Sum rules for the supersymmetric eight-vertex model

TL;DR

This paper derives explicit sum rules and scalar products for eigenvectors of the supersymmetric eight-vertex model's transfer matrix, revealing new algebraic structures and extending eigenvalue problem solutions to inhomogeneous cases.

Contribution

It introduces explicit sum rules and scalar product formulas for the eigenspaces of the supersymmetric eight-vertex model, generalizing to inhomogeneous models.

Findings

Explicit scalar product formulas involving Rosengren and Zinn-Justin polynomials

Sum rules for the doubly-degenerate eigenvalues of the transfer matrix

Extension of eigenvalue problem solutions to inhomogeneous models

Abstract

The eight-vertex model on the square lattice with vertex weights $a,b,c,d$ obeying the relation $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$ is considered. Its transfer matrix with $L=2n+1,\, n\geqslant 0,$ vertical lines and periodic boundary conditions along the horizontal direction has the doubly-degenerate eigenvalue $\Theta_n = (a+b)^{2n+1}$. A basis of the corresponding eigenspace is investigated. Several scalar products involving the basis vectors are computed in terms of a family of polynomials introduced by Rosengren and Zinn-Justin. These scalar products are used to find explicit expressions for particular entries of the vectors. The proofs of these results are based on the generalisation of the eigenvalue problem for $\Theta_n$ to the inhomogeneous eight-vertex model.