+1 ;e i+1 )W (s i ) = P (s i+1 ;e i+1 !s i ;e i )W (s i+1 ); W (s;i;e) = W (s 0 ;e;i): (2.25) Here we have writtenW (s;i;e) =W (s)P (s;i!s 0 ;e). This balance is true for all SSE configurations and entrance legs.W (s;i;e) now is the weight for a transition at vertexs (= 1;:::; 6) from entrance legi (= 0; 1; 2; 3) to exit lege (= 0; 1; 2; 3). It therefore leads to an additional condition that the path always continues through a vertex. It means: X x P (s;e!s x ;x) = 1: (2.26) In terms of weight this requirement translates into X x W (s;e!s x ;x) =W (s); (2.27) eliminating forbidden transitions at each vertex. The above equation forms eight sets of directed equations. The first four of them are shown in Fig. 2.3 and Table 2.1: quadrant I quadrant II quadrant III quadrant IV W 6 =b 1 +a 1 +a 2 W 6 =b 4 +a 4 +a 5 W 6 =b 7 +a 7 +a 8 W 6 =b 10 +a 10 +a 11 W 3 =a 1 +b 2 +a 3 W 3 =a 4 +b 5 +a 6 W 2 =a 7 +b 8 +a 8 W 2 =a 10 +b 11 +a 12 W 4 =a 2 +a 3 +b 3 W 1 =a 5 +a 6 +b 6 W 4 =a 8 +a 9 +b 9 W 1 =a 11 +a 12 +b 12 Table 2.1: First four directed equations are constructed from the detailed balance condi- tion. The other four can be obtained by replacingW 6 byW 5 in the above equations. 24 Figure 2.3: (Color figure) Transitions from one vertex to the others form directed equations. One can obtain the other four directed equations by replacingW 6 withW 5 . To solve these equations, we replaceW i by Eqs. 2.19 and solve the above directed equations for a i . Their solutions are: 25 quadrant I : 8 > > > > > < > > > > > : a 1 = + f j 1 2 (b 1 +b 2 b 3 ) a 2 = +f j 1 2 (b 1 b 2 +b 3 ) a 3 = +f j + 2f i + 1 2 (b 1 +b 2 +b 3 ) (2.28a) quadrant II : 8 > > > > > < > > > > > : a 4 = + +f i 1 2 (b 4 +b 5 b 6 ) a 5 = f i 1 2 (b 4 b 5 +b 6 ) a 6 = +f i + 1 2 (b 4 +b 5 +b 6 ) (2.28b) quadrant III : 8 > > > > > < > > > > > : a 7 = + f i 1 2 (b 7 +b 8 b 9 ) a 8 = +f i 1 2 (b 7 b 8 +b 9 ) a 9 = +f i + 2f j + 1 2 (b 7 +b 8 +b 9 ) (2.28c) quadrant IV : 8 > > > > > < > > > > > : a 10 = + +f j 1 2 (b 10 +b 11 b 12 ) a 11 = f j 1 2 (b 10 b 11 +b 12 ) a 12 = +f j + 1 2 (b 10 +b 11 b 12 ) (2.28d) Here we have used = J b (1 )=4 andf i(j) = h=z i(j) . A constant is chosen so that all weights are positive and the number of non-zero bouncesb i are minimum. This requirement is satisfied when = maxf0; f i ; f j g: (2.29) We have twelve equations and twenty four variables. Basically there are infinite solutions for this set of equations. However, note that if coefficientsb i > 0 means there are bounces along the loop. Bounce processes do not change the configuration of the system, as a result, they do not improve the simulation. We solve the above equations 26 by minimizing all the bounce coefficients as possible. One of the solutions is satisfied withb i = 0: If + f j < 0; b 3 =2( + f j ) (2.30a) If +f j < 0; b 2 =2( +f j ) (2.30b) If f i < 0; b 5 =2( f i ) (2.30c) If + f i < 0; b 9 =2( + f i ) (2.30d) If +f i < 0; b 8 =2( +f i ) (2.30e) If f j < 0; b 11 =2( f j ) (2.30f) 1 2 3 4 5 6 7 8 9 10 11 12 3 4 5 6 7 8 9 10 11 12 8 8 1 2 Spin Spin 1 2 3 4 5 6 7 2 3 4 5 6 7 1 Propagation Level Propagation Level 0 0 Figure 2.4: (Color figure) Schematic example of an operator string configuration on a 8-site S = 1=2 chain with periodic boundary conditions. Only the off-diagonal terms cause a spin-flip. The truncationL is chosen to be larger than the sum of diagonal and off-diagonal termsn = 8. 27 Above, we have found the solutions of directed equations by minimizing all the bounces. A general solution of these equations can be obtained by using linear pro- graming techniques.[2] It is also shown that in some cases minimizing bounces alone does not always lead to more efficient algorithms in terms of autocorrelations of physi- cal observables, because of the nonuniqueness of the bounce-free solutions. Readers are free to choose a method for specific problems Finally, in order to carry out the operator-loop update, a linked list of the vertices with their four spin states is constructed using the current state and the index sequence S L . The list is doubly linked, so that it is possible to move in either direction from any leg of a given vertex to the leg of the next or previous vertex connected to the same spin. Now, we are ready to talk about constructing a program. The diagonal update loop will be built first. Then a directed loop update is implemented by exchanging diagonal and off-diagonal operators. It can best be explained using a graphical interpretation. The principles are following: One of then vertices is first chosen at random, and one of its four legs is randomly selected as the entrance point. One of its legs is then chosen as the exit point from the vertex, according to probabilities of Eqs. 2.28 and Eqs. 2.30. The four possible vertex paths are illustrated in Fig. 2.3. The spins at both the entrance and exit legs are flipped. Note that the entrance and the exit can be the same leg, in which case the net effect is no spin flip; only a reversal of direction of movement in the list. The chosen exit leg points to a leg of another vertex in the linked list, the spin at which is also flipped. From this vertex, an exit leg is again chosen, which points to another vertex, etc. After some number of steps, the exit of the last visited vertex will point to the original entrance point of the update. The loop then closes and as a result all the spins flip along the random path followed in the process. The process can then be regarded as a single loop update [59]. Diagonal updates and loop updates therefore 28 allow us to change the system configurationfji;S L g. And therefore, the system walks from one configuration to another in the configuration space. There are two main questions: (1) How to adjust the length of the operator sequences? (2) How many loop updatesN l are constructed in one Monte Carlo loop? Eq. 2.31 shows that the expansion ordern is proportional to the inverse temperature and internal energy of the system. Therefore the length of the operator sequenceL can be chosen so thatn never reaches the cutoff during the simulation (LN). In practice, L is gradually adjusted after each thermalization step, so thatL = an max , wheren max is the highestn reached after a number of MC loops. We find, in practice,a 1:25 is sufficient. The operator loops are typically of highly varying lengths. Each MC simulation should involve several loop updates, so that a significant fraction of vertices are visited. We, in practice, record the number of visited vertices in loop updates,N l , periodically adjust it so that the average cumulative loop length during one MC simulation is approx- imately 2L. 2.3.3 MeasuringThermalQuantities Energy is the first derivative of the partition function with respect to the inverse temper- ature: hEi = @ @ (lnZ) = 1 Z @ @ ( X X S L n S L (Ln S L )! L! hj L Y i=1 H a(i);b(i) ji ) = 1 Z X X S L n S L n S L (Ln S L )! L! hj L Y i=1 H a(i);b(i) ji = 1 hn S L i W : (2.31) 29 Therefore the average expansion ordern S L NhE b i,hE b i =hH b i is the inter- nal energy per bond. The cut-off for the Taylor expansion is dependent on the inverse temperature and the system sizeN. The specific heat is the second derivative of the partition function with respect to the inverse temperature hCi = 2 @ 2 @ 2 (lnZ) = 2 1 Z @ 2 Z @ 2 1 Z @Z @ 2 = hn 2 S L i W hn S L i 2 W hn S L i W : (2.32) The local magnetization at sitei is given by hS z i i = 1 Z TrfS z i e H g = X f;S L g S z i (0)W (;S L ); (2.33) where we take the average over all steps of the state cycle hS z i i = X f;S L g " 1 L L1 X p=0 S z i (p) # W (;S L ) = * 1 L L1 X p=0 S z i (p) + W : (2.34) Above we denoteS z i (p) =h(p)jS z i j(p)i. The uniform magnetization is hm z u i = 1 N N X i=1 hS z i i = * 1 N N X i=1 S z i (0) + W : (2.35) The spin-spin correlation function between spinsi andj C(i;j) = hS z i S z j i = X f;S L g S z i (0)S z j (0)W (;S L ) = * 1 L L1 X p=0 S z i (p)S z j (p) + W : (2.36) 30 The local bond energy is obtained via E b = J b hS i(b) S j(b) i W = J b hS z i(b) S z j(b) i W + J b 2 hS + i(b) S j(b) +S i(b) S + j(b) i W = C b hH 1;b i W hH 2;b i W = C b hN(b)i W ; (2.37) whereN(b) is the total number of off-diagonal operatorsH 2;b in the operator sequence S L [68]. The uniform and staggered static structure factors are defined as: [72] S jj (0; 0) = 1 N X i;j hS z i S z j i = 1 N * X i S z i (0) ! 2 + W ; (2.38) S jj (;) = 1 N X i;j (1) x j x i +y j y i hS z i S z j i = 1 N * 1 L L1 X p=0 X i (1) x i +y i S z i (p) ! 2 + W : (2.39) Now let’s see how can we measure an imaginary-time independent product A 2 ()A 1 (0) =e H A 2 e H A 1 : (2.40) After Taylor-expanding the exponentials, the ensemble average can be written as hA 2 ()A 1 (0)i = 1 Z X 1 X n=0 1 X m=0 () n () m n!m! hjH n A 2 H m A 1 ji (2.41) = 1 Z X 1 X n=0 n X m=0 () m () nm (nm)!m! hj n Y i=m+1 H l i A 2 m Y j=1 H l j A 1 ji: 31 Consider the first case ofA 1 ;A 2 diagonal hA 2 ()A 1 (0)i = * L1 X p=0 L X m=0 () m () Lm L (L 1)! (Lm)!m! a 1 (p)a 2 (p +m) + W : IntegratinghA 2 ()A 1 (0)i from 0 to gives Kubo integral. Using the periodicity of the propagated states gives in the diagonal case Z 0 dhA 2 ()A 1 (0)i = * L(L + 1) " L1 X p=0 a 1 (p) ! L1 X p=0 a 2 (p) ! + L1 X p=0 a 1 (p)a 2 (p) #+ W : Using the above equation we can measure the uniform and staggered susceptibilities at wave vectorq [69] (Eq.68) (q) = 1 N X i;j e iq(r i r j ) Z 0 dhS z i ()S z j (0)i: (2.42) The uniform susceptibility is (0; 0) = 1 N X i;j * L(L + 1) " L1 X p=0 S z i (p) ! L1 X p=0 S z j (p) ! + L1 X p=0 S z i (p)S z j (p) #+ W = NL(L + 1) * L1 X p=0 X i S z i (p) ! 2 + L1 X p=0 X i S z i (p) ! 2 + W : (2.43) The staggered susceptibility is (;) = 1 N X i;j * L(L + 1) e iq(r i r j ) " L1 X p=0 S z i (p) ! L1 X p=0 S z j (p) ! + L1 X p=0 S z i (p)S z j (p) #+ W = NL(L + 1) * L1 X p=0 X i (1) x i +y i S z i (p) ! 2 + L1 X p=0 X i (1) x i +y i S z i (p) ! 2 + W : (2.44) 32 The imaginary-time dynamical structure factor is defined as [45] S(q;!) = 1 N X i;j Z 0 de iq:(r i r j )i! hS z i ()S z j (0)i: (2.45) The time-averaged structure factor is corresponding with (! = 0) S(q; 0) = 1 N X i;j e iq(r j r i ) * L(L + 1) " L1 X p=0 S z i (p) ! L1 X p=0 S z j (p) ! + L1 X p=0 S z i (p)S z j (p) #+ W = 1 N * 1 L(L + 1) " X i;j e iqr i L1 X p=0 S z i (p) ! e iqr j L1 X p=0 S z j (p) ! + L1 X p=0 e iqr i S z i (p)e iqr j S z j (p) #+ W = 1 N * 1 L(L + 1) 2 4 X i e iqr i L1 X p=0 S z i (p) ! 2 + L1 X p=0 X i e iqr i S z i (p) 2 3 5 + W : The first term in the above summation is X i e iqr i L1 X p=0 S i (p) ! 2 = X i cos(qr i ) L1 X p=0 S z i (p) ! 2 + X i sin(qr i ) L1 X p=0 S z i (p) ! 2 = L1 X p=0 X i cos(qr i )S z i (p) ! 2 + L1 X p=0 X i sin(qr i )S z i (p) ! 2 : The second term in the above summation is * L1 X p=0 X i e iqr i S z i (p) 2 + W = * L1 X p=0 X i cos(qr i )S z i (p) ! 2 + X i sin(qr i )S z i (p) ! 2 + W : The time-averaged structure factor is actually the susceptibility. The staggered correla- tion length is evaluated by the second-moment method,q x = 2=Lx [45] x = 1 q x s S(;; 0) S( +q x ;; 0) 1; (2.46) 33 and the staggered magnetization is defined as p hm 2 s i = p 3S(;)=N. 2.3.4 MeasuringOff-DiagonalQuantities Measuring off-diagonal quantities requires more effort compared with measuring diag- onal quantities. Here we describe how to measure the equal-time structure factorS ? (q). This quantity is often used to study, for example, the Bose-Einstein condensation. First, it is reminder thatS l are hermitian conjugates, (S l ) y =S l and S q = 1 p N X l S l e iqr l : (2.47) ThenhS q S + q i is real becausehS q S + q i y =hS q S + q i. The equal-time structure factor is defined as: [77] S ? (q) = hS q S + q i +hS + q S q i (2.48) = 1 N X i;j hS i S + j +S + i S j i [cos(qr i )cos(qr j ) + sin(qr i )sin(qr j )] Measuring the equal-time structure factor requires insertions of local changes on certain world lines. Therefore, we need to slightly modify the loop update. Instead starting from a random leg, a worm is created at random propagation level on a random spin by inserting raising and lowering operatorsS , see Fig. 2.5. One of these operators is chosen to be the worm head and mobile, while the other one the worm tail is immobile. The worm head moves along the propagation level. When it hits a vertex at an entrance leg, a random number is generated and decides in which leg the worm exits. The exit leg is determined by the weights, Eqs. 2.28, and the bounce weights, Eqs. 2.30. If the worm head passes the initial propagation level, we add 1 to the sum of the structure factor Eq. 2.49. This process is continued and only stops when the worm head bites into 34 3 4 5 6 7 8 9 10 11 12 8 1 2 Spin 1 2 3 4 5 6 7 Propagation Level 0 Figure 2.5: (Color figure) Loop updates are constructed differently in order to measure the Green’s functions. its own tail and the loop closes. Measuring the Green’s functions is performed as long as the loop is still open. 35 Chapter3 CorrelationsinQuantumSpinLadders withSiteandBondDilution 3.1 Introduction Low-dimensional quantum Heisenberg antiferromagnets are known to exhibit non- trivial properties due to quantum fluctuations. A most striking example comes from spin ladders: interpolating between the chain geometry and the square lattice geome- try, they have provided an interesting new playground to explore the physics of low- dimensional strongly correlated electron systems [15]. Specifically, it has been shown that AF Heisenberg spin-1=2 ladders with an even number of legs are quantum spin liq- uids with purely short-range spin correlations. Their spin correlations decay exponen- tially due to a finite singlet-triplet gap in the energy spectrum. In contrast, Heisenberg ladders with an odd number of legs display properties similar to those of single chains at low temperature, namely gapless spin excitations, and power-law spin correlations. This difference between even-leg and odd-leg ladders was predicted by theory