BEIJING HONG Frustrated Spin Systems 2nd Edition H T Diep University of Cergy-Pontoise, France Editor \fa World Scientific NEW JERSEY LONDON SINGAPORE SHANGHAI KONG TAIPEI CHENNAI
CONTENTS Preface of the Second Edition v Preface of the First Edition vii 1 Frustration Exactly Solved Frustrated Models 1 H T Diep and H Giacomini 31 I 1 Frustration: An Introduction 1 111 Definition 2 112 Non-collinear spin configurations 5 12 Frustrated Ising spin systems 8 13 Mapping between Ising models and vertex models 10 131 The 16-vertex model 10 132 The 32-vertex model 12 133 Disorder solutions for two-dimensional Ising models 23 14 Reentrance in exactly solved frustrated Ising spin systems 26 141 Centered square lattice 27 1411 Phase diagram 28 1412 Nature of ordering and disorder solutions 28 142 Kagorne lattice 31 1421 Model with nn and nnn interactions 1422 Generalized Kagome lattice 31 143 Centered honeycomb lattice 37 144 Periodically dilute centered square lattices 40 1441 Model with three centers 44 1442 Model with two adjacent centers 46 1443 Model with one center 47 145 Random-field aspects of the models 48 xiii
xiv Frustrated Spin Systems (2nd Edition) 15 Evidence of partial disorder and reentrance in other frustrated systems 50 16 Conclusion 54 Acknowledgements 57 References 57 2 Properties and Phase Transitions in Frustrated Ising Systems 59 Ojiro Nagai, Tsuyoshi Horiguchi and Seiji Miyashita 21 Introduction 59 22 Ising model on two-dimensional frustrated lattice and on stacked frustrated lattice 62 23 Ising model on antiferromagnetic triangular lattice 65 24 Ising model on stacked antiferromagnetic triangular lattice 71 25 Ising model with large S on antiferromagnetic triangular lattice 76 26 Ising model with infinite-spin on antiferromagnetic triangular lattice 80 27 Ising-like Heisenberg model on antiferromagnetic triangular lattice 82 28 Ising model with infinite-spin on stacked antiferromagnetic triangular lattice 82 29 Phase diagram in spin-magnitude versus temperature for Ising models with spin S on stacked antiferromagnetic triangular lattice 87 210 Effect of antiferromagnetic interaction between next-nearest-neighbor spins in :ry-plane 90 211 Three-dimensional Ising paramagnet 96 212 Concluding remarks 102 Acknowledgements 103 References 103 3 Renormalization Group Approaches to Frustrated Magnets in D=3 107 B Delamotte, D Mouhanna and M Tissier 31 Introduction 107 32 The STA model and generalization 109
e Contents xv 321 The lattice model, its continuum limit and symmetries 109 322 The Heisenberg case 112 323 The XY case 115 324 Generalization 116 33 Experimental and numerical situations 116 331 The XY systems 116 3311 The experimental situation 117 3312 The numerical situation 119 3313 Summary 121 332 The Heisenberg systems 122 3321 The experimental situation 122 3322 The numerical situation 124 3323 Summary 126 333 TheiV = 6STA 126 334 Conclusion 127 34 A brief chronological survey of the theoretical approaches 127 35 The perturbative situation 131 351 The Nonlinear Sigma (NLct) model approach 131 352 The Ginzburg-Landau-Wilson (GLW) model approach 136 3521 The RG flow 136 3522 The three and five-loop results in d = 4 137 3523 The improved three and five-loop results 137 3524 The three-loop results in d = 3 138 3525 The large-iv results 138 3526 The six-loop results in d = 3 139 353 The six-loop results in d = 3 re-examined 141 3531 Conclusion 143 36 The effective average action method 143 361 The effective average action equation 143 362 Properties 148 363 Truncations 149 364 Principle of the calculation 150 365 The 0{N) x 0(2) model 151 3651 The flow equations 153 366 Tests of the method and first results 154 367 The physics in d = 3 according to the NPRG approach 159
