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International Journal of Modern Physics B

MICROCANONICAL MONTE CARLO STUDY OF FIRST-ORDER TRANSITION IN A 2D CLASSICAL XY-MODEL

Smita Ota, Snehadri Bihari Ota

Published Aug 10, 2007• 4 citations
Publisher

Abstract

The microcanonical ensemble given by Boltzmann is used in the computer Monte Carlo simulation of 2D classical XY-model with the modified nearest neighbour interaction potential suggested by Domany, Schick and Swendsen. A relatively simple method to identify first-order transition in computer simulations of a statistical system is described. The critical value of p2 in this XY-model is determined using this method; which is found to increase with system size obeying a power law.

DOI
10.1142/s0217979207037545

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The microcanonical ensemble given by Boltzmann is used in the computer Monte Carlo simulation of 2D classical XY-model with the modified nearest neighbour interaction potential suggested by Domany, Schick and Swendsen. A relatively simple method to identify first-order transition in computer simulations of a statistical system is described. The critical value of p2 in this XY-model is determined using this method; which is found to increase with system size obeying a power law.
We have investigated the first-order transition in the classical two-dimensional (2D) extended XY-spin model using Monte Carlo simulations. The simulations have been carried out on a system with 100 spins in the microcanonical ensemble, which represents a finite-isolated system. The energy as a function of temperature is found to exhibit a 'S'-shape at the first-order transition. We conclude that the observed phenomena at the first-order transition should be interpreted as the equilibrium response of a finite-isolated system.
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.
In this paper, we propose a new generalized XY model with nearest-neighbor interaction which cannot be studied analytically. By adopting both the canonical Monte Carlo simulations and the microcanonical Monte Carlo simulations, the first-order phase transitions in both canonical ensemble and microcanonical ensemble are observed numerically, and the accurate phase-transition temperatures of the model with different parameters are given. From the distinct phase diagrams in two different ensembles, we conclude that the model we propose here exists as an ensemble equivalence.
An $x\ensuremath{-}y$ model with nearest-neighbor interaction $V(\ensuremath{\theta})=2J{1\ensuremath{-}{[{cos}^{2}(\frac{\ensuremath{\theta}}{2})]}^{{p}^{2}}}$ is studied by Monte Carlo simulation. The model has a transition from a paramagnetic to an algebraic (massless) phase. An increase of the parameter ppp, which changes the shape of the potential, brings about a change in the nature of the transition from continuous to first order.
The authors have carried out microcanonical Monte Carlo (MC) simulations for the two-dimensional (2D) classical XY model on a square lattice having 900 spins with periodic boundary conditions. The temperature dependence of the energy, specific heat, mean-square magnetization and vortex density show good agreement with the existing canonical MC simulation results. They thus demonstrate ergodicity in the microcanonical MC simulations for the 2D classical XY model. The maximum of the temperature derivative of the vortex density is found to occur at the temperature corresponding to the maximum of the specific heat.