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TURKISH JOURNAL OF PHYSICS

A Numerical Study of the 2D Potts Model via the Cluster Algorithm

Meral Aydın, Tarık Çelik, Y. Gündüç

Published Jan 1, 1996• 1 citation
Publisher

Abstract

The two-dimensional q-state Potts model is studied with and without external field using a cluster algorithm. Equilibrium averages, fluctuations in the average cluster size and the histograms for energy, order parameter and the average cluster size are obtained as functions of the temperature. Studying the dynamical evolution of clusters through their size distributions and fluctuations provide helpful information on the order of phase transitions.

DOI
10.55730/1300-0101.2653

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The two-dimensional q-state Potts model is studied with and without external field using a cluster algorithm. Equilibrium averages, fluctuations in the average cluster size and the histograms for energy, order parameter and the average cluster size are obtained as functions of the temperature. Studying the dynamical evolution of clusters through their size distributions and fluctuations provide helpful information on the order of phase transitions.
Abstract The two-dimensional Potts Model with 2 to 10 states is studied using a cluster algorithm to calculate fluctuations in cluster size, distributions of cluster sizes as well as commonly used quantities like equilibrium averages and the histograms for energy and the order parameter. Results provide information about the variation of cluster sizes depending on the temperature and the number of states. They also give evidence for first-order transition when energy and the order parameter related measurables are inconclusive on small size lattices.
It is known that finite size scaling theory is well established for the systems undergoing a strong first-order transition [1-3]. However, if the correlation length is larger than the system size, methods using finite size scaling theory may fail to give indication of the first-order transition. The aim in this work is to observe how the system approaches to the first-order regime (where the correlation length is smaller but comparable to the size of the system) in two-dimensional 7-state Potts model which is known to exhibit first-order transition [4,5]. Cluster algorithm [6,7] is used to calculate commonly used equilibrium averages and the energy histogram, as well as cluster size related measurables like the average cluster size, fluctuations in the cluster size and the distribution of the cluster sizes [8,9]. The behavior of these measurables on different sizes varying from 8 \times 8 to 64 \times 64 is observed and the effect of this behavior on the order of the transition is studied.
Dynamical evaluation of the observables changing per iteration in a Monte Carlo simulation possess vital information about the order of the phase transition occuring in the system ~[1]. Being a global operator, average cluster size is a good candidate to study the order of the phase transition. It is expected that study of cluster dynamics yields indicative information ~[1-3]. In this work cluster algorithm ~[4,5] is used to study the dynamical behavior of the average cluster size in 333-dimensional 3-state Potts model ~[6] which is known to exhibit weak first-order transition ~[7]. Cluster size is calculated as the average of 1000 or higher replicas at each Monte Carlo iteration, during first few hundred iterations. Results for different sizes show the first-order character of the phase transition occuring in the model without performing time consuming computations to calculate the equilibrium averages.
The two-dimensional qqq-state Potts model is studied through the variations in cluster size depending on temperature and the number of states qqq, using cluster algorithm. It is observed that cluster related measurables give evidence for the order of the transition at smaller lattice sizes than energy and during early iterations of the Monte Carlo simulation.
The q-state Potts model on a square lattice and in the presence of an external magnetic field that couples to only one state is studied by using extensive Monte Carlo simulations. A hybrid algorithm that combines the single spin-flip Metropolis method with Wolff cluster updates is employed. Single-histogram reweighting techniques and the field-mixing approach are also used to obtain the corresponding universal probability distributions. By analyzing the finite-size scaling behavior of the order parameters and universal probability distributions, the phase diagram in the temperature-versus-external field plane is obtained in the thermodynamic limit for some values of state q≥5. For positive fields and higher values of q, there is a clear first-order transition line that starts at the zero-field transition point and ends at an isolated critical point, which is in the two-dimensional Ising universality class. As the value of state q decreases, the corresponding critical field also decreases. The present results strongly suggest that the critical field vanishes when q→5^{+}. However, for negative fields, another first-order transition line occurs without any critical or multicritical behavior for q≥6. The special case with q=5 undergoes a weak first-order transition at zero field and a line of second-order transitions for negative fields. This second-order transition line is in the same universality class as the q=4 Potts model.