xvi Frustrated Spin Systems (2nd Edition) 3671 The physics in d 3 just below Nc{d): = scaling with a pseudo-fixed point and minimum of the flow 159 3672 Scaling with or without pseudo-fixed point: the Heisenberg and XY cases 162 3673 The integration of the RG flow 164 3674 The Heisenberg case 165 3675 The XY case 167 368 Conclusion 169 37 Conclusion and prospects 170 38 Note added in the 2nd Edition 171 References 175 4 Phase Transitions in Frustrated Vector Spin Systems: Numerical Studies D Loison 181 185 41 Introduction 181 42 Breakdown of Symmetry 182 421 Symmetry in the high-temperature region 183 422 Breakdown of symmetry for ferromagnetic systems 183 423 Breakdown of symmetry for frustrated systems 4231 Stacked triangular antiferromagnetic 4232 bet Helimagnets lattices 185 188 4233 Stacked J\-J<2 square lattices 189 4234 The simple cubic Ji-J? lattice 189 4235 JW2-J3 lattice 190 4236 Villain lattice and fully frustrated simple cubic lattice 190 4237 Face-centered cubic lattice (fee) 191 4238 Hexagonal-close-packed 4239 Pyrochlores lattice (hep) 191 192 42310 Other lattices 192 42311 STAR lattices 192 42312 Dihedral lattices Vw,2 193 42313 Right-handed trihedral lattices V3,3 193 42314 P-hedral lattices VN,P 194 42315 Ising and Potts-Vjvi model 194 42316 Ising and Potts-Vjv,2 model 195
2) P) 2) 1) 1) Contents xvii 42317 Landau-Ginzburg model 195 42318 Cubic term in Hamiltonian 195 42319 Summary 196 43 Phase transitions between two and four dimensions: 2 < d < 4 196 431 0(N)/0(N - breakdown of symmetry 197 4311 Fixed points 197 4312 MCRG and first-order transition 199 4313 Complex fixed point or minimum in the flow 200 4314 Experiment 204 4315 Value of Nc 205 4316 Phase diagram (N,d) 206 4317 Renormalization-Group expansions 206 4318 Short historical review 208 4319 Relations with the Potts model 209 432 0(N)/0(N - 433 Z2 SO(N)/SO{N - breakdown of symmetry for d = 3 210 breakdown of symmetry ford = 3 211 434 Z3 SO(N)/SO{N - breakdown of symmetry for d = 3 212 435 Zq 0{N)/0{N - and other breakdown of symmetry ind=3 212 44 Conclusion 213 45 O(N) frustrated vector spins in d 2 = 214 451 Introduction 214 452 Non-frustrated XY spin systems 214 453 Frustrated XY spin systems: Z2 50(2) 214 454 Frustrated XY spin systems: Z3 50(2) 217 455 Frustrated XY spin systems: Z2 <g> Z2 SO{2) and Z3 Z2 50(2) 218 456 Frustrated Heisenberg spin systems: 50(3) 218 457 Frustrated Heisenberg spin systems: Z2 <8> 50(3), 23 50(3) 219 458 Topological defects for iv > 4 220 46 General conclusions 220 Acknowledgments 220 47 Note added for the 2nd Edition 220 Appendix A: Monte Carlo Simulation 222
instantons xviii Frustrated Spin Systems (2nd Edition) Appendix B: Renormalization Group 226 References 228 5 Two-Dimensional Quantum Antiferromagnets 235 Gregoire Misguich and Claire Lhuillier 51 Introduction 235 52 J1-J2 model on the square lattice 237 521 Classical ground state and spin-wave analysis 237 522 Order by disorder (J2> Ji/2) 238 523 Non-magnetic region (J2 JJ2) 239 5231 Series expansions 240 5232 Exact diagonalizations 242 5233 Quantum Monte Carlo 243 53 Valence-bond crystals 244 531 Definitions 244 532 One-dimensional and quasi one-dimensional examples (spin- \ systems) 245 533 Valence Bond Solids 246 534 Two-dimensional examples of VBC 247 5341 Without spontaneous lattice symmetry breaking 247 5342 With spontaneous lattice symmetry breaking 249 535 Methods 251 536 Summary of the properties of VBC phases 252 54 Large-iV methods 254 541 Bond variables 254 542 SU{N) 255 543 Sp(N) 256 5431 Gauge invariance 257 5432 Mean-field (N = 00 limit) 258 5433 Fluctuations about the mean-field solution 258 5434 Topological effects and spontaneous dimerization 260 5435 Deconfined phases 261 55 Quantum Dimer Models 262 551 Hamiltonian 263 552 Relation with spin-^ models 263
