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arXiv:2304.01824 · Journal of Physics A Mathematical and Theoretical

Construction of determinants for the six-vertex model with domain wall boundary conditions

Mikhail D. Minin, A. G. Pronko, V. Tarasov

Published Jun 9, 2023• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/acdd37

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Abstract. We consider the trigonometric Felderhof model, of free fermions in an external field, on a finite lattice with domain wall boundary conditions. The vertex weights are functions of rapidities and external fields. We obtain a determinant expression for the partition function in the special case where the dependence on the rapidities is eliminated, but for general external field variables. This determinant can be evaluated in product form. In the homogeneous limit, it is proportional to a 2-Toda τ function. Next, we use the algebraic Bethe ansatz factorized basis to obtain a product expression for the partition function in the general case with dependence on all variables. In [1], Korepin introduced domain wall boundary conditions for the six vertex model on a finite square lattice, and proposed recursion relations that determine the corresponding domain wall partition function. In [2], Izergin obtained a determinant solution of Korepin’s recursion relations. At the free fermion point, the
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.
Abstract We study the domain wall partition function Z N for the (Izergin–Korepin) integrable nineteen-vertex model on a square lattice of size N . Z N is a symmetric function of two sets of parameters: horizontal and vertical rapidities. For generic values of the parameter q we derive the recurrence relation for the domain wall partition function relating Z N +1 to , where P N is the proportionality factor in the recurrence, which is a polynomial symmetric in two sets of variables and . After setting the recurrence relation simplifies and we solve it in terms of a Jacobi–Trudi-like determinant of polynomials generated by P N .