and confirmed by experiments in a variety of systems [15]. In this chapter, we focus on disordered spin ladders. For the case of doping with static non-magnetic impurities, one distinguishes between site and bond disorder. Site dilution occurs upon doping the magnetic ions with non-magnetic ones (as for instance substitution of Cu 2+ with Zn 2+ ). This leads to the formation of local moments (LMs) close to the dopant site [75, 68, 42]. Effective residual interactions between the LMs lead 36 to long-range antiferromagnetic ordering[30, 36, 96, 90, 89], experimentally detected in systems such as Sr(Cu 1z Zn z )O 3 (Ref. [3]) and Bi(Cu 1z Zn z ) 2 PO 6 (Ref. [8]). Bond dis- order occurs instead when the dopant ions replace the ions which act as bridges between the magnetic ions. For instance, bond disorder can be introduced in IPA-CuCl 3 by a partial substitution of non-magnetic Br for the likewise non-magnetic Cl , affecting the bond angles in the Cu-halogen-halogen-Cu superexchange pathways [27, 38, 40]. The reduced strength of magnetic interactions on the affected bonds leads to Bose-glass behavior when a strong magnetic field is applied. From the theory side, a number of studies have addressed the effects of site and bond disorder on Heisenberg ladders. Ref. [24] examined the correlation length of randomly site-diluted spin-1/2 Heisenberg 2-leg ladders at weak and intermediate interchain cou- plings, showing an apparent divergence of the spin correlations at low temperatures, due to the presence of the impurities. A related divergence of the staggered suscep- tibility has been reported in Ref. [50]. This behavior has been related to that of the random-exchange Heisenberg model (REHM) [20, 91], describing the effective random interactions between LMs [51, 75]. In a more recent study, three of us have investigated bond-diluted 2-leg ladders [99, 100] and observed enhancement of spin correlations due to bond dilution. However, a general understanding of the mechanisms for enhance- ment of correlations in site- and bond-diluted Heisenberg ladders is still lacking. It is also unclear whether such an enhancement of spin correlations generally exists for even-leg ladders with a number of legs greater than two. Here we present a study of site- and bond-diluted spin-1/2 AF Heisenberg ladders with n=2, 4 and 6 legs. Making use of quantum Monte Carlo simulations, we can address the correlation length of the system down to extremely small temperatures at which the asymptoticT! 0 behavior sets in. Our main findings can be summarized as follows: 37 1. For site-diluted even-leg ladders, we observe a strong enhancement of correla- tions upon doping up ton = 6 legs. In the specific case of 2-leg ladders, we find that the system realizes the physics of the REHM with a discrete distribution of the effective couplings, leading to a non-universal behavior at low temperatures; hence the known predictions for the universal regime of the REHM with a con- tinuous distribution of couplings are expected not to apply to realistic models of doped spin ladders. 2. For bond-diluted ladders, we find that correlations are suppressed by a low level of dilution, due to dilution-induced dimerization. Hence the system is first driven to a gapless phase with short-range correlations. The correlation length becomes logarithmically divergent for vanishing temperatures only beyond a critical dilu- tion. 3.2 ModelandQuantitiesofInterest We study the Heisenberg model onn-leg ladders H = J l n X m=1 L X i=1 p (l) i;m S i;m S i+1;m + J r n1 X m=1 L X i=1 p (r) i;m S i;m S i;m+1 (3.1) whereS i;m is the quantum spin operator at sitei = 1;:::;L along them-th leg. The first term in the Hamiltonian describes interactions along the legs with couplingJ l , whereas the second term represents the rung interaction with couplingJ r . The random variables p (l) i;m ,p (r) i;m express the site or bond dilution: in the case of site dilution,p (l) i;m = i;m i+1;m 38 andp (r) i;m = i;m i;m+1 , where i;m takes value 1 if the site (i;m) is occupied (with prob- ability 1 z) or 0 if it is empty (with probability z). In the case of bond dilution, p (r) i;m = p (l) i;m =2 take values 0 with probabilityz=3 if the bond is not occupied, and take values 1 otherwise. The simulations are performed using the Stochastic Series Expan- sion (SSE) quantum Monte Carlo (QMC) method based on the directed loop algorithm [79]. Periodic boundary conditions (PBC) along the leg direction are used. To access the regime of very low temperatures, at which the asymptotic low-T behavior of the correlation length sets in, we have made use a-doubling scheme [71], allowing us to efficiently access inverse temperatures up to = 4096. The main objective of this chapter is to study the correlation length along the leg direction, which is calculated via the disorder-averaged second-moment estimator: [14]: = L 2 s [S(;)] av [S( + 2=L;)] av 1: (3.2) HereS(q) is the time-averaged structure factor S(q) = 1 N X ij e iq(r i r j ) Z 0 dhS z i ()S z j (0)i; (3.3) withN =nL corresponding to the total number of sites. [:::] av denotes the disorder average, which is performed over 300-600 disorder realizations. 3.3 CorrelationLengthofSite-DilutedLadders 3.3.1 EffectiveInteractionsamongLocalizedMoments Antiferromagnetic even-leg ladders without doping display a rung-singlet ground state [15] with a finite gap to triplet excitations. In this state, then spins on the same rung 39 preferentially form a singlet state, and therefore effectively decouple from the rest of the ladder. This leads to exponentially decaying correlations in the direction of the legs, characterized by a finite correlation length 0 . At low enough concentration of dopants, the main effect of site dilution on an even- leg ladder is that of turning the number of spins on a rung from even to odd: in this situation, the state on the rung turns from a singlet into a doublet, which corresponds to an effectiveS = 1=2 localized moment (LM). ThisS = 1=2 moment remains expo- nentially localized (over a characteristic length 0 ) close to the impurity site [48, 68], but its finite overlap with other LMs leads to an effective interaction which generically decreases exponentially with the distance. In the case of dominant rung interactions, J r J l , the interaction between LMs is appropriately described within second-order perturbation theory (inJ l =J r ) as resulting from the exchange of virtual massive triplets between two LMs [75]. In the case of a 2-leg ladder, in which we can identify the loca- tion of a LM with a spin site next to a missing rung partner (dangling spin), the effective coupling between LMs has the form of a SU(2)-invariant Heisenberg interaction, and at large separation between LMs,jr i r j j 1, the coupling strength takes the form [75] J (e) ij (1) i+j+1 J 2 l exp(jr i r j j= 0 ) p jr i r j j= 0 ; (3.4) where the staggering factor takes value1 if the two LMs belong to the same sublattice and +1 otherwise. Eq. (3.4) holds strictly speaking only in the case in which J l , but a similar exponential decay of the effective LM coupling (without the square-root denominator) has been observed numerically in the case of much stronger leg interaction [48], as e.g. for J l = J r , for which 0:53J l . Moreover the above expression is numerically found to account for the decay of the effective couplings already for moderate distances, 40 jr i r j j & 0 . Hence, towards an effective model of the site-diluted ladders, we will assume in the following that the exponential decay of LM couplings with decay rate 0 remains valid even when perturbation theory is no longer applicable – which is the case of interest. In this chapter, we deliberately choose J r J l to have a sizable correla- tion length 0 , for reasons which will become clear in the following. Our assumption then implies that non-perturbative effects only affect the prefactor to the exponential in Eq. (3.4). In addition, we will assume that the exponential decay present in Eq. (3.4) sets in for distances between LMs of the order of 0 . In the case ofn-leg ladders withn> 2 a detailed theory of effective LM couplings is not available to our knowledge, but for widely spaced vacancies, the extended structure of the localized doublet becomes irrelevant, and for J l J r a perturbation approach analogous to that of 2-leg ladders should be applicable, leading to effective staggering couplings between LMs with exponential dependence on the distance. 3.3.2 StatisticalPropertiesoftheCouplingsfor2-LegLadders In the case of low doping, z 1, the probability distribution of having two nearest- neighboring LMs at a distanced along the leg direction in a 2-leg ladder is given by P (d) = 2z exp(2zd) (3.5) with a corresponding averagehdi = 1=(2z). As found numerically in Ref. [48], the effective coupling has the formJ (e) (d) J 0 exp(d= 0 ) ford& 0 . As an approxi- mation, one may assume that the same exponential behavior survives ford. 0 . Then 41 Figure 3.1: (Color figure). Probability distribution (P ) and cumulative distribution (E) of the effective couplings J (e) between nearest-neighbor localized moments in site- diluted 2-leg ladders, with parametersz = 2%, 0 = 7:5 (upper panels), andz = 2:5%, 0 = 15:3 (lower panels). The red dashed lines show the probability distributions in the continuum approximation according to Eq. (3.8). the probability distribution for the strength of the couplings between nearest neighboring LMs reads P J (e) nn = 1 X d=0 P (d)[J (e) nn J 0 exp(d= 0 )]: (3.6) 42 This result suggests the possibility of extracting analytically the probability distribu- tion for J (e) nn , as done in Ref. [75]. To this end, one may further take the continuum approximation, namely, approximate the summation in Eq. (3.6) by an integral: P J (e) nn Z dlP (l)[J (e) nn J 0 exp(l= 0 )]: (3.7) This allows then to write P J (e) nn 1 2J 0 J 0 J (e) nn ; (3.8) where = 1 2z 0 . Hence, within the continuum approximation the n.n. couplings obey a simple power-law distribution.[75] The average absolute value of the coupling between neighboring LMs takes then the form [75]: [jJ (e) nn j] av = 1 2 J 0 = 2z 0 1 + 2z 0 J 0 : (3.9) However, a critical analysis of the above derivation shows that the continuum approximation is problematic. In fact, it requires that the distancesd giving a significant contribution to the sum in Eq. (3.6) bed 1, which is in contradiction with the fact that P (d) decreases exponentially withd; the characteristic decay length ishdi = (2z) 1 , which forz 2% takes values 20. Fig. 3.1 shows the distribution P (J (e) nn ) determined numerically according to Eq. (3.6), i.e., by sampling the discrete distribution lengthsP (d), forz = 2%, 0 = 7:5 (corresponding toJ r =J l = 1=2, and giving = 0:7), andz = 2:5%, 0 = 15:3 (corre- sponding toJ r =J l = 1=4, and giving = 0:235) – these parameter sets will be relevant for our study of correlations in the following. We notice that the distribution shown in 43 Fig. 3.1 is only quantitatively correct whend& 0 , namely forJ e =J 0 . 1=e, while it is only an approximation otherwise. It is clear that for d & 0 , the distribution of n.n. couplings obtained numerically deviates strongly from the prediction of Eq. (3.8). Over most of its support, the distri- bution of n.n. couplings has a fundamentally discrete structure, due to the fact that the largest values ofJ (e) nn , which also take the largest probabilities, are associated with short separationsd among n.n. LMs. These probabilities grow as a power law of the absolute value of the coupling, opposed to what predicted by Eq. (3.8) for > 0. Obviously a coarse graining on the-peaks of the exact distribution would gradually reconstruct the shape of Eq. (3.8), but given the finite separation between the-peaks even in the ther- modynamic limit, this coarse graining is not justified over a large range ofJ (e) nn . In fact the-peaks become dense only in the limitJ (e) nn ! 0, which corresponds to larger and larger separations among the LMs, and hence to lesser and lesser probabilities. Only in that limit the continuum approximation appears legitimate (in fact numerically we recover the analytical prediction forJ (e) nn . 10 4 J 0 due to the finite width of our bins, J (e) nn = 10 5 J 0 ). In conclusion, for z & 1% and 0 10, P (J (e) nn ) has a remarkable structure: it assigns the largest probability to the range in whichJ (e) nn is a discrete variable, prevent- ing a straightforward approximation of the distribution with a continuous function. This will have significant consequences on the analysis of doped ladder systems in relation with models of randomly coupled spins. 