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arXiv:1703.06250

Domain wall boundary partition function of the six-vertex model with triangular boundary

Kohei Motegi

Published Mar 18, 2017• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.1703.06250

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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 Uq(sl2)U_q(sl_2)Uq​(sl2​) RRR-matrix and a special class of the triangular KKK-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.
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.
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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.
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.