First, it combines discussions of exact solutions, numerical simulations, real-space renormalization and field-theoretic methods, in a way which we hope illuminates the similarities and differences, and the strengths and weak- nesses, of all these approaches to the study of continuous phase transitions. Second, we have tried hard to make the book accessible to students with a good undergraduate background in physics but no knowledge of quantum field theory. Thus, we have taken pains to exclude as much jargon as possible, and to define clearly those technical terms that we do require. We have also tried to make the book as self-contained as possible by covering in boxes and appendices the technical details and mathematical techniques which are necessary for the solution of some of the more difficult problems in the field, but with which the reader may be unfamiliar.

Finally, at the end of each chapter we give several problems, which we hope will help readers to become familiar with the concepts and techniques introduced in that chapter. Complete solutions to all the problems are given at the back of the book.

While writing this book we have drawn freely on many sources, but es- pecially on the books by Daniel Amit (1984), Shang-Keng Ma (1976), Gior- gio Parisi (1988), Eugene Stanley (1971), and Jean Zinn-Justin (1989). In addition we are indebted to many people for helpful comments and suggestions, and for interesting conversations and seminars on the subject of critical phenomena. We would in particular like to thank Eytan Domany for the outstanding series of lectures he gave at Oxford in 1991, and Rob Phillips, Bruce Roberts and Jan von Delft for their careful reading of the manuscript and many helpful criticisms. Thanks are also due to Lawrence Harwood and the Dyson Perrins Laboratory for making Figure 1.8 possible.

Oxford January 1992

JJB NJD AJF MEJN

Contents

1 Introduction 1

• 1.1 Continuous phase transitions and critical points 2
• 1.1.1 Divergences and critical exponents 5
• 1.1.2 Fluctuations and critical opalescence 8
• 1.2 The order parameter 9
• 1.2.1 Liquid-gas transition 11
• 1.2.2 Binary fluids 11
• 1.2.3 Ferromagnetic/paramagnetic transition 12
• 1.2.4 Anti-ferromagnetic/paramagnetic transition 13
• 1.2.5 Helium I/helium II transition 13
• 1.2.6 Conductor/superconductor transitions 14
• 1.2.7 Helium three 15
• 1.3 Correlation functions 16
• 1.4 Universality 21
• 1.5 Thermodynamic potentials 21
• 1.5.1 The Widom and Kadanoff scaling hypotheses 27
• 1.6 Why study phase transitions? 30
• Problems 32

2 Statistical mechanics 33

• 2.1 Thermodynamic quantities 35
• 2.2 Fluctuations and correlation functions 40
• 2.3 Metastability and spontaneous symmetry breaking 47
• 2.3.1 Metastability 47
• 2.3.2 Spontaneous symmetry breaking 48
• Problems 52

3 Models 54

• 3.1 Description of models 55
• 3.1.1 The Ising model 55
• 3.1.2 The lattice gas 56
• 3.1.3 Beta-brass 57
• 3.1.4 The XY and Heisenberg models 57
• 3.1.5 Potts model 58
• 3.1.6 Gaussian and spherical models 58
• 3.1.7 Percolation model 60
• 3.2 Transfer matrices and the Ising ring 61
• 3.2.1 Solution of the Ising ring 62
• 3.2.2 Correlation functions 64
• 3.3 The partition function of the spherical model 66
• 3.4 High-temperature expansions and the Ising model 72
• 3.4.1 High-temperature expansions 72
• 3.4.2 The partition function of the Ising model 73
• 3.4.3 The correlation functions of the Ising model 78
• 3.4.4 Numerical evaluation of high-temperature expansions 80
• Problems 82

4 Numerical simulations 84

• 4.1 Direct evaluation of thermal averages 85
• 4.2 Sampling congurations 87
• 4.2.1 Importance sampling 89
• 4.2.2 General structure of numerical algorithms 91
• 4.3 Monte Carlo methods 92
• 4.3.1 The Metropolis algorithm 94
• 4.4 Molecular dynamics 95
• 4.4.1 Ergodicity and integrability 97
• 4.4.2 From microcanonical to canonical averages 98
• 4.5 Langevin equations 100
• 4.5.1 Comparison of the Langevin and molecular-dynamics Numerical methods 103
• 4.6 Independence of configurations 103
• 4.6.1 Correlations along the path 104
• 4.6.2 Critical slowing down 104
• 4.6.3 The Swendsen-Wang algorithm 106
• 4.6.4 The Wolff algorithm 108
• 4.7 Calculation of critical exponents from simulations 111
• Problems 111

5 Real-space renormalization 113

• 5.1 Renormalizing the lattice 114
• 5.2 Block variables 115
• 5.3 The renormalization of the Hamiltonian 117
• 5.3.1 Fixed points 120
• 5.3.2 The calculation of nu 124
• 5.4 The renormalization of B, M , chi and G_c 127
• 5.4.1 The value of omega 128
• 5.4.2 Non-zero external field 129
• 5.4.3 The renormalization of M , chi and G_c 131
• 5.4.4 Critical exponents for the renormalized model 132
• 5.5 The critical exponents for T = T_c 133
• 5.5.1 The exponent eta 133
• 5.5.2 The exponent delta 134
• 5.6 The critical exponents for T/=T_c 135
• 5.6.1 The exponent beta 136
• 5.6.2 The exponent gamma 137
• 5.6.3 The exponent alpha 137
• 5.7 The scaling laws 140
• 5.8 Bond percolation in two dimensions 143
• 5.9 The Ising model 147
• 5.10 Monte Carlo renormalization 153
• Problems 156