3.3.3 CorrelationsasaFunctionofDoping In both cases ofn = 2 andn> 2 the unfrustrated effective couplings between LMs can lead to the formation of a gapless ground state with algebraically decaying correlations between the sites, and hence an infinite correlation length. When looking at a fixed finite 44 0 0.1 0.2 0.3 0.4 0.5 z 1 10 100 ξ 2-leg ladder 4-leg ladder 6-leg ladder 0 0.1 0.2 0.3 0.4 0.5 z 0 5 10 15 ξ β=1024 β=512 β=256 β=128 β=64 β=32 β=16 (b) (a) Figure 3.2: (Color figure). Correlation length of site-diluted ladders withL = 128 andJ l = J r = J. Upper panel: comparison between 2-leg, 4-leg and 6-leg ladders at inverse temperatureJ = 1024. Lower panel: correlation length of the 2-leg ladder at different temperatures. temperature T the correlation length is necessarily finite for one-dimensional SU(2) invariant systems with short-range interactions, but it is expected to increase because the average coupling between LMs increases withz, as in Eq. (3.9), so that the effective temperatureT=[jJ (e) nn j] av decreases. Despite the increase of correlations due to disorder, the range of correlations is fatally bounded by the fact that any finite amount of site dilution will break an infiniten- leg ladders into finite segments, of characteristic lengthhliz n (corresponding to the inverse linear density of rungs which are fully removed by doping). When considering doped ladders at finite temperatures no geometric bound on the correlation length (nei- ther coming from finite-size effects nor from ladder fragmentation) is present as long as the correlation length satisfies the condition (T )Lhli: (3.10) 45 It is clear that such a condition can only be satisfied when the doping is sufficiently weak. Hence, for any fixed temperature, upon increasing the doping concentration the correlation length(T ;z) will cross over from a growing behavior at low dopingz 1 to a decreasing behavior for z & 0:1, hence going through a maximum for a given optimal doping valuez = z (J r =J l ;n), which is quite stable at sufficiently low tem- peratures. This is indeed seen in Fig. 3.2(a), in which we observe that, for J r = J l , z 0:06 forn = 2, 4, and 6. (It is to be said that, at the low temperature to which the figure refers (T =J l =1024), the correlation length on the doped 4-leg and 6-leg ladders grows to values which are no longer satisfying the condition(T )L, and hence it is limited by the system size (L = 128) explored there.) For any finite value of doping and at arbitrarily low temperatures, the correlation length is upper bounded by the average length of the ladder segments,hli. Yet, as shown in Fig. 3.2(b), at low enough doping the upper bound is reached extremely slowly in tem- perature - in fact, for the 2-leg ladder atT =J l =1024 andJ l =J r , the correlation length is still not saturated to its finiteT = 0 value forz. 0:25. It is easy to understand that this occurs because the average effective coupling [jJ (e) nn j] av decreases asz decreases, leading to exceedingly weak correlations even at extremely low temperatures. This means that for sufficiently low doping, the condition (T )hli is actually satisfied over all numerically accessible low temperatures (and quite possibly also over all experimentally accessible temperatures), so that the segmented nature of the doped ladders become in fact irrelevant. In the following we will focus our discussion on this regime. 46 0.001 0.01 0.1 T 10 100 ξ z=2.0% (J r =0.25, α=0.199, L=384) z=2.6% (J r =0.25, α=0.204, L=512) z=2.0% (J r =0.5, α=0.165, L=256) z=3.9% (J r =0.5, α=0.176, L=384) z=6.4% (J r =0.5, α=0.158, L=256) z=7.4% (J r =0.5, α=0.144 L=256) Figure 3.3: (Color figure). Temperature dependence of the correlation length of site- diluted 2-leg ladders withJ l = 1, and power-law fits at low temperature. 3.3.4 Doped2-LegLaddersandtheRandom-ExchangeHeisenberg Model As discussed in section 3.3.2, from the statistical point of view the system of interacting LMs randomly distributed on the bipartite 2-leg ladder can be approximated by a system ofS = 1=2 spins interacting with randomly distributed couplings, following an intrinsi- cally discrete distribution. In general terms, a doped 2-leg ladder represents a physical realization of the so-called random-exchange Heisenberg model (REHM) [91, 92] , but with a special structure of the coupling distribution. Numerical real-space renormalization group studies [91, 92] show that the REHM colorred for a continuous distribution of couplingsP ( ~ J) has two regimes: Universal regime: if the distribution is not singular, or has a power-law singular- ity P ( ~ J) J with < c (where 0:65 . c . 0:75), the low-temperature thermodynamics of the REHM is dominated by a unique fixed point; the correla- tion length diverges asT! 0 with a universal scaling exponent, T 2 with = 0:22 0:01. This result has been confirmed by quantum Monte Carlo [19] for a non-singular distribution (square box, = 0). 47 Non-universal regime: for a more strongly singular distribution > c the renor- malization group flows to a non-universal fixed point, which depends on the initial coupling distribution. Studies on doped coupled spin-Peierls chains using Real Space Renormalization Group and SSE QMC in this regime also confirm that the exponent depends on the doping concentration[37]. As discussed in Sec. 3.3.2, the distribution of couplings between LMs in a doped 2-leg ladder is intrinsically discrete, approximating a continuous distribution only for vanishing couplings. It is therefore interesting to ask whether the doped 2-leg ladder reproduces the known physics of the REHM model with a continuous distribution of couplings. If a continuous approximation to the coupling distribution is in order, as in Eq. (3.8), the system should exhibit a universal REHM behavior for 2z 0 > 1 c , and a non-universal behavior for 2z 0 < 1 c . In other words, ladders with sufficiently strong dopingz or sufficiently large correlation length 0 in the undoped limit should exhibit universal scaling properties at low temperature. To our knowledge the only numerical study of correlations in site-diluted ladders is Ref. [24], which, making use of quantum Monte Carlo, investigates ladders with J r =J l = 1=2 and dopings z = 0:01, 0.04 and 0.1. Given that 0 7:5 in this case [24], forz = 0:04 and 0.1 one obtains 1 2z 0 = 0:4 and0:5 respectively, which appear to be both safely on the universal side of the REHM with continuous couplings. Indeed a fit to the low-temperature behavior of the correlation length to the scaling law = A(z) +B(z) T 2(z) gives an estimate of = 0:2 0:025 (independent of z) which appears consistent with the predictions of Refs. [91, 92]. 3.3.5 Low-TemperatureScalingoftheCorrelationLength We have performed QMC simulations to extensively study the low-temperature behav- ior of the correlations in site-diluted 2-leg ladders. Our QMC results, however, do not 48 0.001 0.01 0.1 T 1 10 ξ z=0% (Ref. 10) z=1% (Ref. 10) z=4% (Ref. 10) z=10% (Ref. 10) z=0% z=1%, α=0.113 z=4%, α=0.178 z=10% Figure 3.4: (Color figure). Comparison of our data with those by Ref. [24]. Our data refer to ladders withL = 384, andJ l = 2J r = 1. support the conclusion in Ref. [24] about a universal REHM behavior of diluted 2-leg ladders. We investigate doped ladders at similar concentrations with respect to those of Ref. [24], down to significantly lower temperatures - Ref. [24] stops atT = J l =500, while we descend down toT =J l =2048. A collection of our results is shown in Fig. 3.3. In qualitative agreement with what expected from the theory of the REHM model, we also observe the clear onset of a low-temperature power law scaling of the typeT 2 . Yet our central observation is that this power-law scaling exhibits a non-universal, dop- ing dependent exponent =(z). For all the parameter sets we considered, a fit of the low-temperature correlation length provides values of which are systematically below the one exhibited by the universal regime of the REHM with continuous couplings. The data shown in Fig. 3.3 refer to 2-leg ladders withJ r =J l = 1=4 andz = 2:0% and 2:6%, with corresponding 0:39 and 0:235 respectively (here we use 0 = 15:3, which we estimate independently with QMC); and 2-leg ladders with J r =J l = 1=2 (as in Ref. [24]) and z = 2:0%, 3:9%, and 6:4%, corresponding to 0:7, 0:415, and 0:04 respectively. Although all the values of fall below the critical value c , as shown in Fig. 3.3, in fact, all the corresponding data sets for the correlation length display a non-universal power-law scaling. In particular, we observe that data sets with 49 0 0.01 0.02 0.03 0.04 0.05 z 0 0.2 0.4 0.6 J r /J l 0.175 0.142 0.196 0.204 0.199 0.176 0.164 0.154 0.145 0.185 0.167 0.186 0.113 Figure 3.5: (Color figure). Values of the exponent extracted from the low-temperature behavior of site-diluted 2-leg ladders. close values of the parameter (e.g. J r =J l = 1=4, z = 2:0% with 0:39, and J r =J l = 1=2, z = 3:9%, with 0:415) show a clearly different scaling. We also critically revisit the parameter sets explored in Ref. [24]. A comparison to our data, as done in Fig. 3.4, shows that in the data of Ref. [24] the significant scattering due to the statistical uncertainty is masking the correct asymptotic scaling regime at low temperature. A comprehensive summary of our results on the low-temperature scaling of the cor- relation length is provided in Fig. 3.5, where we report the values of the exponent obtained for different doping values andJ r =J l ratios. The values of doping concentra- tions and ratios were chosen such that the average distance between LMshdi is at least larger than 0 . The LMs, therefore, are not overlapping on average and can be described by an effective REHM with couplings obeying Eq. (3.4). Furthermore, for all the points shown in Fig. 3.5, < c , we do not observe a universal value. Indeed, it is found that depends not only on the doping concentration but also on the ratio of couplings J r =J l . And an exponent close to that of the universal REHM regime is only observed in a narrow region aroundJ r =J l 1=4 andz 2%. 50 3.3.6 Discussion The above results lead us to conclude that doped 2-leg ladders do not realize in general the universal physics of the REHM with a continuous coupling distribution. In fact, the coupling distribution for the LMs in doped 2-leg ladders has a fundamentally discrete structure, approximating a continuous one only for vanishing couplings. The weight assigned by the distribution to strong discrete couplings is dominant over that of weak, quasi-continuous couplings in the case of sizable dopingz (suppressing the probability of large separation between the LMs) and of large correlation length 0 1 (increasing the strength of the couplings between LMs, the distance being fixed). Nonetheless, this is the regime in which the parameter = 1 2z 0 is small, and in which one would expect the universal REHM physics to be manifested in the continuum approximation. On the other hand, the quasi-continuous part of the distribution (corresponding to very small couplings) acquires a bigger weight for smallz and 0 , which determine a small average coupling according to Eq. (3.9) and hence increase the probability of the small values of the coupling. In this regime is closer to 1, which leads to too strong a singularity in the probability distribution for the universal regime of the REHM with continuous couplings to be manifested. As a consequence, we generally observe a low-temperature power-law scaling of the correlation length with doping-dependent exponents. As far as the numerical results are concerned, this scaling appears to be the asymptotic one forT ! 0, given that it sets in for temperatures well below the average LM coupling. On the other hand, accord- ing to a real-space renormalization group (RSRG) approach [91], the system at a lower temperature is increasingly sensitive to the part of the distribution related to the weaker couplings; given that this part is the one which best approximates a continuous distribu- tion, one could naturally expect that at very low temperatures the physics of the REHM with continuous couplings be reproduced by the doped 2-leg ladders. This argument 51 would then suggest that, when lowering the temperature, the system will sooner or later attain the universal regime of the REHM with continuous couplings if < c . Nonethe- less one can argue that in a doped 2-leg ladder the RSRG flow has to be necessarily stopped at a finite length, corresponding roughly to the average segment lengthhli. In practice this imposes an upper bound to the correlation length and a lower bound to the temperature through the condition Eq. (3.10); these bounds might prevent one from attaining the low-temperature regime at which universal REHM physics would manifest itself. Finally, our study leaves one open question concerning the role of discreteness of the initial coupling distribution for the physics of the REHM. The numerical RG study of Ref. [92] only addresses initial continuous distributions; a systematic RG study of the flow starting from realistic distributions stemming from the doped-ladder physics would be highly desirable. 3.4 Bond-DilutedLadders In the following, we consider the doping- and temperature dependence of correlations in bond-diluted ladders, which show a marked difference with respect to site dilution. This is due to a fundamental geometric aspect which distinguishes site and bond dilution: diluting a site leaves a single unpaired spin, giving rise to a LM, while eliminating a bond leaves always two unpaired spins, located on different sublattices. The corresponding LMs are therefore interacting with an effective antiferromagnetic coupling, mediated by the the shortest path of bonds connecting them. In the following, we will focus for simplicity on the caseJ l =J r =J. 52 3.4.1 2-LegLadders EvolutionofCorrelationswithDoping Fig. 3.6 shows the evolution of the correlation length at a fixed, low temperatureT = J=1024. A striking difference with respect to the case of site dilution, shown in Fig. 3.6, is that the correlation length does not increase immediately with doping: it remains essentially constant forn = 2, while it even decreases forn> 2. In the case of a 2-leg ladder, we can understand qualitatively the behavior of correla- tions by considering that bond dilution has two fundamentally different effects (sketched in Fig. 3.7(b)): if a rung bond is diluted, two LMs are liberated but still interact antiferromagnet- ically, so that they can screen each other to form a rung singlet at a lower energy J (e) rung (corresponding to the effective coupling between LMs which are rung neigh- bors). This screening is very effective at low dilution because in that case theJ (e) rung interaction is largely dominant over the interaction between two LMs belonging to different rungs, due to the large spacing of two diluted rung bonds – this spacing is 3=z (given that only one bond in three is a rung bond); if a leg bond is diluted, the rung-singlet state on the two adjacent rungs is further reinforced because its connectivity to the other rungs is lowered. The dilution of rung bonds has therefore the effect of lowering locally the gap over the ground state, whereas the dilution of leg bonds has the effect of increas- ing it. (The notion of local gap is associated to the local susceptibility at site i, i = (1=) R 0 d hS i ()S i (0)i. Assuming thathS i ()S i (0)i exp( i ) one obtains that i 1 i , where 1 i is the local gap.) The fact that there are twice as many leg bonds as rung bonds suggests that the second effect dominates over the first. 