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Physical Review Letters

Nonuniversal critical dynamics in Monte Carlo simulations

Robert H. Swendsen, Jian‐Sheng Wang

Published Jan 12, 1987• 2,848 citations
Publisher

Abstract

A new approach to Monte Carlo simulations is presented, giving a highly efficient method of simulation for large systems near criticality. The algorithm violates dynamic universality at second-order phase transitions, producing unusually small values of the dynamical critical exponent.
DOI
10.1103/physrevlett.58.86

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In lattice field theories the partition function is often a very high dimensional integral which can’t be written in a closed form. In order to study the behaviour of the system one needs an efficient algorithm to approximate the high dimensional integral. In this thesis two Markov chain Monte Carlo algorithms the Metropolis algorithm and the Worm algorithm are used to simulate the three dimensional xy-model near its critical point. As the critical point is approached from below in terms of the inverse coupling the correlation length of the system diverges which may lead to the critical slowing down. An efficient algorithm is such that it suffers from the critical slowing down as little as possible which corresponds to a small dynamical critical exponent value. The efficiency of the algorithms is compared by measuring the integrated autocorrelation time of energy for different lattice sizes. The dynamical critical exponent, which controls the computational cost of the simulation, is calculated as a linear least squares fit to the integrated autocorrelation time data. From the simulated data the measured value of the dynamical critical exponent for the Metropolis algorithm is 1.85 and for the Worm algorithm the value is 0.37.