场论、重正化群和临界现象 第3版 英文PDF|Epub|txt|kindle电子书版本网盘下载

场论、重正化群和临界现象 第3版 英文PDF格式电子书版下载

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PART Ⅰ BASIC IDEAS AND TECHNIQUES3

1 Pertinent concepts and ideas in the theory of critical phenomena3

1-1 Description of critical phenomena3

1-2 Scaling and homogeneity5

1-3 Comparison of various results for critical exponents6

1-4 Universality—dimensionality，symmetry8

Exercises9

2 Formulation of the problem of phase transitions in terms of functional integrals11

2-1 Introduction11

2-2 Construction of the Lagrangian12

2-2-1 The real scalar field12

2-2-2 Complex field12

2-2-3 A hypercubic n-vector model13

2-2-4 Two coupled fluctuating fields14

2-3 The parameters appearing in？14

2-4 The partition function，or the generating functional15

2-5 Representation of the Ising model in terms of functional integrals18

2-5-1 Definition of the model and its thermodynamics18

2-5-2 The Gaussian transformation21

2-5-3 The free part22

2-5-4 Some properties of the free theory—a free Euclidean field theory in less than four dimensions26

2-6 Correlation functions including composite operators28

Exercises30

3 Functional integrals in quantum field theory33

3-1 Introduction33

3-2 Functional integrals for a quantum-mechanical system with one degree of freedom34

3-2-1 Schwinger's transformation function34

3-2-2 Matrix elements—Green functions37

3-2-3 The generating functional38

3-2-4 Analytic continuation in time—the Euclidean theory40

3-3 Functional integrals for the scalar boson field theory41

3-3-1 Introduction41

3-3-2 The generating functional for Green functions43

3-3-3 The generating functional as a functional integral44

3-3-4 The S-matrix expressed in terms of the generating functional47

Exercises50

4 Perturbation theory and Feynman graphs53

4-1 Introduction53

4-2 Perturbation expansion in coordinate space54

4-3 The cancellation of vacuum graphs60

4-4 Rules for the computation of graphs60

4-5 More general cases63

4-5-1 The M-vector theory63

4-5-2 Comments on fields with higher spin67

4-6 Diagrammatic expansion in momentum space68

4-7 Perturbation expansion of Green functions withZ composite operators72

4-7-1 In coordinate space72

4-7-2 In momentum space74

4-7-3 Insertion at zero momentum76

Exercises77

5 Vertex functions and symmetry breaking80

5-1 Introduction80

5-2 Connected Green functions and their generating functional82

5-3 The mass operator85

5-4 The Legendre transform and vertex functions86

5-5 The generating functional and the potential91

5-6 Ward-Takahashi identities and Goldstone's theorem94

5-7 Vertex parts for Green functions with composite operators96

Exercises101

6 Expansions in the number of loops and in the number of components103

6-1 Introduction103

6-2 The expansion in the number of loops as a power series104

6-3 The tree（Landau-Ginzburg）approximation105

6-4 The one-loop approximation and the Ginzburg criterion109

6-5 Mass and coupling constant renormalization in the one-loop approximation112

6-6 Composite field renormalization116

6-7 Renormalization of the field at the two-loop level117

6-8 The 0（M）-symmetric theory in the limit of large M126

6-8-1 General remarks126

6-8-2 The origin of the M-dependence of the coupling constant127

6-8-3 Faithful representation of graphs and the dominant terms in Γ（4）128

6-8-4 Γ（2）in the infinite M limit130

6-8-5 Renormalization133

6-8-6 Broken symmetry134

Appendix 6-1 The method of steepest descent and the loop expansion137

Exercises142

7 Renormalization147

7-1 Introduction147

7-2 Some considerations concerning engineering dimensions148

7-3 Power counting and primitive divergences151

7-4 Renormalization of a cutoff φ4 theory157

7-5 Normalization conditions for massive and massless theories159

7-6 Renormalization constants for a massless theory to order two loops161

7-7 Renormalization away from the critical point164

7-8 Counterterms167

7-9 Relevant and irrelevant operators169

7-10 Renormalization of a φ4 theory with an 0（M）symmetry171

7-11 Ward identities and renormalization174

7-12 Iterative construction of counterterms179

Exercises185

8 The renormalization group and scaling in the critical region189

8-1 Introduction189

8-2 The renormalization group for the critical（massless）theory190

8-3 Regularization by continuation in the number of dimensions195

8-4 Massless theory below four dimensions—the emergence of ∈196

8-5 The solution of the renormalization group equation197

8-6 Fixed points，scaling，and anomalous dimensions199

8-7 The approach to the fixed point—asymptotic freedom201

8-8 Renormalization group equation above Tc—identification of v205

8-9 Below the critical temperature—the scaling form of the equation of state208

8-10 The specific heat—renormalization group equation for an additively renormalized vertex210

8-11 The Callan-Symanzik equations212

8-12 Renormalization group equations for the bare theory214

8-13 Renormalization group equations and scaling in the infinite M limit217

Appendix 8-1 General formulas for calculating Feynman integrals222

Exercises223

9 The computation of the critical exponents228

9-1 Introduction228

9-2 The symbolic calculation of the renormalization constants and Wilson functions230