53 0 0.1 0.2 0.3 0.4 0.5 z 1 10 100 ξ 2-leg ladder 4-leg ladder 6-leg ladder Figure 3.6: (Color figure). Correlation length of bond-diluted 2-, 4-, and 6-leg ladders withL = 128,J l =J r =J, and temperatureT =J=1024 Therefore the spin gap of the undoped ladder is locally preserved or even enhanced, lead- ing to short-range correlations even in presence of bond dilution. Nonetheless, standard consideration on rare-event physics lead to conclude that even a very weak bond dilu- tion leads the system to a gapless, Griffiths-like phase. In fact, the gap can close locally if rare regions appear in which only rung bonds are diluted – the limiting case being that of the formation of local strands made of two uncoupledS = 1=2 chains. These exponentially rare regions lead to the closing of the spin gap in the thermodynamic limit; yet global correlations remain short ranged, due to the localized nature of the rare, locally gapless regions. This Griffiths phase is reminiscent of what has been observed by some of us in the Heisenberg model on the inhomogeneously bond-diluted square lattice [99, 100]. In that case, the predominance of correlation-suppressing dilution was guaranteed by the inhomogeneity of doping probabilities (favoring the appearance of ladder-like or dimer-like structures). For a sufficiently large bond dilution z > z (0:07 . z . 0:09), the correlation length starts increasing with increasing z. This implies necessarily that the liberated LMs belonging to different rungs begin to correlate with each other (Fig. 3.7(c), see also Refs. [100, 95]). Such correlations appear because the spacing between diluted 54 Figure 3.7: (Color figure). Sketch of the bond dilution effects on 2-leg ladders. a) pure ladder; b) ladder at low dilution: enhancement of rung singlets by leg-bond dilution, and low-energy singlets formed between LMs after rung-bond dilution; c) ladder at stronger dilution: couplings between LMs belonging to different diluted rungs. 0 0.1 0.2 0.3 0.4 0.5 z 1 10 ξ n=2: random dilution n=4: random dilution n=2: rung bond dilution n=4: outer-rung bond dilution n=2: geometric length n=4: geometric length 0.227 0.219 Figure 3.8: (Color figure). Correlation length for 2-leg and 4-leg ladders with dilution of (center-)rung bonds. The average length of ladders segmentshli is obtained upon homogeneous dilution. rung bonds decreases, and hence the couplings between LMs belonging to different rungs become of the same order ofJ (e) rung . We have isolated the effect of enhancement of correlations through rung-bond dilution by considering uniquely this form of dilution, namely by takingp (r) i;m = 0 with probabilityz, whilep (l) i;m = 1 with probability 1. The resulting correlations r (z) (r = rung) as a function of z for 2- and 4-leg ladders are shown in Fig. 3.8, and they are seen to increase quite fast with dilution (following the approximate form r (z) 0 exp(az) for sufficiently low dilution). Yet in the original 55 system the correlations induced by rung-bond dilution are upper-bounded by the char- acteristic length of the ladder segmentshli, which we have estimated by generalizing an efficient algorithm recently developed for homogeneous percolation [52, 53, 76]. It is found that at finitez,hli a=z 2 . It follows the same asymptotic behavior as from the naive estimatehli 1=z 2 , but with a renormalized factora < 1. Hence the enhance- ment of correlations due to rung-bond dilution competes with this geometrical restric- tion on correlations for sufficiently strong doping. Quite remarkably, we observe that the optimal doping for the enhancement of correlations in the homogeneously doped 2-leg ladder coincides with the value of dilution (z 22%) at which the correlation length of the rung-only diluted ladder equals the average length of ladder segmentshli. Therefore we can conclude that the non-monotonic behavior of the correlation length at intermedi- ate dilution values is completely determined by the competition between the correlation enhancement through rung-bond dilution and the correlation suppression due to ladder fragmentation. Evolutionofcorrelationswithtemperature As discussed in the previous subsection, for z = z correlations start to increase as a function of bond dilution. Fig. 3.9(a) shows the doping dependence of the correla- tion length of 2-leg ladders for higher temperatures than those considered in Fig. 3.6: here we observe that the correlation length does not appear to evolve significantly with temperature for doping valuesz . z , whereas it becomes significantly dependent on temperature forz > z . Moreover, the low-temperature dependence of the correlation length, see Fig. 3.9(b), indicates that in this regime the correlation length grows loga- rithmically with decreasing temperature as A(z) +B(z)j log(T=J)j: (3.11) 56 0 0.1 0.2 0.3 0.4 0.5 z 1 2 3 4 ξ β=1024 β=512 β=256 β=128 β=64 β=32 β=16 10 -3 10 -2 10 -1 T 1 2 3 4 ξ z=0% z=3.1% z=6.25% z=9.4% z=12.5% z=15.6% z=18.7% z=21.9% 0 0.1 0.2 0.3 z 2 2.5 3 3.5 A(z) 0 0.1 0.2 0.3 z 0 0.1 0.2 0.3 B(z) (b) (a) (c) (d) Figure 3.9: (Color figure). (a) Correlation length of bond-diluted 2-leg ladders with L = 128 and J l = J r = J at various temperatures. (b) Temperature dependence of the correlation length of above system. The curves withz > 22% are omitted due to is limited by the geometric length. (c) and (d) Fitting parameters of correlation length A(z) +B(z)j log(T=J)j, as a function of bond dilution. 10 -4 10 -3 10 -2 10 -1 10 0 T 1 2 3 4 5 ξ z=18.75%, random dilution z=21.9%, random dilution z r =6.25%, rung-bond dilution z r =7.3%, rung-bond dilution Figure 3.10: (Color figure). (a) Temperature dependence of correlation lengths for four cases of rung-bond and random dilution. The dependence of the coefficients A(z) and B(z) on bond dilution is shown in Fig. 3.9(c) and (d). WhileA(z) decreases with increasing bond dilution,B(z) increases. 57 In particular B(z) appears to vanish for z . z , and it seems to suggest that a phase transition occurs at finitez = z from a phase with correlations converging to a finite value at low T to a phase with correlations diverging logarithmically with decreasing temperature. Obviously the divergent behavior of correlation length only persists up to lengths of the orderhli, corresponding to the characteristic length of ladder segments, but the exceedingly slow growth of with decreasingT keeps it safely belowhli for all the temperatures and doping values we explored. To gain a deeper understanding on the above results, we study the temperature dependence of rung-bond dilution and random dilution separately. We consider rung- bond dilution with concentrationsz r =z=3 such that the fraction of diluted rung bonds is the same as for the cases of random dilution that we have investigated. This allows us to quantitatively ascertain the separate effect of leg-bond and rung-bond dilution. In presence of rung-bond dilution only, the correlation length is seen to converge to a finite value asT ! 0, even at relatively high concentration. Two examples with concentra- tion z r = 6:25% and 7:3% are shown in Fig. 3.10 (data for higher concentration, not shown here, exhibit a similar behavior). This implies that the system with rung-bond dilution only is gapless (because of the unbounded size of rare regions with e.g. diluted adjacent rung bonds) and it has a finite correlation length, namely it is in a Griffiths-like phase. Adding leg-bond dilution (with concentrations 12% and 14:6% respectively), we recover a randomly bond-diluted ladder (with concentrations 18:75% and 21:9% respec- tively). For large enough dilution the correlation is found to grow logarithmically with decreasing temperature, as in Eq. (3.11), which seems atypical for a Griffiths-like phase. On the other hand, we observe that, for dilution z r = z=3, the correlation length for the rung-bond diluted ladder with concentrationz r is an upper bound to that of the ran- domly diluted ladder with concentrationz, a fact which is somewhat intuitive, given that rung-bond dilution is the main mechanism leading to the enhancement of correlations. 58 Therefore it is a priori unclear whether the logarithmic growth of the randomly diluted ladder persists to even lower temperatures than the ones explored here, because this would eventually lead the correlation length of the randomly diluted ladder to exceed that of the rung-bond diluted one. We can then conclude that the correlation length results are a priori consistent with two different scenarios for the randomly diluted 2-leg ladders. In a first scenario the system transitions from a Griffiths-like phase with a finite correlations for z < z to a new phase with logarithmically diverging correlations, which is reminiscent of the behavior of a system controlled by an infinite-randomness fixed point (IRFP) – see the discussion below. In a second scenario the correlation length exhibits different temper- ature dependences at intermediate temperatures for increasing bond dilution. Yet the correlation length of the randomly diluted ladders converges always to a finite value, because it is expected to be upper-bounded by that of rung-bond diluted ladders, which is seen to converge to a finite value. In this second scenario the system is therefore in a Griffiths-like phase for all the values of bond dilution we explored. In the following we find that the analysis of the temperature dependence of the uniform susceptibility helps clarifying which scenario is the most appropriate. Temperaturedependenceoftheuniformsusceptibility Fig. 3.11(a) shows the uniform susceptibility of a 2-leg ladder for various values of bond dilution. In the undoped system, the susceptibility of 2-leg ladders vanishes exponen- tially at low temperatures, in agreement with well-known previous results, and showing the gapped nature of the spectrum. In contrast, the susceptibility of the doped system diverges following a Curie law for low temperatures. Furthermore, Fig. 3.11(b) shows that the Curie coefficient increases as a function of bond dilution concentration follow- ing a power law, C(z) z 2:66 . One might expect that Curie paramagnetism comes 59 10 -3 10 -2 10 -1 T 10 -5 10 -4 10 -3 10 -2 10 -1 Tχ u z=0% z=3.1% z=6.25% z=9.4% z=12.5% z=15.6% z=18.75% z=21.9% 0.01 0.1 z C(z) (Curie coefficient) C(z)~z 2.66 (a) (b) Figure 3.11: (Color figure). (a) Low-temperature uniform susceptibility of bond-diluted ladders withL = 256 andJ l =J r = 1. (b) The Curie coefficient increases as a power- law function of the bond dilution concentrationC(z)z 2:66 . trivially from spins which have remained isolated after bond dilution. Yet the concen- tration of such spins would scale asz 3 with dilution, given that one needs to dilute at least 3 bonds to decouple one spin from the rest of the ladder. Therefore the paramag- netism observed in the system has a collective nature. In fact, two contributions add to the one of free spins: odd-numbered clusters, obtained after fragmentation of the ladder into two (or more) pieces, have a doublet ground state, behaving as a collective spin 1/2. In our simulations on systems with an even total number of sites, three contiguous diluted bonds lead to an isolated site plus an odd-numbered cluster, so that the latter clusters have a probability z 3 . Alternatively odd-numbered clusters can be obtained by cutting the ladder at two distinct locations into two odd-numbered extended clusters (bigger than one single site). Yet one can easily see that this 60 10 0 10 1 |Log(T)| 10 -5 10 -4 10 -3 10 -2 10 -1 Tχ u,c z=0% z=3.1% z=6.25% z=9.4% z=12.5% z=15.6% z=18.75% z=21.9% 10 -3 10 -2 10 -1 10 0 T (a) (b) Figure 3.12: (Color figure). (a) Uniform susceptibility of even-numbered spin clusters of randomly bond-diluted 2-leg ladders. The solid lines are (a) logarithmic fits of the form u;c =C=(Tj logTj ) and (b) power-law fits u;c =C 0 T 0 1 . would require at least 5 diluted bonds at specific locations, giving rise to a proba- bilityz 5 , and consequently to a very small contribution to the uniform suscep- tibility. Therefore the contribution to collective paramagnetism cannot come from such clusters. clusters with an even number of spins, but a non-equal number of spins on A and B sublattices, can also contribute to a diverging susceptibility. They represent therefore a further candidate for collective paramagnetism. To isolate the second contribution, we calculate the uniform susceptibility coming from even-numbered spin clusters only, u;c . This quantity is shown in Fig. 3.12. We find that even-numbered uniform susceptibility at low temperatures can be equally well described by two fitting laws: a logarithmically-corrected Curie behavior: u;c = C Tj logTj ; (3.12) 61 and a power-law corrected Curie behavior: u;c = C 0 T 1 0 ; (3.13) In both cases we can conclude that the contribution to the uniform susceptibility coming from even-numbered clusters diverges more weakly than for a Curie law, and therefore that our system exhibits a highly non-trivial collective paramagnetism. The fitting coef- ficientsC;, andC 0 ; 0 for both cases are shown in Fig. 3.13. We observe that all these fitting coefficients evolve smoothly withz, apparently contradicting the picture of a pos- sible phase transition suggested by the behavior of the correlation length. In particular the non-universal power-law dependence of Eq. (3.13) is typical of Griffiths behavior, as observed e.g. in anisotropically bond-diluted square lattices [100]. Therefore the uniform susceptibility seems to favor the scenario for which the system is in a Grif- fiths phase for all values of the bond dilution we explored. For completeness we should also mention a third, less likely scenario, for which the correlation length is diverging logarithmically asT ! 