6 Mean-field theory 158

• 6.1 Mean-field theory of the Ising model 159
• 6.2 Mean-field theory of percolation 161
• 6.3 Mean-field theory of the non-ideal gas 162
• 6.4 A variational derivation of mean-field theory 164
• 6.5 Correlation functions in mean-field theory 168
• 6.6 Infinite-range interactions 171
• 6.7 Critical exponents in mean-field theory 173
• 6.7.1 Calculating eta from G_c^(2)(k) 175
• 6.8 What is missing from mean-field theory? 176
• Problems 177

7 The Landau-Ginzburg model 178

• 7.1 Formulation of the Landau-Ginzburg model 178
• 7.2 Landau theory 183
• Problems 187

• 8 Diagrammatic perturbation theory 188
• 8.1 The Gaussian partition function 189
• 8.1.1 Correlation functions in the Gaussian model 192
• 8.2 The partition function for the full Landau-Ginzburg model 195
• 8.2.1 The Feynman rules 195
• 8.2.2 The symmetry factor 199
• 8.3 The Helmholtz free energy of the Landau-Ginzburg model 203
• 8.3.1 Feynman rules in wavevector space 204
• 8.3.2 Vertex functions 211
• 8.4 The Gibbs free energy of the Landau-Ginzburg model 215
• 8.4.1 The rules for finding Gamma[phi] 216
• 8.4.2 The loop expansion 221
• 8.4.3 The one-loop Gibbs free energy 223
• Problems 226

9 Renormalization 228

• 9.1 Mass renormalization 230
• 9.2 Field renormalization 235
• 9.3 Renormalizing the coupling constant 238
• 9.4 Renormalization at higher orders 241
• 9.5 More on field renormalization 245
• 9.6 The Ginzburg criterion 246
• Problems 248

10 The calculation of critical exponents for T>=T_c 249

• 10.1 Ultraviolet and infrared divergences 249
• 10.2 The calculation of gamma 254
• 10.2.1 d = 4 and above 255
• 10.2.2 Below four dimensions 256
• 10.3 The calculation of eta 261
• 10.3.1 d = 4 and above 263
• 10.3.2 Below four dimensions 264
• 10.4 The epsilon-expansion 268
• 10.4.1 Dimensional regularization 270
• 10.4.2 Calculating gamma by dimensional regularization 271
• 10.4.3 Calculating eta by dimensional regularization 273
• 10.4.4 Feynman parameters 273
• 10.4.5 The calculation of eta again 275
• 10.4.6 Calculation of eta by the epsilon-expansion 278
• Problems 281

  11 The renormalization group 282

• 11.1 The renormalization group at T = T_c 283
• 11.2 The exponents eta and delta 293
• 11.2.1 The exponent eta 293
• 11.2.2 The exponent delta 294
• 11.3 The calculation of beta and gamma_1 296
• 11.3.1 The calculation of gamma_1 to order epsilon^2 296
• Problems 298

12 The renormalization group at T /= T_c 299

• 12.1 Expansion about the critical temperature 300
• 12.1.1 Functional Taylor expansions 301
• 12.1.2 Diagrammatic representation of the phi^2 correlation functions 301
• 12.1.3 Wavevector space 302
• 12.1.4 Vertex functions 304
• 12.1.5 Renormalization 305
• 12.1.6 Expanding the renormalized vertex functions 308
• 12.1.7 The validity of the expansion 309
• 12.2 The renormalization group equations 310
• 12.2.1 The exponent nu 311
• 12.2.2 The exponent gamma 312
• 12.2.3 The exponent alpha 312
• 12.3 The renormalization group below T_c 314
• 12.3.1 The exponent beta 315
• 12.4 Calculating gamma_2 to one loop 316
• Problems 318

13 The lower critical dimension 320

• 13.1 Order below T_c 321
• 13.1.1 The case D = 1 321
• 13.1.2 Systems with more than one component 322
• 13.1.3 Goldstone modes 324
• 13.2 The non-linear sigma-model 325
• 13.2.1 The two-point vertex function 329
• 13.2.2 The renormalization group equation 333
• 13.3 The Kosterlitz-Thouless transition 339
• 13.3.1 The two-dimensional Coulomb gas 342
• 13.3.2 General Remarks 350
• Problems 352

14 Universality 353

• 14.1 Perturbing the Gaussian Hamiltonian 355
• 14.1.1 The applicability of these results 361
• 14.2 Perturbing the Landau-Ginzburg Hamiltonian 361
• 14.2.1 The case of three dimensions 362
• 14.2.2 The case of two dimensions 366
• 14.3 Relevance and renormalizability 369
• Problems 373

Appendices

• A The magnetic scattering of neutrons 375
• B The natural variables for thermodynamic potentials 378
• C Magnetic energy 381
• D Connected correlation functions and log Z[J ] 384
• E The Gibbs free energy 386
• F Discrete Fourier transforms 390
• G The method of steepest descent 393
• H Counting closed loops on a square lattice 394
• I Einstein's uctuation theory 398
• J The Gaussian transformation 399
• K The Landau-Ginzburg model and the Ising model 400
• L Functional differentiation and integration 404
• M The Feynman rules for the vertex functions 417
• N Feynman rules for generalized Landau-Ginzburg models 421

Answers 429

References 448

Index 453