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Physical review. B, Solid state

Monte Carlo studies on small Ising systems

H. C. Bolton, David W. R. Gruen

Published May 1, 1977• 5 citations
Publisher

Abstract

Results are presented for the critical temperatures ${T}_{n}$ 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 ${T}_{n}$ were obtained by extrapolating to zero the long-range order below ${T}_{n}$. The extrapolation to the infinite limit $n=\ensuremath{\infty}$ was performed using an expansion in powers of $\frac{1}{n}$ and the final result agrees to 2% with the Onsager value.

DOI
10.1103/physrevb.15.4544

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Extending the Monte Carlo method to dynamic critical phenomena we investigated the time-dependent correlation functions in the two-dimensional one-spin-flip Ising model and the critical behavior of the associated relaxation times. These relaxation times are the following: ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}$, characterizing the approach of the order parameter to equilibrium after a change of temperature $\ensuremath{\Delta}T$ of the system; ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}$ characterizing the slowing down of the order-parameter correlation and autocorrelation functions, respectively; ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}$, characterizing the slowing down of the energy correlation and autocorrelation functions; and finally ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}$, characterizing the cross-correlation function. We give estimates for the associated exponents ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}1.90\ifmmode\pm\else\textpm\fi{}0.10$, and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}\ensuremath{\approx}1.60\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0.95\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0$, which are consistent with the dynamic scaling hypothesis and with exact inequalities. A detailed comparison with recent high-temperature-expansion estimates is performed, and the reliability of the Monte Carlo results is carefully analyzed.

