# Series Expansion Methods for Strongly Interacting Lattice Models

## Jaan Oitmaa, Chris Hamer, Weihong Zheng

## School of Physics, University of new South Wales, Sydney, Australia

Perturbation series expansion methods are sophisticated numerical tools used to provide quantitative calculations in many areas of theoretical physics. This book gives a comprehensive guide to the use of series expansion methods for investigating phase transitions and critical phenomena, and lattice models of quantum magnetism, strongly correlated electron systems and elementary particles. Early chapters cover the classical treatment of critical phenomena through high temperature expansions, and introduce graph theoretical and combinatorial algorithms. The book then discusses high-order linked-cluster perturbation expansions for quantum lattice models, finite temperature expansions, and lattice gauge models. Also included are numerous detailed examples and case studies, and an accompanying resources website, contains programs for implementing these powerful numerical techniques. A valuable resource for graduate students and postdoctoral researchers working in condensed matter and particle physics, this book will also be useful as a reference for specialized graduate courses on series expansion methods.

- Hands on approach, suitable for self-learning
- A comprehensive guide to series expansion methods for lattice models in theoretical physics
- Applications to models in condensed matter theory and particle physics
- Computer programs for implementation of this powerful numerical technique are available at cambridge (fortran 77 version only) or at Source Codes here (both fortran 77 and 90 version)

### Table of Contents:

(last number is the page number)

- Introduction;
- 1.1 Lattice models in theoretical physics 1
- 1.2 Examples and applications 1
- 1.2.1 The Ising model 2
- 1.2.2 The Ising model in a transverse field 4
- 1.2.3 The Heisenberg model 5
- 1.2.4 The Hubbard model 7
- 1.2.5 Lattice gauge models 9

- 1.3 The important questions 10
- 1.4 Series expansion methods 14
- 1.4.1 High-temperature series 15
- 1.4.2 Low-temperature series 16
- 1.4.3 Perturbation expansions at zero temperature 16
- 1.4.4 Thermodynamic perturbation theory 17

- 1.5 Analysis of series 19

- High and low temperature expansions for the Ising Model;
- 2.1 Introduction 26
- 2.2 Graphgeneration and computation of lattice constants 30
- 2.3 A case study: high-temperature susceptibility for the Ising model on the simple cubic lattice 36
- 2.4 Low-temperature expansion 39
- 2.5 Reducing the number of graphs 42
- 2.5.1 Connected graph expansion 43
- 2.5.2 Star graph expansion 45

- 2.6 More on Ising models 47
- 2.6.1 General Spin-S 47
- 2.6.2 Antiferromagnets and Ferrimagnets 48
- 2.6.3 Further Neighbour Interactions 49
- 2.6.4 Continuous Spin Ising Models 51

- Models with continuous symmetry and the free graph expansion;
- 3.1 Introduction 53
- 3.2 The free graph expansion 55
- 3.3 The plane rotator (N = 2) model 66
- 3.4 Analysis of the N = 2 susceptibility 69
- 3.5 Discussion 72
- Quantum spin models at T = 0;
- 4.1 Introduction 74
- 4.2 Linked cluster expansions 74
- 4.3 An example: the transverse field Ising model in one dimension 78
- 4.3.1 Expansion in the disordered phase 78
- 4.3.2 Expansion in the ordered phase 80

- 4.4 Magnetization and susceptibility 82
- 4.5 One-particle excitations 84
- 4.5.1 Block diagonalization 86
- 4.5.2 Linked-cluster expansion 88
- 4.5.3 Example: the transverse field Ising chain 90

- 4.6 The transverse field Ising model in two and three dimensions 92
- 4.6.1 Expansions in The Disordered Phase 93
- 4.6.2 Expansions in the Ordered Phase 95
- 4.6.3 One-Particle Excitations 95

- Quantum antiferromagnets at T = 0;
- 5.1 Introduction: simple antiferromagnets 99
- 5.1.1 The Ising Expansion 99
- 5.1.2 The Energy and Magnetization 101
- 5.1.3 One Magnon Excitations 103

- 5.2 Dimerized systems and quantum phase transitions 106
- 5.2.1 Dimer Expansions and the Alternating Chain 107
- 5.2.2 Coupled Chains and Coupled Planes 109

- 5.3 The J1–J2 square lattice antiferromagnet 112
- 5.3.1 The Néel and Columnar Phases 113
- 5.3.2 The Intermediate Phase 114
- 5.3.3 Summary 116

