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arXiv:1512.07804 · AIP conference proceedings

Probing phase transition order of q-state Potts models using Wang-Landau algorithm

Tasrief Surungan, Y. Komura, Yutaka Okabe

Published Jan 1, 2014• 10 citations
arXivHTMLPDF

Abstract

Phase transitions are ubiquitous phenomena, exemplified by the melting of ice and spontaneous magnetization of magnetic material. In general, a phase transition is associated with a symmetry breaking of a system; occurs due to the competition between coupling interaction and external fields such as thermal energy. If the phase transition occurs with no latent heat, the system experiences continuous transition, also known as second order phase transition. The ferromagnetic q-state Potts model with r extra invisible states, introduced by Tamura, Tanaka, and Kawashima [Prog. Theor. Phys. 124, 381 (2010)], is studied by using the Wang-Landau method. The density of states difference (DOSD), ln g(E + ΔE) − ln g(E), is used to investigate the order of the phase transition and examine the critical value of r changing the second to the first order transition.

DOI
10.1063/1.4897109

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Phase transitions are ubiquitous phenomena, exemplified by the melting of ice and spontaneous magnetization of magnetic material. In general, a phase transition is associated with a symmetry breaking of a system; occurs due to the competition between coupling interaction and external fields such as thermal energy. If the phase transition occurs with no latent heat, the system experiences continuous transition, also known as second order phase transition. The ferromagnetic q-state Potts model with r extra invisible states, introduced by Tamura, Tanaka, and Kawashima [Prog. Theor. Phys. 124, 381 (2010)], is studied by using the Wang-Landau method. The density of states difference (DOSD), ln g(E + ΔE) − ln g(E), is used to investigate the order of the phase transition and examine the critical value of r changing the second to the first order transition.
We study the two-dimensional (q+r)-state ferromagnetic Potts model with q interacting visible states and r noninteracting invisible states on a square lattice. We concentrate on the case q=2 for various r values. The density of states is obtained by the Wang-Landau Monte Carlo simulation method, by which the specific heat, the partition function zeros, and the probability distribution of the internal energy are calculated. There exists a second-order phase transition for r<r_{c}, while a first-order phase transition appears for r>r_{c}, where r_{c} is the critical value. From all analyses, we conclude that r_{c}≈29 for q=2. Calculating the density of visible spins using the Metropolis algorithm, we find that the first-order phase transition is due to the abrupt density change of the visible spins. The critical temperatures are measured to compare with a conjecture.
We use a two-dimensional Wang-Landau sampling algorithm to calculate the density of states for two discrete spin models and then extract their phase diagrams. The first system is an asymmetric Ising model on a triangular lattice with two- and three-body interactions in an external field. An accurate density of states allows us to locate the critical endpoint accurately in a two-dimensional parameter space. We observe a divergence of the spectator phase boundary and of the temperature derivative of the magnetization coexistence diameter at the critical endpoint in quantitative agreement with theoretical predictions. The second model is a Q-state Potts model in an external field H. We map the phase diagram of this model for Q > 8 and observe a first-order phase transition line that starts at the H = 0 phase transition point and ends at a critical point (Tc,Hc), which must be located in a two-dimensional parameter space. The critical field Hc(Q) is positive and increases with Q, in qualitative agreement with previous theoretical predictions.
Randomness is an important subject in the study of phase transition as defect and impurity may present in any real material. The pre-existing ordered phase of a pure system can be affected or even ruined by the presence of randomness. Here we study ferromagnetic Ising model on a square lattice with a presence of randomness in the form of bond dilution. The pure system of this model is known to experience second order phase transition, separating between the high temperature paramagnetic and low-temperature ferromagnetic phase. We used Wang-Landau algorithm of Monte Carlo method to obtain the density of states from which we extract the ensemble average of energy and the specific heat. We observed the signature of phase transition indicated by the diverging peak of the specific heat as system sizes increase. These peaks shift to the lower temperature side as the dilution increases. The lower temperature ordered phase preserves up to certain concentration of dilution and is totally ruined when the bonds no longer percolates.
The phase transitions in the two-dimensional ferro- and antiferromagnetic Potts models with q = 3 states of spin on a triangular lattice are studied using cluster algorithms and the classical Monte Carlo method. Systems with linear sizes L = 20–120 are considered. The method of fourth-order Binder cumulants and histogram analysis are used to discover that a second-order phase transition occurs in the ferromagnetic Potts model and a first-order phase transition takes place in the antiferromagnetic Potts model. The static critical indices of heat capacity (α), magnetic susceptibility (γ), magnetization (β), and correlation radius index (ν) are calculated for the ferromagnetic Potts model using the finite-size scaling theory.
The phase transition in the q -state Potts model with homogeneous ferromagnetic couplings is strongly first order for large q, while it is rounded in the presence of quenched disorder. Here we study this phenomenon on different two-dimensional lattices by using the fact that the partition function of the model is dominated by a single diagram of the high-temperature expansion, which is calculated by an efficient combinatorial optimization algorithm. For a given finite sample with discrete randomness the free energy is a piecewise linear function of the temperature, which is rounded after averaging, however, the discontinuity of the internal energy at the transition point (i.e., the latent heat) stays finite even in the thermodynamic limit. For a continuous disorder, instead, the latent heat vanishes. At the phase transition point the dominant diagram percolates and the total magnetic moment is related to the size of the percolating cluster. Its fractal dimension is found d(f) = ( 5 + square root of 5)/4 and it is independent of the type of the lattice and the form of disorder. We argue that the critical behavior is exclusively determined by disorder and the corresponding fixed point is the isotropic version of the so-called infinite randomness fixed point, which is realized in random quantum spin chains. From this mapping we conjecture the values of the critical exponents as beta=2- d(f), beta(s) =1/2, and nu=1.