EXACT SOLUTION OF THE SIX-VERTEX MODEL WITH DOMAIN WALL BOUNDARY CONDITIONS: CRITICAL LINE BETWEEN DISORDERED AND ANTIFERROELECTRIC PHASES

Abstract

In the present paper 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 → ∞, 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 [Boundary polarization in the six-vertex model, Phys. Rev. E65 (2002) 026126]. 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 non-linear steepest descent method to the corresponding Riemann–Hilbert problem, and the Toda equation for the tau-function.