Spin Systems. Frustrated. \fa. 2nd Edition. H T Diep. World Scientific. University of Cergy-Pontoise, France. Editor HONG SINGAPORE KONG TAIPEI

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1 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

2 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 Definition Non-collinear spin configurations 5 12 Frustrated Ising spin systems 8 13 Mapping between Ising models and vertex models The 16-vertex model The 32-vertex model Disorder solutions for two-dimensional Ising models Reentrance in exactly solved frustrated Ising spin systems Centered square lattice Phase diagram Nature of ordering and disorder solutions Kagorne lattice Model with nn and nnn interactions 1422 Generalized Kagome lattice Centered honeycomb lattice Periodically dilute centered square lattices Model with three centers Model with two adjacent centers Model with one center Random-field aspects of the models 48 xiii

3 xiv Frustrated Spin Systems (2nd Edition) 15 Evidence of partial disorder and reentrance in other frustrated systems 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 Ising model on two-dimensional frustrated lattice and on stacked frustrated lattice Ising model on antiferromagnetic triangular lattice Ising model on stacked antiferromagnetic triangular lattice Ising model with large S on antiferromagnetic triangular lattice Ising model with infinite-spin on antiferromagnetic triangular lattice Ising-like Heisenberg model on antiferromagnetic triangular lattice Ising model with infinite-spin on stacked antiferromagnetic triangular lattice Phase diagram in spin-magnitude versus temperature for Ising models with spin S on stacked antiferromagnetic triangular lattice Effect of antiferromagnetic interaction between next-nearest-neighbor spins in :ry-plane Three-dimensional Ising paramagnet Concluding remarks 102 Acknowledgements 103 References Renormalization Group Approaches to Frustrated Magnets in D=3 107 B Delamotte, D Mouhanna and M Tissier 31 Introduction The STA model and generalization 109

4 e Contents xv 321 The lattice model, its continuum limit and symmetries The Heisenberg case The XY case Generalization Experimental and numerical situations The XY systems The experimental situation The numerical situation Summary The Heisenberg systems The experimental situation The numerical situation Summary TheiV = 6STA Conclusion A brief chronological survey of the theoretical approaches The perturbative situation The Nonlinear Sigma (NLct) model approach The Ginzburg-Landau-Wilson (GLW) model approach The RG flow The three and five-loop results in d = The improved three and five-loop results The three-loop results in d = The large-iv results The six-loop results in d = The six-loop results in d = 3 re-examined Conclusion The effective average action method The effective average action equation Properties Truncations Principle of the calculation The 0{N) x 0(2) model The flow equations Tests of the method and first results The physics in d = 3 according to the NPRG approach 159

5 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 Scaling with or without pseudo-fixed point: the Heisenberg and XY cases The integration of the RG flow The Heisenberg case The XY case Conclusion Conclusion and prospects Note added in the 2nd Edition 171 References Phase Transitions in Frustrated Vector Spin Systems: Numerical Studies D Loison Introduction Breakdown of Symmetry Symmetry in the high-temperature region Breakdown of symmetry for ferromagnetic systems Breakdown of symmetry for frustrated systems 4231 Stacked triangular antiferromagnetic 4232 bet Helimagnets lattices Stacked J\-J<2 square lattices The simple cubic Ji-J? lattice JW2-J3 lattice Villain lattice and fully frustrated simple cubic lattice Face-centered cubic lattice (fee) Hexagonal-close-packed 4239 Pyrochlores lattice (hep) Other lattices STAR lattices Dihedral lattices Vw, Right-handed trihedral lattices V3, P-hedral lattices VN,P Ising and Potts-Vjvi model Ising and Potts-Vjv,2 model 195