0 even for infinitesimal doping (but the prefactorB(z) of the logarithmic divergence becomes lower than our resolution). Therefore the system would be controlled by an IRFP for all doping values explored here; this scenario is somewhat consistent with the fact that a logarithmically corrected Curie law is consistent with the susceptibility at all doping values considered in this section, and that it could be another sign of IRFP behavior. Comparisonwithrandomladders As sketched in Fig. 3.7, the rung LMs in a bond-diluted ladder realize a ladder with weaker random couplings which are at once rung, leg and diagonal ones but without 62 0 0.1 0.2 0.3 10 -5 10 -4 10 -3 χ u,c =C’T β’-1 0 0.1 0.2 0.3 10 -5 10 -4 10 -3 χ u,c =C/(T |log T| β ) 0 0.1 0.2 0.3 z 0 0.02 0.04 0.06 0 0.1 0.2 0.3 z 0 0.1 0.2 0.3 C C’ β β’ Figure 3.13: (Color figure). Fitting coefficients as a function of bond dilution. Left panels are fitting coefficients of the logarithmic function u;c = C=(Tj logTj ). Right panels are fitting coefficients of the power-law function u;c =C 0 T 0 1 . frustration. The rung couplings are randomized by the variety of different local envi- ronments which mediate the effective interaction between the rung LMs; while the leg couplings and the diagonal couplings are obviously randomized (in a correlated way) by the positional disorder of the diluted rung bonds. For ladders withJ l =J r we cannot easily rely on perturbation theory to extract accu- rate expressions for the effective couplings, and an explicit calculation of the disorder statistics goes beyond the purpose of this dissertation. In any instance it is clear that the rung couplings obey a fundamentally different probability distribution with respect to the leg and the diagonal couplings: the rung couplings are always antiferromagnetic and weakly randomized, while the leg and diagonal couplings always take opposite signs and can be either ferromagnetic or antiferromagnetic - they are hence strongly disor- dered. To the best of our knowledge, such a peculiar model of a random ladder has not been investigated before. 63 In a critical quantum spin system the relationship between energy scales ( ) and length scales (T at finite temperature) is governed by the so-called dynamical critical exponent ~ z, T 1=~ z . The logarithmic dependence of the correlation length on tem- perature, Eq. (3.11), implies that ~ z =1: as mentioned before, this behavior, together with the logarithmically corrected Curie law in the susceptibility, could be suggestive of the fact that the bond-diluted ladder at sufficiently strong disorder is governed by an IRFP, similarly to what happens to e.g. random antiferromagneticS = 1=2 chains [17]. Nonetheless, it is not the same IRFP as in random antiferromagnetic chains, given that the correlation length and the susceptibility diverge differently (asj log(T=J)j 2 and as (Tj logTj 2 ) 1 respectively) in the latter. Existing RG studies of disordered ladder systems focus primarily on the case of antiferromagnetic ladders, either with nearest neighbor couplings only or with frustrated ones [104, 46, 28, 29]. The occurrence of extended parameter regions with IRFP is only observed in the case of frustrated ladders. Therefore it is quite remarkable to observe an IRFP-like behavior in our unfrustrated system for a large interval of doping values. On the other hand, the absence of a doc- umented IRFP for unfrustrated random ladders suggest that the logarithmic divergence observed in our system might not be the true asymptotic behavior forT! 0. Nonethe- less, a definite conclusion on such an asymptotic behavior would require a precise deter- mination of the distribution of the effective couplings between LMs, which goes beyond the scopes of this dissertation. 3.4.2 4-Legand6-LegLadders Fig. 3.6 shows that the dependence of the correlation length on bond dilution in 4-leg and 6-leg ladders exhibits a qualitatively different behavior compared to 2-leg ladders. Indeed for 4-leg and 6-leg ladders is initially decreasing with increasing bond dilution. In the case J l = J r = J the correlation length reaches a minimum for z 6:2%, 64 Figure 3.14: (Color figure). Bond-dilution effects on 4-leg ladders: (1) rung-singlet enhancement upon leg-bond dilution; (2) center-rung dilution leading to the formation of two 2-spin rung singlets; (3) outer-rung dilution with formation of a low-energy singlets between a LMs and a 3-spin doublet. and then starts growing with doping, up to a maximum attained for z 22%. In the following we will analyze this phenomenon in details. Similarly to what done for 2-leg ladders, we can distinguish among different effects of doping depending on which bond is diluted; we will focus here on the case of a 4-leg ladder (which is sketched in Fig. 3.14): 1. the dilution of a leg bond leads to the the enhancement of the 4-spin rung singlet on the two adjacent rungs; 2. the dilution of a center rung bond leads to the formation of two 2-spin rung sin- glets on the intact outer rungs; 3. the dilution of an outer rung bond separates an outer LM from three rung spins, which tend to form locally a spin-doublet. This doublet should have then tendency to form again a low-energy singlet with the LM through the effective antiferro- magnetic coupling which binds them. Hence we observe that bond dilution of type 1) and 2) have the tendency to enhance locally the spin gap of the ladder, while dilution of type 3) has the tendency to suppress it. The case of 6-leg ladders is analogous, with the only difference that dilution of 65 the central rung bond leads to the formation of two rung doublets. Therefore we will hereafter focus on the 4-leg case for simplicity. We have disentangled the competing effects of dilution on 4-leg ladders by consid- ering only one at a time. First of all we consider the effect of enhancement of cor- relations due to outer rung-bond dilution only, whose results are shown in Fig. 3.8. Analogously to what has been seen for rung-bond dilution in the 2-leg ladders, outer- rung-bond dilution in 4-leg ladders leads to a strong enhancement of correlations (with a similar exponential scaling or (z) 0 exp(a 0 z), or=outer rung). In homogeneously diluted ladders this effect of enhancement is fundamentally limited by the finiteness of ladder segments (due to leg-bond dilution): indeed a numerical estimate of the average length of ladder segments,hli, shows thathli crosses the correlation length or for a value of z ( 23%) which nicely corresponds to the optimal doping in the homoge- neously bond-diluted ladder. Hence, similarly to 2-leg ladders, we can conclude that the optimal doping is completely dictated by the competing effect of outer-rung-bond dilution and ladder fragmentation. Finally, we study the effect of selective dilution on the correlation length. In Fig. 3.15(a), we investigate the effect of rung-bond and leg-bond dilution in 2-leg lad- ders. In Fig. 3.15(b), we distinguish dilution of outer-rung bonds, center-rung bonds, and leg bonds in 4-leg ladders. As anticipated, latter forms of dilution in above cases lead to a significant suppression of correlations, again with approximate exponential scaling, l;cr (z) 0 exp(b 0 z) (l = leg, cr = center rung). We observe that b 0 a 0 (b a for 2-led ladder), so that the simultaneous suppression and enhancement of correlations due to dilution of different types of bonds are in strong competition. The fact that at low doping the suppression of correlation wins is simply due to combinatorics: in 2-leg ladders (a), rung bonds only hold a fractionf r = 1=3 while leg bonds containf l = 2=3 of all bonds; in 4-leg ladders (b), outer-rung bonds only represent a fractionf or = 2=7 66 0 0.1 0.2 0.3 0.4 0.5 z 1 10 ξ random dilution leg-bond dilution rung-bond dilution average 0 0.1 0.2 0.3 0.4 0.5 z 1 10 0 1 10 random dilution leg-bond dilution central rung-bond dilution outer rung-bond dilution average (a) (b) Figure 3.15: (Color figure). Correlation length with selective bond dilution. Bond dilution is distinguished as: (a) rung-bond and leg-bond dilution in 2-leg ladders; (b) center-rung bond, outer-rung bond and leg bond dilution in 4-leg ladders. The dashed line is the average correlation length from above selectively bond-diluted mechanisms of all the bonds, whereas leg bonds and center-rung bonds are respectively f l = 4=7 andf cr = 1=7. Hence correlation-enhancing bond dilution has a global lower probabil- ity than correlation-suppressing one. In the limit of very low dilution (z 1= 0 ) we can easily imagine to separate a ladder into correlated regions (of characteristic length 0 ) which are statistically independent from each other. Upon dilution, each correlated region will only be affected by a single missing bond on average, with a probabilityf i for thei-th bond type (i = r, l in 2-leg ladders; andi = or, l, cr in 4-leg ladders). Therefore the local correlations will be enhanced or suppressed in the same way as for the case of selective bond doping of typei. Forz 1= 0 it is then tempting to write the correlation length of the homogeneously diluted ladder as a spatial average of different local corre- lation lengths. Assuming that the ratio of structure factors entering in Eq. (3.2) can be written as [S(;)] av [S( + 2=L;)] av X i f i [S(;)] av;i [S( + 2=L;)] av;i (3.14) 67 where [:::] av;i denotes averaging over disorder realizations with dilution of bond ofi-th type only (and with dilution concentrationf i z), we obtain that (z) " X i f i 2 i (f i z) # 1=2 : (3.15) This average correlation length is presented by the dashed lines in Fig. 3.15. It clearly shows a quantitative agreement with random bond dilution results at small dilution. 3.5 Conclusions In this chapter we have discussed the effects of site and bond dilution on the low- temperature correlations of even-leg S = 1=2 ladders with antiferromagnetic Heisen- berg interactions. Site dilution is found to prune rung singlets and thus create local- ized moments which interact through unfrustrated, distance dependent couplings. The Hamiltonian describing the effective interaction between these moments is a random exchange Heisenberg model (REHM), with a gapless spectrum and power-law diverg- ing correlations as temperature decreases to zero. We find that the distribution of the effective couplings has an intrinsically discrete structure, which prevents the system from realizing the universal regime of the REHM with continuous couplings. Further studies would hence be desired to clarify the role of a discrete distribution of effective couplings in determining the fixed point which governs the low temperature physics of this system. Bond dilution, on the other hand, can either enhance or suppress locally the spin gap in even-leg ladders, depending on the location of the bond which is diluted. As a result of these competing effects, weak bond dilution is not enhancing the correlations of pure ladders. In fact, correlations can even be suppressed by disorder, especially 68 in 4-leg and 6-leg ladders. The resulting short-range correlated phase has a gapless, Griffiths nature, due to the appearance of exponentially rare, but locally gapless regions. Beyond a critical concentration, the correlation-enhancing effect of bond dilution leads to a phase in 2-leg ladders, with a logarithmic scaling of the correlation length with decreasing temperature. This behavior may suggests that the low-temperature behavior of the system is governed by an infinite randomness fixed point over an extended range of parameters. Nonetheless, based on the investigation of the uniform susceptibility of the system, and by comparison with existing results on disordered ladders, we argue that this might not be the true asymptotic behavior forT! 0, and that the system is more likely to remain in a Griffiths phase for all values of bond dilution. In order to connect our results to experiments on site-diluted and bond-disordered ladders, a fundamental question to address is the role of finite inter-ladder couplings, which are unavoidable in real materials. IfJ 0 is their energy scale, such couplings drive the system towards a three-dimensional magnetically ordered phase below a tempera- tureT c , which can be estimated via a mean-field approach ask B T c J 0 (T c ;z), where is the correlation length of the uncoupled ladders. For a gapless critical phase, as that induced by site dilution (at any concentration) and by bond dilution (beyond a critical concentration),(T ;z) diverges forT! 0, so that the system orders at finite tempera- ture. On the other hand, the divergence of with decreasing temperature appears to be relatively weak (power law with a small power 0:3 0:4 for site dilution, or loga- rithmic for bond dilution). Hence a difference of several orders of magnitude between the intra-ladder and the interladder couplings, which is common to various ladder mate- rials, allows the asymptotic low-temperature behavior of for the uncoupled doped ladders to be manifested at temperatures which lie well above the critical temperature T c for magnetic ordering. For bond-diluted ladders, moreover, short-range correlations 69 proper of the pure ladder persist up to a critical doping, so that three-dimensional order- ing in coupled ladders should be absent at weak dilution. Increasing the bond dilution leads to an increase of correlations at low temperature, so that the system is expected to undergo a disorder-induced quantum phase transition from a Griffiths phase to a magnet- ically ordered phase (atT = 0), but with exponentially small transition temperatures, T c exp(J=J 0 ), which might be hardly detectable in experiments. As a result the temperature scaling of correlations investigated here for uncoupled ladders is expected to be relevant to the behavior of real ladder materials with non-magnetic doping. 