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Physical review. B, Solid state

Monte Carlo Investigation of Dynamic Critical Phenomena in the Two-Dimensional Kinetic Ising Model

E. Stoll, Kurt Binder, T. Schneider

Published Oct 1, 1973• 181 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.8.3266

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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: \ensuremathτ\ensuremathδ\ensuremathμ\ensuremathΔT{\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}}^{\ensuremath{\Delta}T}\ensuremathτ\ensuremathδ\ensuremathμ\ensuremathΔT​, characterizing the approach of the order parameter to equilibrium after a change of temperature \ensuremathΔT\ensuremath{\Delta}T\ensuremathΔT of the system; \ensuremathτ\ensuremathδ\ensuremathμ\ensuremathδ\ensuremathμ{\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}\ensuremathτ\ensuremathδ\ensuremathμ\ensuremathδ\ensuremathμ​ and \ensuremathτ\ensuremathδ\ensuremathμ\ensuremathδ\ensuremathμA{\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}\ensuremathτ\ensuremathδ\ensuremathμ\ensuremathδ\ensuremathμA​ characterizing the slowing down of the order-parameter correlation and autocorrelation functions, respectively; \ensuremathτ\ensuremathδH\ensuremathδH{\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}\ensuremathτ\ensuremathδH\ensuremathδH​ and \ensuremathτ\ensuremathδH\ensuremathδHA{\ensuremath{\tau}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremathτ\ensuremathδH\ensuremathδHA​, characterizing the slowing down of the energy correlation and autocorrelation functions; and finally \ensuremathτ\ensuremathδ\ensuremathμ\ensuremathδH{\ensuremath{\tau}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}\ensuremathτ\ensuremathδ\ensuremathμ\ensuremathδH​, characterizing the cross-correlation function. We give estimates for the associated exponents \ensuremathΔ\ensuremathδ\ensuremathμ\ensuremathΔT\ensuremath≈\ensuremathΔ\ensuremathδ\ensuremathμ\ensuremathδ\ensuremathμ\ensuremath≈\ensuremathΔ\ensuremathδH\ensuremathδH\ensuremath≈\ensuremathΔ\ensuremathδ\ensuremathμ\ensuremathδH\ensuremath≈1.90\ifmmode±\else\textpm\fi0.10{\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\ensuremathΔ\ensuremathδ\ensuremathμ\ensuremathΔT​\ensuremath≈\ensuremathΔ\ensuremathδ\ensuremathμ\ensuremathδ\ensuremathμ​\ensuremath≈\ensuremathΔ\ensuremathδH\ensuremathδH​\ensuremath≈\ensuremathΔ\ensuremathδ\ensuremathμ\ensuremathδH​\ensuremath≈1.90\ifmmode±\else\textpm\fi0.10, and \ensuremathΔ\ensuremathδ\ensuremathμ\ensuremathδ\ensuremathμA\ensuremath≈1.60\ifmmode±\else\textpm\fi0.10{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\ensuremath{\mu}}^{A}\ensuremath{\approx}1.60\ifmmode\pm\else\textpm\fi{}0.10\ensuremathΔ\ensuremathδ\ensuremathμ\ensuremathδ\ensuremathμA​\ensuremath≈1.60\ifmmode±\else\textpm\fi0.10, \ensuremathΔ\ensuremathδ\ensuremathμ\ensuremathδHA\ensuremath≈0.95\ifmmode±\else\textpm\fi0.10{\ensuremath{\Delta}}_{\ensuremath{\delta}\ensuremath{\mu}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0.95\ifmmode\pm\else\textpm\fi{}0.10\ensuremathΔ\ensuremathδ\ensuremathμ\ensuremathδHA​\ensuremath≈0.95\ifmmode±\else\textpm\fi0.10, \ensuremathΔ\ensuremathδH\ensuremathδHA\ensuremath≈0{\ensuremath{\Delta}}_{\ensuremath{\delta}\mathcal{H}\ensuremath{\delta}\mathcal{H}}^{A}\ensuremath{\approx}0\ensuremathΔ\ensuremathδH\ensuremathδHA​\ensuremath≈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.
We extend a previous Monte Carlo investigation to include the relaxation times associated with the first time moment of the relaxation function and of the relaxation function squared. Moreover, we introduce, in addition to the order-parameter and energy-relaxation functions, that of the order parameter squared. The asymptotic temperature dependence of the associated relaxation times is found to support the validity of rather extended dynamic scaling hypotheses, predicting that all the associated exponents of slowing down are equal and symmetric with respect to Tc{T}_{c}Tc​. We also derive various exact lower bounds and inequalities for these relaxation times and exponents. Our numerical data are found to be consistent with these exact results.
