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Journal of Applied Physics

Hierarchical scaling: An analytical approach to slow relaxations in spin glasses, glasses, and other correlated systems (invited)

Jean Souletie

Published May 15, 1994• 11 citations
Publisher

Abstract

A model has been developed that associates a hierarchical criterion to the usual renormalization of the distances near a phase transition. At times t smaller than τα∝(1−Tc/T)−zν, which diverges at Tc, it is found that the relaxation develops at a scale s(t), which is a power law of the time elapsed: s ∝ tT/zTc. If one sits in the ergodic regime T≳Tc, the system abruptly reaches an equilibrium at s(τα)=ξ(T)=τα1/z. If T<Tc, by contrast, it deviates to logarithmic behavior. The α relaxation is well approximated by a stretched exponential exp(−t/τα)β(T), which traduces the effect of underlying lognormal statistics. The exponent β(T)∝[1−b ln(1−Tc/T)]−1/2 would cancel at Tc if it was possible to wait for the sample to equilibrate at this temperature. The model has the necessary ingredients to describe effects that have been reported in both the ‘‘activated’’ nonergodic regime and in the critical regime of spin glasses, and of other correlated systems like orientational glasses, glass-forming liquids, and vortex glasses. A possible extension of the model is discussed to describe the ‘‘macroscopic quantum effects.’’