70 Chapter4 PhaseDiagramofEven-LegLadder withBondDisorder 4.1 Introduction The effects of doping on spin-gap compounds has attracted considerable interest, fol- lowing the discovery of new families of materials whose low-energy magnetic prop- erties can be described by antiferromagnetic (AF) Heisenberg ladder models[15]. It is known that AF spin-1=2 Heisenberg ladders with an even number of legs are char- acterized by a singlet ground state. This symmetry of the ground state breaks down beyond a sufficiently high applied magnetic field or beyond a critical concentration of random dopants. In the former case, Bose-Einstein condensation (BEC) of spin triplets occurs in a magnetic field window, h c1 < h < h c2 , i.e., in an intermediate field regime which separates the spin-gap phase (h < h c1 ) and a fully polarized phase (h>h c2 ).[31, 38, 82] In the latter case, one distinguishes two types of randomness: site and bond disorder. Site disorder induces local moments, which can give rise to long- range order via an order-by-disorder mechanism. [74, 75, 101] In contrast, the presence of bond disorder can destroy magnetic long-range order through quantum localization, leading to Bose glass formation[18], which has recently been observed in spin gap lad- der compounds[39, 84]. Many interesting phenomena are induced by the simultaneous presence of both magnetic field and randomness.[65, 58] For example, in doped spin gap 71 ladders one observes a very rich phase diagram, with a sequence of magnetic field con- trolled phases, including superfluidity, BEC, Bose glass and full polarization.[38, 58, 97] Specifically, recent experiments on the doped compound IPA-Cu(Cl 0:95 Br 0:05 ) 3 imply the existence of another Bose glass at low applied magnetic fields, before the appear- ance of BEC. While some of these results have been modeled quantitatively using a bond-disordered Heisenberg Hamiltonian [55] it still remains desirable to better under- stand the effects of random doping on local observables. In this chapter, we study how bond disorder affects the quantum phase diagram of even-leg antiferromagnetic Heisenberg ladders in the presence of an applied mag- netic field. Bond disorder occurs when dopant ions replace the ions which act as bridges between the magnetic ions. For instance, bond disorder can be introduced in IPA-CuCl 3 by a partial substitution of nonmagnetic Br for the likewise nonmagnetic Cl , affecting the bond angles in the Cu-halogen-halogen-Cu superexchange pathways [38, 27, 40, 44]. The enhanced strength of magnetic interactions on the affected bonds leads to a Bose glass when a sufficiently strong magnetic field is applied. Making use of quantum Monte Carlo simulations we can address the local bond energies and uniform and staggered magnetization at ultralow temperatures. As discussed below, we find that with increasing magnetic field, BEC and Bose glass phases appear preceding the critical magnetic fields (h c1 andh c2 ) of the pure system. We study the influence of bond impu- rities on the bond energiesE b of its neighbors. This allows us to characterize the effects of bond impurities on the condensation and localization of triplons, which occur in BEC and Bose glass phases respectively. In this context, a ”superfluid” phase corresponds to the delocalization of bosons, and it is characterized by a nonzero magnetizationm ? s perpendicular to the magnetic field. The Bose glass phase occurs due to the localization of triplons. It is characterized by a finite slope of the uniform magnetizationm u and a vanishing order parameterm ? s . 72 4.2 Model We examine the bond-disordered antiferromagnetic Heisenberg model on two- and four- leg ladders described by the Hamiltonian H = X i;j J ij S i S j +h X i S z i ; (4.1) whereS i is the spin-1/2 operator at sitei,h is the applied magnetic field, andJ ij denotes the nearest-neighbor coupling between the spins on sitesi andj, taking the valuesJ = 1 with probability 1p andJ 0 with probabilityp. Periodic boundary conditions are used along the leg direction. The simulations are performed using the stochastic series expan- sion (SSE) quantum Monte Carlo (QMC) method based on the directed loop algorithm [79]. In order to access the regime of very low temperatures, we use a -doubling scheme [71]. This way, maximum inverse temperatures up to = 2048 are obtained. The staggered magnetizationm ? s is calculated from the staggered structure factor, S ? s = 1 N X hi;ji (1) i+j hS x i S x j i; (4.2) usingm ? s = p S ? s =N. The local bond energies are measured byE b = J ij hS i S j i. The staggeredm ? s and uniformm u magnetizations are averaged over at least 700 impurity realizations. An ultra-low temperature T = 1=2048 was chosen in all simulations so that the relevant thermodynamic observables reflect true zero-temperature behavior. The lengths of the two-leg ladders used areL x = 104; 128; 160; 200; 256; 320, and 400. The lengths of the four-leg ladders used areL x = 96; 128; 160; 192; 256; 320, and 400. Their thermodynamic limit, i.e. infinite length, is extrapolated via finite-size scaling. 73 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 5 10 15 20 25 30 P(E b ) all bonds leg bonds rung bonds -0.5 -0.4 -0.3 -0.2 -0.1 0 E b 0 5 10 15 20 P(E b ) all bonds outer-leg bonds inner-leg bonds outer-rung bonds center-rung bonds (a) (b) Figure 4.1: (Color figure) (a) Distribution of local bond energies in pure (a) two-leg and (b) four-leg AF spin-1/2 Heisenberg ladders. For each case, there are different con- tributions due to rung bonds and leg bonds. Simulations are performed at temperature T =J=2048, and lattice lengthL x = 256. 4.3 Bond-DopedHeisenbergLadders Pure antiferromagnetic even-leg ladders display a resonant valence bond (RVB) ground state [15] with a finite gap to triplet excitations, where then spins on the same rung preferentially form local singlets. The value of the spin gap decreases as the number of legs increases.[93] An interesting question that arises concerns how the magnitudes of the bond energies relate to the observed spin gap. Figure 4.1 shows the local bond energy distribution in pure two- and four-leg ladders withJ ij = J. The simulations are performed at a very low temperatureT = J=2048 on ladders with linear sizeL x = 256, measuring the local bond energies. In pure two- leg ladders, the distribution function of these local bond energies shows two peaks, corresponding to two the types of bonds. (The finite width of peaks arises from statistical fluctuations in the Monte Carlo simulations.) The rung-bond energies give rise to the lower-energy peak at E b 0:454J, whereas the leg bonds cause the higher-energy 74 peak atE b 0:350J. This observation is consistent with the fact that pure two-leg ladders display a rung-singlet-dominated ground state. The local bond energies of pure four-leg ladders fall into four categories, corresponding to four types of bonds, located on outer rungs, outer legs, inner legs, and center rungs respectively. They are arranged in ascending order of local bond energies, given byE b 0:388J;0:370J;0:326J, and0:309J. It is noted that the center-rung bonds have the highest energies. In other words, the four rung spins preferentially form effective pairs of weakly coupled singlets on each rung. What happens when the ladders are bond-doped? For sufficiently low bond dilution concentrations, the average distance between the individual bond impurities is large. To understand this regime, it is useful to study the local effects of isolated impurity bonds. Figure 4.2 shows the effect of an isolated impurity on its neighbor bonds. Here, one bond is replaced by a weaker coupling J 0 = J=5, and its position is varied. In the case of two-leg ladders, the impurity bond can be located on either a rung or on a leg. In Figure 4.2(a), it is observed that a leg-bond impurity significantly reduces the magnitude of the bond energy on the opposite leg bond. Additionally, it strongly enhances the bond energies on its two neighbor rungs, indicating that this type of bond impurity strengthens the formation of rung singlets in its immediate neighborhood. In contrast, when the bond impurity is placed on a rung, it reduces the magnitude of the bond energies of the two closest rungs; see Figure4.2(b), but increases the magnitude of the bond energies on the neighboring legs. For the case of four-leg ladders, the preferential formation of pairs of effectively weakly coupled singlets along the four-site rungs has significant consequences when bond doping is introduced. As discussed above, an approximate effective description of the ground state in a pure four-leg ladder would be a product state of two weakly coupled two-leg ladders. When a bond impurity is introduced on a leg, it thus causes an 75 -0.0328 -0.414 -0.418 -0.390 -0.234 -0.393 -0.507 -0.500 -0.414 -0.421 -0.414 -0.416 -0.400 -0.397 -0.059 -0.322 -0.355 -0.387 -0.038 -0.438 -0.433 -0.340 -0.329 -0.372 -0.327 -0.343 -0.267 -0.303 -0.300 -0.424 -0.424 -0.389 -0.395 -0.361 -0.310 -0.390 -0.388 -0.363 -0.329 -0.374 -0.385 -0.327 -0.364 -0.039 -0.306 -0.325 -0.321 -0.420 -0.419 -0.388 -0.386 -0.384 -0.371 -0.421 -0.348 -0.419 -0.355 -0.319 -0.372 -0.320 -0.357 -0.319 -0.352 -0.350 -0.048 -0.377 -0.389 -0.318 -0.398 -0.359 -0.360 -0.395 -0.358 -0.340 -0.337 -0.360 -0.342 -0.341 -0.281 -0.034 -0.397 -0.432 -0.435 -0.401 -0.279 (a) (c) (e) (d) (f) (b) Figure 4.2: (Color figure) Effects of local bond impurities, denoted by dashed lines, on the neighboring bond energies in two- and four-leg ladders. Simulations are performed at temperature T = J=2048, lattice length L x = 256, with the coupling of the bond impurityJ 0 =J=5. effect similar to the two-leg ladder case; see Figs. 4.2(c) and 4.2(e), i.e., the opposite leg bond energy on the two-leg subsystem on which the impurity is placed is reduced while bond energies on neighboring rungs are enhanced. The other two-leg subsystem is barely affected by the leg impurity. In contrast, if the impurity is introduced on an outer rung, it causes a reduction of the rung neighbor bond energies on the same two- leg ladder subsystem; see Figure 4.2(d). It also enhances the bond energies on the neighboring leg bonds, which again is similar to the effect in Fig.4.2(b). Last but not least, when the impurity is placed on a center rung, it enhances the bond energies on the two neighboring rungs. However, its influence compared with the previous case is much less drastic; see Fig. 4.2(f). In randomly bond doped ladders, the distribution of local bond energies displays characteristics stemming from the effects discussed above for isolated impurity bonds. Figure 4.3 shows the distribution of local bond energies of doped (a) two-leg and (b) 76 four-leg ladders. Here, we show three typical examples with (z = 4:2%;J 0 = J=5), (z = 8:3%;J 0 =J=5), and (z = 8:3%;J 0 = 0). For the two-leg ladder, it is observed that, besides the main peaks of the pure case, there are three additional features. For instance, let us consider the case (z = 8:3%;J 0 = J=5). Here, the additional peaks appear at energies0:51J,0:22J, and0:05J. These features originate from affected bonds in the vicinity of the bond impurities. The lowest-energy peak is the contribution of singlets next to leg-bond impurities, the second-lowest-energy peak comes from leg bonds opposite to the impurity, and the third peak stems from the bond impurities themselves on the rung and leg bonds. There is actually another feature in between the two main peaks, which stems from rung singlets next to the rung-bond impurities. In four-leg ladders with (z = 8:9%;J 0 =J=5), the feature at0:42J stems from the rung bonds shown in Figs. 4.2(c), 4.2(e), and 4.2(f) and outer-leg bonds in Fig. 4.2(d). The peak at E b 0:27J corresponds to center leg couplings in Fig. 4.2(c). The slightly lower energy peak atE b 0:28J is a contribution of center rung bonds next to rung-bond impurities; see Fig. 4.2(f). The peak at0:03J is the contribution of bond-impurities. Clearly, the local bond energy distribution is much richer compared with the two-leg ladder case, but the individual features can still be explained using a local description. The strength of local bond energies has direct consequences for the phase diagram when the randomly bond-doped ladders are exposed to an applied magnetic field. Fig- ure 4.4 shows the quantum (low-temperature) phase diagram of pure and doped two and four-leg ladders in the presence of a uniform magnetic fieldh. The uniform magneti- zation is found not to depend significantly on lattice size. Therefore only L x = 256 is shown. The thermodynamic limit of the staggered magnetization is calculated by an 77 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 10 20 30 P(E b ) z=0% z=4.2%, J’=J/5 z=8.3%, J’=J/5 z=8.3%, J’=0 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 E b 0 5 10 15 20 P(E b ) z=0% z=4.5%; J’=J/5 z=8.9%, J’=J/5 z=8.9%, J’=0 (a) (b) Figure 4.3: (Color figure) (a) Distribution of local bond energies in bond-disordered (a) two-leg and (b) four-leg ladders. Simulations are performed at temperature T = J=2048, and lattice lengthL x = 256. extrapolation of the finite-size datam ? s (L) as a function ofL. For instance, in the spin- gap, Bose glass, and fully polarized phases a linear scaling in 1=L is used, whereas in the BEC phase the extrapolation requires the use of higher-order polynomials.[56] The presence of these phases will become clear later. In the pure case, for sufficiently small fields, both the uniform m u and staggered m ? s magnetization remain. They become finite beyond a lower critical field, h c1 = 0:5 for the two-leg ladder and h c1 = 0:16 for the four-leg ladder. Spins in pure two-leg ladders are fully polarized at an upper critical magnetic fieldh c2 = 3, and similarly, the upper critical field for four-leg ladders ish c2 = 3:8.