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Physical review. B, Solid state

Linear and nonlinear critical slowing down in the kinetic Ising model: High-temperature series

Zoltàn Ràcz, Malcolm F. Collins

Published Apr 1, 1976• 62 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.13.3074

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Physical review. B, Condensed matter•May 1, 1993•5 citations
The difference between the critical exponents (\ensuremathΔ(l){\ensuremath{\Delta}}^{(\mathrm{l})}\ensuremathΔ(l) and \ensuremathΔ(nl){\ensuremath{\Delta}}^{(\mathrm{nl})}\ensuremathΔ(nl)) of the linear and nonlinear relaxation times of the order parameter (\ensuremathτ(l){\ensuremath{\tau}}^{(\mathrm{l})}\ensuremathτ(l) and \ensuremathτ(nl){\ensuremath{\tau}}^{(\mathrm{nl})}\ensuremathτ(nl)) is investigated in the two-dimensional one-spin-flip kinetic Ising model. We have calculated the high-temperature series for \ensuremathτ(nl){\ensuremath{\tau}}^{(\mathrm{nl})}\ensuremathτ(nl) up to ninth order and made use of the known series for \ensuremathτ(l){\ensuremath{\tau}}^{(\mathrm{l})}\ensuremathτ(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 \ensuremathΔ(l){\ensuremath{\Delta}}^{(\mathrm{l})}\ensuremathΔ(l). The result, \ensuremathΔ(l)=2.125\ifmmode±\else\textpm\fi0.01{\ensuremath{\Delta}}^{(\mathrm{l})}=2.125\ifmmode\pm\else\textpm\fi{}0.01\ensuremathΔ(l)=2.125\ifmmode±\else\textpm\fi0.01, is higher than previous estimates of 2.0 \ifmmode\pm\else\textpm\fi{} 0.05. The estimate of \ensuremathΔ(nl){\ensuremath{\Delta}}^{(\mathrm{nl})}\ensuremathΔ(nl) is less precise but the result \ensuremathΔ(nl)=1.95\ifmmode±\else\textpm\fi0.15{\ensuremath{\Delta}}^{(\mathrm{nl})}=1.95\ifmmode\pm\else\textpm\fi{}0.15\ensuremathΔ(nl)=1.95\ifmmode±\else\textpm\fi0.15 leads to the conclusion that \ensuremathΔ(l)\ensuremath≠\ensuremathΔ(nl){\ensuremath{\Delta}}^{(\mathrm{l})}\ensuremath{\ne}{\ensuremath{\Delta}}^{(\mathrm{nl})}\ensuremathΔ(l)\ensuremath=\ensuremathΔ(nl) and that the difference between the two is of order \ensuremathβ\ensuremath{\beta}\ensuremathβ (critical index of the order parameter). This is in accord with the scaling prediction.
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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We examine the critical-point singularity of the nonlinear relaxation of purely dissipative systems. Applying the molecular-field approximation to the time-dependent Ginzburg-Landau model, it is shown that the critical exponent of the nonlinear relaxation time (${\ensuremath{\Delta}}^{(\mathrm{nl})}$) can be different from that of the linear one (${\ensuremath{\Delta}}^{(l)}$) even in ergodic systems. From the assumptions that scaling applies and that only the near-equilibrium fluctuations significantly affect the relaxation time, a scaling law ${\ensuremath{\Delta}}^{(\mathrm{nl})}={\ensuremath{\Delta}}^{(l)}\ensuremath{-}\ensuremath{\beta}$ is derived ($\ensuremath{\beta}$ is the critical exponent of the order). On the base of this scaling law we reanalyze the available Monte Carlo calculations, high-temperature series, and the critical-slowing-down experiment on Ni3{\mathrm{Ni}}_{3}Ni3​Mn.
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 ${\mathrm{\ensuremath{\Delta}}}_{\mathit{E}}^{(\mathit{l})}$=${\mathrm{\ensuremath{\Delta}}}_{\mathit{M}}^{(\mathit{l})}$.