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Physical review. B, Condensed matter

Monte Carlo mean-field method for spin systems

E. F. Henriques, Vera B. Henriques, Silvio R. A. Salinas

Published Apr 1, 1995• 6 citations
Publisher

Abstract

We use a Monte Carlo mean-field method proposed by Netz and Berker to analyze the critical behavior of an Ising square lattice. We show that this method demands longer sampling times as compared with the conventional Monte Carlo simulations. Also, similar mean-field results can be obtained from self-consistent analytic calculations for small clusters of spins.

DOI
10.1103/physrevb.51.8621

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We use a Monte Carlo mean-field method proposed by Netz and Berker to analyze the critical behavior of an Ising square lattice. We show that this method demands longer sampling times as compared with the conventional Monte Carlo simulations. Also, similar mean-field results can be obtained from self-consistent analytic calculations for small clusters of spins.
Results are presented for the computer simulation of the time-dependent behavior of the Ising model on a 100 \ifmmode\times\else\texttimes\fi{} 100 lattice. Groups of particles are regarded as samples from an infinite lattice and the interactions between the sample particles and those outside are expressed through new boundary conditions based on extending the concept of mean field in a self-consistent way. These new boundary conditions are explored analytically for a few small samples and a sharp critical point is found in contrast to periodic boundary conditions. The relaxation times for long-range order and nearest-neighbor correlation (energy) for the 100 \ifmmode\times\else\texttimes\fi{} 100 system have been computed and the corresponding critical exponents estimated. The results are ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}=2.30\ifmmode\pm\else\textpm\fi{}0.30$ and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}}=0.38\ifmmode\pm\else\textpm\fi{}0.04$; these are compared with other recent results from computer simulations and series calculations.
We present a self consistent method based on cluster algorithms and Renormalization Group on the lattice to study critical systems numerically. We illustrate it by means of the 2D Ising model. We compute the critical exponents ννν and ηηη and the renormalization group flow of the probability density function of the magnetization. The results, compared to the standard Monte Carlo Renormalization Group proposed by Swendsen [1], are very accurate and the method works faster by a factor which grows monotonically with the lattice size. This allows to simulate larger lattices in reachable computational times.
Results are presented for the critical temperatures Tn{T}_{n}Tn​ of small $n\ifmmode\times\else\texttimes\fi{}n$ Ising clusters using one of the extended mean-field boundary conditions recently introduced by Bolton and Johnson. The spin flips were simulated numerically. The values of Tn{T}_{n}Tn​ were obtained by extrapolating to zero the long-range order below Tn{T}_{n}Tn​. The extrapolation to the infinite limit $n=\ensuremath{\infty}$ was performed using an expansion in powers of 1n\frac{1}{n}n1​ and the final result agrees to 2% with the Onsager value.
The dynamics of Ising spins on a two-dimensional fractal lattice has been studied by Monte Carlo simulations. Both the limits $L>>\ensuremath{\xi}$ and $L<<\ensuremath{\xi}$ were studied where LLL is the size of the sample and $\ensuremath{\xi}$ the correlation length. Exact transfer matrix calculations were also carried out to check that the correct equilibrium values were reached. In both limits the predicted breakdown of conventional dynamic scaling was obtained. Using finite-size scaling $\ensuremath{\tau}\ensuremath{\sim}{L}^{z}$ for $L<<\ensuremath{\xi}$, an effective temperature-dependent critical exponent zzz was obtained consistent with the recent prediction $z\ensuremath{\propto}\frac{1}{T}$ for small TTT.
The Monte Carlo method provides accurate values for the thermal and magnetic properties of magnetic systems of localized moments. The basic ‘importance sampling’ technique is described and a critique of the method is presented. The relative accuracy of the method is demonstrated by direct comparison with both exact results for the Ising square lattice and with other approximate theories. The wide applicability of the Monte Carlo approach is shown through the presentation of results of general thermodynamic properties as well as both critical and tricritical phenomena in systems with diverse Hamiltonians. In addition, the method has been applied to systems which do not undergo cooperative transitions and to some complex models of several real magnetic systems. Recent work involving investigations of the dynamic behavior of the Ising model is also discussed.