Off-equilibrium dynamics in finite-dimensional spin-glass models

Multiple Types of Aging in Active Glasses

Recent experiments and simulations have revealed glassy features in, e.g., cytoplasm, living tissues and dense assemblies of self-propelled colloids. This leads to a fundamental question: how do these nonequilibrium (active) amorphous materials differ from conventional passive glasses, created by lowering temperature or increasing density? To address this we investigate the aging after a quench to an almost arrested state of a model active glass former, a Kob-Andersen glass in two dimensions. Each constituent particle is driven by a constant propulsion force whose direction diffuses over time. Using extensive molecular dynamics simulations we reveal rich aging behavior of this dense active matter system: short persistence times of the active forcing give effective thermal aging; in the opposite limit we find a two-step aging process with active athermal aging at short times and activity-driven aging at late times. We develop a dedicated simulation method that gives access to this longtime scaling regime for highly persistent active forces.

Record-dynamics description of spin-glass thermoremanent magnetization: A numerical and analytical study

Thermoremanent magnetization data for the three-dimensional Edwards-Anderson (EA) spin glass are generated using the waiting time method as a simulational tool and interpreted using record dynamics. We verify that clusters of contiguous spins are overturned by quakes, nonequilibrium events linked to record-sized energy fluctuations, and we show that quaking is a log-Poisson process, i.e., a Poisson process whose average depends on the logarithm of the system age, counted from the initial quench. Our findings compare favorably with experimental thermoremanent magnetization findings and with the spontaneous fluctuation dynamics of the EA model. The logarithmic growth of the size of overturned clusters is related to similar experimental results and to the growing length scale of the spin-spin spatial correlation function. The analysis buttresses the applicability of the waiting time method as a simulational tool, and of record dynamics as a coarse-graining method for aging dynamics.

Strategy for solving difficulties in spin-glass simulations

A spin-glass transition has been investigated for a long time but we have not reached a conclusion yet due to difficulties in the simulation studies. They are slow dynamics, strong finite-size effects, and sample-to-sample dependencies. We found that a size of the spin-glass order reaches a lattice boundary within a very short Monte Carlo step. A competition between the spin-glass order and a boundary condition causes these difficulties. Once the boundary effect was removed, physical quantities exhibited quite normal behaviors. They became self-averaging in a limit of large replica numbers. These findings suggest that the nonequilibrium relaxation method is a good choice for solving the difficulties if a lattice size and a replica number are set sufficiently large. A dynamic scaling analysis on nonequilibrium relaxation functions gave a result that the spin-glass transition and the chiral-glass transition occurs at the same temperature in the Heisenberg model in three dimensions. The estimated critical exponent ν agrees with the experimental result.

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.

Magnetic Field Dependence of Spin Glass Free Energy Barriers

We measure the field dependence of spin glass free energy barriers in a thin amorphous Ge:Mn film through the time dependence of the magnetization. After the correlation length ξ(t,T) has reached the film thickness L=155  Å so that the dynamics are activated, we change the initial magnetic field by δH. In agreement with the scaling behavior exhibited in a companion Letter [M. Baity-Jesi et al., Phys. Rev. Lett. 118, 157202 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.157202], we find that the activation energy is increased when δH<0. The change is proportional to (δH)^{2} with the addition of a small (δH)^{4} term. The magnitude of the change of the spin glass free energy barriers is in near quantitative agreement with the prediction of a barrier model.

Matching Microscopic and Macroscopic Responses in Glasses

We first reproduce on the Janus and Janus II computers a milestone experiment that measures the spin-glass coherence length through the lowering of free-energy barriers induced by the Zeeman effect. Secondly, we determine the scaling behavior that allows a quantitative analysis of a new experiment reported in the companion Letter [S. Guchhait and R. Orbach, Phys. Rev. Lett. 118, 157203 (2017)].PRLTAO0031-900710.1103/PhysRevLett.118.157203 The value of the coherence length estimated through the analysis of microscopic correlation functions turns out to be quantitatively consistent with its measurement through macroscopic response functions. Further, nonlinear susceptibilities, recently measured in glass-forming liquids, scale as powers of the same microscopic length.

