Exact Analyses in Condensed Matter and Statistical Physics:

A Brief Overview of Selected Topics

F. Y. Wu, Center for Advanced Study, Tsinghua University, Beijing

September 16 - October 12, 2005

1. The one-dimensional Hubbard model

1.1 Introduction

1.2 The one-dimensional model

1.3 The Bethe Ansatz

1.4 The ground state

1.5 The integral equation

1.6 The case of a half-filled band

1.7 Absence of a Mott transition

　

2. Knot invariant and statistical mechanics

2.1 Knot invariants

2.2 Reidemeister moves

2.3 Skein relation for oriented knots

2.4 Skein relation for un-oriented knots

2.5 Lattice models

2.6 The Jones polynomial from an edge-interaction model

2.7 Yang-Baxter equation for vertex model

2.8 The bracket polynomial and the non-intersecting string

model

2.9 The Jones polynomial from the bracket polynomial

 

3. Graph theory and statistical mechanice

3.1 Aspects of the graph theory

3.2 The adjacency matrix

3.3 Spanning trees and the Kirchhoff tree matrix

3.4 Spanning trees on regular lattices

3.5 The chromatic polynomial and the Potts model

3.6 The Tutte polynomial

 

4. Dimer statistics

4.1 Perfect matchings - close-packed dimers

4.2 Phaffians

4.3 The Kasteleyn approach

4.4 Solution for the square lattice

4.5 The Temperley bijection between dimer and spanning-tree configurations

4.6 Closed-packed dimers with one boundary vacancy.

 

5. The Ising model

5.1 Brief history of the Ising model

5.2 The one-dimensional system

5.3 The transfer matrix approach

5.4 The combinatorial approach

5.5 The duality relation

5.6 Formulation as a dimer problem

5.7 Solution of the 2-dimensional model

5.8 The Ising lattice gas

5.9 The Yang-Lee zeroes

5.10 The Fisher zeroes

5.11 The spontaneous magnetization

5.12 The correlation function

5.13 The verter model and the free-fermion model

 

6. Random walks

6.1 Elements of random walks on a lattice

6.2 The generating funcitons

6.3 The escape probability

 

7. The q-state Potts model

7.1 Introduction

7.2 Experimental realizations

7.3 The mean-field solution

7.4 Random graph and network communication

7.5 The duality relation and the critical point

7.6 Percolation: the q -> 1 limit

7.7 Spanning trees and forest: the q -> 0 limit

7.8 Potts model as a vertex model

7.9 Critical properties in two dimensions

7.10 Potts model and graph theory