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Physical review. B, Condensed matter

Critical bifurcation point of theopenZ(5)-symmetric spin model

B. Bonnier

Published Jul 1, 1991• 4 citations
Publisher

Abstract

The critical behavior of the general isotropic, ferromagnetic two-dimensional spin system with openZ(5) symmetry is studied with use of high-temperature expansions of its mass gap. On the basis of these expansions we propose a simple analytic representation of the mass gap which naturally reproduces all the different phase transitions exhibited by this model (first order and second order of the Ising and of the Kosterlitz-Thouless types). In addition, the bifurcation point where the soft phases originate is clearly identified with the Fateev-Zamolodchikov value.

DOI
10.1103/physrevb.44.390

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Physical review. B, Condensed matter•Jul 1, 1989•21 citations
The critical behavior of the general isotropic, ferromagnetic two-dimensional spin system with openZ(5) symmetry is studied with use of high-temperature expansions of its mass gap. On the basis of these expansions we propose a simple analytic representation of the mass gap which naturally reproduces all the different phase transitions exhibited by this model (first order and second order of the Ising and of the Kosterlitz-Thouless types). In addition, the bifurcation point where the soft phases originate is clearly identified with the Fateev-Zamolodchikov value.
We study the isotropic, ferromagnetic two-dimensional spin systems with Z(5) symmetry, which, in the general case, depend upon two couplings, a temperature and a parameter \ensuremath{\Omega}, used to interpolate continuously between the clock (\ensuremath{\Omega}=0) and the Potts (\ensuremath{\Omega}=1) models, passing through the integrable Fateev-Zamolodchikov case (\ensuremath{\Omega}=${\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}$\ensuremath{\approxeq}0.618). Two methods, the high-temperature expansion and the exact \ensuremath{\beta} function on finite lattices, are applied within the Hamiltonian formulation to study critical properties. The transitions found are first order (${\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}$\ensuremath{\le}1), second order of the Ising type (\ensuremath{\Omega}\ensuremath{\simeq}${\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}$), or of the Kosterlitz-Thouless type (0\ensuremath{\le}\ensuremath{\Omega}${\mathrm{\ensuremath{\Omega}}}_{\mathit{c}}$) with a varying exponent.
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).
We have performed a Monte Carlo simulation of the Z5{\mathit{Z}}_{5}Z5​ model in the region of the phase diagram where the transition is expected to be first order. Analyzing our results by means of a method recently proposed by Lee and Kosterlitz, we confirm its sensitivity in discriminating between the weak first-order nature of the transition at the five-state Potts point and the second-order one near the Fateev-Zamolodchikov integrability point. In addition, we find the transition to be first order in the intermediate region, as already suggested by previous analysis.
The scaling-limit symmetry of Zq{Z}_{q}Zq​-invariant spin systems is studied by renormalization-group methods based on qualitative bifurcation theory rather than a series expansion in $\ensuremath{\epsilon}$. Under the plausible assumption of the absence of secondary bifurcations, it is shown that an ordinary critical phase with $q\ensuremath{\ne}2 or 4$ must have full SO2S{O}_{2}SO2​ invariance in the scaling limit; moreover, any such asymptotically isotropic critical phase is stable under Zq{Z}_{q}Zq​-invariant perturbations for q>4q>4q>4. Previously obtained results about two-dimensional Kosterlitz-Thouless phases for clock models have a natural interpretation within this group-theoretic bifurcation analysis.
To the lowest order approximation, the method of variational cumulant expansion has been successful in the determination of the critical point of a spin system on the Ising model. In the second order calculation, unphysical phase transition arises. In this letter, we first analyze the origin of unphysical phase transitions in high-order variational cumulant expansion calculations. We then explain the basis on which a conjecture is proposed that to any order m of the variational cumulant expansion, the critical point is given by the bifurcation point of the free energy. The theory is finally verified numerically for the two-dimensional case.
High-temperature series are extended through ${\ensuremath{\beta}}^{17}$ for the susceptibility and for the first and second moments of the correlation functions, in the case of the classical two-dimensional O(2) Heisenberg model (also called the XY model). For the ``true mass gap'' the high-temperature expansion is computed through ${\ensuremath{\beta}}^{12}$. The calculations are performed by a new computer program which solves iteratively the Schwinger-Dyson equations of the model. Ratio-extrapolation, Euler-transform and Pad\'e techniques are used to analyze the series and to discriminate between the recently reproposed conventional power-law form of the critical singularity and the form predicted by Kosterlitz and Thouless. The critical parameters are determined for the Kosterlitz-Thouless form, which is clearly favored by our data.