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arXiv:1001.2189 · Journal of Statistical Mechanics Theory and Experiment

The arctic curve of the domain wall six-vertex model in its antiferroelectric regime

F Colomo, A G Pronko, Paul Zinn-Justin

Published Mar 4, 2010• 38 citations
arXivHTMLPDF

Abstract

An explicit expression for the spatial curve separating the region of ferroelectric order ('frozen' zone) from the disordered one ('temperate' zone) in the six-vertex model with domain wall boundary conditions in its antiferroelectric regime is obtained.
DOI
10.1088/1742-5468/2010/03/l03002

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Abstract We study numerically the density profile in the six-vertex model with domain wall boundary conditions. Using a Monte Carlo algorithm originally proposed by Allison and Reshetikhin we numerically evaluate the inhomogeneous density profiles in the disordered and antiferromagnetic regimes where frozen corners appear. At the free fermion point we present an exact finite-size formula for the density on the horizontal edges that relies on the imaginary time transfer matrix approach. In all cases where exact analytic forms for the density and the arctic curves are known the numerical method shows perfect agreement with them. This also suggests the possibility of its use for accurate quantitative purposes.
We use the Bethe-Peierls method combined with the belief propagation algorithm to study the arctic curves in the six vertex model on a square lattice with domain-wall boundary conditions, and the six vertex model on a rectangular lattice with partial domain-wall boundary conditions. We show that this rather simple approximation yields results that are remarkably close to the exact ones when these are known, and allows one to estimate the location of the phase boundaries with relative little effort in cases in which exact results are not available.
We use the Bethe-Peierls method combined with the belief propagation algorithm to study the arctic curves in the six vertex model on a square lattice with domain-wall boundary conditions, and the six vertex model on a rectangular lattice with partial domain-wall boundary conditions. We show that this rather simple approximation yields results that are remarkably close to the exact ones when these are known, and allows one to estimate the location of the phase boundaries with relative little effort in cases in which exact results are not available.