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Modern Physics Letters A

IRF MODELS ASSOCIATED WITH REPRESENTATIONS OF THE LIE SUPERALGEBRAS gl (m|n) AND sl (m|n)

Tetsuo Deguchi, AKIRA FUJII

Published Dec 7, 1991• 8 citations
Publisher

Abstract

We present two families of exactly solvable interaction round a face (IRF) models associated with representation of the Lie superalgebras gl (m|n) and sl (m|n). These IRF models are the generalizations of integrable spin chains with bosons and fermions. We also presents fusion models associated with higher representation of gl (m|n) and sl (m|n). We introduce restricted IRF models both for gl (m|n) and sl (m|n).

DOI
10.1142/s0217732391003948

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We present two families of exactly solvable interaction round a face (IRF) models associated with representation of the Lie superalgebras gl (m|n) and sl (m|n). These IRF models are the generalizations of integrable spin chains with bosons and fermions. We also presents fusion models associated with higher representation of gl (m|n) and sl (m|n). We introduce restricted IRF models both for gl (m|n) and sl (m|n).
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 corre­ sponding 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
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
In this work we obtain the exact solution of quantum integrable system associated with the Lie superalgebra \mathfrak{gl}(1|1) 𝔤 𝔩 ( 1 | 1 ) , both for periodic and for generic open boundary conditions. By means of the fusion technique we derive a closed set of operator identities among the fused transfer matrices. These identities allow us to determine the complete energy spectrum and the corresponding Bethe Ansatz equations of the model. Our approach furnishes a systematic framework for studying the spectra of quantum integrable models based on Lie superalgebras, in particular when the U(1) U ( 1 ) symmetry is broken. The derivation of the Bethe states from the exact spectrum is also addressed.
Starting from the fusion rules for the algebra SO(5)2 we construct one-dimensional lattice models of interacting anyons with commuting transfer matrices of ‘interactions round the face’ (IRF) type. The conserved topological charges of the anyon chain are recovered from the transfer matrices in the limit of large spectral parameter. The properties of the models in the thermodynamic limit and the low energy excitations are studied using Bethe ansatz methods. Two of the anyon models are critical at zero temperature. From the analysis of the finite size spectrum we find that they are effectively described by rational conformal field theories invariant under extensions of the Virasoro algebra, namely WB2 and WD5, respectively. The latter contains primaries with half and quarter spin. The modular partition function and fusion rules are derived and found to be consistent with the results for the lattice model.