Nonequilibrium relaxation and aging scaling of the Coulomb and Bose glass

We employ Monte Carlo simulations to investigate the nonequilibrium relaxation properties of the two- and three-dimensional Coulomb glass with different long-range repulsive interactions. Specifically, we explore the aging scaling laws in the two-time density autocorrelation function. We find that, in the time window and parameter range accessible to us, the scaling exponents are not universal, depending on the filling fraction and temperature: As either the temperature decreases or the filling fraction deviates more from half filling, the exponents reflect markedly slower relaxation kinetics. In comparison with a repulsive Coulomb potential, appropriate for impurity states in strongly disordered semiconductors, we observe that, for logarithmic interactions, the soft pseudogap in the density of states is considerably broader, and the dependence of the scaling exponents on external parameters is much weaker. The latter situation is relevant for flux creep in the disorder-dominated Bose glass phase of type-II superconductors subject to columnar pinning centers.

Direct dynamical evidence for the spin glass lower critical dimension 2<d(ℓ)<3

A dynamical method is introduced to study the effect of dimensionality on phase transitions. Direct experimental measurements for the lower critical dimension for spin glasses is provided as an example. The method makes use of the spin glass correlation length ξ(t,T). Once nucleated, it can become comparable to sample dimensions in convenient time and temperature ranges. Thin films of amorphous Ge:Mn alloys were prepared with thickness L≈15.5  nm. Conventional behavior is observed as long as ξ(t,T)<L. At the measurement time tco, when ξ(tco,T)≈L, the time dependence is observed to cross over to exponential. These results are interpreted using spin glass dynamics, and are consistent with a lower critical dimension for spin glasses, dℓ, between 2<dℓ<3.

Origin of end-of-aging and subaging scaling behavior in glassy dynamics

Linear response functions of aging systems are routinely interpreted using the scaling variable t(obs)/t(w)(mu), where t(w) is the time at which the field conjugated to the response is turned on or off, and where t(obs) is the "observation" time elapsed from the field change. The response curve obtained for different values of t(w) are usually collapsed using values of mu slightly below one, a scaling behavior generally known as subaging. Recent spin glass thermoremanent magnetization experiments have shown that the value of mu is strongly affected by the form of the initial cooling protocol [G. F. Rodriguez, Phys. Rev. Lett. 91, 037203 (2003)], and even more importantly [G. G. Kenning, Phys. Rev. Lett. 97, 057201 (2006)], that the t(w) dependence of the response curves vanishes altogether in the limit t(obs)>>t(w). The latter result shows that t(obs)/t(w)(mu) scaling of linear response data cannot be generally valid, thereby casting some doubt on the theoretical significance of the exponent mu . In this work, a common mechanism is proposed for the origin of both subaging and end of aging behavior in glassy dynamics. The mechanism combines real and configuration space properties of the state produced by the initial thermal quench which initiates the aging process.

Direct-space investigation of the ultraslow ballistic dynamics of a soft glass

We use light microscopy to investigate the aging dynamics of a glass made of closely packed soft spheres, following a rapid transition from a fluid to a solidlike state. By measuring time-resolved, coarse-grained displacement fields, we identify two classes of dynamical events, corresponding to reversible and irreversible rearrangements, respectively. The reversible events are due to the small, experimentally unavoidable fluctuations of the temperature imposed to the sample, leading to transient thermal expansions and contractions that cause shear deformations. The irreversible events are plastic rearrangements, induced by the repeated shear cycles. We show that the displacement due to the irreversible rearrangements grows linearly with time, both on average and at a local level. The velocity associated with this ballistic motion decreases exponentially with sample age, accounting for the observed slowing down of the dynamics. The displacement field due to the irreversible rearrangements has a vortexlike structure and is spatially correlated over surprisingly long distances.

Nonequilibrium spin-glass dynamics from picoseconds to a tenth of a second

We study numerically the nonequilibrium dynamics of the Ising spin glass, for a time spanning 11 orders of magnitude, thus approaching the experimentally relevant scale (i.e., seconds). We introduce novel analysis techniques to compute the coherence length in a model-independent way. We present strong evidence for a replicon correlator and for overlap equivalence. The emerging picture is compatible with noncoarsening behavior.

