On the transfer matrix of the supersymmetric eight-vertex model. II. Open boundary conditions

Abstract

The transfer matrix of the square-lattice eight-vertex model on a strip with $L\geqslant 1$ vertical lines and open boundary conditions is investigated. It is shown that for vertex weights $a,b,c,d$ that obey the relation $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$ and appropriately chosen $K$-matrices $K^\pm$ this transfer matrix possesses the remarkably simple, non-degenerate eigenvalue $\Lambda_L = (a+b)^{2L}\,\text{tr}(K^+K^-)$. For positive vertex weights, $\Lambda_L$ is shown to be the largest transfer-matrix eigenvalue. The corresponding eigenspace is equal to the space of the ground states of the Hamiltonian of a related XYZ spin chain. An essential ingredient in the proofs is the supersymmetry of this Hamiltonian.