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arXiv:hep-th/9504107 · Journal of Physics A Mathematical and General

Further solutions of critical ABF RSOS models

Yu-Kui Zhou

Published Aug 7, 1995• 2 citations
arXivHTMLPDF

Abstract

The restricted SOS model of Andrews, Baxter and Forrester (1984) has been studied. The finite size corrections to the eigenvalue spectra of the transfer matrix of the model with a more general crossing parameter have been calculated. Therefore the conformal weights and the central charges of the non-unitary or unitary minimal conformal field have been extracted from the finite size corrections.

DOI
10.1088/0305-4470/28/15/014

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The restricted SOS model of Andrews, Baxter and Forrester (1984) has been studied. The finite size corrections to the eigenvalue spectra of the transfer matrix of the model with a more general crossing parameter have been calculated. Therefore the conformal weights and the central charges of the non-unitary or unitary minimal conformal field have been extracted from the finite size corrections.
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
We consider the critical nonunitary minimal model M (3, 5) with integrable boundaries and analyze the patterns of zeros of the eigenvalues of the transfer matrix and then determine the spectrum of the critical theory using the thermodynamic Bethe ansatz ( TBA ) equations. Solving the TBA functional equation satisfied by the transfer matrices of the associated A 4 restricted solid-on-solid Forrester–Baxter lattice model in regime III in the continuum scaling limit, we derive the integral TBA equations for all excitations in the ( r , s ) = (1, 1) sector and then determine their corresponding energies. We classify the excitations in terms of ( m , n ) systems .
The author studies conformal invariance in the RSOS models of Andrews et al (1985) at both the ferromagnetic and antiferromagnetic transitions, using finite-size scaling of the transfer matrix. The author confirms the conjecture of Huse (1984) concerning the conformal classification of the ferromagnetic transitions and gives evidence that the antiferromagnetic transitions lie in the universality class of Z n models. In addition, he shows that conformal results can be applied to anisotropic systems if logarithmic corrections are taken into account.
Abstract Using a corner transfer matrix approach, we compute the bipartite entanglement Rényi entropy in the off-critical perturbations of non-unitary conformal minimal models realised by lattice spin chains Hamiltonians related to the Forrester–Baxter RSOS models (Bianchini et al 2015 J. Stat. Mech. P03010 ) in regime III. This allows to show on a set of explicit examples that the Rényi entropies for non-unitary theories rescale near criticality as the logarithm of the correlation length with a coefficient proportional to the effective central charge. This complements a similar result, recently established for the size rescaling at the critical point (Bianchini et al 2015 J. Phys. A: Math. Theor. 48 04FT01 ), showing the expected agreement of the two behaviours. We also compute the first subleading unusual correction to the scaling behaviour, showing that it is expressible in terms of expansions of various fractional powers of the correlation length, related to the differences between the conformal dimensions of fields in the theory and the minimal conformal dimension. Finally, a few observations on the limit leading to the off-critical logarithmic minimal models of Pearce and Seaton (2012 J. Stat. Mech. P09014 ) are put forward.