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Progress of Theoretical Physics

On the Singularity of Dynamical Response and Critical Slowing Down

Masuo Suzuki

Published Apr 1, 1970• 45 citations
Publisher

Abstract

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.

DOI
10.1143/ptp.43.882

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

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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.

Physical review. B, Solid state•Apr 1, 1976•62 citations
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 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.
Phenomenological arguments are presented to clarify the singularity of static nonlinear response near the critical magnetic (or electric) field Hc. The singularity of nonlinear susceptibility χ(H) is connected with the expansion of the free energy Φ(m) as a function of a quantity (mc - m), where m is the normalized magnetization and mc its value at the critical field; i.e. if Φ(m) = a0 - a1(mc - m) + a2(mc - m)ψ…, then χ(H) ∼1/(Hc - H)γH; γH = (ψ- 2)/(ψ- 1). That is, for 1 ≪ ψ≪ 2, χ(H) is continuous, for ψ= 2, it is finite but discontinuous at H = Hc, and for ψ> 2, it is divergent. The index γH(or γE) is investigated in the one-dimensional exactly soluble magnetic systems at T = 0 and the two-dimensional ferroelectric (KDP) models at finite temperature. The value of γH is zero for a fermion system with repulsive delta-function interaction. γH is equal to 1/2 for the antiferromagnetic Heisenberg model. In the Hubbard model, γH is equal to 1/2 in the case of half-filled band, and γH is always zero other-wise. In the general two-dimensional ferroelectric models, there exists no critical electric field, though the Wu model has a critical field Ec with γE = 1/2.
A semiphenomenological cluster theory is developed for dynamic critical properties, which is not limited to small deviations from equilibrium. Explicit numerical expressions are derived for linear and nonlinear response functions of the kinetic Ising model, which are compatible with dynamic scaling and recent calculations of dynamic critical exponents. An uniaxial magnet (or Ising system) and its critical fluctuations are approximated by a "droplet" or "cluster" model, the critical part of the free energy being the sum of the contributions of clusters each containing lll spins. These clusters are assumed to grow and shrink with a phenomenological rate $\ensuremath{\propto}{l}^{r}$, $r\ensuremath{\approx}1$. These rates are compatible with direct Monte Carlo simulations in two and three dimensions. The resulting cluster reaction and diffusion equation is an approximation to the master equation of the Glauber kinetic Ising model (in which the magnetization is not conserved), and has the same structure as used in nucleation theories. In linear response to space- and time-dependent fields the relaxation times of energy and magnetization are expressed as triple integrals. All relaxation times diverge as ${|1\ensuremath{-}\frac{T}{{T}_{c}}|}^{\ensuremath{-}(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}}$. This treatment is consistent with dynamic scaling and universality, and is more general than the conventional (Van Hove) theory of critical slowing down. Also the (smaller) exponents found for the response to localized variations ("autocorrelation functions") agree with dynamic scaling. The wave-vector dependence of the relaxation times arises from the static correlation function only, and not from the cluster diffusion term. If the equilibrium cluster distribution is assumed to be that of the Fisher droplet model, and for special values of rrr, the (discrete) eigenvalue spectrum and eigenfunctions (generalized Laguerre polynomials) of the cluster reaction equation are found explicitly. Relaxation functions and frequency-dependent susceptibilities are expressed by hypergeometric series, even in the case of nonlinear response. At the critical point, the spectrum becomes continuous and the exponential decay for large times is replaced by power-law behavior. Similarly, it is found that dynamic scaling applies also to the nucleation rate for this model, and the scaling exponent of the nucleation rate is $2\ensuremath{-}\ensuremath{\alpha}+(2\ensuremath{-}r)\ensuremath{\beta}\ensuremath{\delta}$.
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.
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.