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Journal of Physics A Mathematical and General

A new approach to the calculation of the central charge

Gunter Markus Schutz

Published Apr 21, 1992• 1 citation
Publisher

Abstract

The author shows that the central charge c of the Virasoro algebra is determined by the spectrum of the Hamiltonian L 0 +NL 0 corresponding to a partition function which is invariant under the subgroup of modular transformations generated by S. Using this result he discusses in detail a new possibility of determining c for a given system at criticality which turns out to give excellent estimates even if the lattices accessible to numerical calculation are very small. This enables him to predict the central charge of some spin systems. Furthermore his approach to the determination of c leads to new universal functions interpolating between criticality and off-criticality.

DOI
10.1088/0305-4470/25/8/029

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The author shows that the central charge c of the Virasoro algebra is determined by the spectrum of the Hamiltonian L 0 +NL 0 corresponding to a partition function which is invariant under the subgroup of modular transformations generated by S. Using this result he discusses in detail a new possibility of determining c for a given system at criticality which turns out to give excellent estimates even if the lattices accessible to numerical calculation are very small. This enables him to predict the central charge of some spin systems. Furthermore his approach to the determination of c leads to new universal functions interpolating between criticality and off-criticality.
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.
We present a simple method to estimate the central charge of the conformal field theory corresponding to a critical point of a two-dimensional lattice model from Monte Carlo simulations. The main idea is to use the Wang-Landau flat-histogram algorithm, which allows us to obtain the free energy of a lattice model on a torus as a function of torus radii. The central charge is calculated with good precision from a free-energy scaling at the critical point. We apply the method to the Ising, tricritical Ising (Blume-Capel), Potts, and site-diluted Ising models, and we also discuss an estimation of the conformal weights.
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).