Exact solution of the six-vertex model with domain wall boundary conditions. Critical line between disordered and antiferroelectric phases

TL;DR

This paper derives the exact large N asymptotics of the six-vertex model's partition function on the critical line, revealing an infinite order phase transition between disordered and antiferroelectric phases.

Contribution

It provides an explicit asymptotic formula for the partition function at criticality, improving previous physics results and analyzing phase transition nature.

Findings

Asymptotic formula for Z_N involving explicit constants

Identification of an infinite order phase transition

Application of Riemann-Hilbert and Toda methods

Abstract

In the present article we obtain the large $N$ asymptotics of the partition function $Z_N$ of the six-vertex model with domain wall boundary conditions on the critical line between the disordered and antiferroelectric phases. Using the weights $a=1-x,b=1+x,c=2,|x|<1$, we prove that, as $N\rightarrow\infty$, $Z_N=CF^{N^2}N^{1/12}(1+O(N^{-1}))$, where $F$ is given by an explicit expression in $x$ and the $x$-dependency in $C$ is determined. This result reproduces and improves the one given in the physics literature by Bogoliubov, Kitaev and Zvonarev. Furthermore, we prove that the free energy exhibits an infinite order phase transition between the disordered and antiferroelectric phases. Our proofs are based on the large $N$ asymptotics for the underlying orthogonal polynomials which involve a non-analytical weight function, the Deift-Zhou nonlinear steepest descent method to the corresponding Riemann-Hilbert problem, and the Toda equation for the tau-function.