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Journal of Physics A Mathematical and General

Restricted solid-on-solid models connected with simply laced algebras and conformal field theory

Vladimir V. Bazhanov, Nicolai Reshetikhin

Published May 7, 1990• 220 citations
Publisher

Abstract

The spectrum of row-to-row transfer matrices in critical RSOS models of A n type is described explicitly. The thermodynamics of the quantum one-dimensional RSOS model is used to compute values of central charges in corresponding conformal field theories. Generalisation of these results on RSOS related to arbitrary simply laced algebras is discussed.

DOI
10.1088/0305-4470/23/9/012

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•Jan 1, 2013•17 citations
The spectrum of row-to-row transfer matrices in critical RSOS models of A n type is described explicitly. The thermodynamics of the quantum one-dimensional RSOS model is used to compute values of central charges in corresponding conformal field theories. Generalisation of these results on RSOS related to arbitrary simply laced algebras is discussed.
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 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
An equivalence between generalized restricted solid-on-solid models, associated with sets of graphs, and multi-colour loop models is established. As an application the authors consider solvable loop models and, in this way, obtain new solvable families of critical RSOS models. These families can all be classified by the Dynkin diagrams of the simply laced Lie algebras. For one of the RSOS models, labelled by the Lie algebra pair (A L ,A L ) and related to the C 2 (1) vertex model, they give an off-critical extension, which breaks the Z 2 symmetry of the Dynkin diagrams.
We analytically compute the Renyi entropies for the RSOS models, representing a wide class of exactly solvable models with multicritical conformal points described by unitary minimal models and Zn\mathbb{Z}_nZn​ parafermions. The exact expressions allow for an explicit comparison of the expansions around the critical points with the predictions coming from field theory. In this way it is possible to point out the nature of the so-called "unusual corrections", clarifying the link with the operator content, the role of the symmetries and the boundary conditions. By choosing different boundary conditions, we can single out the ground states as well as certain combinations of high energy states. We find that the {\it entanglement spectrum} is given by operators that are not present in the bulk Hamiltonian, although they belong to the same representation of a Virasoro Algebra. In the parafermionic case we observe unexpected logarithmic corrections.