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Physical Review Letters

Inhomogeneous Eight-Vertex System and the One-Dimensional Fermi Gas

Trong Tuong Truong, K. D. Schotte

Published Aug 3, 1981• 20 citations
Publisher

Abstract

It is shown that a proper analytic continuation of eigenvalues of a special inhomogeneous eight-vertex system determines the "phases" and the spectrum of a one-dimensional fermion gas model, and thus provides a particular example of the long suspected connection between factorized $S$ matrices and lattice statistical systems.

DOI
10.1103/physrevlett.47.285

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It is shown that a proper analytic continuation of eigenvalues of a special inhomogeneous eight-vertex system determines the "phases" and the spectrum of a one-dimensional fermion gas model, and thus provides a particular example of the long suspected connection between factorized SSS matrices and lattice statistical systems.
We observe that the exactly solved eight-vertex solid-on-solid model contains an hitherto unnoticed arbitrary field parameter, similar to the horizontal field in the six-vertex model. The parameter is required to describe a continuous spectrum of the unrestricted solid-on-solid model, which has an infinite-dimensional space of states even for a finite lattice. The introduction of the continuous field parameter allows us to completely review the theory of functional relations in the eight-vertex/SOS-model from a uniform analytic point of view.
The full spectrum of transfer matrices of the general eight-vertex model on a square lattice is obtained by numerical diagonalization. The eigenvalue spacing distribution and the spectral rigidity are analysed. In non-integrable regimes we have found eigenvalue repulsion as for the Gaussian orthogonal ensemble in random matrix theory. By contrast, in integrable regimes we have found eigenvalue independence leading to a Poissonian behaviour, and, for some points, level clustering. These first examples from classical statistical mechanics suggest that the conjecture of integrability successfully applied to quantum spin systems also holds for classical systems.