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Bulletin of the American Physical Society

Accelerated histogram-free multicanonical Monte Carlo algorithm for the basis expansion of density of states

Ying Wai Li, Alfred C.K. Farris, Markus Eisenbach

Published Mar 5, 2018• 0 citations
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This talk will review the history of both cluster and histogram methods for Monte Carlo simulations. Cluster methods are based on the famous exact mapping by Fortuin and Kasteleyn from general Potts models onto a percolation representation. I will discuss the Swendsen‐Wang algorithm, as well as its improvement and extension to more general spin models by Wolff. The Replica Monte Carlo method further extended cluster simulations to deal with frustrated systems. The history of histograms is quite extensive, and can only be summarized briefly in this talk. It goes back at least to work by Salsburg et al. in 1959. Since then, it has been forgotten and rediscovered several times. The modern use of the method has exploited its ability to efficiently determine the location and height of peaks in various quantities, which is of prime importance in the analysis of critical phenomena. The extensions of this approach to the multiple histogram method and multicanonical ensembles have allowed information to be obtained over a broad range of parameters. Histogram simulations and analyses have become standard techniques in Monte Carlo simulations.
This paper extends to two and three dimensions the recently proposed renormalized multicanonical sampling procedure [D. Yevick, Int. J. Mod. Phys. C 27, 1650033 (2016)IJMPEO0129-183110.1142/S0129183116500339], which generates the density of states of a system from that of a smaller system. The accuracy and efficiency of the method is additionally enhanced through a transition matrix Monte Carlo method that incorporates multicanonical sampling. The performance of the resulting procedure is investigated through calculations of the specific heat of the standard Ising model for square lattices with up to 128^{2} spins and for cubic lattices with up to 24^{3} spins. A comparison of the scaling properties to those of other algorithms based on the transition matrix Monte Carlo method demonstrates that the relative advantage of the renormalized procedure increases rapidly with system size.
Based on the subgraph expansions for phase transition models, we present a histogram Monte Carlo method to calculate the free energy and other physical quantities for such models as continuous functions of the system parameter. The method does not have any critical slowing down. We then use physical quantities thus obtained in a percolation renormalization group method to calculate critical point, critical exponents, and thermodynamic order parameter. Our method gives quite accurate results and may be applied to many phase transition models.