Off-equilibrium response function in the one-dimensional random-field Ising model

Phase ordering dynamics of the random-field long-range Ising model in one dimension

We investigate the influence of long-range (LR) interactions on the phase ordering dynamics of the one-dimensional random-field Ising model (RFIM). Unlike the usual RFIM, a spin interacts with all other spins through a ferromagnetic coupling that decays as r^{-(1+σ)}, where r is the distance between two spins. In the absence of LR interactions, the size of coarsening domains R(t) exhibits a crossover from pure system behavior R(t)∼t^{1/2} to an asymptotic regime characterized by logarithmic growth: R(t)∼(lnt)^{2}. The LR interactions affect the preasymptotic regime, which now exhibits ballistic growth R(t)∼t, followed by σ-dependent growth R(t)∼t^{1/(1+σ)}. Additionally, the LR interactions also affect the asymptotic logarithmic growth, which becomes R(t)∼(lnt)^{α(σ)} with α(σ)<2. Thus, LR interactions lead to faster growth than for the nearest-neighbor system at short times. Unexpectedly, this driving force causes a slowing down of the dynamics (α<2) in the asymptotic logarithmic regime. This is explained in terms of a nontrivial competition between the pinning force caused by the random field and the driving force introduced by LR interactions. We also study the spatial correlation function and the autocorrelation function of the magnetization field. The former exhibits superuniversality for all σ, i.e., a scaling function that is independent of the disorder strength. The same holds for the autocorrelation function when σ<1, whereas a signature of the violation of superuniversality is seen for σ>1.

Quasideterministic dynamics, memory effects, and lack of self-averaging in the relaxation of quenched ferromagnets

We discuss the interplay between the degree of dynamical stochasticity, memory persistence, and violation of the self-averaging property in the aging kinetics of quenched ferromagnets. We show that, in general, the longest possible memory effects, which correspond to the slowest possible temporal decay of the correlation function, are accompanied by the largest possible violation of self-averaging and a quasideterministic descent into the ergodic components. This phenomenon is observed in different systems, such as the Ising model with long-range interactions, including the mean-field, and the short-range random-field Ising model.

Effects of frustration on fluctuation-dissipation relations

We study numerically the aging properties of the two-dimensional Ising model with quenched disorder considered in our recent paper [Phys. Rev. E 95, 062136 (2017)2470-004510.1103/PhysRevE.95.062136], where frustration can be tuned by varying the fraction of antiferromagnetic interactions. Specifically, we focus on the scaling properties of the autocorrelation and linear response functions after a quench of the model to a low temperature. We find that the interplay between equilibrium and aging occurs differently in the various regions of the phase diagram of the model. When the quench is made into the ferromagnetic phase the two-time quantities are made by the sum of an equilibrium and an aging part, whereas in the paramagnetic phase these parts combine in a multiplicative way. Scaling forms are shown to be obeyed with good accuracy, and the corresponding exponents and scaling functions are determined and discussed in the framework of what is known in clean and disordered systems.

Aging in the three-dimensional random-field Ising model

We studied the nonequilibrium aging behavior of the random-field Ising model in three dimensions for various values of the disorder strength. This allowed us to investigate how the aging behavior changes across the ferromagnetic-paramagnetic phase transition. We investigated a large system size of N=256^{3} spins and up to 10^{8} Monte Carlo sweeps. To reach these necessary long simulation times, we employed an implementation running on Intel Xeon Phi coprocessors, reaching single-spin-flip times as short as 6 ps. We measured typical correlation functions in space and time to extract a growing length scale and corresponding exponents.

Equilibrium structure and off-equilibrium kinetics of a magnet with tunable frustration

We study numerically a two-dimensional random-bond Ising model where frustration can be tuned by varying the fraction a of antiferromagnetic coupling constants. At low temperatures the model exhibits a phase with ferromagnetic order for sufficiently small values of a, a<a_{f}. In an intermediate range, a_{f}<a<a_{a}, the system is paramagnetic, with spin-glass order expected right at zero temperature. For even larger values, a>a_{a}, an antiferromagnetic phase exists. After a deep quench from high temperatures, slow evolution is observed for any value of a. We show that different amounts of frustration, tuned by a, affect the dynamical properties in a highly nontrivial way. In particular, the kinetics is logarithmically slow in phases with ferromagnetic or antiferromagnetic order, whereas evolution is faster, i.e., algebraic, when spin-glass order is prevailing. An interpretation is given in terms of the different nature of phase space.

Coarsening and percolation in a disordered ferromagnet

By studying numerically the phase-ordering kinetics of a two-dimensional ferromagnetic Ising model with quenched disorder (either random bonds or random fields) we show that a critical percolation structure forms at an early stage. This structure is then rendered more and more compact by the ensuing coarsening process. Our results are compared to the nondisordered case, where a similar phenomenon is observed, and they are interpreted within a dynamical scaling framework.

Relaxation processes in a system with logarithmic growth

We discuss relaxation and aging processes in the one- and two-dimensional ABC models. In these driven diffusive systems of three particle types, biased exchanges in one direction yield a coarsening process characterized in the long-time limit by a logarithmic growth of ordered domains that take the form of stripes. From the time-dependent length, derived from the equal-time spatial correlator, and from the mean displacement of individual particles different regimes in the formation and growth of these domains can be identified. Analysis of two-times correlation and response functions reveals dynamical scaling in the asymptotic logarithmic growth regime as well as complicated finite-time and finite-size effects in the early and intermediate time regimes.

