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Wang-Landau simulation of two planar XY-models

Suman Sinha, Soumen Roy

Published Nov 7, 2007• 0 citations
OpenAlex

Abstract

We have performed Monte Carlo simulation of two planar XY-models using the Wang-Landau algorithm. One of these models is the standard XY-model, the other has a modified potential and exhibits a strongly first order phase transition. The first part of the work is an investigation of the growth of the energy histogram fluctuation in these continuous models. We find that these histograms saturate at values proportional to 1 / √ ( ln f), for both models, irrespective of the width of the energy bin used in the simulation. Next, using the number of Monte Carlo sweeps necessary to attain the saturation value of the histograms, we have obtained the density of states and hence some observables for the two models. PACS: 05.10.Ln, 05.70.Fh, 64.60.Cn 1

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We have performed Monte Carlo simulation of two planar XY-models using the Wang-Landau algorithm. One of these models is the standard XY-model, the other has a modified potential and exhibits a strongly first order phase transition. The first part of the work is an investigation of the growth of the energy histogram fluctuation in these continuous models. We find that these histograms saturate at values proportional to 1 / √ ( ln f), for both models, irrespective of the width of the energy bin used in the simulation. Next, using the number of Monte Carlo sweeps necessary to attain the saturation value of the histograms, we have obtained the density of states and hence some observables for the two models. PACS: 05.10.Ln, 05.70.Fh, 64.60.Cn 1
The effect of inclusion of higher-order interactions in the XY model on critical properties is studied by Monte Carlo simulations. It is found that an increasing number of the higher-order terms in the Hamiltonian modifies the shape of the potential, which beyond a certain value leads to the change of the nature of the transition from continuous to first order. The evidence for the first-order transition is provided in the form of the finite-size scaling and the energy histogram analysis. A rough phase diagram is presented as a function of the number of the higher-order interaction terms.
Using the Wang-Landau algorithm combined with the analytic method, the density of states of a two-dimensional XY model on square lattices of sizes 16x16, 24x24, and 32x32 is accurately calculated. Thermodynamic quantities, such as internal energy, free energy, entropy, and specific heat are obtained from the resulted density of states by numerical integration. It is found that both the density of states and the extensive thermodynamic quantities obey the scaling law. From the entropy curve, symptoms of phase transition are observed. A general method of calculation of the density of states of continuous models by simulation combined with analytical method is proposed.
Monte Carlo simulation has been performed in a two-dimensional modified XY -model first proposed by Domany [Phys. Rev. Lett. 52, 1535 (1984)] The cluster algorithm of Wolff has been used and multiple histogram reweighting is performed. The first-order scaling behavior of the quantities such as specific heat and free-energy barrier are found to be obeyed accurately. While the lowest-order correlation function was found to decay to zero at long distance just above the transition, the next-higher-order correlation function shows a nonzero plateau.
Phase transitions and critical phenomena are the most universal phenomena in nature.To understand the phase transitions and critical phenomena of a given system as a continuous function of temperature and to obtain the partition function zeros in the complex temperature plane indicating most effectively phase transitions and critical phenomena, we need to calculate the density of states.Currently, Wang-Landau Monte Carlo algorithm is one of the most efficient Monte Carlo methods to calculate the approximate density of states.Using Wang-Landau Monte Carlo algorithm, the density of states for the Ising model on L x L square lattices (L = 4 ~ 32) with periodic boundary conditions is obtained, and the partition function zeros of the Ising model are evaluated in the complex temperature plane.By examining the behavior of the first partition function zero (partition function zero closest to the positive real axis), phase transitions and critical phenomena can be much more accurately analyzed.The approximate first zeros of the Ising ferromagnet, obtained from Wang-Landau algorithm, are quite close to the exact ones, indicating that it is a reliable method for calculating the density of states and the first partition function zeros.
Monte Carlo simulations using the recently proposed Wang–Landau algorithm are performed to the q = 8 state Potts model in two dimension with various degrees of randomness. We systematically studied the effect of quenched bond randomness to system which has first-order phase transition. All simulations and measurements were done from pure case r = 1 to r = 0.4. Physical quantities such as energy density and ground-state entropy were evaluated at all temperatures. We have also obtained probability distributions of energy to monitor softening of transitions. It appears quite feasible to simulate spin systems with quenched bond randomness by Wang–Landau algorithm.
The concept of the partition function zeros is one of the most efficient methods for investigating the phase transitions and the critical phenomena in various physical systems. Estimating the partition function zeros requires information on the density of states Ω( E ) as a function of the energy E . Currently, the Wang-Landau Monte Carlo algorithm is one of the best methods for calculating Ω( E ). The partition function zeros in the complex temperature plane of the Ising model on an L × L square lattice ( L = 10 ~ 80) with a periodic boundary condition have been estimated by using the Wang-Landau Monte Carlo algorithm. The efficiency of the Wang-Landau Monte Carlo algorithm and the accuracies of the partition function zeros have been evaluated for three different, 5%, 10%, and 20%, flatness criteria for the histogram H ( E ).