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Journal of Physics A Mathematical and General

Hunting for the central charge of the Virasoro algebra: six- and eight-state spin models

G von Gehlen, V. Rittenberg

Published Aug 21, 1986• 16 citations
Publisher

Abstract

In two dimensions conformal invariance has important implications for the finite-size scaling properties of the spectra of transfer matrices and quantum chains at the critical point. Some relations between the finite-size scaling amplitudes are obtained which can be used as a test of conformal invariance and implicitly to distinguished between second-order and first-order phase transitions. The numerical values of the scaling amplitudes give the scale dimensions of various operators. Six- and eight-state self-dual quantum chains with cubic symmetries are considered at the critical point for three values of the coupling constant. The systems are found to be conformal invariant and estimates for several critical exponents are obtained. Based on the approximate values of the critical exponent one tries to find the values of the corresponding central charges of the Virasoro algebras.

DOI
10.1088/0305-4470/19/12/028

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In two dimensions conformal invariance has important implications for the finite-size scaling properties of the spectra of transfer matrices and quantum chains at the critical point. Some relations between the finite-size scaling amplitudes are obtained which can be used as a test of conformal invariance and implicitly to distinguished between second-order and first-order phase transitions. The numerical values of the scaling amplitudes give the scale dimensions of various operators. Six- and eight-state self-dual quantum chains with cubic symmetries are considered at the critical point for three values of the coupling constant. The systems are found to be conformal invariant and estimates for several critical exponents are obtained. Based on the approximate values of the critical exponent one tries to find the values of the corresponding central charges of the Virasoro 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.
The ground-state properties of a quantum spin chain with three-spin couplings are investigated using finite-size calculations. The results are compared with the predictions of conformal invariance for finite systems at the critical point. The critical point. The critical exponents eta E and eta S of energy density and spin correlation functions are calculated and estimated to be eta E approximately=1.30 and eta S approximately=0.25. The results suggest that this model is conformally invariant with a central charge very close to or equal to one.
This paper is concerned with the critical properties of a family of self-dual two-dimensional Z(N) models whose bulk free energy is exactly known at the self-dual point. The author's analysis is performed by studying the finite-size behaviour of the corresponding one-dimensional quantum Hamiltonians which also possess an exact solution at their self-dual point. By exploring finite-size scaling ideas and the conformal invariance of the critical infinite system the author calculates, for N up to 8, the critical temperature and critical exponents as well as the central charge associated with the underlying conformal algebra. The results strongly suggest that the Z(N) quantum field theory of Zamolodchikov and Fateev is the underlying field theory associated with these statistical mechanical systems. The author also tests for the Z(5) case, the conjecture that these models correspond to the bifurcation points in the phase diagram of the General Z(N) spin model, where a massless phase originates.