统计场论：英文版.第1卷2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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1 From Brownian motion to Euclidean fields1

1.1 Brownian motion1

1.1.1 Random walks1

1.1.2 The sum over paths9

1.1.3 The dimension two of Brownian curves12

1.2 Euclidean fields21

1.2.1 Free fields22

1.2.2 Interacting fields and random walks25

1.2.3 Self-avoiding walks and the limit n→031

1.2.4 Comparison with the high temperature expansion33

1.2.5 The one-dimensional case37

1.A Lattices43

Notes47

2 Grassmannian integrals and the two-dimensional Ising model48

2.1 Grassmannian integrals48

2.1.1 Anticommuting variables48

2.1.2 Integrals52

2.2 The two-dimensional Ising model58

2.2.1 Duality59

2.2.2 Transfer matrix62

2.2.3 Fermionic representation63

2.2.4 Free energy67

2.2.5 Spontaneous magnetization72

2.2.6 Correlation function in the high temperature phase78

2.2.7 Surface tension85

2.3 Critical continuous theory91

2.3.1 Effective action91

2.3.2 Correlation functions95

2.A Quadratic differences and Painlevé equations98

Notes104

3 Spontaneous symmetry breaking,mean field107

3.1 Mean field approximation108

3.1.1 Dielectric constant of a polarizable medium108

3.1.2 Classical spin model with a finite symmetry group112

3.1.3 Continuous symmetry group118

3.1.4 The Bethe approximation121

3.1.5 Critical exponents125

3.2 Lee-Yang zeroes131

3.2.1 The Lee-Yang theorem132

3.2.2 The one-dimensional case135

3.2.3 General properties136

3.2.4 Zeroes in the temperature plane139

3.3 Large n limit140

3.3.1 Saddle point method141

3.3.2 Factorization145

3.3.3 Coupling to an external field148

3.4 Corrections to mean field151

3.4.1 Laplace transform153

Notes159

4 Scaling transformations and the XY-model162

4.1 Scaling laws.Real space renormalization162

4.1.1 Homogeneity and scale invariance162

4.1.2 Recurrence relations in real space168

4.1.3 Examples and approximations176

4.2 The XY-model193

4.2.1 High temperature behaviour195

4.2.2 Low temperature expansion.Vortices197

4.2.3 The Villain action204

4.2.4 Correlations207

4.2.5 Renormalization flow213

4.A Two-dimensional systems with continuous symmetry219

4.A.1 Magnetization inequality219

4.A.2 Correlation inequality222

4.B Phenomenological renormalization224

Notes230

5 Continuous field theory and the renormalization group233

5.1 The Lagrangian and dimensional analysis233

5.1.1 Introduction233

5.1.2 Generating functionals and dimensional analysis236

5.2 The perturbative method242

5.2.1 Diagrammatic series242

5.2.2 Loop expansion246

5.2.3 Evaluation of integrals and dimensional continuation250

5.2.4 Group theoretical factors259

5.2.5 Power counting261

5.2.6 Perturbative renormalization264

5.3 The renormalization group270

5.3.1 Renormalization flow270

5.3.2 Critical exponents280

5.3.3 From the Gaussian ultraviolet fixed point to the infrared critical point in dimension less than four282

5.3.4 Correlation functions at the critical point284

5.3.5 Expansion near the critical point290

5.3.6 Scaling laws below the critical temperature296

5.4 Corrections to scaling laws301

5.4.1 Deviation from the critical point in dimension lower than four301

5.4.2 Logarithmic corrections in dimension four303

5.4.3 Irrelevant operators307

5.5 Numerical results310

5.5.1 ε-expansion of critical exponents311

5.5.2 Equation of state313

5.5.3 Amplitude ratios314

5.5.4 Three-dimensional results317

5.A Multicritical points317

Notes326

6 Lattice gauge fields328

6.1 Generalities328

6.1.1 Presentation328

6.1.2 The continuous limit332

6.1.3 Order parameter and Elitzur's theorem341

6.1.4 Duality345

6.2 Structure of the phase diagram352

6.2.1 Mean field approximation352

6.2.2 Corrections to mean field and restoration of gauge invariance359

6.2.3 Discrete groups:1/d expansion362

6.2.4 Continuous groups:computation of corrections365

6.3 Strong coupling expansions371

6.3.1 Convergence371

6.3.2 Character expansions374

6.3.3 Free energy380

6.3.4 String tension and roughening transition385

6.3.5 Mass spectrum390

6.4 Lattice fermions393

6.4.1 The doubling problem393

6.4.2 The Nielsen-Ninomiya theorem396

6.4.3 Staggered fermions399

Notes401

Index405

7 Diagrammatic methods405

8 Numerical simulations456

9 Conformal invariance501

10 Disordered systems and fermionic methods646

11 Random geometry738