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Journal of Physics A Mathematical and General

The spectra of quantum chains with free boundary conditions and Virasoro algebras

G von Gehlen, V. Rittenberg

Published Jul 11, 1986• 37 citations
Publisher

Abstract

At the critical point the spectra of quantum chains with periodic and twisted boundary conditions are described by irreducible representations of two Virasoro algebras with the same central charge. The authors show that in the case of free boundary conditions, the spectra can be understood in terms of irreducible representations of a single Virasoro algebra. For the Ising and the three-state Potts models, the corresponding irreducible representations are identified.

DOI
10.1088/0305-4470/19/10/014

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At the critical point the spectra of quantum chains with periodic and twisted boundary conditions are described by irreducible representations of two Virasoro algebras with the same central charge. The authors show that in the case of free boundary conditions, the spectra can be understood in terms of irreducible representations of a single Virasoro algebra. For the Ising and the three-state Potts models, the corresponding irreducible representations are identified.
The spectrum of the quantum Ising chain is studied in the finite-size scaling limit for periodic and antiperiodic boundary conditions. The finite-size corrections are computed for all energy gaps. The results of the model are used to test the Privman-Fisher universality hypothesis and conformal theory. At the critical point, it is shown how to build the spectrum from the irreducible representations of the Virasoro algebra. In addition, the author finds evidence for an unsuspected symmetry for systems both non-critical and of finite size.
The central charge of the Virasoro algebra of the one-dimensional dimerised XY model with arbitrary boundary conditions is obtained at criticality. The authors obtain c=1-12 tau 2 where tau (0<or= tau <or=1/2) characterises the boundary conditions. This shift of c yields relations between the XY model and both the q-state Potts model and the O(N) model. They show that the phenomenological renormalisation group equation, which is used to estimate the critical value of the parameter of the system, has no real solution for the range tau >1/26. This corresponds to the non-unitary Potts model (0<or=q<2) and the O(N) one (-2<or=N<1).
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.