6 2) P) 2) 1) 1) Contents xvii Landau-Ginzburg model Cubic term in Hamiltonian Summary Phase transitions between two and four dimensions: 2 < d < (N)/0(N - breakdown of symmetry Fixed points MCRG and first-order transition Complex fixed point or minimum in the flow Experiment Value of Nc Phase diagram (N,d) Renormalization-Group expansions Short historical review Relations with the Potts model (N)/0(N Z2 SO(N)/SO{N - breakdown of symmetry for d = breakdown of symmetry ford = Z3 SO(N)/SO{N - breakdown of symmetry for d = Zq 0{N)/0{N - and other breakdown of symmetry ind= Conclusion O(N) frustrated vector spins in d 2 = Introduction Non-frustrated XY spin systems Frustrated XY spin systems: Z2 50(2) Frustrated XY spin systems: Z3 50(2) Frustrated XY spin systems: Z2 <g> Z2 SO{2) and Z3 Z2 50(2) Frustrated Heisenberg spin systems: 50(3) Frustrated Heisenberg spin systems: Z2 <8> 50(3), 23 50(3) Topological defects for iv > General conclusions 220 Acknowledgments Note added for the 2nd Edition 220 Appendix A: Monte Carlo Simulation 222

7 instantons xviii Frustrated Spin Systems (2nd Edition) Appendix B: Renormalization Group 226 References Two-Dimensional Quantum Antiferromagnets 235 Gregoire Misguich and Claire Lhuillier 51 Introduction J1-J2 model on the square lattice Classical ground state and spin-wave analysis Order by disorder (J2> Ji/2) Non-magnetic region (J2 JJ2) Series expansions Exact diagonalizations Quantum Monte Carlo Valence-bond crystals Definitions One-dimensional and quasi one-dimensional examples (spin- \ systems) Valence Bond Solids Two-dimensional examples of VBC Without spontaneous lattice symmetry breaking With spontaneous lattice symmetry breaking Methods Summary of the properties of VBC phases Large-iV methods Bond variables SU{N) Sp(N) Gauge invariance Mean-field (N = 00 limit) Fluctuations about the mean-field solution Topological effects and spontaneous dimerization Deconfined phases Quantum Dimer Models Hamiltonian Relation with spin-^ models 263

8 Contents xix Square lattice Transition 265 graphs and topological sectors 5532 Staggered VBC for V/J > Columnar crystal for V < Plaquette phase Rokhsar-Kivelson point Hexagonal lattice Triangular lattice RVB liquid at the RK point Topological order Solvable QDM on the kagome lattice Hamiltonian RK ground state Ising pseudo-spin variables Dimer-dimer correlations Visons excitations Spinous deconfinement Z2 gauge theory A QDM with an extensive ground state entropy Multiple-spin exchange models Physical realizations of multiple-spin interactions 5611 Nuclear magnetism of solid 3He Wigner crystal Cuprates Two-leg ladders MSE model on the square lattice RVB phase of the triangular J2-J4 MSE Non-planar classical ground states Absence of Neel LRO Local singlet-singlet correlations absence of lattice symmetry breaking Topological degeneracy and Lieb-Schultz-Mattis Theorem Deconfined spinons Other models with MSE interactions 290 Antiferromagnets on the kagome lattice Ising model Classical Heisenberg models on the kagome lattice 291

9 XX Frustrated Spin Systems (2nd Edition) 573 Nearest-neighbor RVB description of the spin-^ kagome antiferromagnet Spin-5 Heisenberg model on the kagome lattice: numerics Ground-state energy per spin Correlations Spin gap Singlet gap Entanglement entropy and signature of a Z2 liquid Spin liquids on the kagome lattice and Projective symmetry groups Competing phases Valence Bond Crystals U(l) Dirac Spin Liquid Spontaneously breaking the time-reversal symmetry, "crural" spin liquids Experiments in compounds with kagome-like lattices Conclusions 304 References One-Dimensional Quantum Spin Liquids 321 P Lecheminant 61 Introduction Unfrustrated spin chains Spin-1/2 Heisenberg chain Haldane's conjecture Haldane spin liquid: spin-1 Heisenberg chain General spin-5 case Two-leg spin ladder Non-Haldane spin liquid Frustration effects Semiclassical analysis Spin liquid phase with massive deconfined spinons Field theory of spin liquid with incommensurate correlations Extended criticality stabilized by frustration 359