[90] Figure 4.5(a) provides a schematic picture to explain the observation of the phase diagram found from the QMC simulations. At zero field, quantum spin fluctuations in the pure two-leg Heisenberg ladder destroy conventional magnetic order. The RVB ground state can be approximated by a product state of rung spins forming singlets, separated from the lowest triplet excitation by a minimum excitation energy .[15] This is a nonmagnetic spin liquid state. The spin gap is reduced as the magnetic 78 0 1 2 3 4 h 0 0.1 0.2 m s ⊥ n=2, z=0% n=2, z=4.2% n=4, z=0% n=4, z=4.5% 0 0.1 0.2 0.3 0.4 0.5 m u 0.12 0.14 0.16 0.18 0.2 0.22 h 0 0.05 0.1 0.3 0.4 0.5 0.6 0.7 0.8 h 0 0.02 0.04 0.06 0.08 0.1 0 0.005 0.01 0 0.02 0.04 0.06 (a) (b) (c) Figure 4.4: (Color figure) Staggered and uniform magnetization of pure and bond- disordered even-leg ladders. Bond couplings are randomly replaced by couplings J 0 = J=5. The values of the magnetization are averaged over at least 700 realizations for each lattice size. The uniform magnetization is measured at lattice lengthL x = 256. The thermodynamic limit of the staggered magnetization, i.e., infinite length, is extrapo- lated via finite-size scaling usingL x = 104; 128; 160; 200; 256; 320, and 400 for two-leg ladders and L x = 96; 128; 160; 192; 256; 320, and 400 for four-leg ladders. The error bars fall within the symbol size. field increases. It closes when the magnetic field reaches the lower critical valueh c1 , at which the system experiences a quantum phase transition from the spin liquid phase to a BEC of magnons.[55] This phase is characterized by nonzero uniform and staggered magnetizations. Ultimately, at large applied magnetic fields all the spins are polarized along the z direction, i.e., for h > h c2 . At this upper critical magnetic field, the sys- tem transitions from a BEC to a polarized paramagnet, and there is no longer coherence between spins. The staggered magnetization vanishes and the uniform magnetization saturates atm u = 1=2. In the doped case, these phase diagrams are modified by the presence of bond disorder. The staggered magnetization of doped two and four-leg ladders are shown for respective doping concentrations z = 4:2% and 4:5%. The bond couplings J 79 are randomly replaced by J 0 = J=5. This particular choice is motivated by recent experiments,[40] but the results shown here are generic. Interestingly, close to both critical fields h 0 < h . h c1 and h 00 < h < h c2 [Figs. 4.4(b) and 4.4(c)] we observe a finite slope of the uniform magnetization but a vanishing staggered magnetization. These are the characteristics of Bose glass phases. We observe the phase transition of these Bose glass phases to BEC at a magnetic field smaller thanh c1 for two-leg ladders and greater thanh c1 for four-leg ladders. Observations of disorder-induced phases near the originally critical fields, recently reported in three-dimensional dimer systems, [55] suggest that BEC and Bose glass phases appear near the higher critical fields. Our find- ing differs in that for the bond-doped two-leg ladder the Bose glass phases occur at both critical fields, depending on the magnitude of bond disorder. However the existence of Bose glass phases at both critical magnetic fields has recently been observed in the compound IPA-Cu(Cl 0:95 Br 0:05 ) 3 . [27] Below, we discuss mechanisms derived from our model study which offer explanations for the experimental observations. In Fig. 4.5(b), we show the five phases which appear in the QMC data of doped two-leg ladders. At sufficiently small fields the system stays in the spin liquid regime I. A finite magnetic field strength is required to overcome the lowest singlet-triplet gap of bond-impurity couplings. As discussed above, the impurity rung bonds are weaker, and therefore singlets on these bonds break first when an external magnetic field is applied to the system. Beyond this first critical field, the uniform magnetization becomes finite with increasing magnetic field while the staggered magnetization stays zero. This magnetic field is sufficiently strong to gradually polarize spins on the impurity rung bonds, but not strong enough to polarize the spins in the bulk. During this phase, the field-induced triplons stay localized on the impurity-rung bonds. There is no coherence among them. The system therefore forms a Bose glass phase which is manifested in region II. At magnetic field, h h c1 , singlets on the bulk rungs break, triplons form 80 h 1/2 h 1/2 (a) pure (b) randomly doped I II III IV V Figure 4.5: (Color figure) Schematic illustration of the response of the uniform and staggered magnetizations to an applied magnetic field. In the pure case (a), the system remains in an RVB spin fluid state up to a lower critical fieldh c 1 . Then it undergoes a BEC transition into the regimeh c 1 <h<h c2 . All spins are fully polarized beyond the saturation fieldh c2 . In the doped case (b), a field scan reveals the following sequence of phases: I: RVB spin liquid; II: Bose glass phase; III: BEC; IV: Bose glass phase; V: fully polarized phase. on these rung bonds, and they interact with each other. We note that, for any small amount of dilution, impurities in a two-leg ladder separate rung bonds into clusters. Rung couplings on these clusters are generally greater than that in the pure case. Hence, magnetic fields higher than h c1 are needed to polarize these rung bonds; Fig.4.4(c). However, the above argument can not be applied to four-leg ladders because, at small dilution, rung bonds are not separated into clusters. Therefore a magnetic field smaller than h c1 is sufficient to polarize these rung bonds; Fig.4.4(b). The system undergoes a BEC phase transition, regime III. Recall that there are still basically unrenormalized singlets next to the leg-bond impurities that are untouched. Their bond energies are relatively large compared with the original bond energies; see Fig. 4.2(a). Although the 81 magnetic field has fully polarized the original singlets in this regime, it is not sufficiently strong to break these impurity induced higher-energy singlets. The system therefore undergoes another Bose glass phase transition into regime IV . Finally, above h c2 , all spins are finally polarized, regime V . This sequence is observed in the QMC data shown in Fig. 4.4. 4.4 Conclusions To summarize, we have studied the effects of bond impurities in even-leg antiferromag- netic spin-1/2 Heisenberg ladders. We find that, depending on the location of the bond impurities, bond energies on neighboring bonds are either enhanced or reduced. In gen- eral, bond impurities enhance bond energies connected with them by the same spins, and reduce the opposite couplings. This effect is related to the so-called edge disorder effect. [105] We identify various types of impurity-induced bond energy shifts in two- and four-leg ladders. In light of these results, we demonstrate the emergence of BEC and Bose glass phases close to both critical fields of the pure system. These results can be used to explain recent observations of a disorder-induced Bose glass phase in IPA-Cu(Cl 0:95 Br 0:05 ) 3 . [27] 82 Chapter5 Conclusions In this dissertation the physical properties and quantum phases of even-leg spin lad- ders under influence of geometric randomness were studied using numerical simulation techniques. Here we summarize our results and present potential directions for future research. In the first part, we discussed the effects of site and bond dilution on the low- temperature correlations of even-leg S = 1=2 ladders with antiferromagnetic Heisen- berg interactions. Site dilution was found to prune rung singlets and thus create local- ized moments which interact through unfrustrated, distance dependent couplings. The Hamiltonian describing the effective interaction between these moments is a random exchange Heisenberg model (REHM), with a gapless spectrum and power-law diverg- ing correlations as temperature decreases to zero. We find that the distribution of the effective couplings has an intrinsically discrete structure, which prevents the system from realizing the universal regime of the REHM, which is only realized in the limit of a continuous distribution of the effective couplings. Bond dilution, on the other hand, can either enhance or suppress locally the spin gap in even-leg ladders, depending on the location of the bond which is diluted. As a result of these competing effects, weak random bond dilution is not enhancing the correlations of pure ladders. In fact, correlations can even be suppressed by disorder, especially in 4-leg and 6-leg ladders. The resulting short-range correlated phase has a gapless, Griffiths nature, due to the appearance of exponentially rare, but locally gapless regions. Beyond a critical concentration, the correlation-enhancing effect of bond dilution leads 83 to a phase in 2-leg ladders, with a logarithmic scaling of the correlation length with decreasing temperature. This behavior suggests that the low-temperature behavior of the system is governed by an infinite randomness fixed point over an extended range of parameters. Nonetheless, based on the investigation of the uniform susceptibility of the system, and by comparison with existing results on disordered ladders, we argued that this may not be the true asymptotic behavior for T ! 0, and that the system is more likely to remain in a Griffiths phase for all values of bond dilution. In the second part, we studied the effects of individual bond impurities in even- leg antiferromagnetic spin-1/2 Heisenberg ladders. We found that these effects vary strongly, depending on their positions on legs and rungs. In general, bond impurities enhance bond energies connected with them by the same spins, and reduce the opposite couplings. These effects are related to the so-called edge disorder effect. We also iden- tified various types of impurity-induced bond energy shifts in two- and four-leg ladders. In light of these results, we demonstrated the emergence of BEC and Bose glass phases close to both critical fields of the pure system. So far we have focused on the study of doped even-leg ladders. There are still numerous open questions related to doped odd-leg ladders, see Figure 5.1(a). Below, we will describe some physical properties of clean odd-leg ladders and the effects of single site or bond impurity on its neighboring couplings. Based on these observations, we depict physical properties and quantum phases of site- and bond-doped odd-leg ladders. In contrast to even-leg ladders, there is a zero-energy mode even in clean odd-leg systems. In isotropic clean odd-leg ladders, two spins on a given rung participate in a gapped state, whereas the third remaining spin on this rung forms a doublet. Such doublets interact along the leg in a formation of a spin-1=2 chain with effective couplings J eff . [5] The three-leg ladder in this case can be mapped onto a single Heisenberg chain, with no gap and power-law correlations. [33] When this system is in an applied 84 -0.365 3-leg Ladder -0.378 -0.367 -0.377 -0.319 Single Bond Impurity -0.424 -0.324 -0.421 -0.340 -0.361 -0.360 -0.343 -0.380 -0.427 -0.325 -0.012 -0.380 -0.352 -0.0155 -0.465 -0.465 -0.329 -0.393 -0.392 -0.331 -0.263 -0.363 -0.365 -0.397 -0.397 -0.323 -0.390 -0.390 -0.353 -0.390 -0.390 -0.352 -0.017 -0.325 -0.392 -0.391 -0.394 -0.394 -0.357 Single Site Impurity -0.483 -0.374 -0.482 -0.312 -0.341 -0.344 -0.313 -0.371 -0.458 -0.375 -0.372 -0.408 -0.408 -0.356 -0.407 -0.408 -0.375 -0.376 -0.374 -0.376 (a) (b) (c) (d) (e) (f) Figure 5.1: (Color figure) Effects of local site/bond impurities on the neighboring local bond energies in three-leg ladders. Site impurities are denoted by blue dots with no connections. Bond impurities are denoted by dashed lines. Simulations are performed at temperatureT =J=2048, lattice lengthL x = 256, with the couplings of the impurity bondsJ 0 =J=10. magnetic field, its magnetization increases as the external field is increased. However, at an intermediate field, there is a plateau in the magnetization curve, [10, 11] related to spin locking. When the field is sufficiently large to overcome the spin-gap of the dimers along the rungs, the magnetization continues to increase and saturates at very high fields. In site-diluted three-leg ladders, depending where the site impurity is, dilution can have very different effects. If the impurity site is at the boundary, it induces a local rung singlet, see Figure 5.1(b). If it is in the center, it induces two local moments at the neighboring outer legs, see Figure 5.1(c). These two induced moments can again interact with an effective antiferromagnetic coupling which is mediated by the shortest path of bonds connecting them. The combination of all above effects can shorten the spin-spin correlations of the original system, and dramatically change its ground state. 85 For the bond-diluted three-leg ladders, there are three possibilities for local dilution: If the bond impurity is on one of the rungs, it induces one singlet and one doublet, see Figure 5.1(e). If it is on either the middle or the outer leg, it only enhances the neigh- boring bond couplings, see Figure 5.1(d) and (f). Here the phenomena, in principle, is very similar to that of the edge disorder effect,[105] where the bond impurities enhance the local bond energies connected with them by the same spins, and reduce the opposite couplings. So we expect that the ground state of the system is dramatically affected by the dilution. In both site- and bond-diluted systems with an applied magnetic field, new plateaus can emerge due to the presence of the induced dimers and local moments. 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Abstract (if available)

