Field Theory, the Renormalization Group, and Critical Phenomena

The following sections are included:

• CONTENTS

• PREFACE

• PREFACE TO THE THIRD EDITION

• GENERAL SOURCES AND REFERENCES

The following sections are included:

• DESCRIPTION OF CRITICAL PHENOMENA

• SCALING AND HOMOGENEITY

• COMPARISON OF VARIOUS RESULTS FOR CRITICAL EXPONENTS

• UNIVERSALITY — DIMENSIONALITY, SYMMETRY

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION

• CONSTRUCTION OF THE LAGRANGIAN
  • The Real Scalar Field
  • Complex Field
  • A Hypercubic n-Vector Model
  • Two Coupled Fluctuating Fields

• THE PARAMETERS APPEARING IN ℒ

• THE PARTITION FUNCTION, OR THE GENERATING FUNCTIONAL

• REPRESENTATION OF THE ISING MODEL IN TERMS OF FUNCTIONAL INTEGRALS
  • Definition of the Model and its Thermodynamics
  • The Gaussian Transformation
  • The Free Part
  • Some Properties of the Free Theory – a Free Euclidean Field Theory in Less than Four Dimensions

• CORRELATION FUNCTIONS INCLUDING COMPOSITE OPERATORS

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION

• FUNCTIONAL INTEGRALS FOR A QUANTUM-MECHANICAL SYSTEM WITH ONE DEGREE OF FREEDOM
  • Schwinger's Transformation Function
  • Matrix Elements — Green Functions
  • The Generating Functional
  • Analytic Continuation in Time — the Euclidean Theory

• FUNCTIONAL INTEGRALS FOR THE SCALAR BOSON FIELD THEORY
  • Introduction
  • The Generating Functional for Green Functions
  • The Generating Functional as a Functional Integral
  • The S-Matrix Expressed in Terms of the Generating Functional

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION

• PERTURBATION EXPANSION IN COORDINATE SPACE

• THE CANCELLATION OF VACUUM GRAPHS

• RULES FOR THE COMPUTATION OF GRAPHS

• MORE GENERAL CASES
  • The M-vector theory
  • Comments on Fields with Higher Spin

• DIAGRAMMATIC EXPANSION IN MOMENTUM SPACE

• PERTURBATION EXPANSION OF GREEN FUNCTIONS WITH COMPOSITE OPERATORS
  • In Coordinate Space
  • In Momentum Space
  • Insertion at Zero Momentum

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION

• CONNECTED GREEN FUNCTIONS AND THEIR GENERATING FUNCTIONAL

• THE MASS OPERATOR

• THE LEGENDRE TRANSFORM AND VERTEX FUNCTIONS

• THE GENERATING FUNCTIONAL AND THE POTENTIAL

• WARD–TAKAHASHI IDENTITIES AND GOLDSTONE'S THEOREM

• VERTEX PARTS FOR GREEN FUNCTIONS WITH COMPOSITE OPERATORS

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION

• THE EXPANSION IN THE NUMBER OF LOOPS AS A POWER SERIES

• THE TREE (LANDAU–GINZBURG) APPROXIMATION

• THE ONE-LOOP APPROXIMATION AND THE GINZBURG CRITERION

• MASS AND COUPLING CONSTANT RENORMALIZATION IN THE ONE-LOOP APPROXIMATION

• COMPOSITE FIELD RENORMALIZATION

• RENORMALIZATION OF THE FIELD AT THE TWO-LOOP LEVEL

• THE 0(M)-SYMMETRIC THEORY IN THE LIMIT OF LARGE M
  • General Remarks
  • The Origin of the M-Dependence of the Coupling Constant
  • Faithful Representation of Graphs and the Dominant Terms in Γ⁽⁴⁾
  • Γ⁽²⁾ in the Infinite M Limit
  • Renormalization
  • Broken Symmetry

