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Physical review. B, Solid state

Critical slowing down in the two-dimensional one-spin-flip Ising model

T. Schneider, E. Stoll

Published Aug 1, 1974• 11 citations
Publisher

Abstract

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 ${T}_{c}$. 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.

DOI
10.1103/physrevb.10.959

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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.

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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.
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.
Exact lower bounds for the exponent of critical slowing down of the time-dependent order-parameter correlation and autocorrelation functions have been derived by Abe and Hatano and by Halperin, respectively. Moreover, Suzuki proved an exact inequality among the exponents characterizing the critical slowing down of the order parameter and energy fluctuations. In the present work we extend these results to include lower bounds for the exponents associated with the critical energy fluctuations and an inequality among the exponents characterizing the critical slowing down of the time-dependent autocorrelation functions. This is achieved by extending Mori's generalized Langevin-equation approach to the case of two variables, namely the order-parameter fluctuations and the energy fluctuations. The general results based on this approach hold for all Hermitian kinetic Ising models. Exact results are found revealing the failure of the extended conventional theory of slowing down. It is shown that this theory does not even hold in the mean-field approximation, which is expected to be exact at dimensionality d=4d=4d=4.
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.
We review the current state of knowledge of dynamic critical phenomena in simple magnetic models and present high statistics Monte Carlo data for critical slowing down in Ising and Potts models in two and three dimensions. By fitting the relaxation functions to multi-exponential decays at the critical temperature and applying finite size scaling we are able to extract very accurate estimates for the dynamical critical exponent z. Our results are compared with previous numerical estimates and with the predictions of renormalization group theory.
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 ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathit{l})}$=${\mathrm{\ensuremath{\Delta}}}_{\mathit{M}}^{(\mathit{l})}$.
Slow dynamics in the ± J Ising model on a square lattice is studied by means of Monte Carlo simulation, in which the distribution function of (logarithms of) relaxation times, P ( x ) with x =ln τ , is obtained by analyzing individual spin auto-correlation functions one by one. At each temperature in the \lqGriffiths phase', namely in the range above the spin-glass transition temperature T c (=0) but below the Curie temperature of the corresponding pure ferromagnet Tpurec, P ( x ) exhibits a characteristic crossover behavior at around x c (=ln τ c ) which diverges toward T c . The contributions to the branch x ≤ x c are interpreted due to the ordinary critical slowing down process associated with T c (critical dynamics). The behavior of P ( x ) at x ≤ x c , on the other hand, is quite consistent with the picture of relaxation processes of large, rare clusters related to the expected Griffiths singularity at Tpurec (cluster dynamics).