The critical dynamics of the two dimensional Kinetic Ising Model have been studied using Monte Carlo Real Space Renormalization Group techniques. Time dependent correlation functions were calculated for square lattices of 16 x 16 and 32 x 32 spins, using periodic boundary conditions wth 60,000 and 15,000 Monte Carlo steps per spin, respectively;The critical exponent z was estimated using a matching process for a variety of renormalization group operators. The estimates for z were surprisingly sensitive to small changes in the value of the renormalization group operator, with physically reasonable operators giving estimates for z less than the rigorous lower bound for z of z(,o) = 1.75. The estimates for z from a given renormalization group operator showed poor convergence properties. Attempts to estimate z from the scaling form of the correlation function were inconclusive, as were attempts to estimate the static exponent (eta);Possible explanations of these unexpected results are discussed. The most plausible source of the complex behavior seen in this work is the discrete nature of time in the Monte Carlo computer simulation, which generates phenomena not present in standard theoretical treatments of the Kinetic Ising Model.
We study the critical relaxation of the two-dimensional Ising model from a fully ordered configuration by series expansion in time ttt and by Monte Carlo simulation. Both the magnetization (m)(m)(m) and energy series are obtained up to 12th order. An accurate estimate from series analysis for the dynamical critical exponent zzz is difficult but compatible with 2.2. We also use Monte Carlo simulation to determine an effective exponent, ${z}_{\mathrm{eff}}(t)=\ensuremath{-}\frac{1}{8}d\mathrm{ln}t/d\mathrm{ln}m$, directly from a ratio of three-spin correlation to mmm. Extrapolation to $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ leads to an estimate $z=2.169\ifmmode\pm\else\textpm\fi{}0.003$.
We report the results of a Monte Carlo investigation of the (equilibrium) time-displaced correlation functions for the magnetization and energy of a simple cubic Ising model as a function of time, temperature, and lattice size. The simulations were carried out on a CDC CYBER 205 supercomputer employing a high-speed, vectorized multispin coding program and using a total of 5\ifmmode\times\else\texttimes\fi{}1012{10}^{12}1012 Monte Carlo spin-flip trials. We used L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L lattices with periodic boundary conditions and L as large as 96. The short-time and long-time behaviors of the correlation functions are analyzed by fits to a sum of exponential decays, and the critical exponent z for the largest relaxation time is extracted using a finite-size-scaling analysis. Our estimate z=2.04\ifmmode\pm\else\textpm\fi{}0.03 resolves an intriguing contradiction in the literature; it satisfies the theoretical lower bound and is in agreement with the prediction obtained by \ensuremath{\epsilon} expansion. We also consider various small systematic errors that typically occur in the analysis of relaxation functions and show how they can lead to spurious results if sufficient care is not exercised.
The energy relaxation in the two-dimensional Ising model, with temperature greater than Tc{\mathit{T}}_{\mathit{c}}Tc​ and in a spin system having a 5120\ifmmode\times\else\texttimes\fi{}5120 lattice, is studied using supercomputer-conducted Monte Carlo simulation. The critical exponents of the linear and nonlinear relaxation times of the internal energy are respectively estimated to be ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathit{l})}$=2.064\ifmmode\pm\else\textpm\fi{}0.026 and ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathrm{nl})}$=1.132\ifmmode\pm\else\textpm\fi{}0.080, being consistent with the Racz scaling law ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathit{l})}$-${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathrm{nl})}$=1-\ensuremath{\alpha}, where \ensuremath{\alpha} is the critical exponent of the specific heat. Based on the author's previous result in which the critical exponent of the linear relaxation for magnetization was determined as ${\mathrm{\ensuremath{\Delta}}}_{\mathit{M}}^{(\mathit{l})}$=2.076\ifmmode\pm\else\textpm\fi{}0.005, it is strongly suggested that ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathit{l})}$=${\mathrm{\ensuremath{\Delta}}}_{\mathit{M}}^{(\mathit{l})}$.