Six-Vertex Model with Domain Wall Boundary Conditions and One-Matrix Model

TL;DR

This paper reformulates the six-vertex model with domain wall boundary conditions as a one-matrix model, enabling exact finite-size expressions and thermodynamic limit analysis of the bulk free energy across different phases.

Contribution

It provides an exact matrix model representation of the six-vertex model with DWBC, allowing comprehensive analysis of its free energy in various phases.

Findings

Exact finite-lattice partition function as a hermitean one-matrix model.

Thermodynamic limit computed using matrix model techniques.

Explicit expressions for bulk free energy in the anti-ferroelectric phase involving elliptic theta functions.

Abstract

The partition function of the six-vertex model on a square lattice with domain wall boundary conditions (DWBC) is rewritten as a hermitean one-matrix model or a discretized version of it (similar to sums over Young diagrams), depending on the phase. The expression is exact for finite lattice size, which is equal to the size of the corresponding matrix. In the thermodynamic limit, the matrix integral is computed using traditional matrix model techniques, thus providing a complete treatment of the bulk free energy of the six-vertex model with DWBC in the different phases. In particular, in the anti-ferroelectric phase, the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta functions.