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Journal of Physics C Solid State Physics

Non-linear critical slowing down for the time-dependent Ising model

Noam White

Published Apr 14, 1976• 7 citations
Publisher

Abstract

Reports the results of a calculation of a high-temperature expansion for the non-linear relaxation time for a square Ising model with zero applied field. The author has also attempted to estimate the corresponding critical exponent.
DOI
10.1088/0022-3719/9/7/003

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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.
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The difference between the critical exponents (${\ensuremath{\Delta}}^{(\mathrm{l})}$ and ${\ensuremath{\Delta}}^{(\mathrm{nl})}$) of the linear and nonlinear relaxation times of the order parameter (${\ensuremath{\tau}}^{(\mathrm{l})}$ and ${\ensuremath{\tau}}^{(\mathrm{nl})}$) is investigated in the two-dimensional one-spin-flip kinetic Ising model. We have calculated the high-temperature series for ${\ensuremath{\tau}}^{(\mathrm{nl})}$ up to ninth order and made use of the known series for ${\ensuremath{\tau}}^{(\mathrm{l})}$ up to twelfth order. The series are analyzed by the ratio and the Pad\'e-approximant methods. The correlation between the critical-point and critical-exponent estimates in the Pad\'e approximants allows an improvement in the determination of ${\ensuremath{\Delta}}^{(\mathrm{l})}$. The result, ${\ensuremath{\Delta}}^{(\mathrm{l})}=2.125\ifmmode\pm\else\textpm\fi{}0.01$, is higher than previous estimates of 2.0 \ifmmode\pm\else\textpm\fi{} 0.05. The estimate of ${\ensuremath{\Delta}}^{(\mathrm{nl})}$ is less precise but the result ${\ensuremath{\Delta}}^{(\mathrm{nl})}=1.95\ifmmode\pm\else\textpm\fi{}0.15$ leads to the conclusion that ${\ensuremath{\Delta}}^{(\mathrm{l})}\ensuremath{\ne}{\ensuremath{\Delta}}^{(\mathrm{nl})}$ and that the difference between the two is of order $\ensuremath{\beta}$ (critical index of the order parameter). This is in accord with the scaling prediction.