The Functional Method for the Domain-Wall Partition Function

TL;DR

This paper reviews and extends the functional method for calculating domain-wall partition functions, providing new formulas and connecting it with existing approaches for models like the six-vertex and elliptic solid-on-solid models.

Contribution

The paper introduces a new crossing-symmetrized sum formula for the elliptic model's partition function and demonstrates how the functional method unifies and extends previous techniques.

Findings

Derived a linear functional equation for the partition function.

Rewritten the solution as a symmetrized sum and multiple-integral formula.

Presented a new formula for the elliptic model's partition function with $2^L$ terms.

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 domain-wall 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.