10 Contents xxi 6341 Critical phases with SU(N) quantum criticality Chirally stabilized critical spin liquid Concluding remarks Note added for the 2nd Edition 370 References Spin Ice 383 Steven T Bramwell, Michel J P Gingras and Peter C W Holdsworth 71 Introduction Prom Water Ice to Spin Ice Pauling's model Why is the zero point entropy not zero? Generalizations of Pauling's model Wannier's model Anderson's model Vertex models Possibility of realizing magnetic vertex models Spin ice Definition of the spin ice model and its application to Ho2Ti Identification of spin ice materials Basic properties of the spin ice materials Spin ice as a frustrated magnet Frustration and underconstraining (111) Pyrochlore models Properties of the Zero Field Spin Ice State Experimental properties Heat capacity: zero point entropy Low field magnetic susceptibility: spin freezing Spin arrangement observed by neutron scattering Microscopic theories and experimental tests Near-neighbour spin ice model: successes and failures 407

11 xxii Frustrated Spin Systems (2nd Edition) 7322 The problem of treating the dipolar interaction The Ewald Monte Carlo Mean-field theory The loop Monte Carlo Application of the dipolar model to neutron scattering results How realistic is the dipolar model? Field-Induced Phases Theory Near neighbour model Dipolar model Magnetization measurements above T = IK Bulk measurements at low temperature [111] Direction [110] Direction [100] Direction [211] Direction Powder measurements Neutron scattering results [110] Direction [100], [111] and [211] Directions Kagome ice Basic Kagome ice model and mappings Experimental results: specific heat Theory of the Kagome ice state: Kastelyn transition Spin Dynamics of the Spin Ice Materials Experimental quantities of interest Correlation functions and neutron scattering Fluctuation-dissipation theorem and AC-susceptibility Spectral shape function Exponential relaxation Differences between Ho2Ti207 and Dy2Ti Relaxation at high temperature, T ~ 15 K and above AC-susceptibility (AC-x) 452

12 \ Contents xxiii 7532 Neutron spin echo (NSE) Origin of the 15 K AC-susceptibility peak Relaxation in the range 1 K < T < 15 K AC-susceptibility: phenomenological model 7542 AC-susceptibility: towards a microscopic 457 model Spin dynamics in the spin ice regime below IK Slow relaxation Evidence for residual dynamics in the frozen state Doped spin ice Spin ice under pressure Spin Ice Related Materials Rare earth titanates Other pyrochlores related to spin ice Conclusions 465 Acknowledgments Note added for the 2nd Edition 467 References Experimental Studies of Frustrated Pyrochlore Antiferromagnets 475 Bruce D Gaulin and Jason S Gardner 81 Introduction Pyrochlore Lattices Neutron Scattering Techniques Cooperative Paramagnetism in TbaTi The Spin Glass Ground State in Y2Mo Composite Spin Degrees of Freedom and Spin-Peierls-like Ground State in the Frustrated Spinel ZnCr Conclusions and Outlook 504 References Recent Progress in Spin Glasses 509 N Kawashima and H Rieger 91 Two Pictures Mean-field picture Droplet picture 514

13 Introduction xxiv Frustrated Spin Systems (2nd Edition) 92 Equilibrium Properties of Two-Dimensional Ising Spin Glasses Zero-temperature transition? Droplet argument for Gaussian-Coupling models 923 Droplets in Gaussian-Coupling models: numerics Finite-temperature transition? Equilibrium Properties of Three-Dimensional Models Finite temperature transition? Universality class Low-temperature phase of the ±J model Low-temperature phase of the Gaussian-Coupling model Effect of magnetic fields Sponge-like excitations TNT picture of a new scaling length Arguments supporting the droplet picture Models in Four or Higher Dimensions Aging A growing length scale during aging? Two time quantities: isothermal aging More complicated temperature protocols Violation of the Fluctuation-Dissipation theorem Hysteresis in spin glasses Equilibrium Properties of Classical XY and Heisenberg Spin Glasses Continuous spin models in three dimensions Continuous spin models in higher dimensions Potts spin glasses Weak Disorder Phase diagram of the discrete spin models Dynamical properties The renormalization group approach for the discrete models The location of the multi-critical point Phase diagram of the random XY model in two dimensions Quantum Spin Glasses Random transverse Ising models 581

14 Dissipative Dynamics Contents xxv 982 Mean-field theory Mean-field theory 984 Mean-field theory effects Heisenberg quantum spin glasses Finite dimensions Mean-field model Summary and Remaining Problems 602 Acknowledgments 604 References 605 Index 615

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