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Journal of Physics A Mathematical and General

The one-dimensional dimerised spin-1/2 XY chain with arbitrary boundary conditions and related unitary and non-unitary models

G. Spronken, M. Avignon

Published Jan 7, 1989• 0 citations
Publisher

Abstract

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).

DOI
10.1088/0305-4470/22/1/020

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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).
An analytical real space renormalisation group transformation is obtained for the one-dimensional spin-1/2 antiferromagnetic dimerised XY model. Using the method of Sarker (1984) the recursion relations for the parameters are derived from the mapping between blocks of sizes L and L/b, for arbitrary size L and scale factor b. The results for the gap and for the critical exponent are shown to converge rapidly toward their exact values. The dynamical exponent has 1/L corrections (b approximately 1) instead of the usual 1/ln L ones (b=L). The differential renormalisation group equations are obtained explicitly. The authors show that the non-analyticity of the scaling function (for the density of energy) stems from a dangerous irrelevant variable, the antiferromagnetic coupling.
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.
Recent results concerning non-universality in the dynamical critical behavior of the one-dimensional Ising model, are generalised to several formulations of the Potts model, using arguments based on domain wall motion. By applying real-space renormalisation group methods, the authors also study the dynamical Ising model, with single flips, both for a period distribution of nearest-neighbor exchanges and for the random chain (with nonzero couplings). They show that the critical dynamical exponent is z=1+J M /J m where J M (J m ) is the greatest (smallest) of the exchanges.