Michel Pleimling

Ageing and the Glass Transition

Ageing, Rejuvenation and Memory: The Example of Spin-Glasses.- About the Nature of the Structural Glass Transition: An Experimental Approach.- Glassy Behaviours in A-Thermal Systems, the Case of Granular Media: A Tentative Review.- to Simulation Techniques.- From Urn Models to Zero-Range Processes: Statics and Dynamics.- Field-Theory Approaches to Nonequilibrium Dynamics.In this paper, we review the general features of the out-of-equilibrium dynamics of spin glasses. We use this example as a guideline for a brief description of glassy dynamics in other disordered systems like structural and polymer glasses, colloids, gels etc. Starting with the simplest experiments, we discuss the scaling laws used to describe the isothermal aging observed in spin glasses after a quench down to the low temperature phase (these scaling laws are the same as established for polymer glasses). We then discuss the rejuvenation and memory effects observed when a spin glass is submitted to temperature variations during aging, and show some examples of similar phenomena in other glassy systems. The rejuvenation and memory effects and their implications are analyzed from the point of view of both energy landscape pictures and of real space pictures. We highlight the fact that both approaches point out the necessity of hierarchical processes involved in aging. We introduce the concept of a slowly growing and strongly temperature dependent dynamical correlation length, which is discussed at the light of a large panel of experiments.

Aging, Phase Ordering, and Conformal Invariance

In a variety of systems which exhibit aging, the two-time response function scales as R(t,s) approximately s(-1-a)f(t/s). We argue that dynamical scaling can be extended towards conformal invariance, thus obtaining the explicit form of the scaling function f. This quantitative prediction is confirmed in several spin systems, both for T<T(c) (phase ordering) and T = T(c) (nonequilibrium critical dynamics). The 2D and 3D Ising models with Glauber dynamics are studied numerically, while exact results are available for the spherical model with a nonconserved order parameter, both for short-ranged and long-ranged interactions, as well as for the mean-field spherical spin glass.

Critical phenomena at perfect and non-perfect surfaces

The effect of imperfections on surface critical properties is studied for Ising models with nearest-neighbour ferromagnetic couplings on simple cubic lattices. In particular, results of Monte Carlo simulations for flat, perfect surfaces are compared to those for flat surfaces with random, “weak” or “strong”, interactions between neighbouring spins in the surface layer, and for surfaces with steps of monoatomic height. Surface critical exponents at the ordinary transition, in particular \(\),are found to be robust against these perturbations.

Phenomenology of aging in the Kardar-Parisi-Zhang equation.

We study aging during surface growth processes described by the one-dimensional Kardar-Parisi-Zhang equation. Starting from a flat initial state, the systems undergo simple aging in both correlators and linear responses, and its dynamical scaling is characterized by the aging exponents a=-1/3, b=-2/3, λ(C)=λ(R)=1, and z=3/2. The form of the autoresponse scaling function is well described by the recently constructed logarithmic extension of local scale invariance.

Two-time autocorrelation function in phase-ordering kinetics from local scale invariance

The time-dependent scaling of the two-time autocorrelation function of spin systems without disorder undergoing phase-ordering kinetics is considered. Its form is shown to be determined by an extension of dynamical scaling to a local scale invariance which turns out to be a new version of conformal invariance. The predicted autocorrelator is in agreement with Monte Carlo data on the autocorrelation function of the 2D kinetic Ising model with Glauber dynamics quenched to a temperature below criticality.

Surface criticality at a dynamic phase transition.

In order to elucidate the role of surfaces at nonequilibrium phase transitions, we consider kinetic Ising models with surfaces subjected to a periodic oscillating magnetic field. Whereas, the corresponding bulk system undergoes a continuous nonequilibrium phase transition characterized by the exponents of the equilibrium Ising model, we find that the nonequilibrium surface exponents do not coincide with those of the equilibrium critical surface. In addition, in three space dimensions, the surface phase diagram of the nonequilibrium system differs markedly from that of the equilibrium system.

Anisotropic Scaling and Generalized Conformal Invariance at Lifshitz Points

The behaviour of the 3D axial next-nearest-neighbor Ising model at the uniaxial Lifshitz point is studied using Monte Carlo techniques. A new variant of the Wolff cluster algorithm permits the analysis of systems far larger than in previous studies. The Lifshitz point critical exponents are alpha = 0.18(2), beta = 0.238(5), and gamma = 1.36(3). Data for the spin-spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance, which is a generalization of conformal invariance to the anisotropic scaling at the Lifshitz point.

Cyclic competition of four species: Mean-field theory and stochastic evolution

Generalizing the cyclically competing three-species model (often referred to as the rock-paper-scissors game), we consider a simple system of population dynamics without spatial structures that involves four species. Unlike the previous model, the four form alliance pairs which resemble partnership in the game of Bridge. In a finite system with discrete stochastic dynamics, all but 4 of the absorbing states consist of coexistence of a partner-pair. From a master equation, we derive a set of mean-field equations of evolution. This approach predicts complex time dependence of the system and that the surviving partner-pair is the one with the larger product of their strengths (rates of consumption). Simulations typically confirm these scenarios. Beyond that, much richer behavior is revealed, including complicated extinction probabilities and non-trivial distributions of the population ratio in the surviving pair. These discoveries naturally raise a number of intriguing questions, which in turn suggests a variety of future avenues of research, especially for more realistic models of multispecies competition in nature.

Crossing the Coexistence Line at Constant Magnetization

Using Monte Carlo histogram methods, the microcanonical caloric curve is computed for the Ising model in two and three dimensions with fixed magnetization. Whereas the signatures of a possible first order phase transition are clearly visible for large systems, intriguing finite size effects are revealed for smaller system sizes. The behaviour of the caloric curve is studied in a systematic way. Furthermore, results for the thermal stability of three-dimensional droplets of minority spins inside the two-phase region are presented. The effect of the percolation transition on the stability of these droplets is discussed.

Mobility and asymmetry effects in one-dimensional rock-paper-scissors games.

As the behavior of a system composed of cyclically competing species is strongly influenced by the presence of fluctuations, it is of interest to study cyclic dominance in low dimensions where these effects are the most prominent. We here discuss rock-paper-scissors games on a one-dimensional lattice where the interaction rates and the mobility can be species dependent. Allowing only single site occupation, we realize mobility by exchanging individuals of different species. When the interaction and swapping rates are symmetric, a strongly enhanced swapping rate yields an increased mixing of the species, leading to a mean-field-like coexistence even in one-dimensional systems. This coexistence is transient when the rates are asymmetric, and eventually only one species will survive. Interestingly, in our spatial games the dominating species can differ from the species that would dominate in the corresponding nonspatial model. We identify different regimes in the parameter space and construct the corresponding dynamical phase diagram.