On the transfer matrix of the supersymmetric eight-vertex model. I. Periodic boundary conditions

Abstract

The square-lattice eight-vertex model with vertex weights obeying the relation and periodic boundary conditions is considered. It is shown that the transfer matrix of the model for L = 2n + 1 vertical lines and periodic boundary conditions along the horizontal direction possesses the doubly degenerate eigenvalue . This proves a conjecture by Stroganov from 2001. The proof uses the supersymmetry of a related XYZ spin-chain Hamiltonian. The eigenstates of the transfer matrix corresponding to are shown to be the ground states of the spin-chain Hamiltonian. Moreover, for positive vertex weights is the largest eigenvalue of the transfer matrix.