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

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.