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

TL;DR

This paper analyzes the transfer matrix of the supersymmetric eight-vertex model with open boundaries, revealing a simple eigenvalue structure and its relation to the ground states of a supersymmetric XYZ spin chain.

Contribution

It demonstrates that under certain conditions, the transfer matrix has a unique, simple eigenvalue linked to the ground states of a supersymmetric XYZ spin chain.

Findings

The transfer matrix has a non-degenerate eigenvalue for specific vertex weights.

This eigenvalue corresponds to the ground state space of a related Hamiltonian.

Supersymmetry plays a key role in the eigenvalue structure and proofs.

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.