Contents xix 279 280 553 Square lattice 265 5531 Transition 265 graphs and topological sectors 5532 Staggered VBC for V/J > 1 266 5533 Columnar crystal for V < 0 267 5534 Plaquette phase 267 5535 Rokhsar-Kivelson point 268 554 Hexagonal lattice 269 555 Triangular lattice 270 5551 RVB liquid at the RK point 271 5552 Topological order 271 556 Solvable QDM on the kagome lattice 272 5561 Hamiltonian 272 5562 RK ground state 273 5563 Ising pseudo-spin variables 274 5564 Dimer-dimer correlations 275 5565 Visons excitations 276 5566 Spinous deconfinement 278 5567 Z2 gauge theory 279 557 A QDM with an extensive ground state entropy Multiple-spin exchange models 280 561 Physical realizations of multiple-spin interactions 5611 Nuclear magnetism of solid 3He 280 5612 Wigner crystal 282 5613 Cuprates 283 562 Two-leg ladders 283 563 MSE model on the square lattice 285 564 RVB phase of the triangular J2-J4 MSE 285 5641 Non-planar classical ground states 286 5642 Absence of Neel LRO 286 5643 Local singlet-singlet correlations absence of lattice symmetry breaking 287 5644 Topological degeneracy and Lieb-Schultz-Mattis Theorem 287 5645 Deconfined spinons 289 565 Other models with MSE interactions 290 Antiferromagnets on the kagome lattice 290 571 Ising model 291 572 Classical Heisenberg models on the kagome lattice 291
XX Frustrated Spin Systems (2nd Edition) 573 Nearest-neighbor RVB description of the spin-^ kagome antiferromagnet 292 574 Spin-5 Heisenberg model on the kagome lattice: numerics 294 5741 Ground-state energy per spin 294 5742 Correlations 295 5743 Spin gap 295 5744 Singlet gap 295 5745 Entanglement entropy and signature of a Z2 liquid 296 5746 Spin liquids on the kagome lattice and Projective symmetry groups 297 575 Competing phases 299 5751 Valence Bond Crystals 299 5752 U(l) Dirac Spin Liquid 300 5753 Spontaneously breaking the time-reversal symmetry, "crural" spin liquids 300 576 Experiments in compounds with kagome-like lattices 301 58 Conclusions 304 References 306 6 One-Dimensional Quantum Spin Liquids 321 P Lecheminant 61 Introduction 321 62 Unfrustrated spin chains 324 621 Spin-1/2 Heisenberg chain 324 622 Haldane's conjecture 327 623 Haldane spin liquid: spin-1 Heisenberg chain 329 624 General spin-5 case 333 625 Two-leg spin ladder 335 626 Non-Haldane spin liquid 341 63 Frustration effects 345 631 Semiclassical analysis 345 632 Spin liquid phase with massive deconfined spinons 348 633 Field theory of spin liquid with incommensurate correlations 355 634 Extended criticality stabilized by frustration 359
Contents xxi 6341 Critical phases with SU(N) quantum criticality 360 6342 Chirally stabilized critical spin liquid 364 64 Concluding remarks 367 65 Note added for the 2nd Edition 370 References 371 7 Spin Ice 383 Steven T Bramwell, Michel J P Gingras and Peter C W Holdsworth 71 Introduction 384 72 Prom Water Ice to Spin Ice 387 721 Pauling's model 387 722 Why is the zero point entropy not zero? 389 723 Generalizations of Pauling's model 390 7231 Wannier's model 390 7232 Anderson's model 391 7233 Vertex models 392 7234 Possibility of realizing magnetic vertex models 392 724 Spin ice 394 7241 Definition of the spin ice model and its application to Ho2Ti207 394 7242 Identification of spin ice materials 396 7243 Basic properties of the spin ice materials 396 725 Spin ice as a frustrated magnet 399 7251 Frustration and underconstraining 399 7252 (111) Pyrochlore models 400 73 Properties of the Zero Field Spin Ice State 401 731 Experimental properties 401 7311 Heat capacity: zero point entropy 401 7312 Low field magnetic susceptibility: spin freezing 405 7313 Spin arrangement observed by neutron scattering 406 732 Microscopic theories and experimental tests 407 7321 Near-neighbour spin ice model: successes and failures 407
