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Physical review. B, Condensed matter

Testing weak first-order transitions in the two-dimensional Z5 -symmetric spin model

K. Rouidi, Y. Leroyer

Published Jan 1, 1992• 5 citations
Publisher

Abstract

We have performed a Monte Carlo simulation of the ${\mathit{Z}}_{5}$ 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.

DOI
10.1103/physrevb.45.1013

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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.
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 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.
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.
In this paper we study the phase diagram of the five-state Potts antiferromagnet on the bisected-hexagonal lattice. This question is important since Delfino and Tartaglia recently showed that a second-order transition in a five-state Potts antiferromagnet is allowed, and the bisected-hexagonal lattice had emerged as a candidate for such a transition on numerical grounds. By using high-precision Monte Carlo simulations and two complementary analysis methods, we conclude that there is a finite-temperature first-order transition point. This one separates a paramagnetic high-temperature phase, and a low-temperature phase where five phases coexist. This phase transition is very weak in the sense that its latent heat (per edge) is two orders of magnitude smaller than that of other well-known weak first-order phase transitions.
We have used the Monte Carlo renormalization group method to study the order-disorder phase transition of the three and four dimensional, three-state Potts model. In both cases, the transition appears to be first order. However, no evidence for a discontinuity fixed point associated with these transitions was found. Instead, the existence of a nearby second order fixed point just beyond the first order transition could be inferred from the RG flow in the ordered as well as in the disordered state. These results imply that the thermodynamic quantities are analytically continuable into the metastable regions, and that apart from discontinuities caused by the intersection of the two free-energy branches, no additional singularities occur at the transition.