Many scientists and non-scientists are familiar with fractals, abstract self-similarentities which resemble the shapes of clouds or mountain landscapes. Fewer arefamiliar with the concepts of scale-invariance and universality which underliethe ubiquity of these shapes. Such properties may emerge from the collectivebehavior of simple underlying constituents, and are studied through statisticalfield theories constructed easily on the basis of symmetries. This book demon-strates how such theories are formulated, and studied by innovative methodssuch as the renormalization group.

Preface

1 Collective behavior, from particles to fields

1.1 Introduction

1.2 Phonons and elasticity

1.3 Phase transitions

1.4 Critical behavior

Problems

2 Statistical fields

2.1 Introduction

2.2 The Landau-Ginzburg Hamiltonian

2.3 Saddle point approximation, and mean-field theory

2.4 Continuous symmetry breaking and Goldstone modes

2.5 Discrete symmetry breaking and domain walls

Problems

3 Fluctuations

3.1 Scattering and fluctuations

3.2 Correlation functions and susceptibilities

3.3 Lower critical dimension

3.4 Comparison to experiments

3.5 Gaussian integrals

3.6 Fluctuation corrections to the saddle point

3.7 The Ginzburg criterion

Problems

4 The scaling hypothesis

4.1 The homogeneity assumption

4.2 Divergence of the correlation length

4.3 Critical correlation functions and self-similarity

4.4 The renormalization group (conceptual)

4.5 The renormalization group (formal)

4.6 The Gaussian model (direct solution)

4.7 The Gaussian model (renormalization group)

Problems

5 Perturbative renormalizafion group

5.1 Expectation values in the Gaussian model

5.2 Expectation values in perturbation theory

5.3 Diagrammatic representation of perturbation theory

5.4 Susceptibility

5.5 Perturbative RG (first order)

5.6 Perturbative RG (second order)

5.7 The e-expansion

5.8 Irrelevance of other interactions

5.9 Comments on the e-expansion

Problems

6 Lattice systems

6.1 Models and methods

6.2 Transfer matrices

6.3 Position space RG in one dimension

6.4 The Niemeijer-van Leeuwen cumulant approximation

6.5 The Migdal-Kadanoff bond moving approximation

6.6 Monte Carlo simulations

Problems

7 Series expansions

7.1 Low-temperature expansions

7.2 High-temperature expansions

7.3 Exact solution of the one-dimensional Ising model

7.4 Self-duality in the two-dimensional Ising model

7.5 Dual of the three-dimensional Ising model

7.6 Summing over phantom loops

7.7 Exact free energy of the square lattice Ising model

7.8 Critical behavior of the two-dimensional Ising model

Problems

8 Beyond spin waves

8.1 The nonlinear tr model

8.2 Topological defects in the XY model

8.3 Renormalization group for the Coulomb gas

8.4 Two-dimensional solids

8.5 Two-dimensional melting

Problems

9 Dissipative dynamics

9.1 Brownian motion of a particle

9.2 Equilibrium dynamics of a field

9.3 Dynamics of a conserved field

9.4 Generic scale invariance in equilibrium systems

9.5 Non-equilibrium dynamics of open systems

9.6 Dynamics of a growing surface

10 Directed paths in random media

10.1 Introduction

10.2 High-T expansions for the random-bond Ising model

10.3 The one-dimensional chain

10.4 Directed paths and the transfer matrix

10.5 Moments of the correlation function

10.6 The probability distribution in two dimensions

10.7 Higher dimensions

10.8 Random signs

10.9 Other realizations of DPRM

10.10 Quantum interference of strongly localized electrons

10.11 The locator expansion and forward scattering paths

10.12 Magnetic field response

10.13 Unitary propagation

10.14 Unitary averages

Solutions to selected problems

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Index