xxii Frustrated Spin Systems (2nd Edition) 7322 The problem of treating the dipolar interaction 409 7323 The Ewald Monte Carlo 413 7324 Mean-field theory 417 7325 The loop Monte Carlo 420 7326 Application of the dipolar model to neutron scattering results 429 7327 How realistic is the dipolar model? 429 74 Field-Induced Phases 430 741 Theory 431 7411 Near neighbour model 431 7412 Dipolar model 434 742 Magnetization measurements above T = IK 434 743 Bulk measurements at low temperature 435 7431 [111] Direction 435 7432 [110] Direction 439 7433 [100] Direction 439 7434 [211] Direction 440 7435 Powder measurements 441 744 Neutron scattering results 442 7441 [110] Direction 442 7442 [100], [111] and [211] Directions 444 745 Kagome ice 444 7451 Basic Kagome ice model and mappings 445 7452 Experimental results: specific heat 447 7453 Theory of the Kagome ice state: Kastelyn transition 448 75 Spin Dynamics of the Spin Ice Materials 449 751 Experimental quantities of interest 449 7511 Correlation functions and neutron scattering 449 7512 Fluctuation-dissipation theorem and AC-susceptibility 450 7513 Spectral shape function 450 7514 Exponential relaxation 451 752 Differences between Ho2Ti207 and Dy2Ti207 452 753 Relaxation at high temperature, T ~ 15 K and above 452 7531 AC-susceptibility (AC-x) 452
\ Contents xxiii 7532 Neutron spin echo (NSE) 452 7533 Origin of the 15 K AC-susceptibility peak 455 754 Relaxation in the range 1 K < T < 15 K 456 7541 AC-susceptibility: phenomenological model 7542 AC-susceptibility: towards a microscopic 457 model 457 755 Spin dynamics in the spin ice regime below IK 459 7551 Slow relaxation 459 7552 Evidence for residual dynamics in the frozen state 460 756 Doped spin ice 461 757 Spin ice under pressure 462 76 Spin Ice Related Materials 462 761 Rare earth titanates 463 762 Other pyrochlores related to spin ice 464 77 Conclusions 465 Acknowledgments 466 78 Note added for the 2nd Edition 467 References 468 8 Experimental Studies of Frustrated Pyrochlore Antiferromagnets 475 Bruce D Gaulin and Jason S Gardner 81 Introduction 476 82 Pyrochlore Lattices 477 83 Neutron Scattering Techniques 479 84 Cooperative Paramagnetism in TbaTi207 481 85 The Spin Glass Ground State in Y2Mo207 492 86 Composite Spin Degrees of Freedom and Spin-Peierls-like Ground State in the Frustrated Spinel ZnCr204 501 87 Conclusions and Outlook 504 References 505 9 Recent Progress in Spin Glasses 509 N Kawashima and H Rieger 91 Two Pictures 510 911 Mean-field picture 511 912 Droplet picture 514
Introduction xxiv Frustrated Spin Systems (2nd Edition) 92 Equilibrium Properties of Two-Dimensional Ising Spin Glasses 516 921 Zero-temperature transition? 516 922 Droplet argument for Gaussian-Coupling models 923 Droplets in Gaussian-Coupling models: numerics 518 518 924 Finite-temperature transition? 521 93 Equilibrium Properties of Three-Dimensional Models 521 931 Finite temperature transition? 522 932 Universality class 523 933 Low-temperature phase of the ±J model 525 934 Low-temperature phase of the Gaussian-Coupling model 529 935 Effect of magnetic fields 535 936 Sponge-like excitations 536 937 TNT picture of a new scaling length 537 938 Arguments supporting the droplet picture 538 94 Models in Four or Higher Dimensions 539 95 Aging 541 951 A growing length scale during aging? 541 952 Two time quantities: isothermal aging 546 953 More complicated temperature protocols 551 954 Violation of the Fluctuation-Dissipation theorem 556 955 Hysteresis in spin glasses 559 96 Equilibrium Properties of Classical XY and Heisenberg Spin Glasses 562 961 Continuous spin models in three dimensions 562 962 Continuous spin models in higher dimensions 567 963 Potts spin glasses 568 97 Weak Disorder 569 971 Phase diagram of the discrete spin models 570 972 Dynamical properties 572 973 The renormalization group approach for the discrete models 573 974 The location of the multi-critical point 576 975 Phase diagram of the random XY model in two dimensions 578 98 Quantum Spin Glasses 580 981 Random transverse Ising models 581
Dissipative Dynamics Contents xxv 982 Mean-field theory 589 983 Mean-field theory 984 Mean-field theory effects 593 597 985 Heisenberg quantum spin glasses 599 9851 Finite dimensions 600 9852 Mean-field model 600 99 Summary and Remaining Problems 602 Acknowledgments 604 References 605 Index 615