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arXiv:cond-mat/0610647 · Physical Review Letters

Wang-Landau Multibondic Cluster Simulations for Second-Order Phase Transitions

Bernd A. Berg, Wolfhard Janke E

Published Jan 22, 2007• 14 citations
arXivHTMLPDF

Abstract

For a second-order phase transition the critical energy range of interest is larger than the energy range covered by a canonical Monte Carlo simulation at the critical temperature. Such an extended energy range can be covered by performing a Wang-Landau recursion for the spectral density followed by a multicanonical simulation with fixed weights. But in the conventional approach one loses the advantage due to cluster algorithms. A cluster version of the Wang-Landau recursion together with a subsequent multibondic simulation improves for 2D and 3D Ising models the efficiency of the conventional Wang-Landau or multicanonical approach by power laws in the lattice size. In our simulations real gains in CPU time reach 2 orders of magnitude.

DOI
10.1103/physrevlett.98.040602

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For a second-order phase transition the critical energy range of interest is larger than the energy range covered by a canonical Monte Carlo simulation at the critical temperature. Such an extended energy range can be covered by performing a Wang-Landau recursion for the spectral density followed by a multicanonical simulation with fixed weights. But in the conventional approach one loses the advantage due to cluster algorithms. A cluster version of the Wang-Landau recursion together with a subsequent multibondic simulation improves for 2D and 3D Ising models the efficiency of the conventional Wang-Landau or multicanonical approach by power laws in the lattice size. In our simulations real gains in CPU time reach 2 orders of magnitude.
At second-order phase transitions the critical energy range covered by a canonical Monte Carlo simulation close to the critical temperature is often smaller than the energy range needed for reliable reweighting analyses of certain observables. Such an extended energy range can be covered by performing a Wang–Landau recursion for the spectral density followed by a multicanonical simulation with fixed weights. But in the conventional approach based on local update rules one loses the advantage due to non-local cluster algorithms which are well known to drastically reduce critical slowing down. We develop a cluster version of the Wang–Landau recursion together with a subsequent multibondic simulation and show for 2D and 3D Ising models that the efficiency of the conventional Wang–Landau multicanonical approach can be improved by power laws in the lattice size. In our simulations real gains in CPU time reach two orders of magnitude.
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature/energy range around the critical point.By combining the replica-exchange algorithm with cluster updates and an adaptive routine to find the range of interest, we introduce a new flexible and powerful method for systematic investigations of critical phenomena.As a result, we gain two further orders of magnitude in the performance for 2D and 3D Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.
In finite-size scaling analyses of Monte Carlo simulations of second-order phase transitions one often needs an extended temperature range around the critical point. By combining the parallel-tempering algorithm with cluster updates and an adaptive routine to find the temperature window of interest, we introduce a flexible and powerful method for systematic investigations of critical phenomena. As a result, we gain one to two orders of magnitude in the performance for two- and three-dimensional Ising models in comparison with the recently proposed Wang-Landau recursion for cluster algorithms based on the multibondic algorithm, which is already a great improvement over the standard multicanonical variant.
Using the ``multihit'' Swendsen-Wang cluster flipping method, we performed extensive Monte Carlo simulations to investigate the critical behavior of the two-dimensional (2D) eight-state random-bond Potts model. We applied finite-size-scaling techniques to extract the critical exponents for two different sets of bond strengths, from which we concluded that the transition is second order with critical exponents for both sets falling into same universality class, that of a 2D Ising model. A variation of the Lee-Kosterlitz method for determining the order of a phase transition was also applied. The double-peaked structure of the specific heat, which was found in some of the bond configurations, was also studied by simulation on periodic arrangements of strong and weak bonds.
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.
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 ).