PART Ⅰ BASIC IDEAS AND TECHNIQUES 3
1 Pertinent concepts and ideas in the theory of critical phenomena 3
1-1 Description of critical phenomena 3
1-2 Scaling and homogeneity 5
1-3 Comparison of various results for critical exponents 6
1-4 Universality—dimensionality，symmetry 8
Exercises 9
2 Formulation of the problem of phase transitions in terms of functional integrals 11
2-1 Introduction 11
2-2 Construction of the Lagrangian 12
2-2-1 The real scalar field 12
2-2-2 Complex field 12
2-2-3 A hypercubic n-vector model 13
2-2-4 Two coupled fluctuating fields 14
2-3 The parameters appearing in？ 14
2-4 The partition function，or the generating functional 15
2-5 Representation of the Ising model in terms of functional integrals 18
2-5-1 Definition of the model and its thermodynamics 18
2-5-2 The Gaussian transformation 21
2-5-3 The free part 22
2-5-4 Some properties of the free theory—a free Euclidean field theory in less than four dimensions 26
2-6 Correlation functions including composite operators 28
Exercises 30
3 Functional integrals in quantum field theory 33
3-1 Introduction 33
3-2 Functional integrals for a quantum-mechanical system with one degree of freedom 34
3-2-1 Schwinger's transformation function 34
3-2-2 Matrix elements—Green functions 37
3-2-3 The generating functional 38
3-2-4 Analytic continuation in time—the Euclidean theory 40
3-3 Functional integrals for the scalar boson field theory 41
3-3-1 Introduction 41
3-3-2 The generating functional for Green functions 43
3-3-3 The generating functional as a functional integral 44
3-3-4 The S-matrix expressed in terms of the generating functional 47
Exercises 50
4 Perturbation theory and Feynman graphs 53
4-1 Introduction 53
4-2 Perturbation expansion in coordinate space 54
4-3 The cancellation of vacuum graphs 60
4-4 Rules for the computation of graphs 60
4-5 More general cases 63
4-5-1 The M-vector theory 63
4-5-2 Comments on fields with higher spin 67
4-6 Diagrammatic expansion in momentum space 68
4-7 Perturbation expansion of Green functions withZ composite operators 72
4-7-1 In coordinate space 72
4-7-2 In momentum space 74
4-7-3 Insertion at zero momentum 76
Exercises 77
5 Vertex functions and symmetry breaking 80
5-1 Introduction 80
5-2 Connected Green functions and their generating functional 82
5-3 The mass operator 85
5-4 The Legendre transform and vertex functions 86
5-5 The generating functional and the potential 91
5-6 Ward-Takahashi identities and Goldstone's theorem 94
5-7 Vertex parts for Green functions with composite operators 96
Exercises 101
6 Expansions in the number of loops and in the number of components 103
6-1 Introduction 103
6-2 The expansion in the number of loops as a power series 104
6-3 The tree（Landau-Ginzburg）approximation 105
6-4 The one-loop approximation and the Ginzburg criterion 109
6-5 Mass and coupling constant renormalization in the one-loop approximation 112
6-6 Composite field renormalization 116
6-7 Renormalization of the field at the two-loop level 117
6-8 The 0（M）-symmetric theory in the limit of large M 126
6-8-1 General remarks 126
6-8-2 The origin of the M-dependence of the coupling constant 127
6-8-3 Faithful representation of graphs and the dominant terms in Γ（4） 128
6-8-4 Γ（2）in the infinite M limit 130
6-8-5 Renormalization 133
6-8-6 Broken symmetry 134
Appendix 6-1 The method of steepest descent and the loop expansion 137
Exercises 142
7 Renormalization 147
7-1 Introduction 147
7-2 Some considerations concerning engineering dimensions 148
7-3 Power counting and primitive divergences 151
7-4 Renormalization of a cutoff φ4 theory 157
7-5 Normalization conditions for massive and massless theories 159
7-6 Renormalization constants for a massless theory to order two loops 161
7-7 Renormalization away from the critical point 164
7-8 Counterterms 167
7-9 Relevant and irrelevant operators 169
7-10 Renormalization of a φ4 theory with an 0（M）symmetry 171
7-11 Ward identities and renormalization 174
7-12 Iterative construction of counterterms 179
Exercises 185
8 The renormalization group and scaling in the critical region 189
8-1 Introduction 189
8-2 The renormalization group for the critical（massless）theory 190
8-3 Regularization by continuation in the number of dimensions 195
8-4 Massless theory below four dimensions—the emergence of ∈ 196
8-5 The solution of the renormalization group equation 197
8-6 Fixed points，scaling，and anomalous dimensions 199
8-7 The approach to the fixed point—asymptotic freedom 201
8-8 Renormalization group equation above Tc—identification of v 205
8-9 Below the critical temperature—the scaling form of the equation of state 208
8-10 The specific heat—renormalization group equation for an additively renormalized vertex 210
8-11 The Callan-Symanzik equations 212
8-12 Renormalization group equations for the bare theory 214
8-13 Renormalization group equations and scaling in the infinite M limit 217