DOI
10.1063/1.355673

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A model has been developed that associates a hierarchical criterion to the usual renormalization of the distances near a phase transition. At times t smaller than τα∝(1−Tc/T)−zν, which diverges at Tc, it is found that the relaxation develops at a scale s(t), which is a power law of the time elapsed: s ∝ tT/zTc. If one sits in the ergodic regime T≳Tc, the system abruptly reaches an equilibrium at s(τα)=ξ(T)=τα1/z. If T<Tc, by contrast, it deviates to logarithmic behavior. The α relaxation is well approximated by a stretched exponential exp(−t/τα)β(T), which traduces the effect of underlying lognormal statistics. The exponent β(T)∝[1−b ln(1−Tc/T)]−1/2 would cancel at Tc if it was possible to wait for the sample to equilibrate at this temperature. The model has the necessary ingredients to describe effects that have been reported in both the ‘‘activated’’ nonergodic regime and in the critical regime of spin glasses, and of other correlated systems like orientational glasses, glass-forming liquids, and vortex glasses. A possible extension of the model is discussed to describe the ‘‘macroscopic quantum effects.’’
Within the framework of a model which associates Kadanoff's renormalization with a hierarchical condition, we justify dynamic scaling (τ/τ ≈ (ξ/ξ0)z) and non-exponential relaxations (m(t) = t −1/zv at T = T C). In the presence of some disorder the same model predicts β → α transition from a short-time regime where we observe activated T In t scaling for short-range effects, to a regime where we have stretched exponential behaviour and dynamic scaling (τ ≈ τ0(1 – T C/T)−zv) for long-range effects.
Assuming that hierarchical procedure governs the evolution from one to the next step of Kadanoff's renormalization in the presence of some disorder one finds that the dynamic response is described by a log-normal distribution of the relaxation times. There are two natural limits to the model which restitute either the dynamic scaling or the activated dynamic hypotheses. This provides the basis for a simple justification to the success of the usual fits by a stretched exponential or by a Cole-Cole law which are commonly made to the non-exponential responses observed in the time or in the frequency domains in spin glasses. We provide expressions for the temperature dependences of the parameters which control these laws. We also propose a model for freezing which is based on the idea that ∂\partial∂ ln τ/∂(1/T)\tau/\partial(1/T)τ/∂(1/T) should be continuous at the freezing point. We discuss available data in spin glasses but our conclusions should be very general.
By use of Monte Carlo simulations in 10002{1000}^{2}10002 and 1003{100}^{3}1003 Ising systems in the para phase, magnetic relaxation is shown to have a Kohlrausch-type stretched-exponential behavior, m(t)\ensuremath{\sim}exp(-t/ \ensuremath{\tau}${)}^{\mathrm{\ensuremath{\alpha}}}$, 01, for ttc{\mathit{t}}_{\mathit{c}}tc​ and normal Debye relaxation with \ensuremath{\alpha}=1 for t>tc{\mathit{t}}_{\mathit{c}}tc​; the crossover time tc{\mathit{t}}_{\mathit{c}}tc​ diverges near the transition point Tc{\mathit{T}}_{\mathit{c}}Tc​. The average relaxation time \ensuremath{\tau} shows normal critical slowing down: \ensuremath{\tau}\ensuremath{\sim}(T-Tc{\mathit{T}}_{\mathit{c}}Tc​${)}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}\mathit{z}}$, \ensuremath{\nu}z\ensuremath{\simeq}1.8 and 1.1 in dimensions d=2 and 3, respectively. We find \ensuremath{\alpha}\ensuremath{\simeq}0.33 for d=2 and 0.4 for d=3. These are compared with previous observations of stretched-exponential relaxation and critical slowing-down behavior of Ising critical dynamics.
As an analysis of equilibrium phase transitions, the nonequilibrium relaxation method is extended to the spin glass (SG) transition. The $\ifmmode\pm\else\textpm\fi{}J$ Ising SG model is analyzed for three-dimensional (cubic) lattices up to the linear size of L=127L=127L=127 and for four-dimensional (hypercubic) lattice up to L=41.L=41.L=41. These sizes of systems are quite large as compared with those calculated, so far, by equilibrium simulations. As a dynamical order parameter, we calculate the clone correlation function (CCF) ${Q(t,t}_{\mathrm{w}})\ensuremath{\equiv}[〈{S}_{i}^{(1)}{(t+t}_{\mathrm{w}}){S}_{i}^{(2)}{(t+t}_{\mathrm{w}}){〉}^{\mathrm{F}}],$ which is a spin correlation of two replicas produced after the waiting time tw{t}_{\mathrm{w}}tw​ from a simple starting state. It is found that the CCF shows an exponential decay in the paramagnetic phase, and a power-law decay after aginglike development $(t\ensuremath{\gg}{t}_{\mathrm{w}})$ in the SG phase. This provides a reliable upper bound of the transition temperature Tg.{T}_{\mathrm{g}}.Tg​. It is also found that a scaling relation, ${Q(t,t}_{\mathrm{w}}{)=t}_{\mathrm{w}}^{\ensuremath{-}{\ensuremath{\lambda}}_{q}}\overline{q}{(t/t}_{\mathrm{w}}),$ holds just around the transition point providing the lower bound of Tg.{T}_{\mathrm{g}}.Tg​. Together with these two bounds, we propose a new dynamical way for the estimation of Tg{T}_{\mathrm{g}}Tg​ from much larger systems. In the SG phase, the power-law behavior of the CCF for $t\ensuremath{\gg}{t}_{\mathrm{w}}$ suggests that the SG phase in short-range Ising models has a rugged phase space.
A simple scaling description of the ordering transition in random-field Ising systems is developed and supported by renormalization-group arguments in terms of a zero-temperature critical fixed point. The main prediction is that the characteristic relaxation time \ensuremath{\tau} will diverge extremely rapidly as the critical point is approached: \ensuremath{\tau}\ensuremath{\sim}exp(${\ensuremath{\xi}}^{\mathrm{theta}}$) with \ensuremath{\xi} the correlation length and theta the ``violation of hyperscaling'' exponent (d-theta)\ensuremath{\nu}=2-\ensuremath{\alpha}. Recent experiments which exhibit onset of hysteresis in a very narrow temperature range are discussed.
We investigate the scaling relation between the physical entanglement time τent and the modular time τmod in the dynamics of the integrable Heisenberg XXX spin chain. Contrary to the hypothesis of a stable universal exponent γ ≈ 2, we find no evidence for a time-independent power-law scaling. Instead, the effective exponent γ(t) exhibits a systematic drift toward smaller values at longer evolution times, following a robust power-law relaxation γ(t) −γ∞ ∼ t−α with α ≈ 0.65, supported by an excellent data collapse with relative spread below 6%. Remarkably, the amplitude of this relaxation is independent of system size N within the accessible window N ∈ {14,16,18,20}. We interpret this behavior as an effective renormalization group flow of the scaling exponent in time, suggesting that γ is not a fixed scaling dimension but a running quantity controlled by a marginally irrelevant operator. Our results demonstrate that previously reported values γ ≈ 2 originate from preasymptotic regimes rather than a true scaling law, and reveal a dynamical mechanism by which simple scaling breaks down in quantum entanglement evolution.