The Six and Eight-Vertex Models Revisited

David A. Lavis, G. M. Bell

M. Tanhayi-Ahari, Shahin Rouhani

We show that holomorphic Parafermions exist in the eight vertex model. This is done by extending the definition from the six vertex model to the eight vertex model utilizing a parameter redefinition. These Parafermions exist on the critical plane and integrable cases of the eight vertex model. We show that for the case of staggered eight vertex model, these Parafermions correspond to those of the Ashkin-Teller model. Furthermore, the loop representation of the eight vertex model enabled us to show a connection with the O(n) model which is in agreement with the six vertex limit found as a special case of the O(n) model.

Giuliano Niccoli, V. Terras

Abstract We study the inhomogeneous eight-vertex model (or equivalently the XYZ Heisenberg spin-1/2 chain) with all kinds of integrable quasi-periodic boundary conditions: periodic, -twisted, -twisted or -twisted. We show that in all these cases but the periodic one with an even number of sites , the transfer matrix of the model is related, by the vertex-IRF transformation, to the transfer matrix of the dynamical six-vertex model with antiperiodic boundary conditions, which we have recently solved by means of Sklyanin’s separation of variables approach. We show moreover that, in all the twisted cases, the vertex-IRF transformation is bijective. This allows us to completely characterize, from our previous results on the antiperiodic dynamical six-vertex model, the twisted eight-vertex transfer matrix spectrum (proving that it is simple) and eigenstates. We also consider the periodic case for odd. In this case we can define two independent vertex-IRF transformations, both not bijective, and by using them we show that the eight-vertex transfer matrix spectrum is doubly degenerate, and that it can, as well as the corresponding eigenstates, also be completely characterized in terms of the spectrum and eigenstates of the dynamical six-vertex antiperiodic transfer matrix. In all these cases we can adapt to the eight-vertex case the reformulations of the dynamical six-vertex transfer matrix spectrum and eigenstates that had been obtained by T – Q functional equations, where the Q -functions are elliptic polynomials with twist-dependent quasi-periods. Such reformulations enable one to characterize the eight-vertex transfer matrix spectrum by the solutions of some Bethe-type equations, and to rewrite the corresponding eigenstates as the multiple action of some operators on a pseudo-vacuum state, in a similar way as in the algebraic Bethe ansatz framework.

David A. Lavis, G. M. Bell

Barry M. McCoy

Abstract This chapter derives the matrix TQ functional equation for the eight-vertex model. The Bethe's equation for the eigenvalues of Q is derived from the matrix TQ equation. A numerical study is made of the eigenvalues of Q, and the TQ equation is used to compute the free energy of the eight-vertex model. Results on the excitations, order parameters and correlation functions of the six- and eight-vertex models and the related XYZ spin chain are presented.

K. Fabricius, Barry M. McCoy