Dilute one-dimensional spin glasses with power law decaying interactions

We introduce a diluted version of the one-dimensional spin-glass model with interactions decaying in probability as an inverse power of the distance. In this model, varying the power corresponds to changing the dimension in short-range models. The spin-glass phase is studied in and out of the range of validity of the mean-field approximation in order to discriminate between different theories. Since each variable interacts only with a finite number of others the cost for simulating, the model is drastically reduced with respect to the fully connected version, and larger sizes can be studied. We find both static and dynamic indications in favor of the so-called replica symmetry breaking theory.

Superaging in two-dimensional random ferromagnets

We study the aging properties, in particular the two-time autocorrelations, of the two-dimensional randomly diluted Ising ferromagnet below the critical temperature via Monte Carlo simulations. We find that the autocorrelation function displays additive aging C(t,t{w})=C{st}(t)+C{ag}(t,t{w}), where the stationary part Cst} decays algebraically. The aging part shows anomalous scaling C{ag}(t,t{w})=C[h(t)h(t{w})], where h(u) is a nonhomogeneous function excluding a t/t{w} scaling.

Signature of the ground-state topology in the low-temperature dynamics of spin glasses

We numerically address the issue of how the ground-state topology is reflected in the finite temperature dynamics of the +/-J Edwards-Anderson spin glass model. In this system a careful study of the ground-state configurations allows us to classify spins into two sets: solidary and nonsolidary spins. We show that these sets quantitatively account for the dynamical heterogeneities found in the mean flipping time distribution at finite low temperatures. The results highlight the relevance of taking into account the ground-state topology in the analysis of the finite temperature dynamics of spin glasses.

Universal aging properties at a disordered critical point

We investigate, analytically near the dimension d(uc) =4 and numerically in d=3 , the nonequilibrium relaxational dynamics of the randomly diluted Ising model at criticality. Using the exact renormalization-group method to one loop, we compute the two times t, t(w) correlation function and fluctuation dissipation ratio (FDR) for any Fourier mode of the order parameter, of finite wave vector q . In the large time separation limit, the FDR is found to reach a nontrivial value X(infinity) independently of (small) q and coincide with the FDR associated to the total magnetization obtained previously. Explicit calculations in real space show that the FDR associated to the local magnetization converges, in the asymptotic limit, to this same value X(infinity). Through a Monte Carlo simulation, we compute the autocorrelation function in three dimensions, for different values of the dilution fraction p at T(c) (p) . Taking properly into account the corrections to scaling, we find, according to the renormalization-group predictions, that the autocorrelation exponent lambda(c) is independent of p . The analysis is complemented by a study of the nonequilibrium critical dynamics following a quench from a completely ordered state.

Domain growth in Ising systems with quenched disorder

We present results from extensive Monte Carlo (MC) simulations of domain growth in ferromagnets and binary mixtures with quenched disorder. These are modeled by the random-bond Ising model and the dilute Ising model with either nonconserved (Glauber) spin-flip kinetics or conserved (Kawasaki) spin-exchange kinetics. In all cases, our MC results are consistent with power-law growth with an exponent theta(T, epsilon) which depends on the quench temperature T and the disorder amplitude epsilon. Such exponents arise naturally when the coarsening domains are trapped by energy barriers that grow logarithmically with the domain size. Our MC results show excellent agreement with the predicted dependence of theta(T, epsilon).

Autocorrelation functions for phase separation in ternary mixtures

We present numerical and analytical results for the autocorrelation functions which characterize domain growth in ternary mixtures. The numerical results are obtained from Monte Carlo simulations of the spin-1 Blume-Emery-Griffiths model with spin-exchange kinetics. Further, we model the autocorrelation functions using an approach based on the continuous-time random walk formalism. The aging property of these functions is related to the time dependence of the domain-size distribution. Our analytical results are found to be in good agreement with the numerical data.

