Preface

Chapter 1: Phase transitions and critical phenomena

1. Phase and phase diagram
2. Phase transitions
3. Critical phenomena
4. Scale transformation and renormalization group
5. Ising model and related systems

Chapter 2: Mean-field theories

1. Mean-field approximation
2. Critical exponents of the mean-field theory
3. Landau theory
4. Landau theory of the tricritical point
5. Infinite-range model
6. Variational method
7. Antiferromagnetic Ising model
8. Bethe approximation
9. Correlation function
10. Limit of applicability of the mean-field approximation
11. Dynamic critical phenomena

Chapter 3: Renormalization group and scaling

1. Coarse-graining and scale transformations
2. Parameter space and renormalization group equation
3. Renormalization group flow near a fixed point and universality
4. Scaling law and critical exponents
5. Scaling law for correlation functions and hyperscaling
6. A simple example: One-dimensional Ising model
7. Mean-field theory and scaling law
8. Scaling dimension and scaling law
9. Scaling and anomalous dimension
10. Data analysis by scaling law and finite-size scaling
11. Crossover phenomena
12. Dynamic scaling law

Chapter 4: Implementation of renormalization group

1. Real-space renormalization group for arbitrary dimensions
2. Momentum-space renormalization group: epsilon=4-d expansion
3. Real-space renormalization group for a quantum system

Chapter 5: Statistical field theory

1. From bits to fields
2. Continuum limit and field theory
3. Hubbard-Stratonovich transformation
4. Integrating out degrees of freedom: Coarse graining
5. Phenomenological Landau-Ginzburg approach
6. Symmetry and its breakdown
7. Nambu-Goldstone modes
8. Topological defects

Chapter 6: Conformal field theory

1. From scale invariance to conformal symmetry
2. Conformal transformations
3. Primary and quasi-primary operators
4. Energy-momentum tensor and the Ward identity
5. Virasoro algebra
6. Gaussian theory
7. Operator formalism
8. Unitary representation of the Virasoro algebra
9. Ising model
10. Finite-size effects

Chapter 7: Kosterlitz-Thouless transition

1. Peierls argument
2. Lower critical dimension of the XY model
3. Mermin-Wagner theorem: Absence of spontaneous magnetization
4. Kosterlitz-Thouless transition
5. Interaction energy of vortex pairs
6. Renormalization group analysis
7. Lattice gauge theory and Elitzur's theorem

Chapter 8: Random systems

1. Random fields
2. Spin glass
3. Diluted ferromagnet and percolation

Chapter 9: Exact solutions and related topics

1. One-dimensional Ising model
2. One-dimensional n-vector model
3. Spherical model
4. One-dimensional quantum XY model
5. Two-dimensional Ising model
6. Zeros of the partition function

Chapter 10: Duality

1. Classical duality
2. High- and low-temperature series expansions
3. Duality by Fourier transformation
4. Quantum duality

Chapter 11: Numerical methods

1. Master equation
2. Monte Carlo simulation
3. Numerical transfer matrix method

Appendixes

1. Saddle-point method
2. Expressing the susceptibility in terms of correlation functions
3. Rushbrooke's inequality
4. Cumulants
5. Renormalization group equations from the epsilon expansion
6. Symmetry and Noether's theorem
7. Basics of group theory and Lie algebras
8. Basics of homotopy theory
9. Restrictions on the type of conformal mappings
10. Properties of the energy-momentum tensor
11. Energy-momentum tensor of the Gaussian theory
12. Existence of spontaneous magnetization in the two-dimensional Ising model
13. Quantum version of the Mermin-Wagner theorem
14. Replica symmetric solution of the SK model
15. Integral for the partition function of the n-vector model
16. Multiple Gaussian integral and lattice Green function
17. Jordan-Wigner transformation
18. Proof of Theorem 9.1
19. Poisson summation formula
20. Sample codes for Monte Carlo simulation of the Ising model

Solutions to exercises

For further reading

Index