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Journal of Physics A Mathematical and General

Finite-size scaling spectra in the six-states quantum chains

Gunter Markus Schutz

Published Mar 21, 1989• 4 citations
Publisher

Abstract

The finite-size scaling spectra of the six-states quantum chains with D 6 symmetry are studied numerically in the massless phase. At least three values of the central charge c of the Virasoro algebra are found: c=1, 1.25, and 1.3. On the self-dual line, one finds, in the c=1 region, N=2 superconformal invariance. In the c=1.25 region there is a line where N=1 superconformal and Zamolodchikov-Fateev symmetry (1985) is observed. The occurrence of higher symmetries is also discussed.

DOI
10.1088/0305-4470/22/6/023

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The finite-size scaling spectra of the six-states quantum chains with D 6 symmetry are studied numerically in the massless phase. At least three values of the central charge c of the Virasoro algebra are found: c=1, 1.25, and 1.3. On the self-dual line, one finds, in the c=1 region, N=2 superconformal invariance. In the c=1.25 region there is a line where N=1 superconformal and Zamolodchikov-Fateev symmetry (1985) is observed. The occurrence of higher symmetries is also discussed.
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.
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.
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.