Spin glass behavior was first seen in metallic alloys with magnetic impurities dispersed randomly in the main non-magnetic component. Due to the simultaneous presence of quenched disorder and competing ferromagnetic and antiferromagnetic interactions, the nonequilibrium behaviors of spin glasses are intricate and difficult to understand. Experimental research on spin glass materials indicate extremely slow dynamics, suggesting that the direct numerical simulation using local spin flips will take impractical time scales to replicate the nonequilibrium behaviors. This thesis introduces efficient algorithms and heuristics for the numerical simulation of simple models for spin glasses and discusses significant simulation results. The results provide insights into understanding the nonequilibrium behaviors, especially aging and memory, through studying microscopic constituent configurations. The methodology and conclusion presented in this thesis can be potentially extended to address general questions about disordered materials at nonequilibrium. Chapter 1 provides a thorough overview of spin glasses. A detailed introduction on complex nonequilibrium behaviors, including slow dynamics and memory effects, seen in typical experiments of spin glass materials, is presented. A general discussion about the characterization and categorization of memory effects found in many out-of-equilibrium systems is highlighted. Two prevalent theoretical models on spin glasses, the droplet and replica symmetry breaking theory, are described. Lastly, algorithm development and simulation techniques used to study spin glasses are covered. In Chapter 2, a patchwork heuristic is applied to accelerate the relaxation of spin glass models in dimensions
d=1 and
d=2 with the goal of investigating the dynamics of loss and recovery of spin configurations. A well-designed simulation protocol, consists of a aging and recovery stage, is used to mimic the temperature cycling experiment, where the temperature variations are replaced by the variations of the spin coupling realization. Eventual recovery of memory upon coarsening is seen in two-dimensional Ising spin glasses and one-dimensional clock models, while one-dimensional Ising spin systems do not exhibit configuration memory. A central result is that the coarsening recovery length scale
sc grows faster than the aging length scale
ℓ as a power law relation
sc∼ℓ2.58. In Chapter 3, the microscopic dependence of spin glass equilibration on sample history, i.e., the initial configuration and the specific noise history of stochastic dynamics, is examined. The patchwork dynamics is used to simulate the glassy dynamics over a wide range of length scales. The dependence of the nearest-neighbor spin correlations at long time on initial spin configurations and on noise histories are compared. Most of the local spin correlations are independent of the details of evolution, due to the formation of rigid domains. The correlations on a fractal set of domain walls are found to have distinct dependence on the noise history and the initial state. The initial ``nature'' of the state is retained in the domain walls under coarsening more than the ``nurture'' of the noise, while the location of the domain walls at each scale is determined by the frozen disorder (``environment''). We also provide evidence that coarsening with local dynamics at finite temperature gives persistence of overlaps roughly consistent with coarsening at zero temperature, at least over short distances, supporting the use of zero-temperature heuristics to address finite-temperature dynamics and memory.
A patchwork method is used to study the dynamics of loss and recovery of an initial configuration in spin glass models in dimensions
d=1 and
d=2. The patchwork heuristic is used to accelerate the dynamics to investigate how models might reproduce the remarkable memory effects seen in experiment. Starting from a ground-state configuration computed for one choice of nearest-neighbor spin couplings, the sample is aged up to a given scale under new random couplings, leading to the partial erasure of the original ground state. The couplings are then restored to the original choice and patchwork coarsening is again applied, in order to assess the recovery of the original state. Eventual recovery of the original ground state upon coarsening is seen in two-dimensional Ising spin glasses and one-dimensional clock models, while one-dimensional Ising spin systems neither lose nor gain overlap with the ground state during the recovery stage. The recovery for the two-dimensional Ising spin glasses suggests scaling relations that lead to a recovery length scale that grows as a power of the aging length scale.
Disorder is present in many systems in nature and in many different versions. For instance, the dislocations of a crystal lattice, or the randomness of the interaction between magnetic moments. One of the most studied examples is that of spin glasses because they are simple to model but keep most of the very complex features that many disordered systems have. The frustration of the ground state configuration is responsible for the existence of a gap less spectrum of excitations and a rugged and complex free-energy landscape which bring about a very slow relaxation towards the equilibrium state. The main concern of the thesis has been to study what the properties of the typical excitation, i.e. those excitations that are large and contribute dominantly to the physics in the frozen phase. The existence of these large excitations brings about large fluctuations of the order parameter, and we have shown in these theses that this feature can be exploited to study the transition of any spin glass model. Moreover, we have shown that the information about these excitations can be extracted from the statistics of the lowest lying excitations. This is because due to the random nature of spin glasses, the physics obtained from averaging over the whole spectrum of excitations of an infinite sample is equivalent to averaging over many finite systems where only the ground state and the first excitation are considered. The novelty of this approach is that we do not need to make any assumption on what are typical excitations like because we can compute them exactly using numerical methods. Finally, we have investigated the dynamics and more specifically the link between the problem of chaos and the rejuvenation phenomena observed experimentally. Rejuvenation means that when lowering the temperature the aging process restarts again from scratch. This is potentially linked with the chaos assumption which states that equilibrium configurations at two different properties are not correlated. Chaos is a large scale phenomenon possible if entropy fluctuations are large. However, in this thesis we have shown that the response to temperature changes can be large in the absence of chaos close to a localization transition where the Boltzmann weight condenses in a few states. This has been observed in simulation of the Sinai model in which this localization is realized dynamically. In this model, since at low temperatures the system gets trapped in the very deep states, the dynamics is only local, so that only small excitations contribute to the rejuvenation signal that we have been able to observe. Thus, in agreement with the hierarchical picture, rejuvenation is possible even in the absence of chaos and reflects the start of the aging process of small length scales.