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arXiv:2509.11797 · SciPost Physics

Modified rational six vertex model on a rectangular lattice: New formula, homogeneous and thermodynamic limits

Matthieu Cornillault, S. Belliard

Published Mar 9, 2026• 0 citations
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Abstract

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.

DOI
10.21468/scipostphys.20.3.076

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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.
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.
We show factorization formulas for a class of partition functions of rational six vertex model. First we show factorization formulas for partition functions under triangular boundary. Further, by combining the factorization formulas with the explicit forms of the generalized domain wall boundary partition functions by Belliard-Pimenta-Slavnov, we derive factorization formulas for partition functions under trapezoid boundary which can be viewed as a generalization of triangular boundary. We also discuss an application to emptiness formation probabilities under trapezoid boundary which admit determinant representations.
The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite nnn by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an n×nn\times nn×n Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large nnn asymptotics of the six-vertex model with DWBC. The solution is based on the Riemann-Hilbert approach. In this paper we review asymptotic results obtained in different regions of the phase diagram.
The partition function of the six-vertex model on a square lattice with domain wall boundary conditions (DWBC) is rewritten as a Hermitian one-matrix model or a discretized version of it (similar to sums over Young diagrams), depending on the phase. The expression is exact for finite lattice size, which is equal to the size of the corresponding matrix. In the thermodynamic limit, the matrix integral is computed using traditional matrix model techniques, thus providing a complete treatment of the bulk free energy of the six-vertex model with DWBC in the different phases. In particular, in the antiferroelectric phase, the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta functions.
We review the (algebraic-)functional method devised by Galleas and further developed by Galleas and the author.We first explain the method using the simplest example: the computation of the partition function for the six-vertex model with domainwall boundary conditions.At the heart of the method lies a linear functional equation for the partition function.After deriving this equation we outline its analysis.The result is a closed expression in the form of a symmetrized sum -or, equivalently, multiple-integral formula -that can be rewritten to recover Izergin's determinant.Special attention is paid to the relation with other approaches.In particular we show that the Korepin-Izergin approach can be recovered within the functional method.We comment on the functional method's range of applicability, and review how it is adapted to the technically more involved example of the elliptic solid-on-solid model with domain walls and a reflecting end.We present a new formula for the partition function of the latter, which was expressed as a determinant by Tsuchiya-Filali-Kitanine.Our result takes the form of a 'crossing-symmetrized' sum with 2 L terms featuring the elliptic domain-wall partition function, which appears to be new also in the limiting case of the six-vertex model.Further taking the rational limit we recover the expression obtained by Frassek using the boundary perimeter Bethe ansatz.