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Journal of Physics A Mathematical and General

Operator content of the cyclic solid-on-solid models

D Kim, Paul A. Pearce

Published May 7, 1989• 23 citations
Publisher

Abstract

The operator content of the cyclic solid-on-solid models is derived under general boundary conditions by mapping the row-to-row transfer matrix onto the six-vertex model and XXZ Heisenberg chain. For the shift boundary conditions, the dimensions and spins of the primary operators are derived analytically from finite-size corrections to the scaling spectra. The full operator content agrees with results for a quantum chain model having the same local symmetry.

DOI
10.1088/0305-4470/22/9/029

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The operator content of the cyclic solid-on-solid models is derived under general boundary conditions by mapping the row-to-row transfer matrix onto the six-vertex model and XXZ Heisenberg chain. For the shift boundary conditions, the dimensions and spins of the primary operators are derived analytically from finite-size corrections to the scaling spectra. The full operator content agrees with results for a quantum chain model having the same local symmetry.
The fusion model of the L-state cyclic solid-on-solid (CSOS) model is considered at criticality. Each Z L -charge sector of the fusion CSOS model is related to a sum of total spin sectors of the fusion vertex model with a seam on which the vertex weights are modified by a phase factor. The latter in turn becomes the higher-spin XXZ quantum chain with a twisted boundary condition in the extremely anisotropic limit. Using the known operator content of the higher-spin XXZ chain, the authors deduce that of the fusion CSOS model. The modular invariant partition function of the fusion CSOS at fusion level k is a sum of products of the Z(k) parafermionic and free boson sectors with the effective coupling g'=L 2 (k-lambda/pi) for L odd and g"=L 2 (1/k-lambda/pi)/4 for L even, where lambda is the crossing parameter. Where L is odd and a multiple of k, or when L/2 is a multiple of k for L even, the modular invariant partition function becomes a simple product of the Z(k) parafermionic and the Gaussian partition function.
The authors are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.
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 .