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Journal of the Physical Society of Japan

Critical Slowing Down in the Kinetic Ising Model

Hideo Yahata, Masuo Suzuki

Published Dec 1, 1969• 143 citations
Publisher

Abstract

A simple dynamical model of interacting Ising spins is discussed near the critical point. It is shown that all the moments are positive and never vanish even at the critical point. Lower and upper bounds of the susceptibility are obtained, which yield the non-vanishing of the real part of the susceptibility. This gives a simple proof for the poly-dispersive nature of our system. The method of high temperature expansions for the dynamical susceptibility is presented and is applied to the two-dimensional Ising model up to the ninth order, the results of which show that the critical index of slowing down (the relaxation time τ (1) ∝( T - T c ) - Δ ; Δ =2±0.05) is different from that of the static susceptibility.

DOI
10.1143/jpsj.27.1421

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A simple dynamical model of interacting Ising spins is discussed near the critical point. It is shown that all the moments are positive and never vanish even at the critical point. Lower and upper bounds of the susceptibility are obtained, which yield the non-vanishing of the real part of the susceptibility. This gives a simple proof for the poly-dispersive nature of our system. The method of high temperature expansions for the dynamical susceptibility is presented and is applied to the two-dimensional Ising model up to the ninth order, the results of which show that the critical index of slowing down (the relaxation time τ (1) ∝( T - T c ) - Δ ; Δ =2±0.05) is different from that of the static susceptibility.
We study dynamic properties of the critical phenomena of the two-dimensional kinetic Ising model above and below the critical temperature Tc. In particular, we study the critical slowing-down (T > Tc) and the symmetry breaking (T ≪ Tc) through the temperature- and size-dependences of relaxation time of magnetization and we compare the dependences with those of static quantities such as the susceptibility. We find intrinsic dynamical critical behaviour on both sides of Tc with a critical region as wide as the one in the static critical behaviour. We also discuss the dynamic properties as an eigenvalue problem of the time-evolution operator (the Liouville operator) and thier scaling properties.
Phenomenological arguments on the critical slowing down is presented and “similarity law” is proposed on the indices of the critical slowing down. The similarity law is confirmed in linear spin chains near the critical field and in the kinetic Ising model near the critical temperature. It is exactly shown in the linear spin chains that the critical index of slowing down is different from that of the static susceptibility and that the dynamical susceptibility has a logarithmic singularity with respect to the frequency at the critical field and at zero temperature.
A markoffian stochastic model of interacting Ising spins is discussed near its critical point. The long time behavior of relaxation of its magnetization and energy is investigated near T c by the use of high-temperature series expansion and the ratio method. The numerical estimate of the critical exponent Δ M M of slowing down of magnetization is made, the results of which show Δ M M ≃2 for the simple square and triangular lattices, and Δ M M ≃1.4 for the simple cubic lattice. The exponent Δ E E of slowing down of energy is also estimated, which gives the result Δ E E ≃2 for the simple square and triangular lattices. The method to derive high-temperature expansion of an n -spin correlation function <σ 1 σ 2 …σ n > for an Ising model of spin 1/2 by the use of an equation of detailed balance is presented and applied to these estimates.
Based on the phenomenological approach, loci of relaxation time and magnetic dispersion maxima near the critical regime in a spin-1/2 mean-field Ising model were performed. The shift in temperature (T) of relaxation time (τ) maximum was detected and its behavior near the second-order transition points are presented at different magnetic field values (h) and different lattice coordination numbers (q). An expression for the dynamic (or complex) susceptibility (χ = χ1 - iχ2) is also derived. The temperature dependence of the magnetic dispersion (χ1) and magnetic absorption (χ2) factors have been studied near the critical regime. It is found that the maximum of χ1 as a function of frequency (ω) and kinetic coefficient (L) obeying an approximately exponential increases and decreases in T- ω and T-L planes near the critical region.
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.