Introduction to Frustrated Magnetism (eBook)

Introduction to Frustrated Magnetism 3
Preface 5
Contents 9
Contributors 23
Part I Basic Concepts in Frustrated Magnetism 27
Chapter 1 Geometrically Frustrated Antiferromagnets: Statistical Mechanics and Dynamics 28
1.1 Introduction 28
1.2 Models 30
1.3 Some Experimental Facts 31
1.4 Classical Ground State Degeneracy 33
1.5 Order by Disorder 35
1.6 Ground State Correlations 39
1.7 Dynamics 42
1.8 Final Remarks 46
References 46
Chapter 2 Introduction to Quantum Spin Liquids 48
2.1 Introduction 48
2.2 Basic Building Blocks of VBC and RVB Physics: The Valence Bonds 52
2.3 Valence-Bond Crystals 54
2.3.1 Zeroth-Order VBC Wave Function 55
2.3.2 Quantum Fluctuations in VBCs 56
2.3.3 VBC Excitations 57
2.4 Resonating-Valence-Bond Spin Liquids 58
2.5 VBCs or RVB Spin Liquids on Kagoméand Pyrochlore Lattices? 61
2.6 Conclusion 63
References 64
Part II Probing Frustrated Magnets 67
Chapter 3 Neutron Scattering and Highly Frustrated Magnetism 68
3.1 Introduction 68
3.2 What Neutron Scattering Measures 70
3.2.1 Scattering Triangle 70
3.2.2 Partial Differential Cross Section 71
3.2.3 Relation to Sample Properties 72
3.2.4 Scattering from Atomic Magnetic Moments 73
3.2.5 Orientation Factor and Form Factor 73
3.2.6 General Expression for the Neutron Scattering 74
3.2.7 Real Experiments 75
3.2.8 Powder Averaging 75
3.2.9 Static Approximation 75
3.2.10 Wavevector Dependent Magnetic Moment and Susceptibility 76
3.2.11 Fully Ordered Magnet 77
3.2.12 Magnet with Full or Partial Disorder 78
3.2.13 Validity of the Static Approximation 78
3.2.14 Generalised Susceptibility 79
3.2.15 Neutron Spectroscopy 80
3.3 Typical Neutron Scattering Patterns 81
3.3.1 Scattering Plane 81
3.3.2 Free Energy 81
3.3.3 Ideal Paramagnet 83
3.3.4 Conventional Magnet Above TC 83
3.3.5 Conventional Magnet Below TC 84
3.3.6 Cooperative Paramagnet 85
3.3.7 Absent Pinch Points 86
3.3.8 Dynamical Signature of Cooperative Paramagnetism 87
3.4 Experimental Results 88
3.4.1 Cooperative Paramagnet States 88
3.4.1.1 Spin Ice 88
3.4.1.2 Pyrochlore and Spinel Antiferromagnets 90
3.4.1.3 Powder Experiments 92
3.4.2 Ordered States 93
3.4.2.1 Ising Pyrochlore: FeF3 93
3.4.2.2 XY Pyrochlore: Er2Ti2O7 93
3.4.2.3 Dipolar Ordering: Gd2Sn2O7 94
3.4.2.4 Partial Order: Gd2Ti2O7 95
3.4.2.5 Kagomé Lattice: Jarosite Materials 95
3.4.3 Excited States 97
3.4.3.1 Ho2Ti2O7: Crystal Field States 97
3.4.3.2 Er2Ti2O7: Weakly Dispersed Crystal Field States 98
3.4.3.3 Potassium Iron Jarosite: Spin Wave Dispersions 99
3.5 Conclusions 99
References 100
Chapter 4 NMR and SR in Highly Frustrated Magnets 102
4.1 Basic Aspects of NMR and SR Techniques 102
4.1.1 Line Shift and Line Width 103
4.1.2 Nuclear and Muon Spin-Lattice Relaxation Rate 1/T1 106
4.1.3 SR: The Static Case 108
4.1.4 SR: The Dynamic Case 111
4.2 From Zero- to Three-Dimensional Frustrated Magnets 114
4.2.1 Molecular Magnets 114