9-3 The ∈expansion of the critical exponents233

9-4 The nature of the fixed points—universality237

9-5 Scale invariance at finite cutoff238

9-6 At the critical dimension—asymptotic infrared freedom240

9-7 ∈expansion for the Callan-Symanzik method243

9-8 ∈expansion of the renormalization group equations for the bare functions247

9-9 Dimensional regularization and critical phenomena248

9-10 Renormalization by minimal subtraction of dimensional poles250

9-11 The calculation of exponents in minimal subtraction255

Appendix 9-1 Calculation of some integrals with cutoff257

9-2 One-loop integrals in dimensional regularization260

9-3 Two-loop integrals in dimensional regularization263

Exercises266

PART Ⅱ FURTHER APPLICATIONS AND DEVELOPMENTS273

1 Introduction273

2 Beyond leading scaling275

2-1 Corrections to scaling in aφ4 theory275

2-2 Finite-size scaling277

2-3 Anomalous dimensions of high composite operators280

2-4 Corrections due to irrelevant operators288

2-5 Next-to-leading terms in the scaling region293

2-6 The operator product expansion295

2-7 Computation of next-to-leading terms in ∈-expansion297

Appendix 2-1 Renormalized equations of motion300

Exercises306

3 Universality revisited310

3-1 Renormalization scheme independence of critical exponents310

3-2 The universal form of the equation of state311

3-3 The equation of state to order ∈314

3-4 Two scale factor universality—universal ratios of amplitudes316

Exercises320

4 Critical behavior with several couplings323

4-1 Introduction323

4-2 More than one coupling constant—cubic anisotropy324

4-3 Runaway trajectories328

4-4 First order transitions induced by fluctuations:the Coleman-Weinberg mechanism330

4-5 Geometrical description of the Coleman-Weinberg phenomenon337

Exercises339

5 Crossover phenomena342

5-1 Introduction342

5-2 Crossover in magnetic systems interacting quadratically and the Harris criterion for relevance of random dilution343

5-3 The crossover exponent at a bicritical point:scale invariance with quadratic symmetry breaking346

5-4 The crossover function at a bicritical point:a case study of renormalization group analysis in the presence of two lengths350

Exercises361

6 Critical phenomena near two dimensions364

6-1 An alternative field theory for the Heisenberg model—the low temperature phase364

6-2 Perturbation theory for the non-linear sigma model368

6-2-1 The free propagator and infrared regularization369

6-2-2 Disposing of the measure369

6-2-3 The interactions370

6-2-4 The expansion of Γ（2）α370

6-3 Renormalization group treatment of the non-linear sigma model372

6-4 Scaling behavior and critical exponents376

Appendix 6-1 Renormalization of the non-linear sigma model378

Exercises380

PART Ⅲ NONPERTURBATIVE AND NUMERICAL METHODS385

1 Real space methods385

1-1 Introduction385

1-1-1 Lattice models386

1-1-2 Brief visit to high temperature expansion390

1-1-3 High order expansions and critical behavior394

1-2 Real space renormalization group395

1-2-1 The 1-d Ising model400

1-2-2 2-d Ising model403

1-2-3 General case409

1-3 At and around a fixed point414

1-3-1 Scaling of the correlation functions418

1-3-2 Renormalized trajectory424

1-4 The large M model427

1-4-1 Path integral and saddle point428

1-4-2 The propagator430

1-4-3 Factorization431

1-4-4 Gap equation433

1-4-5 The exponent v434

1-4-6 Example of real-space RG-transformation436

Exercises439

2 Finite size scaling446

2-1 Introduction446

2-1-1 Geometry and boundary conditions449

2-1-2 The finite size scaling ansatz452

2-2 The RG derivation of finite size scaling456

2-2-1 Logarithmic specific heat460

2-2-2 Order parameter probability461

2-2-3 Corrections to scaling464

2-2-4 First-order phase transitions469

2-3 Applications of FSS470

2-3-1 Finite lattice correlation length471

2-3-2 Extrapolations to infinite volume474

2-3-3 Working at the critical point476

Exercises480

3 Monte Carlo methods.Numerical field theory486

3-1 Introduction486

3-1-1 Motivations486

3-1-2 Static Monte Carlo methods:first example487

3-1-3 Problems with uniform sampling490

3-2 Dynamic Monte Carlo492

3-2-1 Methods for the Ising model496

3-2-2 Methods for the O（3）non-linear σ-model497

3-3 Data analysis499

3-3-1 General considerations499

3-3-2 Practical recipes503

3-4 Cluster methods509

3-4-1 Discrete spins510

3-4-2 Performance513

3-4-3 Continuous spins514

3-4-4 Final remark515

Exercises516

Appendix A Sample Programs521

A-1 Static Monte Carlo Integration521

A-2 Simulation of 2-D Ising Model521

A-3 Autocorrelation Analysis521

A-4 Data Analysis521

Author Index521

Subject Index527