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arXiv:2310.05850 · SciPost Physics

Modified rational six vertex model on the rectangular lattice

S. Belliard, Rodrigo A. Pimenta, Никита Андреевич Славнов

Published Jan 9, 2024• 4 citations
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Abstract

We consider a rational six vertex model on a rectangular lattice with boundary conditions that generalize the usual domain wall type. We find that the partition function of the inhomogeneous version of this model is given by a modified Izergin determinant. The proofs are based on the quantum inverse scattering method and its representation theory together with elementary linear algebra.

DOI
10.21468/scipostphys.16.1.009

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We consider a rational six vertex model on a rectangular lattice with boundary conditions that generalize the usual domain wall type. We find that the partition function of the inhomogeneous version of this model is given by a modified Izergin determinant. The proofs are based on the quantum inverse scattering method and its representation theory together with elementary linear algebra.
We continue the work of [S. Belliardet al., SciPost Phys. 16, 009 (2024)] studying the modified rational six vertex model. We find another formula of the partition function for the inhomogeneous model, in terms of a determinant that mix the modified Izergin one and a Vandermonde one. This expression enables us to compute the partition function in the homogeneous limit for the rectangular lattice, and then to study the thermodynamic limit. It leads to a new result, we obtain the first order of free energy with boundary effects in the thermodynamic limit.
The six-vertex model on an N × N square lattice with domain wall boundary conditions is considered. A Fredholm determinant representation for the partition function of the model is given. The kernel of the corresponding integral operator is of the so-called integrable type, and involves classical orthogonal polynomials. From this representation, a ‘reconstruction’ formula is proposed, which expresses the partition function as the trace of a suitably chosen quantum operator, in the spirit of corner transfer matrix and vertex operator approaches to integrable spin models.
Abstract We consider the problem of construction of determinant formulas for the partition function of the six-vertex model with domain wall boundary conditions that depend on two sets of spectral parameters. In the pioneering works of Korepin and Izergin a determinant formula was proposed and proved using a recursion relation. In later works, equivalent determinant formulas were given by Kostov for the rational case and by Foda and Wheeler for the trigonometric case. Here, we develop an approach in which the recursion relation is replaced by a system of algebraic equations with respect to one of the two sets of spectral parameters. We prove that this system has a unique solution. The result can be easily given as a determinant parametrized by an arbitrary basis of polynomials. In particular, the choice of the basis of Lagrange polynomials with respect to the remaining set of spectral parameters leads to the Izergin–Korepin representation, and the choice of the monomial basis leads to the Kostov and Foda–Wheeler representations.
In this note, we consider the six-vertex model with domain wall boundary conditions, defined on an M×M lattice, in the inhomogeneous case where the partition function depends on 2M inhomogeneities λj and μk. For a particular choice of the set of λj we find a new determinant representation for the partition function, which allows evaluation of the bulk free energy in the thermodynamic limit. This provides a new connection between two types of determinant formulas. We also show in a special case that spin correlations on the horizontal line going through the center coincide with the ones for periodic boundary conditions.