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Journal of Physics A Mathematical and General

Superconformal invariance in the Ashkin-Teller quantum chain with free boundary conditions

G von Gehlen, V. Rittenberg

Published Nov 11, 1986• 16 citations
Publisher

Abstract

The finite-size limit of the lower part of the spectrum of the Ashkin-Teller chain with free boundary conditions is studied numerically and interpreted from the point of view of conformal invariance. Several irreducible representations of the Virasoro algebra with the central charge c=1 are identified. For two special values of the coupling constant, higher degeneracies occur and the whole spectrum can be understood in terms of a few irreducible representations of the N=2 superconformal algebra.

DOI
10.1088/0305-4470/19/16/016

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Journal of Physics A Mathematical and General•Mar 21, 1989•4 citations
The finite-size limit of the lower part of the spectrum of the Ashkin-Teller chain with free boundary conditions is studied numerically and interpreted from the point of view of conformal invariance. Several irreducible representations of the Virasoro algebra with the central charge c=1 are identified. For two special values of the coupling constant, higher degeneracies occur and the whole spectrum can be understood in terms of a few irreducible representations of the N=2 superconformal algebra.
Based on numerical analysis, the authors conjecture the operator content of the finite-size limit of the spectra of the Ashkin-Teller model with free boundary conditions. The same operator content is obtained from a Hamiltonian with a four-fermion interaction and a U(1) Kac-Moody Sugawara structure. For some special values of the coupling constant the model exhibits N=2 superconformal and Zamolodchikov-Fateev invariance. The operator content in these cases is expressed in terms of irreducible representations of the corresponding algebras.
The authors study the finite-size effects for the fourth-state one-dimensional quantum chain introduced by Kohmoto et al. (1981) using different boundary conditions. From the corrections to the ground-state energy for periodic boundary conditions they verify that the central charge of the Virasoro algebra is equal to one in the whole domain of criticality. Their numerical study suggests a new relation between the corrections to the ground-state energy with free boundary conditions and with periodic boundary conditions. Various critical exponents including the surface ones and those corresponding to the para-fermions are determined.
For pt. I, see ibid., vol. 20, no.8, p.L479-85 (1987). The authors present the operator content of the Ashkin-Teller quantum chain with eight boundary conditions corresponding to the eight elements of the group D 4 which gives the global symmetry of the system. This operator content is a conjecture based on an extensive numerical study and was proven to be correct at the Ising decoupling point. For the values of the coupling constant where N=2, superconformal invariance was seen for free boundary conditions. The situation is now more complex. One finds sectors which have N=2 and N=1 superconformal invariance and sectors which have no superconformal invariance at all. If, however, they combine the 'wrong' sectors of the spectra which correspond to two different values of the coupling constant, a new symmetry shows up. They also analyse the operator content at the Fateev-Zamolodchikov points. Here the 'wrong' sectors can be described by a few new primary fields.