• Appendix 6-1 THE METHOD OF STEEPEST DESCENT AND THE LOOP EXPANSION

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION

• SOME CONSIDERATIONS CONCERNING ENGINEERING DIMENSIONS

• POWER COUNTING AND PRIMITIVE DIVERGENCES

• RENORMALIZATION OF A CUTOFF φ⁴ THEORY

• NORMALIZATION CONDITIONS FOR MASSIVE AND MASSLESS THEORIES

• RENORMALIZATION CONSTANTS FOR A MASSLESS THEORY TO ORDER TWO LOOPS

• RENORMALIZATION AWAY FROM THE CRITICAL POINT

• COUNTERTERMS

• RELEVANT AND IRRELEVANT OPERATORS

• RENORMALIZATION OF A φ⁴ THEORY WITH AN 0(M) SYMMETRY

• WARD IDENTITIES AND RENORMALIZATION

• ITERATIVE CONSTRUCTION OF COUNTERTERMS

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION

• THE RENORMALIZATION GROUP FOR THE CRITICAL (MASSLESS) THEORY

• REGULARIZATION BY CONTINUATION IN THE NUMBER OF DIMENSIONS

• MASSLESS THEORY BELOW FOUR DIMENSIONS – THE EMERGENCE OF ∈

• THE SOLUTION OF THE RENORMALIZATION GROUP EQUATION

• FIXED POINTS, SCALING, AND ANOMALOUS DIMENSIONS

• THE APPROACH TO THE FIXED POINT – ASYMPTOTIC FREEDOM

• RENORMALIZATION GROUP EQUATION ABOVE T_c — IDENTIFICATION OF ν

• BELOW THE CRITICAL TEMPERATURE – THE SCALING FORM OF THE EQUATION OF STATE

• THE SPECIFIC HEAT – RENORMAUZATION GROUP EQUATION FOR AN ADDITIVELY RENORMALEED VERTEX

• THE CALLAN–SYMANZIK EQUATIONS

• RENORMALIZATION GROUP EQUATIONS FOR THE BARE THEORY

• RENORMALIZATION GROUP EQUATIONS AND SCALING IN THE INFINITE M LIMIT

• Appendix 8-1 GENERAL FORMULAS FOR CALCULATING FEYNMAN INTEGRALS

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION

• THE SYMBOLIC CALCULATION OF THE RENORMALIZATION CONSTANTS AND WILSON FUNCTIONS

• THE ∈ EXPANSION OF THE CRITICAL EXPONENTS

• THE NATURE OF THE FIXED POINTS – UNIVERSALITY

• SCALE INVARIANCE AT FINITE CUTOFF

• AT THE CRITICAL DIMENSION – ASYMPTOTIC INFRARED FREEDOM

• ∈ EXPANSION FOR THE CALLAN–SYMANZIK METHOD

• ∈ EXPANSION OF THE RENORMALIZATION GROUP EQUATIONS FOR THE BARE FUNCTIONS

• DIMENSIONAL REGULARIZATION AND CRITICAL PHENOMENA

• RENORMALIZATION BY MINIMAL SUBTRACTION OF DIMENSIONAL POLES

• THE CALCULATION OF EXPONENTS IN MINIMAL SUBTRACTION

• Appendix 9-1 CALCULATION OF SOME INTEGRALS WITH CUTOFF

• Appendix 9-2 ONE-LOOP INTEGRALS IN DIMENSIONAL REGULARIZATION

• Appendix 9-3 TWO-LOOP INTEGRALS IN DIMENSIONAL REGULARIZATION

• EXERCISES

• REFERENCES

As was mentioned in the preface, the application of field theoretic methods, and in particular of the renormalization group analysis, to critical phenomena and to other problems is far from being a closed subject. Part of the effort in this field, in addition to the natural search for new horizons in the depth of understanding of field theory and of its possible new applications, is devoted to the assimilation of the immense culture that has developed around the theory of critical phenomena in the last few years using a variety of techniques…

The following sections are included:

