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arXiv:hep-lat/9601018

Study of Cluster Distributions in Two-dimensional q-State Potts Model

Meral Aydın, Y. Gündüç, Tarık Çelik

Published Jan 26, 1996• 0 citations
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Abstract

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.

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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.
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.
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.
Abstract The thermodynamics of the q‐state Potts model is calculated by the cluster approximation. With two, three, and four particles as the basic cluster the free energy is obtained for any q. It is found that the transition is of the first order for all two‐dimensional and three‐dimensional Potts models. In the case of planar lattices for q = 3 and q = 4 it is found that the critical point approaches the Curie‐Weiss temperature but the discontinuity of the order parameter remains almost equal to that obtained by the mean‐field approximation.