4.2.2 Antiferromagnets on a Square Lattice with Competing Interactions: The J1-J2 Model 115
4.2.3 Magnetic Frustration on a Triangular Lattice 118
4.2.4 SR and NMR in the Spin-1/2 Kagomé Lattice ZnCu3(OH)6Cl2 120
4.2.5 The Problem of + Relaxation in Some Kagomé Lattices 121
4.2.6 Persistent Dynamics and Lattice Distortions in the Pyrochlore Lattice 124
References 126
Chapter 5 Optical Techniques for Systems with Competing Interactions 129
5.1 Introduction 129
5.2 Inelastic Light-Scattering 130
5.3 Inelastic Phonon Light-Scattering 132
5.4 Inelastic Magnetic, Quasielastic, and ElectronicLight Scattering 133
5.5 The IR Experiment 137
5.6 Spins, Phonons, and Light 138
5.7 Spin–Phonon Interaction in Cr Spinels 140
5.8 Exciton–Magnon Absorption in KCuF3 144
References 146
Part III Frustrated Systems 151
Chapter 6 The Geometries of Triangular Magnetic Lattices 152
6.1 Introduction 152
6.2 Two-Dimensional Structures 153
6.2.1 Planes of Edge-Sharing Triangles 153
6.2.2 Planes of Corner-Sharing Triangles 157
6.3 Three-Dimensional Structures 162
6.4 Note on Synthesis of the Compounds 172
6.5 Conclusion 172
References 173
Chapter 7 Highly Frustrated Magnetism in Spinels 176
7.1 Introduction 176
7.2 Spinel Structure 177
7.3 Basic Electronic Configuration 178
7.4 Uniqueness of the Spinel as a Frustrated Magnet 178
7.5 Materials Overview of Spinels 180
7.6 Frustration in Selected Spinels 182
7.6.1 Pyrochlore Antiferromagnets in Spinel Oxides – B-site Frustration 182
7.6.1.1 ACr2O4 (A = Zn, Cd, and Hg) 182
7.6.1.2 AV2O4 [A = Mg, Zn, and Cd] 186
7.6.1.3 MgTi2O4 188
7.6.2 Frustrated Spins on Spinel A Sites 188
7.6.3 Frustrated Magnets based on Cation-ordered Spinels: The Hyper-Kagomé Lattice of Na4Ir3O8 189
7.6.4 Charge Frustration in Mixed-valent Spinels 192
7.7 Summary 193
References 194
Chapter 8 Experimental Studies of Pyrochlore Antiferromagnets 197
8.1 Introduction 197
8.2 The Cubic Pyrochlores 198
8.3 The Spin Liquid Ground State in Tb2Ti2O7 200
8.4 Ordered Ground States in Tb2Ti2O7 205
8.5 Structural Fluctuations in the Spin Liquid State of Tb2Ti2O7 210
8.6 Magnetic Order and Fluctuations in Tb2Sn2O7 215
8.6.1 Phase Transitions and Fluctuations in Gd2Ti2O7 and Gd2Sn2O7 218
8.7 Conclusions 223
References 224
Chapter 9 Kagomé Antiferromagnets: Materials Vs. Spin Liquid Behaviors 227
9.1 A Short Theoretical Survey: What would be the Ideal Kagomé Antiferromagnet? 228
9.2 The Jarosites 230
9.2.1 Synthesis and the Jarosite Crystal Structure: Idealized and Disordered 230
9.2.1.1 Néel States and Short-ranged Order 233
9.2.2 Fe jarosites: S=52 Kagomé Antiferromagnets 235
9.2.2.1 Is Hydronium Jarosite a Topological Spin Glass? 235
9.2.3 Cr Jarosites- S=32 Kagomé Antiferromagnets 237
9.2.4 Conclusion 238
9.3 Pyrochlore Slabs 238
9.3.1 Synthesis 238
9.3.2 Magnetic Network 239
9.3.3 Generic Physics 240
9.3.4 Non-magnetic Defects 242
9.3.5 Concluding Remarks 244
9.4 Towards S = 1/2 Ideal Compounds 245
