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.

arXiv comment21 pages, v2: added a theorem on the largest transfer-matrix eigenvalue

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Cited by in corpus (5)

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• Eigenstate entanglement entropy in the integrable spin-$\frac{1}{2}$ XYZ model
• Boundary emptiness formation probabilities in the six-vertex model at $\Delta =-\frac{1}{2}$
• Sum rules for the supersymmetric eight-vertex model
• Lattice SUSY for the DiSSEP at ${\lambda }^2=1$ (and ${\lambda }^2=-3$)