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arXiv:cond-mat/9904136 · Physical Review Letters

Glassy Time-Scale Divergence and Anomalous Coarsening in a Kinetically Constrained Spin Chain

Peter Sollich A, M. R. Evans

Published Oct 18, 1999• 143 citations
arXivHTMLPDF

Abstract

We analyze the out-of-equilibrium behavior of an Ising spin chain with an asymmetric kinetic constraint after a quench to a low temperature $T$. In the limit $T\ensuremath{\rightarrow}0$, we provide an exact solution of the resulting coarsening process. The equilibration time exhibits a ``glassy'' divergence ${t}_{\mathrm{eq}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{exp}(\mathrm{const}/{T}^{2})$ (popular as an alternative to the Vogel-Fulcher law), while the average domain length grows with a temperature-dependent exponent, $\overline{d}\ensuremath{\sim}{t}^{T\mathrm{ln}2}$. We show that the equilibration time ${t}_{\mathrm{eq}}$ also sets the time scale for the linear response of the system at low temperatures.

DOI
10.1103/physrevlett.83.3238

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We analyze the out-of-equilibrium behavior of an Ising spin chain with an asymmetric kinetic constraint after a quench to a low temperature TTT. In the limit T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0, we provide an exact solution of the resulting coarsening process. The equilibration time exhibits a ``glassy'' divergence teq=exp(const/T2){t}_{\mathrm{eq}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{exp}(\mathrm{const}/{T}^{2})teq​=exp(const/T2) (popular as an alternative to the Vogel-Fulcher law), while the average domain length grows with a temperature-dependent exponent, d‾\ensuremath∼tTln2\overline{d}\ensuremath{\sim}{t}^{T\mathrm{ln}2}d\ensuremath∼tTln2. We show that the equilibration time teq{t}_{\mathrm{eq}}teq​ also sets the time scale for the linear response of the system at low temperatures.
We find a breakdown of the critical dynamic scaling in the coarsening dynamics of an antiferromagnetic XYXYXY model on the kagome lattice when the system is quenched from disordered states into the Kosterlitz-Thouless phases at low temperatures. There exist multiple growing length scales; the length scales of the average separation between fractional vortices are found to be not proportional to the length scales of the quasiordered domains. They are instead related through a nontrivial power-law relation. The length scale of the quasiordered domains (as determined from the optimal collapse of the correlation functions for the order parameter $\text{exp}[3i\ensuremath{\theta}(r)]$) does not follow a simple power-law growth but exhibits an anomalous growth with time-dependent effective growth exponent. The breakdown of the critical dynamic scaling is accompanied by unusual relaxation dynamics in the decay of the fractional $(3\ensuremath{\theta})$ vortices, where the decay of the vortex numbers is characterized by an exponential function of the logarithmic powers in time.
We investigate the relaxation dynamics of a one-dimensional Ising chain via Glauber kinetic Monte Carlo simulations, when the system is cooled slowly from infinite temperature to zero temperature with different cooling protocols. The main quantity of interest is the excess defect density that represents the total defect density minus the equilibrium defect density at varying temperatures. We find that, for three classes of cooling protocols, the time dependence of the excess defect density for various cooling speed shows a dynamic scaling behavior that largely encompasses the Kibble-Zurek mechanism as well as Krapivsky's calculation of the final defect density at zero temperature. We also find distinct features in the behavior of the dynamic scaling when the temperature approaches in a power-law fashion to zero temperature and the excess defect density reaches a peak at finite temperature, where the scaling of the excess defect density at its peak and that at zero temperature exhibits different asymptotic behavior due to different logarithmic corrections. This characteristic behavior can be attributed to the exponential divergence of the relaxation time near zero temperature. While the qualitative theoretical predictions by Krapivsky on the asymptotic exponents are confirmed, the asymptotic predictions for amplitudes overestimate by up to 40% the simulation results.
