Series expansion methods for strongly interacting lattice models

Contents......Page 5
Preface......Page 8
1.2 Examples and applications......Page 11
1.2.1 The Ising model......Page 12
1.2.2 The Ising model in a transverse fiel......Page 14
1.2.3 The Heisenberg model......Page 15
1.2.4 The Hubbard model......Page 17
1.2.5 Lattice gauge models......Page 19
1.3 The important questions......Page 20
1.4 Series expansion methods......Page 24
1.4.1 High-temperature series......Page 25
1.4.3 Perturbation expansions at zero temperature......Page 26
1.4.4 Thermodynamic perturbation theory......Page 27
1.5 Analysis of series......Page 29
2.1 Introduction......Page 36
2.2 Graph generation and computation of lattice constants......Page 40
2.3 A case study: high-temperature susceptibility for the Ising model on the simple cubic lattice......Page 46
2.4 Low-temperature expansion......Page 49
2.5 Reducing the number of graphs......Page 52
2.5.1 Connected graph expansion......Page 53
2.5.2 Star graph expansion......Page 55
2.6.1 General spin-S......Page 57
2.6.2 Antiferromagnets and ferrimagnets......Page 58
2.6.3 Further neighbour interactions......Page 59
2.6.4 Continuous spin Ising models......Page 61
3.1 Introduction......Page 63
3.2 The free graph expansion......Page 65
Rules for unrenormalized free energy......Page 68
3.3 The plane rotator (N = 2) model......Page 76
3.4 Analysis of the N = 2 susceptibility......Page 79
3.5 Discussion......Page 82
4.2 Linked cluster expansions......Page 84
4.3.1 Expansion in the disordered phase......Page 88
4.3.2 Expansion in the ordered phase......Page 90
4.4 Magnetization and susceptibility......Page 92
4.5 One-particle excitations......Page 94
4.5.1 Block diagonalization......Page 96
4.5.2 Linked cluster expansion......Page 98
4.5.3 Example: the transverse fiel Ising chain......Page 100
4.6 The transverse fiel Ising model in two and three dimensions......Page 102
4.6.1 Expansions in the disordered phase......Page 103
4.6.3 One-particle excitations......Page 105
5.1.1 The Ising expansion......Page 109
5.1.2 The energy and magnetization......Page 111
5.1.3 One-magnon excitations......Page 114
5.2 Dimerized systems and quantum phase transitions......Page 116
5.2.1 Dimer expansions and the alternating chain......Page 117
5.2.2 Coupled chains and coupled planes......Page 119
5.3 The J1–J2 square lattice antiferromagnet......Page 122
5.3.1 The Neel´ and columnar phases......Page 123
5.3.2 The intermediate phase......Page 124
5.3.3 Summary......Page 126
5.4.1 The CAVO system......Page 128
5.4.2 The SCBO system......Page 130
5.5 Open questions......Page 132
6.1 Introduction......Page 134
6.2 Two-spin correlators for the Heisenberg antiferromagnet......Page 135
6.3.1 Formalism......Page 136
6.3.2 Calculation of structure factors......Page 138
6.3.3 Example: the transverse fiel Ising chain......Page 139
6.3.4 The S = 1/2 square lattice antiferromagnet......Page 141
6.4.1 Linked cluster expansion......Page 144
6.4.2 Finite lattice approach to the two-particle Schrödinger equation......Page 146
6.4.3 Example: the transverse fiel Ising chain......Page 147
6.4.4 The J1–J2–δ chain......Page 152
6.5 Two-particle structure factors......Page 155
6.5.1 The alternating Heisenberg chain......Page 156
6.6 Summary and further work......Page 157
7.1 Introduction......Page 160
7.2 Derivation of high-temperature series......Page 161
7.2.1 The moment method......Page 162
7.2.2 The linked cluster method......Page 167
7.2.3 The staggered susceptibility......Page 172
7.3 The cubic (SC and BCC) lattices......Page 175
7.4.2 Further neighbour interactions......Page 178
7.5 Perturbation expansions at finit T......Page 179
7.5.1 The alternating S = 1/2 antiferromagnetic chain......Page 181
7.6 Further applications......Page 185
7.7 Fitting to experimental data......Page 186
8.1 Introduction......Page 189
8.2 The Hubbard model......Page 190
8.2.1 t/U expansions at zero temperature......Page 191
8.2.2 Ising expansions......Page 195
8.2.3 Dimer expansions......Page 200
8.2.4 t/U expansions at finit temperature......Page 202
8.3 The t-J model......Page 207
8.3.1 Expansions at zero temperature......Page 208
8.3.2 Variable electron density......Page 213
8.3.3 High-temperature expansions......Page 215
8.4 Further topics and possibilities......Page 219
9.1 Quantum chromodynamics......Page 221
9.2 The path integral approach to fiel theory......Page 224
9.3 Euclidean lattice gauge theory......Page 227
9.4 Confinemen and phase structure on the lattice......Page 229
9.5 Renormalization theory and the continuum limit......Page 231
9.6 Monte Carlo simulations......Page 232
9.7 Including fermions on the lattice......Page 235
9.8 The Hamiltonian lattice formulation......Page 237
9.9 Conclusions......Page 238
10.1 Strong coupling expansions for Euclidean lattice Yang–Mills theory......Page 240
10.1.1 Group theory considerations......Page 241
10.1.2 The free energy......Page 243
10.1.3 The string tension......Page 248
10.1.4 Glueball masses......Page 251
10.2.1 Group theory considerations......Page 254
10.2.2 Ground state energy......Page 255
10.2.3 The string tension......Page 258
10.2.4 Glueball masses......Page 259
10.3.1 The lattice Schwinger model......Page 261
10.3.2 Hamiltonian strong-coupling expansions in (3 + 1)D......Page 265
10.4 The t-expansion......Page 269
10.4.1 Results for dynamical fermions......Page 271
10.5 Conclusions......Page 273
11.1.1 Percolation......Page 275
11.1.2 Random bond Ising models......Page 277
11.1.3 Ising spin glasses......Page 281
11.1.4 Further topics......Page 283
11.2.1 The ‘finit lattice’ method of Enting......Page 284
11.2.2 The Schwinger–Dyson equation method......Page 286
11.2.3 The continuous unitary transformation method......Page 288
Appendix 1: some graph theory ideas......Page 293
Appendix 2: the ‘pegs in holes’ algorithm......Page 296
Appendix 3: free graph expansion technicalities......Page 298
Appendix 4: matrix perturbation theory......Page 301
Appendix 5: matrix block diagonalization......Page 304
Appendix 6: the moment–cumulant expansion......Page 307
Appendix 7: integral equation approach to the two-particle Schrodinger¨ equation......Page 309
Appendix 8: correspondences between fiel theory and statistical mechanics......Page 314
Appendix 9: computer programs......Page 317
References......Page 321
Index......Page 334