9.4.1 Volborthite 245
9.4.2 Herbertsmithite: ``An end to the Drought of Quantum Spin Liquids '' 248
9.5 Other Compounds 253
9.5.1 Organic Materials 253
9.5.2 Y0.5Ca0.5BaCo4O7 254
9.5.3 Langasites 254
9.6 Conclusion 255
References 256
Part IV Specific Effects in Frustrated Magnets 259
Chapter 10 Magnetization Plateaus 260
10.1 Introduction 260
10.2 Mechanisms for Formation of Magnetization Plateaus 261
10.2.1 Spin Gap 262
10.2.2 Quantized Plateaus 263
10.2.3 Order by Disorder 264
10.2.4 Superfluid-Insulator Transition 265
10.2.5 `Quantum' Plateaus 266
10.2.6 High-Order Plateaus 268
10.2.7 Transition into Plateaus 269
10.3 Experimental Observation of Magnetization Plateaus 270
10.3.1 `Classical' Plateaus in Triangular and Pyrochlore Lattices 271
10.3.2 SrCu2(BO3)2 and the Shastry–Sutherland Model 274
10.3.3 `Quantum' Plateaux and Spin Superstructure in SrCu2(BO3)2 277
10.3.4 Phase Diagram of SrCu2(BO3)2 280
10.3.5 RB4: A New Family of Shastry–Sutherland System 282
10.4 Conclusion 283
References 283
Chapter 11 Spin-Lattice Coupling in Frustrated Antiferromagnets 287
11.1 Introduction 287
11.2 Spin-Driven Jahn–Teller Effect in a Tetrahedron 288
11.2.1 Generalized Coordinates and Forces 289
11.2.2 Four S = 1/2 Spins on a Tetrahedron 291
11.2.3 Four Classical Spins on a Tetrahedron 293
11.2.4 Color Notation and Other Useful Analogies 294
11.2.5 Spin–Jahn–Teller Effect on a Triangle 294
11.3 Models with Local Phonon Modes 296
11.3.1 Half-Magnetization Plateau in ACr2O4 Spinels 297
11.4 Collective Spin–Jahn–Teller Effect on the Pyrochlore Lattice 298
11.5 Collective Jahn–Teller Effect in CdCr2O4 300
11.5.1 Spiral Magnetic Order in CdCr2O4 301
11.5.2 Theory of Spiral Magnetic Order 302
11.6 Summary and Open Questions 307
References 308
Chapter 12 Spin Ice 310
12.1 Introduction 310
12.2 Water Ice, Pauling Entropy, and Anderson Model 311
12.2.1 Water Ice and Pauling Model 311
12.2.2 Cation Ordering in Inverse Spinels and Antiferromagnetic Pyrochlore Ising Model 313
12.3 Discovery of Spin Ice 315
12.3.1 Rare-Earth Pyrochlore Oxides: Generalities 315
12.3.2 Microscopic Hamiltonian: Towards an Effective Ising Model 316
12.3.3 Discovery of Spin Ice in Ho2Ti2O7 321
12.3.4 Nearest-Neighbor Ferromagnetic "426830A 111 "526930B Ising Model and Pauling's Entropy 322
12.3.4.1 Nearest-Neighbor Spin-Ice Model 322
12.3.4.2 Pauling Entropy of the Nearest-Neighbor Spin Ice Model 323
12.3.5 Residual Entropy of Dy2Ti2O7 and Ho2Ti2O7 324
12.4 Dipolar Spin-Ice Model 326
12.4.1 Competing Interactions in the Dipolar Spin-Ice Model 326
12.4.2 Mean-Field Theory 329
12.4.3 Loop Monte Carlo Simulations and Phase Diagram of Dipolar Spin Ice 333
12.4.4 Origin of Ice Rules in Dipolar Spin Ice 335
12.5 Current Research Topics in Spin Ices and Related Materials 336
12.5.1 Magnetic-Field Effects 336
12.5.1.1 Field Parallel to [,,(+ )], [11] 337
12.5.1.2 Field Parallel to [1 0] 337
12.5.1.3 Field Parallel to [111] 338
