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Journal of Physics A Mathematical and General

The critical behaviour of self-dual Z(N) spin systems: finite-size scaling and conformal invariance

Francisco Castilho Alcaraz

Published Jun 21, 1987• 48 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/20/9/035

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arXiv:1409.3915•Physical Review E•Oct 1, 2014•16 citations
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.
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 are concerned with the critical properties of the anisotropic Heisenberg chain, or XXZ model, with arbitrary integer or half-integer spin. The eigenspectra of these Hamiltonians, with periodic boundaries, are calculated for finite chains by solving numerically their associated Bethe ansatz equations. The resulting spectra are found to be in accord with the predictions of conformal invariance and the operator content is identified, for lattices with an even and odd number of sites. The results for spin 1 and spin 3/2 indicate that the conformal anomaly c for these models, in the gapless regime, has the value c=3S/(1+S), independent of the anisotropy, and the exponents vary continuously with the anisotropy as in the eight-vertex model. The operator content of these models indicate that the underlying field theory governing these critical spin-S models is described by composite fields formed by the product of Gaussian and Z(N) fields, with N=2S. Finally some of the irrelevant operators which produce the leading finite-size corrections of the eigenenergies are identified.
We have investigated the dynamic critical behavior of the two-dimensional Z(5)-symmetric spin model by using short-time Monte Carlo (MC) simulations. We have obtained estimates of some critical points in its rich phase diagram and included, among the usual critical lines the study of first-order (weak) transition by looking into the order-disorder phase transition. In addition, we also investigated the soft-disorder phase transition by considering empiric methods. A study of the behavior of β/νz along the self-dual critical line has been performed and special attention has been devoted to the critical bifurcation point, or Fateev-Zamolodchikov (FZ) point. First, by using a refinement method and taking into account simulations out of equilibrium, we were able to localize parameters of this point. In a second part of our study, we turned our attention to the behavior of the model at the early stage of its time evolution in order to find the dynamic critical exponent z as well as the static critical exponents β and ν of the FZ point on square lattices. The values of the static critical exponents and parameters are in good agreement with the exact results, and the dynamic critical exponent z≈2.28 very close to the four-state Potts model (z≈2.29).