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

Abstract

The square-lattice eight-vertex model with vertex weights $a,b,c,d$ obeying the relation $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$ 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 ${\Theta }_n=(a+b{)}^{2n+1}$. 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 ${\Theta }_n$ are shown to be the ground states of the spin-chain Hamiltonian. Moreover, for positive vertex weights ${\Theta }_n$ is the largest eigenvalue of the transfer matrix.

21 pages, v2: added a theorem on the largest transfer-matrix eigenvalue