场论、重正化群和临界现象 第3版 英文PDF电子书下载

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