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Journal of the Physical Society of Japan

Virasoro Algebra, von Neumann Algebra and Critical Eight-Vertex SOS Models

Atsuo Kuniba, Yasuhiro Akutsu, Miki Wadati

Published Oct 15, 1986• 31 citations
Publisher

Abstract

It is shown that ( r -1)-state restricted eight-vertex SOS model (8VSOS) by Andrews, Baxter and Forrester is equivalent to q -state Potts model with q =4·cos 2 (π/ r ) ( r =4, 5, 6···) at criticality. The equivalence follows from the Temperley-Lieb algebra shared by the models which is nothing but C * -algebra A q , n defined by Jones in von Neumann algebra theory. This implies the correspondence of indices for subfactors of type II 1 factors and discrete series of Virasoro algebra which labels the universality classes of the 8VSOS models.

DOI
10.1143/jpsj.55.3285

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It is shown that ( r -1)-state restricted eight-vertex SOS model (8VSOS) by Andrews, Baxter and Forrester is equivalent to q -state Potts model with q =4·cos 2 (π/ r ) ( r =4, 5, 6···) at criticality. The equivalence follows from the Temperley-Lieb algebra shared by the models which is nothing but C * -algebra A q , n defined by Jones in von Neumann algebra theory. This implies the correspondence of indices for subfactors of type II 1 factors and discrete series of Virasoro algebra which labels the universality classes of the 8VSOS models.
A lattice Virasoro algebra is constructed for the Baxter eight-vertex model. The operator L0{\mathrm{L}}_{0}L0​ is obtained from the logarithm of the corner transfer matrix and is given by the first moment of the XYZ spin-chain Hamiltonian. The algebra is valid even when the Hamiltonian includes a mass term, in which case it represents lattice coordinate transformations which distinguish between even and odd sublattices. We apply the quantum inverse scattering method to demonstrate that the Virasoro algebra follows from the Yang-Baxter relations.
An analog of the Yang-Baxter Algebra (YBA) is defined in face language. The operators tαα′,ββ′(θ) introduced here enjoy all essential properties of the vertex language YBA. Using this face YBA an algebraic Bethe Ansatz (BA) is constructed for SOS models (unrestricted IRF models). The face dual of the six-vertex model and the critical ABF model are worked out explicitly. Eigenvectors and eigenvalues of the transfer matrix are found and the corresponding BA equations derived and compared with the six vertex BAE.
In the present paper we study scaling field theories related to the class of exactly solvable noncritical models of statistical mechanics which are known as RSOS (restricted solid-on-solid) or IRF (interaction round a face) models.These models are related to simple Lie algebras and they generalize the integrable version of RSOS model proposed by Andrews, Baxter and Forrester. 1 > The models related to gl(n) were found by Jimbo, Miwa and Okado, 2 > the models related to other classical Lie algebras were obtained by the same authors in Ref. 3) and by Pasquier in Ref. 4) for Dn-case with a special coupling constant.Critical properties of JMO model were found in Ref. 5) where the authors obtain the explicit formulas for magnetization in the model and compute its critical behavior.It was established there that the critical behavior of these models was described by coset conformal field theories. 6>. 7 > Therefore it is natural to expect that corresponding scaling field theories will be integrable perturbations of these conformal field theories.s>In this paper we describe the thermodynamics of the noncritical magnetic chain.Then we find the ground state, excitations over the ground state and the scattering amplitudes of excitations.By getting the scaling limit over the ground state of the model we compute the central charges of corresponding field theories, the dimensions of perturbations, the spectrum and scattering amplitudes of excitations.Under certain assumptions we generalize these results for any Lie algebra of A-D-E type.In certain simplest cases this gives known S-matrices like Es, E1, Es scalar S-matrices. 9> In certain other models matrix elements of the S-matrix are given by Pasquier's solutions 4 > of the Yang-Baxter equation.We do not discuss here the relation of RSOS models to integrable models of vertex type. 23>' 25 > Partially the relation was explained in Refs.4) and 26).The relation between the spectrum of row-to-row transfer matrices in RSOS and in vertex models will be given in a separate publication.For some of the scaling models the same answer was obtained by Bernard and