PART Ⅰ BASIC IDEAS AND TECHNIQUES

1 Pertinent concepts and ideas in the theory of critical phenomena

1-1 Description of critical phenomena

1-2 Scaling and homogeneity

1-3 Comparison of various results for critical exponents

1-4 Universality—dimensionality，symmetry

Exercises

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

2-1 Introduction

2-2 Construction of the Lagrangian

2-2-1 The real scalar field

2-2-2 Complex field

2-2-3 A hypercubic n-vector model

2-2-4 Two coupled fluctuating fields

2-3 The parameters appearing in？

2-4 The partition function，or the generating functional

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

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

2-5-2 The Gaussian transformation

2-5-3 The free part

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

2-6 Correlation functions including composite operators

Exercises

3 Functional integrals in quantum field theory

3-1 Introduction

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

3-2-1 Schwinger's transformation function

3-2-2 Matrix elements—Green functions

3-2-3 The generating functional

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

3-3 Functional integrals for the scalar boson field theory

3-3-1 Introduction

3-3-2 The generating functional for Green functions

3-3-3 The generating functional as a functional integral

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

Exercises

4 Perturbation theory and Feynman graphs

4-1 Introduction

4-2 Perturbation expansion in coordinate space

4-3 The cancellation of vacuum graphs

4-4 Rules for the computation of graphs

4-5 More general cases

4-5-1 The M-vector theory

4-5-2 Comments on fields with higher spin

4-6 Diagrammatic expansion in momentum space

4-7 Perturbation expansion of Green functions withZ composite operators

4-7-1 In coordinate space

4-7-2 In momentum space

4-7-3 Insertion at zero momentum

Exercises

5 Vertex functions and symmetry breaking

5-1 Introduction

5-2 Connected Green functions and their generating functional

5-3 The mass operator

5-4 The Legendre transform and vertex functions

5-5 The generating functional and the potential

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

5-7 Vertex parts for Green functions with composite operators

Exercises

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

6-1 Introduction

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

6-3 The tree（Landau-Ginzburg）approximation

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

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

6-6 Composite field renormalization

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

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

6-8-1 General remarks

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

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

6-8-4 Γ（2）in the infinite M limit

6-8-5 Renormalization

6-8-6 Broken symmetry

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

Exercises

7 Renormalization

7-1 Introduction

7-2 Some considerations concerning engineering dimensions

7-3 Power counting and primitive divergences

7-4 Renormalization of a cutoff φ4 theory

7-5 Normalization conditions for massive and massless theories

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

7-7 Renormalization away from the critical point

7-8 Counterterms

7-9 Relevant and irrelevant operators

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

7-11 Ward identities and renormalization

7-12 Iterative construction of counterterms

Exercises

8 The renormalization group and scaling in the critical region

8-1 Introduction

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

8-3 Regularization by continuation in the number of dimensions

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

8-5 The solution of the renormalization group equation

8-6 Fixed points，scaling，and anomalous dimensions

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

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

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

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

8-11 The Callan-Symanzik equations

8-12 Renormalization group equations for the bare theory

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

Appendix 8-1 General formulas for calculating Feynman integrals

Exercises

9 The computation of the critical exponents

9-1 Introduction

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

9-3 The ∈expansion of the critical exponents

9-4 The nature of the fixed points—universality

9-5 Scale invariance at finite cutoff

9-6 At the critical dimension—asymptotic infrared freedom

9-7 ∈expansion for the Callan-Symanzik method

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

9-9 Dimensional regularization and critical phenomena

9-10 Renormalization by minimal subtraction of dimensional poles

9-11 The calculation of exponents in minimal subtraction

Appendix 9-1 Calculation of some integrals with cutoff

9-2 One-loop integrals in dimensional regularization

9-3 Two-loop integrals in dimensional regularization

Exercises

PART Ⅱ FURTHER APPLICATIONS AND DEVELOPMENTS

1 Introduction

2 Beyond leading scaling

2-1 Corrections to scaling in aφ4 theory

2-2 Finite-size scaling

2-3 Anomalous dimensions of high composite operators

2-4 Corrections due to irrelevant operators

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

2-6 The operator product expansion

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

Appendix 2-1 Renormalized equations of motion

Exercises

3 Universality revisited

3-1 Renormalization scheme independence of critical exponents

3-2 The universal form of the equation of state

3-3 The equation of state to order ∈

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

Exercises

4 Critical behavior with several couplings

4-1 Introduction

4-2 More than one coupling constant—cubic anisotropy

4-3 Runaway trajectories

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

4-5 Geometrical description of the Coleman-Weinberg phenomenon

Exercises

5 Crossover phenomena

5-1 Introduction

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

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

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

Exercises

6 Critical phenomena near two dimensions

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

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

6-2-1 The free propagator and infrared regularization

6-2-2 Disposing of the measure

6-2-3 The interactions

6-2-4 The expansion of Γ（2）α

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

6-4 Scaling behavior and critical exponents

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

Exercises

PART Ⅲ NONPERTURBATIVE AND NUMERICAL METHODS

1 Real space methods

1-1 Introduction

1-1-1 Lattice models

1-1-2 Brief visit to high temperature expansion

1-1-3 High order expansions and critical behavior

1-2 Real space renormalization group

1-2-1 The 1-d Ising model

1-2-2 2-d Ising model

1-2-3 General case

1-3 At and around a fixed point

1-3-1 Scaling of the correlation functions

1-3-2 Renormalized trajectory

1-4 The large M model

1-4-1 Path integral and saddle point

1-4-2 The propagator

1-4-3 Factorization

1-4-4 Gap equation

1-4-5 The exponent v

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

Exercises

2 Finite size scaling

2-1 Introduction

2-1-1 Geometry and boundary conditions

2-1-2 The finite size scaling ansatz

2-2 The RG derivation of finite size scaling

2-2-1 Logarithmic specific heat

2-2-2 Order parameter probability

2-2-3 Corrections to scaling

2-2-4 First-order phase transitions

2-3 Applications of FSS

2-3-1 Finite lattice correlation length

2-3-2 Extrapolations to infinite volume

2-3-3 Working at the critical point

Exercises

3 Monte Carlo methods.Numerical field theory

3-1 Introduction

3-1-1 Motivations

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

3-1-3 Problems with uniform sampling

3-2 Dynamic Monte Carlo

3-2-1 Methods for the Ising model

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

3-3 Data analysis

3-3-1 General considerations

3-3-2 Practical recipes

3-4 Cluster methods

3-4-1 Discrete spins

3-4-2 Performance

3-4-3 Continuous spins

3-4-4 Final remark

Exercises

Appendix A Sample Programs

A-1 Static Monte Carlo Integration

A-2 Simulation of 2-D Ising Model

A-3 Autocorrelation Analysis

A-4 Data Analysis

Author Index

Subject Index