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Physical review. B, Condensed matter

Critical dynamics of the two-dimensional kinetic Ising model: High-temperature series analysis of the autorelaxation time

Jos Rogiers, Joseph O. Indekeu

Published Apr 1, 1990• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.41.6998

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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 \ensuremathΔA{\mathrm{\ensuremath{\Delta}}}_{\mathit{A}}\ensuremathΔA​ 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.
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.
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.
In this work we study the Glauber dynamics of the one-dimensional Ising model with nearest-neighbor and next-nearest-neighbor interactions, for which an approximate solution of the magnetization per site is obtained. When the dynamical critical exponent z is investigated following the treatment of Cordery, Sarker, and Tobochnik [Phys. Rev. B 24, 5402 (1981)], our observation shows that its upper-bound value is the same as the known value, thus implying that z is independent of the range of the interaction. We also suggest a high-temperature expansion approximation which is then used to solve the two-dimensional Glauber dynamics governed by a master equation; this solution is compared with that of the decoupling method. The time-delayed correlation function is also calculated.
First, a new method proposed by Baker and Rushbrooke is used to study the simple ferromagnetic Ising model at and below the Curie temperature. Of course, the properties of the Ising model are already well known, so that the main aim here is to assess the potential and reliability of the new method, since it has wide applicability to other models which have not been otherwise studied. Between 8 and 16 coefficients of exact high-temperature expansions for fixed values of the magnetization are derived for various two- and three-dimensional lattices. A Pad\'e-approximant analysis of these expansions at the critical isotherm and magnetic phase boundary enables us to estimate the critical exponents $\ensuremath{\beta}$, ${\ensuremath{\gamma}}^{\ensuremath{'}}$, and $\ensuremath{\delta}$, and plot the spontaneous magnetization. The results are in good agreement with previous calculations. Secondly, an analysis of the exact series expansions provides no support for the conjecture that the phase boundary is a line of essential singularities. However, the same expansions strongly suggest the existence of a "spinodal" curve, whose properties are in reasonable agreement with the predictions of various heuristic arguments (based essentially upon analyticity at the phase boundary and one-phase homogeneity in the critical region). Finally, structure and a mild extension of the proven analyticity of the free energy are used to show the $\ensuremath{\Delta}\ensuremath{\le}{\ensuremath{\Delta}}^{\ensuremath{'}}$, $\ensuremath{\gamma}\ensuremath{\le}{\ensuremath{\gamma}}^{\ensuremath{'}}$.