12.5.1.4 Field Parallel to [100] 338
12.5.2 Dynamical Properties and Role of Disorder 339
12.5.3 Beyond the Dipolar Spin-Ice Model 339
12.5.4 Metallic Spin Ice 339
12.5.5 Artificial Spin Ice 340
12.5.6 Stuffed Spin Ice 340
12.5.7 Quantum Mechanics, Dynamics, and Order in Spin Ices 340
12.5.8 Coulomb Phase, Monopoles and Dirac Strings in Spin Ices 341
12.6 Conclusion 342
References 343
Chapter 13 Spin Nematic Phases in Quantum Spin Systems 347
13.1 Introduction and Materials 347
13.2 Multipolar States of a Single Spin 349
13.3 Competition Between Dipoles and Quadrupoles 352
13.3.1 The Bilinear–Biquadratic Model 352
13.3.2 Energy Spectra of Small Clusters 354
13.4 Quadrupolar Ordering in S = 1 Systems 356
13.4.1 Variational Phase Diagram 356
13.4.1.1 Ferromagnetic and Ferroquadrupolar Phases 358
13.4.1.2 Antiferroquadrupolar Phase on the Square Lattice 359
13.4.1.3 Three-Sublattice Phases on the Triangular Lattice 359
13.4.2 One- and Two-Magnon Instability of the Fully Polarized State 362
13.4.3 Spin-Wave Theory for the Ferroquadrupolar Phase 363
13.4.3.1 Technical Details 365
13.4.3.2 Long-Wavelength Limit 368
13.4.3.3 Instability to an Antiferromagnetic State at = -/2 369
13.4.4 Numerical Approach 369
13.5 From Chains to the Square Lattice 371
13.6 Nematic Ordering in S = 1/2 Systems 373
13.7 Conclusions 375
References 376
Part V Advanced Theoretical Methods and Concepts in Frustrated Magnetism 379
Chapter 14 Schwinger Bosons Approaches to Quantum Antiferromagnetism 380
14.1 SU(N) Heisenberg Models 380
14.2 Schwinger Representation of SU(N) Antiferromagnets 381
14.2.1 Bipartite Antiferromagnet 382
14.2.2 Non-bipartite (Frustrated) Antiferromagnets 383
14.3 Mean Field Hamiltonian 384
14.3.1 Mean Field Equations 386
14.4 The Mean Field Antiferromagnetic Ground State 388
14.5 Staggered Magnetization in the Layered Antiferromagnet 390
References 392
Chapter 15 Variational Wave Functions for Frustrated Magnetic Models 393
15.1 Introduction 393
15.2 Symmetries of the Wave Function: General Properties 396
15.3 Symmetries in the Two-dimensional Case 398
15.3.1 The Marshall–Peierls Sign Rule 400
15.3.2 Spin Correlations 401
15.4 Connection with the Bosonic Representation 402
15.5 Antiferromagnetic Order 404
15.6 Numerical Results 406
15.6.1 One-dimensional Lattice 406
15.6.2 Two-dimensional Lattice 410
15.7 Other Frustrated Lattices 416
15.8 Conclusions 418
References 419
Chapter 16 Quantum Spin Liquids and Fractionalization 421
16.1 Introduction 421
16.2 What is a Spin Liquid? 423
16.2.1 Absence of Magnetic Long-Range Order(Definition 1) 423
16.2.2 Absence of Spontaneously Broken Symmetry (Definition 2) 423
16.2.3 Fractional Excitations (Definition 3) 424
16.2.3.1 What is a Fractional Excitation? 424
16.2.3.2 What is the Connection Between Gauge Theories and Fractional Spin Liquids? 426
16.2.3.3 Topological Order 428
16.2.4 Half-odd-integer Spins and the Lieb-Schultz-Mattis-Hastings Theorem 429
