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Journal of Applied Physics

Monte Carlo studies of magnetic critical phenomena using histogram techniques (invited)

Alan M. Ferrenberg, D. P. Landau, Pawel Peczak

Published Apr 15, 1991• 5 citations
Publisher

Abstract

Recent developments in the use of histograms (measured discrete probability distributions) now make it possible to determine critical properties with high accuracy using Monte Carlo (MC) simulations combined with finite-size scaling analyses. The advantages and limitations of the use of histograms for extracting the maximum information from MC simulation data are described. Results from extensive MC studies of the critical properties of two important magnetic systems, the d=3 ferromagnetic Ising and Heisenberg models, are presented. It is shown that the careful use of histogram techniques can provide results comparable or superior to these obtained with other numerical methods while maintaining all the advantages of standard MC techniques.

DOI
10.1063/1.348790

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Recent developments in the use of histograms (measured discrete probability distributions) now make it possible to determine critical properties with high accuracy using Monte Carlo (MC) simulations combined with finite-size scaling analyses. The advantages and limitations of the use of histograms for extracting the maximum information from MC simulation data are described. Results from extensive MC studies of the critical properties of two important magnetic systems, the d=3 ferromagnetic Ising and Heisenberg models, are presented. It is shown that the careful use of histogram techniques can provide results comparable or superior to these obtained with other numerical methods while maintaining all the advantages of standard MC techniques.
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The static critical properties of classical, isotropic Heisenberg ferromagnets have been studied numerically. Using an improved cluster algorithm we performed extensive Monte Carlo simulations on L×L×L simple cubic and body-centered cubic systems with L≤40 over a temperature range covering the critical region. Temperature dependences of thermodynamic quantities were determined with an optimized multiple-histogram method, and both critical temperatures and critical exponents were extracted from finite-size scaling of several different thermodynamic quantities. It was found that within their respective error bars the critical exponents for both ferromagnets agree with each other and with field theoretical predictions.
Using extensive Monte Carlo simulations, we study the equilibrium properties of the simple-cubic, classical Heisenberg ferromagnet. We employ very long runs for L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices to obtain high-precision data for the magnetization probability distribution. Using finite-size scaling for L\ensuremath{\le}24 and an optimized multiple-histogram data analysis, we obtain an accurate value of the inverse critical temperature J/kB{\mathit{k}}_{\mathit{B}}kB​Tc{\mathit{T}}_{\mathit{c}}Tc​=0.6929\ifmmode\pm\else\textpm\fi{}0.0001, which is higher than previously accepted estimates. Calculated values of various static exponents are in excellent agreement with renormalization-group and \ensuremath{\epsilon}-expansion predictions.
The probability distribution of the order parameter is analyzed in order to obtain the criticality of magnetic systems. Monte Carlo simulations have been employed by using single spin flip Metropolis algorithm aided by finite-size scaling and histogram reweighting techniques. A method is proposed to obtain this probability distribution even when the transition temperature of the model is unknown. A test is performed on the two-dimensional spin-1/2 and spin-1 Ising model and the results show that the present procedure can be quite efficient and accurate to describe the criticality of the system.
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.