Sum rules for the supersymmetric eight-vertex model

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 $L=2n+1,n⩾0,$ 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.

arXiv commentV2: added section 4.4; 38 pages, no figures

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