16.3 Mean Fields and Gauge Fields 430
16.3.1 Fermionic Representation of Heisenberg Models 430
16.3.2 Local SU(2) Gauge Invariance 432
16.3.3 Mean-field (Spin-liquid) States 432
16.3.4 Gauge Fluctuations 436
16.3.4.1 Projective Symmetry Group and Invariant Gauge Group 437
16.3.4.2 Two Simple Examples of IGG 439
16.3.4.3 PSG Beyond the mean-field Approximation 439
16.4 Z2 Spin Liquids 441
16.4.1 Short-range RVB Description 441
16.4.2 Z2 Gauge Theory, Spinon Deconfinement, and Visons 442
16.4.3 Examples 444
16.4.4 How to Detect a Gapped Z2 Liquid 445
16.5 Gapless (Algebraic) Liquids 446
16.6 Other Spin Liquids 446
16.7 Conclusion 447
References 447
Chapter 17 Quantum Dimer Models 450
17.1 Introduction 450
17.2 How Quantum Dimer Models Arise 451
17.2.1 Link Variables and Hard Constraints 451
17.2.2 The Origin of Constraints 452
17.2.3 Tunable Constraints 453
17.2.4 Adding Quantum Dynamics 454
17.3 The Quantum Dimer Model Hilbert Space 456
17.3.1 Topological Invariants 456
17.3.2 Topological Order 458
17.3.3 Fractionalisation 459
17.4 QDM Phase Diagrams 460
17.4.1 General Structure of Phase Diagrams 460
17.4.1.1 The Rokhsar–Kivelson Point 461
17.4.1.2 Columnar Phase 461
17.4.1.3 Staggered Phase 462
17.4.2 Z2 RVB Liquid Phase 462
17.4.3 U(1) RVB Liquid Phase 464
17.4.4 Deconfined Critical Points 465
17.4.5 Valence Bond Crystals 465
17.4.5.1 Plaquette Phase 465
17.4.5.2 More Complex Crystals 466
17.4.5.3 Even more Complex Crystals: Cantor Deconfinement 466
17.4.6 Summary of Phase Diagrams 468
17.5 The Rokhsar–Kivelson Point 469
17.5.1 Ground-state Wavefunction 469
17.5.2 Fractionalisation and Deconfinement 470
17.5.3 Spatial Correlations 470
17.5.4 Excited States 471
17.5.5 A Special Liquid Point or part of a Liquid Phase? 472
17.6 Resonons, Photons, and Pions: Excitations in the Single mode Approximation 473
17.7 Dualities and Gauge Theories 475
17.7.1 Emergence of the QDM 476
17.7.2 Continuum Limit of the Gauge Theory 477
17.8 Height Representation 478
17.9 Numerical Methods 483
17.10 Dimer Phases in SU(2) Invariant Models 484
17.10.1 Overlap Expansion 485
17.10.2 Decoration 486
17.10.3 Large-N 487
17.10.4 Klein Models: SU(2) Invariant Spin Liquids 488
17.11 Outlook 488
17.11.1 Hopping Fermions … 489
17.11.2 … and much more 489
References 490
Chapter 18 Numerical Simulations of Frustrated Systems 493
18.1 Overview of Methods 493
18.2 Classical Monte Carlo 493
18.3 Quantum Monte Carlo 497
18.3.1 Stochastic Series Expansion (SSE) 497
18.3.2 Green-function Monte Carlo 499
18.4 Series Expansions 500
18.4.1 High-temperature Series 500
18.4.2 T = 0 Perturbative Expansions for Ground- and Excited-state Properties 501
18.5 Density-Matrix Renormalization Group (DMRG) 501
18.5.1 Finite T 502
18.5.2 Dynamical Response Functions 502
18.5.3 DMRG in two and more Dimensions 503
18.6 Exact Diagonalization (ED) 503
18.6.1 Basis Construction 504
18.6.2 Coding of Basis States 505
