Factorization of rational six vertex model partition functions

Abstract

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.

Figures (12)

• Figure 1: The $R$-matrix acting on $V_j \otimes V_k$. The horizontal and vertical line represents $V_j$ and $V_k$ respectively, and to each line variable $u$ and $v$ is associated.
• Figure 2: Nonzero matrix elements of the rational $R$-matrix.
• Figure 3: Yang-Baxter relation \ref{['YBE']}.
• Figure 4: Unitarity relation \ref{['unitarity']}.
• Figure 5: Partition functions under triangular boundary $Z_n(u_1,u_2,\dots,u_n)$\ref{['sumhalftwist']}. At the bottom boundary, each state is a mixture of $|1 \rangle$ and $|2 \rangle$ given by $|s \rangle=s_1|1 \rangle+s_2|2 \rangle$. At the right boundary, each state is given by $\langle e |=e_1 \langle 1 |+e_2 \langle 2 |$.

Theorems & Definitions (10)

• Proposition 3.1
• proof
• Theorem 4.1
• Theorem 4.2
• Lemma 4.3
• proof
• Theorem 4.4
• proof
• Theorem 4.5
• proof