- 5.4 Other systems 118
- 5.4.1 The CAVO system 118
- 5.4.2 The SCBO system 120

- 5.5 Open questions 122
- Correlators, dynamical structure factors and multiparticle excitations;
- 6.1 Introduction 124
- 6.2 Two-spin correlators for the Heisenberg antiferromagnet 125
- 6.3 Dynamical and static structure factors 126
- 6.3.1 Formalism 126
- 6.3.2 Calculation of Structure Factors 128
- 6.3.3 Example: The Transverse Field Ising Chain 129
- 6.3.4 The S = 1/2 Square Lattice Antiferromagnet 131

- 6.4 Two-particle and multi-particle excitations 134
- 6.4.1 Linked-Cluster Expansion 134
- 6.4.2 Finite-Lattice Approach to the 2-particle Schrödinger Equation 136
- 6.4.3 Example: The Transverse Field Ising Chain 137
- 6.4.4 The J1-J2-
*d*Chain 142

- 6.5 Two-particle structure factors 145
- 6.5.1 The Alternating Heisenberg Chain 146

- 6.6 Summary and further work 147
- Quantum spin models at finite temperature;
- 7.1 Introduction 150
- 7.2 Derivation of high-temperature series 151
- 7.2.1 The Moment Method 152
- 7.2.2 The Linked-Cluster Method 157
- 7.2.3 The Staggered Susceptibility 162

- 7.3 The cubic (SC and BCC) lattices 165
- 7.4 Generalizations 168
- 7.4.1 Exchange Anisotropy 168
- 7.4.2 Further Neighbour Interactions 168
- 7.4.3 Higher Spin 169

- 7.5 Perturbation expansions at finite T 169
- 7.5.1 The Alternating S = 1/2 Antiferromagnetic Chain 171

- 7.6 Further applications 175
- 7.7 Fitting to experimental data 176
- Electronic models;
- 8.1 Introduction 179
- 8.2 The Hubbard model 180
- 8.2.1 t/U Expansions at Zero Temperature 181
- 8.2.2 Ising Expansions 185
- 8.2.3 Dimer Expansions 190
- 8.2.4 t/U Expansions at Finite Temperature 192

- 8.3 The t–J model 197
- 8.3.1 Expansions at Zero Temperature 198
- 8.3.2 Variable Electron Density 203
- 8.3.3 High Temperature Expansions 205

- 8.4 Further topics and possibilities 209
- Review of lattice gauge theory;
- 9.1 Quantum chromodynamics 211
- 9.2 The pathinte gral approachto field theory 214
- 9.3 Euclidean lattice gauge theory 217
- 9.4 Confinement and phase structure on the lattice 219
- 9.5 Renormalization theory and the continuum limit 221
- 9.6 Monte Carlo simulations 222
- 9.7 Including fermions on the lattice 225
- 9.8 The Hamiltonian lattice formulation 227
- 9.9 Conclusions 228
- Series expansions for lattice gauge models;
- 10.1 Strong coupling expansions for Euclidean lattice
Yang–Mills theory 230
- 10.1.1 Group Theory Considerations. 231
- 10.1.2 The Free Energy 233
- 10.1.3 The String Tension 238
- 10.1.4 Glueball Masses 241

- 10.2 Strong coupling expansions in Hamiltonian Yang–Mills theory 244
- 10.2.1 Group Theory Considerations 244
- 10.2.2 Ground State Energy 245
- 10.2.3 The String Tension 248
- 10.2.4 Glueball Masses 249

- 10.3 Models withdynamical fermions 251
- 10.3.1 The Lattice Schwinger Model 251
- 10.3.2 Hamiltonian Strong-Coupling Expansions in (3+1)D 255

- 10.4 The t-expansion 259
- 10.4.1 Results for Dynamical Fermions 261

- 10.5 Conclusions 263
- Additional topics;
- 11.1 Disordered systems 265
- 11.1.1 Percolation 265
- 11.1.2 Random Bond Ising Models 267
- 11.1.3 Ising Spin Glasses 271
- 11.1.4 Further Topics 273

- 11.2 Other series expansion methods 274
- 11.2.1 The Finite-Lattice Method of Enting 274
- 11.2.2 The Schwinger-Dyson Equation Method 276
- 11.2.3 The Continuous Unitary Transformation Method 278

- Appendix 1: some graphth eory ideas 283
- Appendix 2: the pegs in holes algorithm 286
- Appendix 3: free graphe xpansion technicalities 288
- Appendix 4: matrix perturbation theory 291
- Appendix 5: matrix block diagonalization 294
- Appendix 6: the moment–cumulant expansion 297
- Appendix 7: integral equation approachto the two-particle Schrodinger equation 299
- Appendix 8: correspondences between field theory and statistical mechanics 304
- Appendix 9: computer programs 307
- Bibliography 311
- Index 324