18.6.3 Symmetrized Basis States 506
18.6.4 Hamiltonian 508
18.6.5 Eigensolvers 509
18.6.5.1 Lanczos Algorithm 509
18.6.6 Implementation Details and Performance Aspects 511
18.6.7 Observables 512
18.6.7.1 Tower-of-States Analysis – the Nuclear Magnetism of bcc 3He 512
18.6.7.2 Detection of Exotic Order 514
18.6.8 Dynamical Response Functions 515
18.6.9 Time Evolution 516
18.6.10 Finite Temperatures 517
18.7 Miscellaneous Further Methods 518
18.7.1 Classical Spin Dynamics (Molecular Dynamics) 518
18.7.2 Coupled-Cluster Method 518
18.7.3 Dynamical Mean-Field Theory (DMFT) 519
18.7.4 Contractor Renormalization (CORE) 519
18.7.5 SR-RVB Calculations 519
18.8 Source Code Availability 520
References 521
Chapter 19 Exact Results in Frustrated Quantum Magnetism 524
19.1 Introduction 524
19.1.1 Dimer Model 525
19.2 Exact Results in Spin-1/2 Heisenberg Models 526
19.2.1 Exact Ground States in Coupled Triangular Cluster Models 527
19.2.1.1 Sawtooth–Chain Model 529
19.2.1.2 Majumdar–Ghosh Model 530
19.2.1.3 Two-Dimensional Shastry–Sutherland Model 531
19.2.2 Exact Ground States in Coupled Tetrahedral Cluster Models 533
19.2.2.1 Frustrated Ladder Model 534
19.2.3 Realization of Exact Ground States 535
19.3 Exact Results in Frustrated Spin-1/2 Models with Four-Spin Interactions 537
19.3.1 General Ladder Model with Four-Spin Interactions 537
19.3.1.1 One-Dimensional Shastry–Sutherland Model 540
19.3.1.2 Generalized AKLT Model 541
19.3.2 Two-Dimensional Model with Four-Spin Interactions 542
19.3.2.1 Generalized Two-Dimensional J1–J2 Model 542
19.3.2.2 Two-Dimensional J1–J2 Model with Products of Three-Spin Projectors 543
19.4 Conclusion 545
References 546
Chapter 20 Strong-Coupling Expansion and Effective Hamiltonians 548
20.1 Introduction 548
20.2 Strong-Coupling Expansion 549
20.2.1 Second-Order Perturbation Theory 550
20.2.2 High-Order Perturbation Theory 550
20.2.3 Examples 551
20.3 Alternative Approaches Yielding Effective Hamiltonians 558
20.3.1 Canonical Transformation 558
20.3.2 Continuous Unitary Transformation 559
20.3.3 Contractor Renormalization 566
20.4 Conclusions 567
References 569
Part VI Frustration, Charge Carriers and Orbital Degeneracy 571
Chapter 21 Mobile Holes in Frustrated Quantum Magnets and Itinerant Fermions on Frustrated Geometries 572
21.1 Introduction 572
21.2 Doping Holes in Frustrated Quantum Magnets 573
21.2.1 The Holon–Spinon Deconfinement Scenario 573
21.2.2 Single Hole Doped in Frustrated Mott Insulators 574
21.2.3 Hole Pairing and Superconductivity 577
21.3 Doped Quantum Dimer Model 578
21.3.1 Origin of the Quantum Dimer Model 578
21.3.2 Phase Diagrams at Zero Doping 580
21.3.3 Connection to the XXZ Magnet on the Checkerboard Lattice 580
21.3.4 Bosonic Doped Quantum Dimer Model 582
21.3.5 Non-Frobenius Doped Quantum Dimer Model on the Square Lattice 583
21.4 Mott Transition on the Triangular Lattice 584
21.4.1 Frustration in Itinerant Electron Systems 584
