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The 6-Vertex Model with Fixed Boundary Conditions

Nicolai Reshetikhin

Published Aug 8, 2007• 17 citations
Publisher

Abstract

We study the 6-vertex model with fixed boundary conditions.In the thermodynamical limit there is a formation of the limit shape.We collect most of the known results about the analytical properties of the free energy of the model as the function of electric fields and study the asymptotical behavior near singularities.We also study the asymptotic of limit shapes and the structure of correlation functions in the bulk Bethe Ansatz: 75 years later October 19-21 2006, Brussels,

DOI
10.22323/1.038.0012

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We study the 6-vertex model with fixed boundary conditions.In the thermodynamical limit there is a formation of the limit shape.We collect most of the known results about the analytical properties of the free energy of the model as the function of electric fields and study the asymptotical behavior near singularities.We also study the asymptotic of limit shapes and the structure of correlation functions in the bulk Bethe Ansatz: 75 years later October 19-21 2006, Brussels,
We study the dependence of entropy [per lattice site] of six-vertex model on boundary conditions. We start with lattices of finite size and then proceed to thermodynamic limit. We argue that the six-vertex model with periodic, anti-periodic and mixed boundary conditions produce the same free-energy in the thermodynamic limit. We have found fixed boundary conditions such that the entropy varies continuously from zero to its value for periodic boundary condition. We have also shown that the physical quantities of the six-vertex model at the isotropic point does not change in the case of singular toroidal boundary.
Abstract In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime Δ<1\Delta <1Δ<1 Δ<1 . As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when a=b=1a=b=1a=b=1 a=b=1 and c≥1c\ge 1c≥1 c≥1 , and the rotational invariance of the six-vertex model and the Fortuin–Kasteleyn percolation.
We consider the six-vertex model with domain wall boundary conditions. We choose the inhomogeneities as solutions of the Bethe Ansatz equations. These equations have many solutions, so we can consider a wide variety of inhomogeneities. For certain choices of the inhomogeneities we study arrow correlation functions on the horizontal line going through the centre. In particular, we obtain a multiple integral representation for the emptiness formation probability that generalizes the known formulae for XXZ antiferromagnets.
We consider the six-vertex model with domain wall boundary conditions. We choose the inhomogeneities as solutions of the Bethe Ansatz equations. The Bethe Ansatz equations have many solutions, so we can consider a wide variety of inhomogeneities. For certain choices of the inhomogeneities we study arrow correlation functions on the horizontal line going through the centre. In particular we obtain a multiple integral representation for the emptiness formation probability that generalizes the known formulæ for XXZ antiferromagnets. 1
Abstract We compute the exact partition function of the isotropic 6-vertex model on a cylinder geometry with free boundary conditions, for lattices of intermediate size, using Bethe ansatz and algebraic geometry. We perform the computations in both the open and closed channels. We also consider the partial thermodynamic limits, whereby in the open (closed) channel, the open (closed) direction is kept small while the other direction becomes large. We compute the zeros of the partition function in the two partial thermodynamic limits, and compare with the condensation curves.