SCALING AND RENORMALIZATION IN STATISTICAL PHYSICS

TABLE OF CONTENTS

1. Phase transitions in simple systems

   • 1.1 Phase diagrams
   • 1.2 Simple models
   • Exercises

2. Mean field theory

   • 2.1 The mean field free energy
   • 2.2 Critical exponents
   • 2.3 Mean field theory for the correlation function
   • 2.4 Corrections to mean field theory
   • Exercises

3. The renormalization group idea

   • 3.1 Block spin transformations
   • 3.2 One-dimensional Ising model
   • 3.3 General theory
   • 3.4 Scaling behaviour of the free energy
   • 3.5 Critical exponents
   • 3.6 Irrelevant eigenvalues
   • 3.7 Scaling for the correlation functions
   • 3.8 Scaling operators and scaling dimensions
   • 3.9 Critical amplitudes
   • Exercises

4. Phase diagrams and fixed points

   • 4.1 Ising model with vacancies
   • 4.2 Cross-over behaviour
   • 4.3 Cross-over to long range behaviour
   • 4.4 Finite-size scaling
   • 4.5 Quantum critical behaviour
   • Exercises

5. The perturbative renormalization group

   • 5.1 The operator product expansion
   • 5.2 The perturbative renormalization group
   • 5.3 The Ising model near four dimensions
   • 5.4 The Gaussian fixed point
   • 5.5 The Wilson--Fisher fixed point
   • 5.6 Logarithmic corrections in d=4
   • 5.7 The O(n) model near four dimensions
   • 5.8 Cubic symmetry breaking
   • Exercises

6. Low dimensional systems

   • 6.1 The lower critical dimension
   • 6.2 The two-dimensional XY model
   • 6.3 The solid-on-solid model
   • 6.4 Renormalization group analysis
   • 6.5 The O(n) model in 2+epsilon dimensions
   • Exercises

7. Surface critical behaviour

   • 7.1 Mean field theory
   • 7.2 The extraordinary and special transitions
   • 7.3 Renormalization group approach
   • Exercises

8. Random systems

   • 8.1 Quenched and annealed disorder
   • 8.2 The Harris criterion
   • 8.3 Perturbative approach to the random fixed point
   • 8.4 Percolation
   • 8.5 Random fields
   • Exercises

9. Polymer statistics

   • 9.1 Random walk model
   • 9.2 The Edwards model and the Flory formula
   • 9.3 Mapping to the O(n) model
   • 9.4 Finite concentration
   • 9.5 Other applications
   • Exercises

10. Critical dynamics

    • 10.1 Continuum models
    • 10.2 Discrete models
    • 10.3 Dynamic scaling
    • 10.4 Response functional formalism
    • 10.5 Other dynamic universality classes
    • 10.6 Directed percolation
    • Exercises

11. Conformal symmetry

    • 11.1 Conformal transformations
    • 11.2 Simple consequences of conformal symmetry
    • 11.3 The stress tensor
    • 11.4 Further developments
    • 11.5 The c-theorem
    • Exercises

• Appendix: Gaussian integration
• Selected bibliography
• Index