Introduction to solvable lattice models in statistical and mathematical physics

TL;DR

This paper reviews integrable lattice models, especially the six-vertex model, discussing their mathematical structures, solutions via Bethe ansatz, phase transition behavior, and connections to conformal field theory, emphasizing graphical methods.

Contribution

It provides a comprehensive review of solvable lattice models, highlighting the graphical approach and detailed derivation of solutions like the free energy and critical phenomena.

Findings

Derivation of free energy using Bethe ansatz

Analytic demonstration of critical singularity near phase transition

Connection of the six-vertex model to conformal field theory with c=1

Abstract

Some features of integrable lattice models are reviewed for the case of the six-vertex model. By the Bethe ansatz method we derive the free energy of the six-vertex model. Then, from the expression of the free energy we show analytically the critical singularity near the phase transition in the anti-ferroelectric regime, where the essential singularity similar to the Kosterlitz-Thouless transition appears. We discuss the connection of the six-vertex model to the conformal field theory with c=1. We also introduce various exactly solvable models defined on two-dimensional lattices such as the chiral Potts model and the IRF models. We show that the six-vertex model has rich mathematical structures such as the quantum groups and the braid group. The graphical approach is emphasized in this review. We explain the meaning of the Yang-Baxter equation by its diagram. Furthermore, we can understand the defining relation of the algebraic Bethe ansatz by the graphical representation. We can thus easily translate formulas of the algebraic Bethe ansatz into those of the statistical models. As an illustration, we show explicitly how we can derive Baxter's expressions from those of the algebraic Bethe ansatz.