21.4.2 Mott Transition in Organic Compounds with Triangular Geometry 584
21.4.3 Mott Transition in the Triangular-Lattice Hubbard Model 585
21.5 Ordering Phenomena at Commensurate Fermion Densities on Frustrated Geometries 588
21.5.1 Bond Order Waves from Nesting Properties of the Fermi surface 589
21.5.2 Metal–Insulator Transitions and Frustrated Charge Order 590
21.5.3 Away from Commensurability: Doping the Resonating-Singlet-Pair Crystal 592
21.6 Summary 593
References 593
Chapter 22 Metallic and Superconducting Materials with Frustrated Lattices 596
22.1 Introduction 596
22.2 Materials Overview 599
22.2.1 Pyrochlore Lattice 599
22.2.1.1 Spinel Compounds 599
22.2.1.2 Mn Metals and LiV2O4 602
22.2.1.3 Pyrochlore Oxides 603
22.2.2 Triangular and Kagomé Lattices 605
22.2.2.1 NaxCoO2 and Its Hydrate 605
22.2.2.2 Ag2MO2 and AgNiO2 606
22.2.2.3 AB6O11 608
22.2.3 Organic Conductors with Triangular Lattice 608
22.2.3.1 -(BEDT-TTF)2X 610
22.2.3.2 - and -(BEDT-TTF)2X 612
22.3 Theoretical Background 613
22.3.1 RVB Spin State and RVB Superconductivity 613
22.3.1.1 Square- and Triangular-Lattice t-J Model 614
22.3.1.2 Triangular-Lattice Heisenberg Model 615
22.3.2 Triangular-Lattice Hubbard Model 616
22.3.3 Extended Hubbard Model for Organic Conductors 618
22.4 Superconducting Compounds 620
22.4.1 Pyrochlore Lattice: Cd2Re2O7 and AOs2O6 620
22.4.2 Triangular Lattice: NaxCoO2 and Its Hydrate 625
22.4.3 Anisotropic Triangular Lattice: Organic Superconductivity 629
22.5 Summary 630
References 630
Chapter 23 Frustration in Systems with Orbital Degrees of Freedom 637
23.1 Introduction 637
23.2 Orbital Degrees of Freedom 638
23.2.1 Orbitals and Their Energy Scales 638
23.2.2 Comparing Orbital and Spin Degrees of Freedom 640
23.3 Orbital Interactions and Orbital Models 642
23.3.1 Crystal-Field Splitting of Orbitals 642
23.3.2 Jahn–Teller Deformation 642
23.3.3 Jahn–Teller-Mediated Orbital–Orbital Interactions 644
23.3.4 Superexchange-Mediated Orbital–OrbitalInteractions 646
23.4 Symmetry and Symmetry-Breaking in Orbital Models 646
23.4.1 Types of Symmetry in Orbital Models 646
23.4.2 Examples of Intermediate Symmetries in Orbital Systems 647
23.4.3 A Theorem on Dimensional Reduction 650
23.4.4 Consequences of the Theorem for Orbital (and Spin) Orders and Excitations 653
23.5 Order by Disorder in Classical Orbital Models 654
23.6 Connection with Quantum Computation 657
23.6.1 Kitaev's Honeycomb Model 657
23.6.2 Kitaev's Toric Code model 659
23.6.3 Recent Discussions of Quantum Computing Realizations 660
23.7 Spin-Orbital Frustration 660
23.7.1 General Structure of Spin-Orbital Superexchange Models 660
23.7.2 Spin-Orbital Models for eg Perovskites 661
23.7.3 Spin-Orbital Superexchange for t2g Perovskites 665
23.7.4 Spin-Orbital Frustration on a Triangular Lattice 669
23.7.5 Spin-Orbital Frustration in Spinels 672
23.8 Spin-Orbital Entanglement 673
References 676
Index 679