We present a collection of simulations of the Edwards–Anderson lattice spin glass at T=0 to elucidate the nature of low-energy excitations over a range of dimensions that reach from physically realizable systems to the mean-field limit. Using heuristic methods, we sample the ground states of instances to determine their energies while eliciting excitations through manipulating boundary conditions. We exploit the universality of the phase diagram of bond-diluted lattices to make such a study in higher dimensions computationally feasible. As a result, we obtain a variety of accurate exponents for domain wall stiffness and finite-size corrections, which allow us to examine their dimensional behavior and their connection with predictions from mean-field theory. We also provide an experimentally testable prediction for the thermal-to-percolative crossover exponent in dilute lattice Ising spin glasses.
We study the phase-space geometry and the low-temperature relaxation of short-range Gaussian Ising spin-glasses, by finding all the configurations and their interconnection within a ``pocket'' of states surrounding a low-lying energy minimum. Thermalization within the pocket is modeled by a master equation, which is solved numerically, yielding the time-dependent propagator. The findings show that the relaxation proceeds by a series of local equilibrations in nested ``valleys,'' and support the dynamical predictions of hierarchical relaxation models.
The Ising spin glass in two dimensions is a model system with extraordinarily rich behavior. Although it does not show a spin-glass phase at finite temperatures, it exhibits many of the salient features of fully fledged spin glass systems. The authors use recently developed mappings to graph-theoretic problems together with highly efficient implementations of combinatorial optimization algorithms to determine exact ground states for systems with up to $10,000\ifmmode\times\else\texttimes\fi{}10,000$ spins. These and further results from new algorithms for fully periodic lattices and for systems with degeneracies allow for the determination of critical exponents with unprecedented accuracy.
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.
As an analysis of equilibrium phase transitions, the nonequilibrium relaxation method is extended to the spin glass (SG) transition. The $\ifmmode\pm\else\textpm\fi{}J$ Ising SG model is analyzed for three-dimensional (cubic) lattices up to the linear size of
L=127 and for four-dimensional (hypercubic) lattice up to
L=41. These sizes of systems are quite large as compared with those calculated, so far, by equilibrium simulations. As a dynamical order parameter, we calculate the clone correlation function (CCF) ${Q(t,t}_{\mathrm{w}})\ensuremath{\equiv}[〈{S}_{i}^{(1)}{(t+t}_{\mathrm{w}}){S}_{i}^{(2)}{(t+t}_{\mathrm{w}}){〉}^{\mathrm{F}}],$ which is a spin correlation of two replicas produced after the waiting time
tw from a simple starting state. It is found that the CCF shows an exponential decay in the paramagnetic phase, and a power-law decay after aginglike development $(t\ensuremath{\gg}{t}_{\mathrm{w}})$ in the SG phase. This provides a reliable upper bound of the transition temperature
Tg. It is also found that a scaling relation, ${Q(t,t}_{\mathrm{w}}{)=t}_{\mathrm{w}}^{\ensuremath{-}{\ensuremath{\lambda}}_{q}}\overline{q}{(t/t}_{\mathrm{w}}),$ holds just around the transition point providing the lower bound of
Tg. Together with these two bounds, we propose a new dynamical way for the estimation of
Tg from much larger systems. In the SG phase, the power-law behavior of the CCF for $t\ensuremath{\gg}{t}_{\mathrm{w}}$ suggests that the SG phase in short-range Ising models has a rugged phase space.
Non-exponential slow dynamics in the `Griffiths phase' of the three-dimensional (3D) ± J Ising spin-glass model is studied by means of Monte Carlo simulation. The distribution of relaxation times averaged over samples, \bar P ( x ) with x = lnτ, is calculated from spin auto-correlation functions simulated. It is found that \bar P ( x ) consists of two branches. The cross-over value between the branches, x c , diverges as T → T c , T c being the spin-glass transition temperature. Therefore τ c (= e x c ) is regarded as the critical relaxation time associated with the onset of the spin-glass order at T c , and the branch of \bar P ( x ) at x < x c is interpreted due to the ordinary critical slowing down process ( critical dynamics ). The other branch at x > x c causes the unusual slow dynamics peculiar to spin glasses. Various aspects of \bar P ( x ) of this branch examined are consistent with the following picture; compact regions with less frustrations than the system as a whole are thermally fluctuating with very small frequencies, and independently of each others ( cluster dynamics ).