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arXiv:q-alg/9504027 · Journal of Physics A Mathematical and General

Bethe ansatz for higher spin eight-vertex models

Takashi Takebe

Published Dec 7, 1995• 28 citations
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Abstract

A generalization of the eight-vertex model by means of higher spin representations of the Sklyanin algebra is investigated by the quantum inverse scattering method and the algebraic Bethe ansatz. Under the well known string hypothesis low-lying excited states are considered and scattering phase shifts of two physical particles are calculated. The S-matrix of two-particle states is shown to be proportional to Baxter`s (1972) elliptic R-matrix with a different elliptic modulus from the original one.

DOI
10.1088/0305-4470/28/23/019

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A generalization of the eight-vertex model by means of higher spin representations of the Sklyanin algebra is investigated by the quantum inverse scattering method and the algebraic Bethe ansatz. Under the well known string hypothesis low-lying excited states are considered and scattering phase shifts of two physical particles are calculated. The S-matrix of two-particle states is shown to be proportional to Baxter`s (1972) elliptic R-matrix with a different elliptic modulus from the original one.
This work deals on the one hand with how to use the representation theory of a quantum group ( [14]) to investigate a statistical mechanical model and on the other hand with how to solve the statistical mechanical model by Sklyanin's method of separation of variables.[47].More concretely, we use the elliptic quantum group E y,'I)(sl2) established by Felder in [29, 25] to investigate the SOS eight-vertex model established by Date, Jimbo, Miwa and Okado [10] with antiperiodic boundary conditions which are the reason that Bethe ansatz fails and we have to use Sklyanin's method of separation of variables [47].The SOS eight-vertex model is a face-model version of Baxter's original eight-vertex model [3].It is related to the elliptic quantum group E y,'I)(sl2), since we rediscover its Boltzmann weights We(c, b, a, dlz) by a suitably discretized version of the R-matrix Re(z,.\)defining E Y,'I)(sl2)' This relation reads
We compute the operator which creates the missing degenerate states in the algebraic Bethe ansatz of the eight-vertex model at roots of unity and relate it to the concept of an elliptic current operator. We find that in sharp contrast with the corresponding formalism in the six-vertex model at roots of unity [12] the current operator is not nilpotent with the consequence that in the construction of degenerate eigenstates of the transfer matrix an arbitrary number of exact strings can be added to the set of regular Bethe roots. Thus the original set of free parameters { s , t } of an eigenvector of the transfer matrix T is enlarged to become { s , t , λ c ,1 , ..., λ c , n } with arbitrary string centres λ c , j and arbitrary n .