• CORRECTIONS TO SCALING IN A φ⁴ THEORY

• FINITE-SIZE SCALING

• ANOMALOUS DIMENSIONS OF HIGH COMPOSITE OPERATORS

• CORRECTIONS DUE TO IRRELEVANT OPERATORS

• NEXT-TO-LEADING TERMS IN THE SCALING REGION

• THE OPERATOR PRODUCT EXPANSION

• COMPUTATION OF NEXT-TO-LEADING TERMS IN ∈-EXPANSION

• Appendix 2-1 RENORMALIZED EQUATIONS OF MOTION

• EXERCISES

• REFERENCES

The following sections are included:

• RENORMALIZATION SCHEME INDEPENDENCE OF CRITICAL EXPONENTS

• THE UNIVERSAL FORM OF THE EQUATION OF STATE

• THE EQUATION OF STATE TO ORDER ∈

• TWO SCALE FACTOR UNIVERSALITY – UNIVERSAL RATIOS OF AMPLITUDES

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION

• MORE THAN ONE COUPLING CONSTANT – CUBIC ANISOTROPY

• RUNAWAY TRAJECTORIES

• FIRST ORDER TRANSITIONS INDUCED BY FLUCTUATIONS: THE COLEMAN-WEINBERG MECHANISM

• GEOMETRICAL DESCRIPTION OF THE COLEMAN-WEINBERG PHENOMENON

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION

• CROSSOVER IN MAGNETIC SYSTEMS INTERACTING QUADRATICALLY AND THE HARRIS CRITERION FOR RELEVANCE OF RANDOM DILUTION

• THE CROSSOVER EXPONENT AT A BICRITICAL POINT: SCALE INVARIANCE WITH QUADRATIC SYMMETRY BREAKING

• THE CROSSOVER FUNCTION AT A BICRITICAL POINT: A CASE STUDY OF RENORMALIZATION GROUP ANALYSIS IN THE PRESENCE OF TWO LENGTHS

• EXERCISES

• REFERENCES

The following sections are included:

• AN ALTERNATIVE FIELD THEORY FOR THE HEISENBERG MODEL – THE LOW TEMPERATURE PHASE

• PERTURBATION THEORY FOR THE NON-LINEAR SIGMA MODEL
  • The Free Propagator and Infrared Regularization
  • Disposing of the Measure
  • The Interactions
  • The Expansion of

• RENORMALIZATION GROUP TREATMENT OF THE NON-LINEAR SIGMA MODEL

• SCALING BEHAVIOR AND CRITICAL EXPONENTS

• Appendix 6-1 RENORMALIZATION OF THE NON-LINEAR SIGMA MODEL

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION
  • Lattice Models
  • Brief Visit to High Temperature Expansion
  • High Order Expansions and Critical Behavior

• REAL SPACE RENORMALIZATION GROUP
  • The 1-d Ising Model
  • 2-D Ising Model
  • General Case

• AT AND AROUND A FIXED POINT
  • Scaling of the Correlation Functions
  • Renormalized Trajectory

• THE LARGE M MODEL
  • Path Integral and Saddle Point
  • The Propagator
  • Factorization
  • Gap Equation
  • The Exponent ν
  • Example of Real-Space RG-Transformation

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION
  • Geometry and Boundary Conditions
  • The Finite Size Scaling Ansatz

• THE RG DERIVATION OF FINITE SIZE SCALING
  • Logarithmic Specific Heat
  • Order Parameter Probability
  • Corrections to Scaling
  • First-Order Phase Transitions

• APPLICATIONS OF FSS
  • Finite Lattice Correlation Length
  • Extrapolations to Infinite Volume
  • Working at the Critical Point

• EXERCISES

• REFERENCES

The following sections are included:

• INTRODUCTION
  • Motivations
  • Static Monte Carlo Methods: First Example
  • Problems with Uniform Sampling

• DYNAMIC MONTE CARLO
  • Methods for the Ising Model
  • Methods for the O(3) Non-Linear σ -Model

• DATA ANALYSIS
  • General Considerations
    • Dangers of Autocorrelations
  • Practical Recipes

• CLUSTER METHODS
  • Discrete Spins
  • Performance
  • Continuous Spins
  • Final Remark

• EXERCISES

• REFERENCES

The following sections are included:

• AUTHOR INDEX

• SUBJECT INDEX