Aging and equilibrium fluctuations for domain growth in ternary mixtures

We present numerical and analytical results for the autocorrelation functions which characterize domain growth in ternary mixtures. The numerical results are obtained from Monte Carlo studies of the spin-1 Blume-Emery-Griffiths model with spin-exchange kinetics. We formulate a stochastic model, which accounts for both aging and equilibrium contributions to the autocorrelation functions.

Spin anisotropy and slow dynamics in spin glasses

We report on an extensive study of the influence of spin anisotropy on spin glass aging dynamics. New temperature cycle experiments allow us to compare quantitatively the memory effect in four Heisenberg spin glasses with various degrees of random anisotropy and one Ising spin glass. The sharpness of the memory effect appears to decrease continuously with the spin anisotropy. Besides, the spin glass coherence length is determined by magnetic field change experiments for the first time in the Ising sample. For three representative samples, from Heisenberg to Ising spin glasses, we can consistently account for both sets of experiments (temperature cycle and magnetic field change) using a single expression for the growth of the coherence length with time.

Microscopic description of an Ising spin glass near the percolation threshold

Monte Carlo results using a microscopic model to describe FexZn(1-x)F2 indicate that its spin-glass phase at x=0.25 and zero magnetic field is characterized by the presence of antiferromagnetic fractal domains, separated by random vacancies and strongly correlated in time. The effective local random-field distribution corroborates this glassy behavior, which emerges irrespective of ab initio competing interactions and is a consequence of the fractal domain structure near the percolation threshold, x(p)=0.24. The aging properties of the system are in agreement with predictions of short-range stochastic spin-glass models and with the droplets model for spin glass close to percolation.

Domain growth by isothermal aging in 3D Ising and Heisenberg spin glasses

Nonequilibrium dynamics of three-dimensional model spin glasses, the Ising system Fe0.50Mn0.50TiO3 and the Heisenberg-like system Ag(11 at % Mn), has been investigated by measurements of the isothermal time decay of the low frequency ac susceptibility after a quench from the paramagnetic to the spin-glass phase. It is found that the relaxation data measured at different temperatures can be scaled according to predictions from the droplet scaling model, provided that critical fluctuations are accounted for in the analyses.

Real-space application of the mean-field description of spin-glass dynamics

The out of equilibrium dynamics of finite dimensional spin glasses is considered from a point of view going beyond the standard "mean-field theory" versus "droplet picture" debate of the past decades. The main predictions of both theories concerning the spin-glass dynamics are discussed. It is shown, in particular, that predictions originating from mean-field ideas concerning the violations of the fluctuation-dissipation theorem apply quantitatively, provided one properly takes into account the role of a spin-glass coherence length, which plays a central role in the droplet picture. Dynamics in a uniform magnetic field is also briefly discussed.

Aging of the zero-field-cooled magnetization in Ising spin glasses: experiment and numerical simulation

Growth of the zero-field-cooled magnetization (ZFCM) under continuous heating with and without an intermittent stop(s) is studied on Ising spin glasses both experimentally and numerically. Despite the large difference between time scales of the experiment and the simulation, the ZFCM behavior observed in the two systems can be quantitatively interpreted by means of a common set of the scaling expressions based on the droplet picture. The results strongly suggest that the spin-glass coherence length reached by the laboratory time scales is about a hundred lattice spacings or less. Within this length scale no signature of the chaos effect (rejuvenation) has been found in the ZFCM measured.

Dynamic ultrametricity in spin glasses

We investigate the dynamics of spin glasses from the "rheological" point of view, in which aging is suppressed by the action of small, nonconservative forces. The different features can be expressed in terms of the scaling of relaxation times with the magnitude of the driving force, which plays the role of the critical parameter. Stated in these terms, ultrametricity loses much of its mystery and can be checked rather easily. This approach also seems a natural starting point to investigate what would be the real-space structures underlying the hierarchy of time scales. We study in detail the appearance of this many-scale behavior in a mean-field model, in which dynamic ultrametricity is clearly present. A similar analysis is performed on numerical results obtained for a three-dimensional spin glass: In that case, our results are compatible with either that dynamic ultrametricity is absent or that it develops so slowly that even in experimental time-windows it is still hardly observable.