We study the kinetics after a low temperature quench of the one-dimensional Ising model with long range interactions between spins at distance r r decaying as r^{-\alpha} r − α . For \alpha =0 α = 0 , i.e. mean field, all spins evolve coherently quickly driving the system towards a magnetised state. In the weak long range regime with \alpha >1 α > 1 there is a coarsening behaviour with competing domains of opposite sign without development of magnetisation. For strong long range, i.e. 0<\alpha <1 0 < α < 1 , we show that the system shows both features, with probability P_\alpha (N) P α ( N ) of having the latter one, with the different limiting behaviours \lim _{N\to \infty}P_\alpha (N)=0 lim N → ∞ P α ( N ) = 0 (at fixed \alpha<1 α < 1 ) and \lim _{\alpha \to 1}P_\alpha (N)=1 lim α → 1 P α ( N ) = 1 (at fixed finite N N ). We discuss how this behaviour is a manifestation of an underlying dynamical scaling symmetry due to the presence of a single characteristic time \tau _\alpha (N)\sim N^\alpha τ α ( N ) ∼ N α .
We study the low temperature dynamics of a two-dimensional short-range spin system with uniform ferromagnetic interactions, which displays glassiness at low temperatures despite the absence of disorder or frustration. The model has a dual description in terms of free defects subject to dynamical constraints, and is an explicit realization of the "hierarchically constrained dynamics" scenario for glassy systems. We give a number of exact results for the statics of the model, and study in detail the dynamical behavior of one-time and two-time quantities. We also consider the role played by the configurational entropy, which can be computed exactly, in the relation between fluctuations and response.
The nonequilibrium dynamics of classical random Ising spin chains with nonconserved magnetization are studied using an asymptotically exact real space renormalization group (RSRG). We focus on random field Ising model (RFIM) spin chains with and without a uniform applied field, as well as on Ising spin glass chains in an applied field. For the RFIM we consider a universal regime where the random field and the temperature are both much smaller than the exchange coupling. In this regime, the Imry-Ma length that sets the scale of the equilibrium correlations is large and the coarsening of domains from random initial conditions (e.g., a quench from high temperature) occurs over a wide range of length scales. The two types of domain walls that occur diffuse in opposite random potentials, of the form studied by Sinai, and domain walls annihilate when they meet. Using the RSRG we compute many universal asymptotic properties of both the nonequilibrium dynamics and the equilibrium limit. We find that the configurations of the domain walls converge rapidly toward a set of system-specific time-dependent positions that are independent of the initial conditions. Thus the behavior of this nonequilibrium system is pseudodeterministic at long times because of the broad distributions of barriers that occur on the long length scales involved. Specifically, we obtain the time dependence of the energy, the magnetization, and the distribution of domain sizes (found to be statistically independent). The equilibrium limits agree with known exact results. We obtain the exact scaling form of the two-point equal time correlation function <S0(t)Sx(t)> and the two-time autocorrelations <S0(t('))S0(t)>. We also compute the persistence properties of a single spin, of local magnetization, and of domains. The analogous quantities for the +/-J Ising spin glass in an applied field are obtained from the RFIM via a gauge transformation. In addition to these we compute the two-point two-time correlation function <S0(t)Sx(t)><S0(t('))Sx(t('))> which can in principle be measured by experiments on spin-glass-like systems. The thermal fluctuations are studied and found to be dominated by rare events; in particular all moments of truncated equal time correlations are computed. Physical properties which are typically measured in aging experiments are also studied, focusing on the response to a small magnetic field which is applied after waiting for the system to equilibrate for a time t(w). The nonequilibrium fluctuation-dissipation ratio X(t,t(w)) is computed. We find that for (t-t(w)) approximately t(alpha)(w) with alpha;<1, the ratio equal to its equilibrium value X=1, although time translational invariance does not hold in this regime. For t-t(w) approximately t(w) the ratio exhibits an aging regime with a nontrivial X=X(t/t(w)) not equal to 1, but the behavior is markedly different from mean field theory. Finally the distribution of the total magnetization and of the number of domains is computed for large finite size systems. General issues about convergence toward equilibrium and the possibilities of weakly history-dependent evolution in other random systems are discussed.
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