Appendix 8-1 General formulas for calculating Feynman integrals 222
Exercises 223
9 The computation of the critical exponents 228
9-1 Introduction 228
9-2 The symbolic calculation of the renormalization constants and Wilson functions 230
9-3 The ∈expansion of the critical exponents 233
9-4 The nature of the fixed points—universality 237
9-5 Scale invariance at finite cutoff 238
9-6 At the critical dimension—asymptotic infrared freedom 240
9-7 ∈expansion for the Callan-Symanzik method 243
9-8 ∈expansion of the renormalization group equations for the bare functions 247
9-9 Dimensional regularization and critical phenomena 248
9-10 Renormalization by minimal subtraction of dimensional poles 250
9-11 The calculation of exponents in minimal subtraction 255
Appendix 9-1 Calculation of some integrals with cutoff 257
9-2 One-loop integrals in dimensional regularization 260
9-3 Two-loop integrals in dimensional regularization 263
Exercises 266
PART Ⅱ FURTHER APPLICATIONS AND DEVELOPMENTS 273
1 Introduction 273
2 Beyond leading scaling 275
2-1 Corrections to scaling in aφ4 theory 275
2-2 Finite-size scaling 277
2-3 Anomalous dimensions of high composite operators 280
2-4 Corrections due to irrelevant operators 288
2-5 Next-to-leading terms in the scaling region 293
2-6 The operator product expansion 295
2-7 Computation of next-to-leading terms in ∈-expansion 297
Appendix 2-1 Renormalized equations of motion 300
Exercises 306
3 Universality revisited 310
3-1 Renormalization scheme independence of critical exponents 310
3-2 The universal form of the equation of state 311
3-3 The equation of state to order ∈ 314
3-4 Two scale factor universality—universal ratios of amplitudes 316
Exercises 320
4 Critical behavior with several couplings 323
4-1 Introduction 323
4-2 More than one coupling constant—cubic anisotropy 324
4-3 Runaway trajectories 328
4-4 First order transitions induced by fluctuations:the Coleman-Weinberg mechanism 330
4-5 Geometrical description of the Coleman-Weinberg phenomenon 337
Exercises 339
5 Crossover phenomena 342
5-1 Introduction 342
5-2 Crossover in magnetic systems interacting quadratically and the Harris criterion for relevance of random dilution 343
5-3 The crossover exponent at a bicritical point:scale invariance with quadratic symmetry breaking 346
5-4 The crossover function at a bicritical point:a case study of renormalization group analysis in the presence of two lengths 350
Exercises 361
6 Critical phenomena near two dimensions 364
6-1 An alternative field theory for the Heisenberg model—the low temperature phase 364
6-2 Perturbation theory for the non-linear sigma model 368
6-2-1 The free propagator and infrared regularization 369
6-2-2 Disposing of the measure 369
6-2-3 The interactions 370
6-2-4 The expansion of Γ（2）α 370
6-3 Renormalization group treatment of the non-linear sigma model 372
6-4 Scaling behavior and critical exponents 376
Appendix 6-1 Renormalization of the non-linear sigma model 378
Exercises 380
PART Ⅲ NONPERTURBATIVE AND NUMERICAL METHODS 385
1 Real space methods 385
1-1 Introduction 385
1-1-1 Lattice models 386
1-1-2 Brief visit to high temperature expansion 390
1-1-3 High order expansions and critical behavior 394
1-2 Real space renormalization group 395
1-2-1 The 1-d Ising model 400
1-2-2 2-d Ising model 403
1-2-3 General case 409
1-3 At and around a fixed point 414
1-3-1 Scaling of the correlation functions 418
1-3-2 Renormalized trajectory 424
1-4 The large M model 427
1-4-1 Path integral and saddle point 428
1-4-2 The propagator 430
1-4-3 Factorization 431
1-4-4 Gap equation 433
1-4-5 The exponent v 434
1-4-6 Example of real-space RG-transformation 436
Exercises 439
2 Finite size scaling 446
2-1 Introduction 446
2-1-1 Geometry and boundary conditions 449
2-1-2 The finite size scaling ansatz 452
2-2 The RG derivation of finite size scaling 456
2-2-1 Logarithmic specific heat 460
2-2-2 Order parameter probability 461
2-2-3 Corrections to scaling 464
2-2-4 First-order phase transitions 469
2-3 Applications of FSS 470
2-3-1 Finite lattice correlation length 471
2-3-2 Extrapolations to infinite volume 474
2-3-3 Working at the critical point 476
Exercises 480
3 Monte Carlo methods.Numerical field theory 486
3-1 Introduction 486
3-1-1 Motivations 486
3-1-2 Static Monte Carlo methods:first example 487
3-1-3 Problems with uniform sampling 490
3-2 Dynamic Monte Carlo 492
3-2-1 Methods for the Ising model 496
3-2-2 Methods for the O（3）non-linear σ-model 497
3-3 Data analysis 499
3-3-1 General considerations 499
3-3-2 Practical recipes 503
3-4 Cluster methods 509
3-4-1 Discrete spins 510
3-4-2 Performance 513
3-4-3 Continuous spins 514
3-4-4 Final remark 515
Exercises 516
Appendix A Sample Programs 521
A-1 Static Monte Carlo Integration 521
A-2 Simulation of 2-D Ising Model 521
A-3 Autocorrelation Analysis 521
A-4 Data Analysis 521
Author Index 521
Subject Index 527