Non-Porod behavior in systems with rough morphologies

Many experiments yield multi-scale morphologies which are smooth on some length scales and fractal on others. Accurate statements about morphological properties, e.g., roughness exponent, fractal dimension, domain size, interfacial width, etc. are obtained from the correlation function and structure factor. In this paper, we present structure factor data for two systems: (a) droplet-in-droplet morphologies of double-phase-separating mixtures; and (b) ground-state morphologies in dilute anti-ferromagnets. An important characteristic of the scattering data is a non-Porod tail, which is associated with scattering off rough domains and interfaces.

Nonequilibrium steady state and induced currents of a mesoscopically glassy system: interplay of resistor-network theory and Sinai physics

We introduce an explicit solution for the nonequilibrium steady state (NESS) of a ring that is coupled to a thermal bath, and is driven by an external hot source with log-wide distribution of couplings. Having time scales that stretch over several decades is similar to glassy systems. Consequently there is a wide range of driving intensities where the NESS is like that of a random walker in a biased Brownian landscape. We investigate the resulting statistics of the induced current I. For a single ring we discuss how sign of I fluctuates as the intensity of the driving is increased, while for an ensemble of rings we highlight the fingerprints of Sinai physics on the distribution of the absolute value of I.

Scaling in the aging dynamics of the site-diluted Ising model

We study numerically the phase-ordering kinetics of the two-dimensional site-diluted Ising model. The data can be interpreted in a framework motivated by renormalization-group concepts. Apart from the usual fixed point of the nondiluted system, there exist two disorder fixed points, characterized by logarithmic and power-law growth of the ordered domains. This structure gives rise to a rich scaling behavior, with an interesting crossover due to the competition between fixed points, and violation of superuniversality.

Nonuniversal disordered Glauber dynamics

We consider the one-dimensional Glauber dynamics with coupling disorder in terms of bilinear fermion Hamiltonians. Dynamic exponents embodied in the spectrum gap of these latter are evaluated numerically by averaging over both binary and Gaussian disorder realizations. In the first case, these exponents are found to follow the nonuniversal values of those of plain dimerized chains. In the second situation their values are still nonuniversal and subdiffusive below a critical variance above which, however, the relaxation time is suggested to grow as a stretched exponential of the equilibrium correlation length.

Crossover in growth law and violation of superuniversality in the random-field Ising model

We study the nonconserved phase-ordering dynamics of the d=2,3 random-field Ising model, quenched to below the critical temperature. Motivated by the puzzling results of previous work in two and three dimensions, reporting a crossover from power-law to logarithmic growth, together with superuniversal behavior of the correlation function, we have undertaken a careful investigation of both the domain growth law and the autocorrelation function. Our main results are as follows: We confirm the crossover to asymptotic logarithmic behavior in the growth law, but, at variance with previous findings, we find the exponent in the preasymptotic power law to be disorder dependent, rather than being that of the pure system. Furthermore, we find that the autocorrelation function does not display superuniversal behavior. This restores consistency with previous results for the d=1 system, and fits nicely into the unifying scaling scheme we have recently proposed in the study of the random-bond Ising model.

Nonlinear response of the trap model in the aging regime: exact results in the strong-disorder limit

We study the dynamics in the one-dimensional disordered trap model with a broad distribution of trapping times p(tau) approximately 1/tau(1+mu), when an external force is applied from the very beginning at t=0, or only after a waiting time t(w), in the linear as well as in the nonlinear response regime. Using a real-space renormalization procedure that becomes exact in the limit of strong disorder mu-->0, we obtain explicit results for many observables, such as the diffusion front, the mean position, the thermal width, the localization parameters and the two-particle correlation function. In particular, the scaling functions for these observables give access to the complete interpolation between the unbiased case and the directed case. Finally, we discuss in detail the various regimes that exist for the average position in terms of the two times and the external field.

Localization properties of the anomalous diffusion phase in the directed trap model and in the Sinai diffusion with a bias

We study the localization properties of the anomalous diffusion phase x approximately t(micro) with 0<micro<1, which exists both in the Sinai diffusion with a small bias, and in the related directed trap model presenting a broad distribution of trapping times p(tau) approximately 1/tau(1+micro). Our starting point is the real space renormalization method, in which the whole thermal packet is considered to be in the same renormalized valley at large time: this assumption is asymptotically exact only in the limit of vanishing bias micro-->0 and corresponds to the Golosov localization. For finite micro we thus generalize the usual real space renormalization method to allow for the spreading of the thermal packet over many renormalized valleys. Our construction allows one to compute exact series expansions in micro for all observables: to compute observables at order micro (n), it is sufficient to consider in each sample a spreading of the thermal packet onto at most (1+n) traps. So our approach provides a description of the structure of the thermal packet sample by sample, and a full statistical characterization of the important traps at a given order in micro. For the directed trap model, we show explicitly up to order micro(2) how to recover the exact expressions for the diffusion front, the thermal width, and the localization parameter Y2. We then use our method to derive exact results for the localization parameters Y(k) for arbitrary k, the correlation function of two particles, and the generating function of thermal cumulants. We then explain how these results apply to the Sinai diffusion with bias by deriving the quantitative mapping between the large-scale renormalized descriptions of the two models. Finally we study the internal structure of the effective "traps" for the Sinai model via path-integral methods.