arXiv:cond-mat/0205592 · Journal of Physics A Mathematical and General

Cumulants of the three-state Potts model and of nonequilibrium models with C3v symmetry

T$acirc$nia Tom$eacute$, Alberto Petri

Published Jun 21, 2002• 24 citations

Abstract

The critical behaviour of two-dimensional stochastic lattice gas models with C 3 v symmetry is analysed. We study the cumulants of the order parameter for the three-state (equilibrium) Potts model and for two irreversible models whose dynamic rules are invariant under the symmetry operations of the point group C 3 v . By means of extensive numerical analysis of the phase transition we show that irreversibility does not affect the critical behaviour of the systems. In particular, we find that the Binder reduced fourth-order cumulant takes a universal value U * which is the same for the three-state Potts model and for the irreversible models. The same universal behaviour is observed for the reduced third-order cumulant.

DOI: 10.1088/0305-4470/35/26/302

Related papers

Universal non-equilibrium scaling of cumulants across a critical point

Leon J. Sieke, Mattis Harhoff, Sören Schlichting +1 more

We study the critical dynamics of a scalar field theory with Z 2 symmetry in the dynamic universality class of Model A in two and three spatial dimensions with classical-statistical lattice simulations. In particular, we measure the non-equilibrium behavior of the system under a quench protocol in which the symmetry-breaking external field is changed at a constant rate through the critical point. Using the well-established Kibble-Zurek scaling theory we compute non-equilibrium scaling functions of cumulants of the order parameter up to fourth order. Together with the static critical exponents and the dynamic critical exponent, these fully describe the universal non-equilibrium evolution of the system near the critical point. We further extend the analysis to include finite-size effects and observe good collapse of our data onto two-dimensional universal non-equilibrium and finite-size scaling functions.

arXiv:2411.10266Nuclear Physics BJan 23, 20252 citations

Random and Out-of-Equilibrium Potts models

Christophe Chatelain

A summary of my research activity is presented in this thesis. The first chapter starts with a short presentation of the known critical properties of the pure Potts model, mainly in two dimensions. The influence of disorder coupled to the energy density is then discussed in the general context of temperature-driven phase transitions. A physical explanation for the smoothing of the transition, due to Imry and Wortis, is presented and then applied to explain the phase diagram of the homogeneous random two and three-dimensional Potts models. It is then extended to the Potts model on irregular graphs and with layered randomness. In the third section, the critical properties at the random fixed point are examined in the light of the series expansion obtained by Renormalisation Group. The multifractal spectrum of scaling dimensions is discussed in the two-dimensional case and a new symmetry is proposed. The predictions of conformal invariance for profiles and correlation functions are compared with numerical data for the two-dimensional random Potts model in strip or square geometries. Finally, the geometrical properties of interfaces induced by symmetry-breaking boundary conditions are analysed in the context of the recent Stochastic Schramm-Loewner Evolution theory. The second chapter is devoted to the aging properties of homogeneous or frustrated Potts models submitted to a quench below or at their critical temperature. The scaling theory of two-time functions, as well as the more recent Local Scale Invariance theory , are presented in the first section. The violation of the Fluctuation-Dissipation theorem is described and the interpretation of the so-called Fluctuation-Dissipation Ratio as an out-of-equilibrium effective temperature is discussed. These ideas are applied to different spin models. First, one-dimensional systems with the Ising symmetry are considered. The equivalence of the KDH model with a gas of immobile particles subjected to pair-annihilation is exploited to obtain new analytical results. The question of the universality of dynamical exponents and ratios is then addressed in the two-dimensional case. Several lattices and several models in different universality classes are compared. A new model with the same symmetry as the three-state Potts model but with an irreversible dynamics is introduced. Its phase diagram in the steady state is determined and its aging properties on the critical line are studied. Finally, the aging of the Fully-Frustrated Ising and XY models are examined. Emphasis is put on the presence of logarithmic corrections due to the existence of topological defects. Two applications of the Jarzynski relation to the Potts model are presented at the end of the chapter. A short third chapter introduces the numerical simulations employed to analyse different set of experimental data. Monte Carlo simulations were used to reproduced some magnetic phase transitions in layered nanostructures. Spectroscopic data of surface reconstructions were compared to the electronic band structure obtained by computing the first eigenvectors of the Bloch-Schrödinger equation with a projective technique combined with a multiscale approach.

Dec 17, 20120 citations

Computer simulation of phase transitions in low-dimensional Potts models

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev, М. А. Магомедов

The Monte Carlo method is used to study phase transitions in the two-dimensional Potts model with the number of states spin q=3 on square and hexagonal lattices. Considered systems with linear dimensions Lx L=N, L=21/102. The obtained numerical data indicate that in the considered In the Potts model, a second-order phase transition is observed in accordance with the analytical theory. The Binder cumulants method of the fourth order determines the values of the critical points of the Potts model on different lattices. Keywords: Phase transitions, Potts model, Monte Carlo, thermodynamic parameters, critical temperature.

Physics of the Solid StateJan 1, 20220 citations

Critical behaviour of a two-dimensional bonded lattice model

A. P. Young, David A. Lavis

The Bell-Lavis model (1970) of a two-dimensional bonded lattice fluid is investigated by real space renormalisation group methods. All the fixed points in a restricted parameter subspace are obtained, but their stability is analysed in the full parameter space of the system. Within the subspace melting is predicted to occur via a second-order transition in the same universality class as the 3-state ferromagnetic Potts model. This contrasts with the predictions of mean-field theory. There is however a weakly relevant eigenvalue whose corresponding eigenvector is in a direction out of the subspace, so the fixed point may not be a critical point for the full model. A second result of interest is that the fixed point, found by Schick and Griffiths (1977) for the 3-state antiferromagnetic Potts model on a triangular lattice, also describes the transition in the more general case of an antiferromagnetic spin-one Ising model (the Blume-Emery-Griffiths model) on the same lattice. Some interesting symmetry properties of the subspace are also discussed.

Journal of Physics A Mathematical and GeneralFeb 1, 197925 citations

Universal Gaussian behavior of driven lattice gases at short times

Valerio Volpati, Urna Basu, Sergio Caracciolo A +1 more

The dynamic and static critical behaviors of driven and equilibrium lattice gas models are studied in two spatial dimensions. We show that in the short-time regime immediately following a critical quench, the dynamics of the transverse anisotropic order parameter, its autocorrelation, and Binder cumulant are consistent with the prediction of a Gaussian, i.e., noninteracting, effective theory, both for the nonequilibrium lattice gases and, to some extent, their equilibrium counterpart. Such a superuniversal behavior is observed only at short times after a critical quench, while the various models display their distinct behaviors in the stationary states, described by the corresponding, known universality classes.

arXiv:1708.07035Physical review. ENov 27, 20176 citations

Phase transitions in two-dimensional ferromagnetic Potts model with q = 3 on a triangular lattice

Akai Kurbanovich Murtazaev, Albert Babaevich Babaev, G. Ya. Ataeva

Phase transitions in the two-dimensional ferromagnetic Potts model with the number of spin states q = 3 on a triangular lattice were studied using Monte-Carlo method. The systems of linear size L = 20–120 were considered. Using the method of the fourth-order Binder cumulants, it was demonstrated that the second order phase transition is observed in the two-dimensional ferromagnetic Potts model. The static critical exponents for the heat capacity α, susceptibility γ, magnetization β and correlation radius ν were calculated based on the finite-dimensional scaling theory.

Low Temperature PhysicsFeb 1, 201310 citations