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arXiv:cond-mat/9712198 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Nonequilibrium relaxation of the two-dimensional Ising model: Series-expansion and Monte Carlo studies

Jian‐Sheng Wang, Chee Kwan Gan

Published Jun 1, 1998• 20 citations
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Abstract

We study the critical relaxation of the two-dimensional Ising model from a fully ordered configuration by series expansion in time $t$ and by Monte Carlo simulation. Both the magnetization $(m)$ and energy series are obtained up to 12th order. An accurate estimate from series analysis for the dynamical critical exponent $z$ 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 $m$. Extrapolation to $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}$ leads to an estimate $z=2.169\ifmmode\pm\else\textpm\fi{}0.003$.

DOI
10.1103/physreve.57.6548

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Physical review. B, Condensed matter•Apr 1, 1988•32 citations
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, zeff(t)=\ensuremath−18dlnt/dlnm{z}_{\mathrm{eff}}(t)=\ensuremath{-}\frac{1}{8}d\mathrm{ln}t/d\mathrm{ln}mzeff​(t)=\ensuremath−81​dlnt/dlnm, directly from a ratio of three-spin correlation to mmm. Extrapolation to t\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}t\ensuremath→​\ensuremath∞ leads to an estimate z=2.169\ifmmode±\else\textpm\fi0.003z=2.169\ifmmode\pm\else\textpm\fi{}0.003z=2.169\ifmmode±\else\textpm\fi0.003.
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 ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathit{l})}$=${\mathrm{\ensuremath{\Delta}}}_{\mathit{M}}^{(\mathit{l})}$.
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.
We derive twenty nontrivial terms of the high-temperature series expansion for the linear relaxation time τ of the time-displaced correlation function C(t) = ⟨ m (0) m(t) ⟩ of the magnetization m(t) in the two-dimensional nearest-neighbour ferromagnetic Ising model on the square lattice. We study the dynamics introduced by Glauber and compute the longest (characteristic) relaxation time of C(t) . We analyse the series by using unbiased and biased methods, such as the ratio method, Padé approximants and generalized differential approximants. It is reassuring that all the methods yield compatible results providing the estimate for the dynamical critical exponent: z = 2.183 ± 0.005.
From the non-equilibrium critical relaxation study of the two-dimensional Ising model, the dynamical critical exponent z is estimated to be 2.165 ± 0.010 for this model. The relaxation in the ordered phase of this model is consistent with exp (−√t/τ) behavior. The interface energy of the three-dimensional Ising model is studied and the critical exponent of the correlation length ν and the critical amplitude of the surface tension σ0 are estimated to be 0.6250 ± 0.0025 and 1.42 ± 0.04, respectively. A dynamic Monte Carlo renormalization group method is successfully applied to the equilibrium properties of the three-dimensional Ising model.
For the two-dimensional nearest-neighbor kinetic Ising model without conservation laws, we derive on the square lattice the high-temperature series expansion of the autorelaxation time, to the eleventh order. With use of ratio methods and Pad\'e approximants, we obtain for the dynamical critical exponent ${\mathrm{\ensuremath{\Delta}}}_{\mathit{A}}$ the value 2.09\ifmmode\pm\else\textpm\fi{}0.03, which implies the value 2.34\ifmmode\pm\else\textpm\fi{}0.03 for the linear relaxation exponent z. This value is about 10% larger than several other recent estimates. This discrepancy is possibly due to the relatively small number of nonzero coefficients in the series expansion.
The relaxation of the two-dimensional Ising model above Tc{T}_{c}Tc​ for lattices with size up to 10 240\ifmmode\times\else\texttimes\fi{}10 240 is studied by the Monte Carlo simulation in vectorized super-spin-coding on the HITAC S-810/20 supercomputer. The present estimation of the dynamic critical exponent z gives z=2.076\ifmmode\pm\else\textpm\fi{}0.005, which is equal to the critical exponent ${\ensuremath{\Delta}}^{(l)}$ of the linear relaxation time in a two-dimensional system. Also, the critical exponent, ${\ensuremath{\Delta}}^{(\mathrm{nl})}$ of the nonlinear relaxation time is obtained: ${\ensuremath{\Delta}}^{(\mathrm{nl})=1.932\ifmmode\pm\else\textpm\fi{}0.018}$. These results support the Racz scaling law ${\ensuremath{\Delta}}^{(l)}$=${\ensuremath{\Delta}}^{(\mathrm{nl})+\mathrm{\ensuremath{\beta}}}$ (\ensuremath{\beta} is the critical exponent of the order parameter).