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International Journal of Modern Physics C

Cluster Study in the Three-Dimensional 3-State Potts Model

M. Aydin, T. Çelik, Y. Gündüç

Published Oct 1, 1997• 2 citations
Publisher

Abstract

The three-dimensional 3-state Potts model has been investigated by examining the average cluster size distributions. Cluster distributions give evidence for the weak first-order nature of the transition for lattice sizes as small as 163, where the energy histogram method fails to exhibit the true nature of the transition.
DOI
10.1142/s0129183197000941

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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.
We provide a convincing numerical demonstration of the first-order nature of the transition in the three-state ferromagnetic Potts model in three dimensions by combining Monte Carlo simulations of the bulk free-energy barrier between the ordered and disordered states with finite-size scaling. From the histogram of internal energy and the number of clusters we extrapolate in q, the number of states, and find qc{\mathit{q}}_{\mathit{c}}qc​=2.45\ifmmode\pm\else\textpm\fi{}0.1. We also find that the free-energy barrier \ensuremath{\Delta}F(q,L)\ensuremath{\sim}(q-qc{\mathit{q}}_{\mathit{c}}qc​)2{)}^{2})2L2.3{\mathit{L}}^{2.3}L2.3 and conclude that there is no essential singularity in the correlation length or latent heat in contrast to two dimensions.