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arXiv:1805.05949 · International Journal of Applied Mathematics & Statistics/International journal of applied mathematics and statistics

Analytic structures of unitary RSOS models with integrable boundary conditions

Omar El Deeb

Published May 15, 2018• 1 citation
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Abstract

In this paper, we consider the unitary critical restricted-solid-on-solid (RSOS) lattice $\mathcal{M}(5,6)$ model with integrable boundary conditions. We introduce its commuting double row transfer matrix satisfying the universal functional relations, and we use it in order to study the analytic structure of the transfer matrix eigenvalues and plot representative zero configurations of sample eigenvalues of the transfer matrix. We finally conclude with a comparative analysis with the critical and tricritical Ising models with integrable boundary conditions.

DOI
10.48550/arxiv.1805.05949

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In this paper, we consider the unitary critical restricted-solid-on-solid (RSOS) lattice M(5,6)\mathcal{M}(5,6)M(5,6) model with integrable boundary conditions. We introduce its commuting double row transfer matrix satisfying the universal functional relations, and we use it in order to study the analytic structure of the transfer matrix eigenvalues and plot representative zero configurations of sample eigenvalues of the transfer matrix. We finally conclude with a comparative analysis with the critical and tricritical Ising models with integrable boundary conditions.
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 .
We derive exact inversion identities satisfied by the transfer matrix of inhomogeneous interaction-round-a-face (IRF) models with arbitrary boundary conditions using the underlying integrable structure and crossing properties of the local Boltzmann weights. For the critical restricted solid-on-solid (RSOS) models these identities together with some information on the analytical properties of the transfer matrix determine the spectrum completely and allow to derive the Bethe equations for both periodic and general open boundary conditions.
By considering the continuum scaling limit of the A4 RSOS lattice model of Andrews-Baxter-Forrester with integrable boundaries, we derive excited state TBA equations describing the boundary flows of the tricritical Ising model. Fixing the bulk weights to their critical values, the integrable boundary weights admit two parameters ξ and η which play the role of the perturbing boundary fields φ1,3 and φ1,2 and induce the renormalization group flow between boundary fixed points. The boundary TBA equations determining the RG flows are derived in the [Formula: see text] example. The induced map between distinct Virasoro characters of the theory are specified in terms of distribution of zeros of the double row transfer matrix.
We consider the L-state cyclic solid-on-solid lattice models under a class of open boundary conditions. The integrable boundary face weights are obtained by solving the reflection equations. Functional relations for the fused transfer matrices are presented for both periodic and open boundary conditions. The eigenspectra of the unfused transfer matrix is obtained from the functional relations using the analytic Bethe ansatz. For a special case of crossing parameter , the finite-size corrections to the eigenspectra of the critical models are obtained, from which the corresponding conformal dimensions follow. The calculation of the surface free energy away from criticality yields two surface specific heat exponents, and , where coprime to L. These results are in agreement with the scaling relations 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 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.