Physical review. B, Solid state•Oct 1, 1973•181 citations
Results are presented for the critical temperatures Tn{T}_{n}Tn​ of small n\ifmmode×\else\texttimes\finn\ifmmode\times\else\texttimes\fi{}nn\ifmmode×\else\texttimes\fin 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∞n=\ensuremath{\infty}n=\ensuremath∞ was performed using an expansion in powers of 1n\frac{1}{n}n1​ and the final result agrees to 2% with the Onsager value.
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.
Two Ising systems are investigated with the two extended-mean-field boundary conditions recently considered by Bolton and Johnson. The systems are NNN spins on a line with boundary spins on either side and at each end. The exact solutions are obtained by the use of the modified-transfer-matrix method. A general formula for the critical temperature is found for each system. The critical temperature is found to take an asymptotic form similar to that obtained by Ferdinand and Fisher for the two-dimensional planar infinite Ising lattice. The thermodynamic limits of the results are discussed.
Finite Ising $N\ifmmode\times\else\texttimes\fi{}N$ ($N=1\ensuremath{-}5$) and $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ ($N=1\ensuremath{-}8$) systems with one of the three self-consistent extended mean-field boundary conditions all around and on both sides, respectively, are studied. The boundary conditions used are: (aaa) extended mean-field boundary condition using average, abbreviated EA (in which each boundary spin takes some average value); (bbb) extended mean-field boundary condition using probability, abbreviated EP (in which each boundary spin takes the values \ifmmode\pm\else\textpm\fi{}1 with certain probabilities) and (ccc) extended Bethe-Peierls, EBP (in which each boundary spin is assumed to be acted on by some effective field). The self-consistent equations are then derived by letting the average magnetization of the internal spins of the system be equal to that of the boundary spins. The finite systems considered therefore represent some high-order systematic generalization of the familiar molecular-field and Bethe-Peierls approximations. The equations for the critical temperature are formulated graphically in terms of some high-temperature expansion series which are calculated to the ninth and tenth orders. Two of the series in the equations of $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ for $N\ensuremath{\rightarrow}\ensuremath{\infty}$ are directly related to the layer and local susceptibilities considered by Binder and Hohenberg for the Ising infinite half plane. Critical temperatures obtained numerically with the use of truncation and extrapolation techniques. Results on the systems with EA and EP indicate that the shifts of their critical temperatures can be described by a power law ${N}^{\ensuremath{-}\ensuremath{\lambda}}$ where $\ensuremath{\lambda}=\frac{1}{{\ensuremath{\gamma}}_{1}}$ and ${\ensuremath{\gamma}}_{1}(\ensuremath{\approx}\frac{11}{8})$ is the critical exponent of the layer susceptibility. This is different from the value $\ensuremath{\lambda}=1$ which is believed true for two-dimensional finite systems with periodic or free boundary conditions. No conclusive remark can be made to systems with EBP through they suggest some even higher value (\ensuremath{\ge} 2).
In this paper, we theoretically study the critical properties of the classical spin-1 Ising model using two approaches: 1) the analytical low-temperature series expansion and 2) the numerical Metropolis Monte Carlo technique. Within this analysis, we discuss the critical behavior of one-, two- and three-dimensional systems modeled by the first-neighbor spin-1 Ising model for different types of exchange interactions. The comparison of the results obtained according the Metropolis Monte Carlo simulations allows us to highlight the limits of the widely used mean-field theory approach. We also show, via a simple transformation, that for the special case where the bilinear and bicubic terms are set equal to zero in the Hamiltonian the partition function of the spin-1 Ising model can be reduced to that of the spin-1/2 Ising model with temperature dependent external field and temperature independent exchange interaction times an exponential factor depending on the other terms of the Hamiltonian and confirm this result numerically by using the Metropolis Monte Carlo simulation. Finally, we investigate the dependence of the critical temperature on the strength of long-range interactions included in the Ising Hamiltonian comparing it with that of the first-neighbor spin-1/2 Ising model.
Using a cluster-flipping Monte Carlo algorithm combined with a generalization of the histogram reweighting scheme of Ferrenberg and Swendsen [Phys. Rev. Lett. 61, 2635 (1988); 63, 1195 (1989)], we have studied the equilibrium properties of the thermal random-field Ising model on a cubic lattice in three dimensions. We have equilibrated systems of L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L spins, with values of L up to 32, and for these systems the cluster-flipping method appears to a large extent to overcome the slow equilibration seen in single-spin-flip methods. From the results of our simulations we have extracted values for the critical exponents and the critical temperature and randomness of the model by finite size scaling. For the exponents we find \ensuremath{\nu}=1.02\ifmmode\pm\else\textpm\fi{}0.06, \ensuremath{\beta}=0.06\ifmmode\pm\else\textpm\fi{}0.07, \ensuremath{\gamma}=1.9\ifmmode\pm\else\textpm\fi{}0.2, and \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=2.9\ifmmode\pm\else\textpm\fi{}0.2, where \ensuremath{\nu}, \ensuremath{\beta}, \ensuremath{\gamma}, and \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{} govern the critical singularities in the correlation length, magnetization, and connected and disconnected susceptibilities, respectively. \textcopyright{} 1996 The American Physical Society.
Extending the Monte Carlo method to dynamic critical phenomena we investigated the time-dependent correlation functions in the two-dimensional one-spin-flip Ising model and the critical behavior of the associated relaxation times. These relaxation times are the following: ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}$, characterizing the approach of the order parameter to equilibrium after a change of temperature $\ensuremath{\Delta}T$ of the system; ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}$ characterizing the slowing down of the order-parameter correlation and autocorrelation functions, respectively; ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}$ and ${\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}$, characterizing the slowing down of the energy correlation and autocorrelation functions; and finally ${\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}$, characterizing the cross-correlation function. We give estimates for the associated exponents ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}\ensuremath{\approx}1.90\ifmmode\pm\else\textpm\fi{}0.10$, and ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}\ensuremath{\approx}1.60\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0.95\ifmmode\pm\else\textpm\fi{}0.10$, ${\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0$, which are consistent with the dynamic scaling hypothesis and with exact inequalities. A detailed comparison with recent high-temperature-expansion estimates is performed, and the reliability of the Monte Carlo results is carefully analyzed.