Abstract The presence of disorder can deeply affect the critical behavior of quantum spin systems. Disorder can localize the long-wavelength modes developed at the quantum phase transition of the clean systems, and thus introduce glassy phases. Furthermore, it has been shown that many novel phases can be induced by the simultaneous presence of both magnetic field and geometric randomness. For instance, in doped spin-gap ladders one observes a very rich phase diagram, with a sequence of magnetic field controlled phases, including superfluidity, Bose-Einstein condensate, Bose glass and full polarization. Geometric randomness can also induce local magnetic moments giving rise to long-range order through an order-by-disorder mechanism in a gapped quantum-disordered ground state. In this dissertation, we investigate the physical properties and phase diagrams of doped even-leg ladders. Even-leg ladders are the minimal system in which we can study these effects: pruned spins, bond vs. rung vacancies, and order-by-disorder. Furthermore, such minimal even-leg ladders do not have any intrinsic low-energy degrees of freedom, because the pure system is gapped. Thus the effects of impurity vacancies, introducing new low-energy features, can be cleanly studied. ❧ In the first part, we uncover the effects of quenched disorder in the form of site and bond dilution on the physics of the S=1/2 antiferromagnetic Heisenberg model in even-leg ladders. Site dilution is found to prune rung singlets and thus create localized moments which interact via a random, unfrustrated network of effective couplings, realizing a random-exchange Heisenberg model in one spatial dimension. This system exhibits a power-law diverging correlation length as the temperature decreases. Contrary to previous claims, we observe that the scaling exponent is non-universal, i.e., doping dependent. This finding can be explained by the discrete nature of the values taken by the effective exchange couplings in the doped ladder. Bond dilution of even-leg ladders leads to a more complex evolution with doping. Correlations are weakly enhanced in 2-leg ladders, and are even suppressed for low dilution in the case of 4-leg and 6-leg ladders. We clarify the different aspects of correlation enhancement and suppression due to bond dilution by isolating the contributions of rung-bond dilution and leg-bond dilution. ❧ In the second part, we investigate random bond disorder in antiferromagnetic spin-1/2 Heisenberg ladders. We find that the effects of individual bond impurities vary strongly, depending on their positions on legs and rungs. We initially focus on how the distribution of local bond energies depends on the impurity concentration. Then we study how the phase diagram of even-leg ladders is affected by random bond doping. We observe Bose glass phases in two regimes (h' < h ≲ h{c₁} and h'' < h <h{c₂}) and a Bose-Einstein Condensate in between. Their presence is discussed in relation to the local bond energies. ❧ To study the physical properties and phase diagram of antiferromagnetic spin-1/2 Heisenberg ladders, we apply the stochastic series expansion quantum Monte Carlo method. This algorithm has been proved to be a very powerful technique to investigate the low-temperature properties of quantum spin systems at large scales. Details of the method are discussed before we explore the physical results.

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Creator Trinh, Kien Trong (author)

Core Title Phase diagram of disordered quantum antiferromagnets

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 10/24/2014

Defense Date 08/12/2014

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag bond dilution,Bose-Einstein condensate,collective effects,collective excitations,correlation length,effects of disorder,exchange interactions,Griffiths phase,Heisenberg ladders,Heisenberg model,OAI-PMH Harvest,quantum Monte Carlo,quantum phase transition,quantum spin liquids,site dilution,spin chain models,spin ladders,spin-glass and other random models,stochastic series expansion,valence bond phases and related phenomena

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Language English

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Advisor Haas, Stephan W. (committee chair), Bickers, Nelson Eugene (committee member), Haselwandter, Christoph (committee member), Nakano, Aiichiro (committee member), Takahashi, Susumu (committee member)

Creator Email ktrinh@usc.edu,trongkien@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-508093

Unique identifier UC11298379

Identifier etd-TrinhKienT-3030.pdf (filename),usctheses-c3-508093 (legacy record id)

Legacy Identifier etd-TrinhKienT-3030.pdf

Dmrecord 508093

Document Type Dissertation

Format application/pdf (imt)

Rights Trinh, Kien Trong

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions 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...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA