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arXiv:1801.09635 · Symmetry Integrability and Geometry Methods and Applications

The Functional Method for the Domain-Wall Partition Function

Jules Lamers

Published Jun 26, 2018• 4 citations

Abstract

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.

DOI: 10.3842/sigma.2018.064

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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.

We introduce and study the domain wall boundary partition function of the integrable six-vertex model with triangular boundary. We first formulate the domain wall boundary partition function with triangular boundary by using the

U_q(sl_2)

R

-matrix and a special class of the triangular

K

-matrix. By using its graphical representation, we make the Izergin-Korepin analysis with the help of the Yang-Baxter relation and the reflection equation to give a characterization of the partition function. The explict form of the symmetric function representing the partition function is presented by showing that it satisfies all the required properties.

In this work we investigate the possibility of using the reflection algebra as a source of functional equations. More precisely, we obtain functional relations determining the partition function of the six-vertex model with domain-wall boundary conditions and one reflecting end. The model's partition function is expressed as a multiple-contour integral that allows the homogeneous limit to be obtained straightforwardly. Our functional equations are also shown to give rise to a consistent set of partial differential equations satisfied by the partition function.

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.

We present a method to analyze the wavefunctions of six-vertex models by extending the Izergin–Korepin analysis originally developed for domain wall boundary partition functions. First, we apply the method to the case of the basic wavefunctions of the XXZ-type six-vertex model. By giving the Izergin–Korepin characterization of the wavefunctions, we show that these wavefunctions can be expressed as multiparameter deformations of the quantum group deformed Grothendieck polynomials. As a second example, we show that the Izergin–Korepin analysis is effective for analysis of the wavefunctions for a triangular boundary and present the explicit forms of the symmetric functions representing these wavefunctions. As a third example, we apply the method to the elliptic Felderhof model which is a face-type version and an elliptic extension of the trigonometric Felderhof model. We show that the wavefunctions can be expressed as one-parameter deformations of an elliptic analog of the Vandermonde determinant and elliptic symmetric functions.