The authors study here low-energy linear excitations in XY and Heisenberg spin glasses at zero temperature, considering systems defined on sparse random regular graphs. The localization properties of the soft modes are linked to the spin glass transition in a field. The authors discover a new kind of low-energy excitation, called a ``Concentrated and Delocalized Low Energy Mode'' (CDLEM), given that this excitation is restricted to a sub-extensive number of vertices spread over the largest possible distances.
The authors study here low-energy linear excitations in XY and Heisenberg spin glasses at zero temperature, considering systems defined on sparse random regular graphs. The localization properties of the soft modes are linked to the spin glass transition in a field. The authors discover a new kind of low-energy excitation, called a ``Concentrated and Delocalized Low Energy Mode'' (CDLEM), given that this excitation is restricted to a sub-extensive number of vertices spread over the largest possible distances.
The work of this thesis concerns the problem of linear low energy excitations of vector spin glass models. An analytical and numerical study is carried out, considering a fully connected random-field Heisenberg model at zero temperature, a fully-connected vector p-spin glass model and a sparse random-field Heisenberg model. We test these models against the low temperature behavior of finite dimensional glassy systems, in particular we show that they posses phases where the density of states is gapless with quasi-localised modes. In the case of the sparse model, we show that the density of states follows a quartic law at low frequency, consistently with several recent measures of this quantity that can be found in the literature of computer glasses. In all the three models, the spin glass transition is characterised in terms of the behavior of the softest excitations. We found that in the fully connected models the zero temperature spin glass transition in a field is a delocalisation transition of the softest modes. In the sparse case, a weaker form of delocalisation appears at the transition. These results broaden our understanding of the zero temperature critical point, by showing how spin glass ordering affects the way the system responds to small magnetic perturbations.
This thesis addresses the surprising features of zero-temperature statics and dynamics of several spin glass models, including correlations between soft spins that arise spontaneously during avalanches, and the discovery of localized states that involve the presence of two-level systems. It also presents the only detailed historiographical research on the spin glass theory. Despite the extreme simplicity of their definition, spin glasses display a wide variety of non-trivial behaviors that are not yet fully understood. In this thesis the author sheds light on some of these, focusing on both the search for phase transitions under perturbations of Hamiltonians and the zero-temperature properties and responses to external stimuli. After introducing spin glasses and useful concepts on phase transitions and numerics, the results of two massive Monte Carlo campaigns on three-dimensional systems are presented: The first of these examines the de Almeida Thouless transition, and proposes a new finite-size scaling ansatz, which accelerates the convergence to the thermodynamic limit. The second reconstructs the phase diagram of the Heisenberg spin glass with random exchange anisotropy
The transverse dynamics of a vector spin glass chain at zero temperature is studied by a numerical scaling method which provides a new physical insight into the nature of the spin excitations. In the range of low energies considered we find that the states are all localized with the coupling between spins V decaying asymptotically for large lengths separating the sites l , like V ∼ exp[- l /ξ], where ξ measures the localization length. A dispersion relation is found between the energies and the associated inverse localization lengths of the form ω ∝ (1/ξ) z with the dynamic exponent z = 1.501 ± 0.044. This implies that the system at zero temperature has nontrivial dynamic critical behaviour, in agreement with our recent analytical results; it moreover gives for the first time a clear interpretation for the characteristic length of the spin fluctuations as the localization length.
An Ising (spin- 1 / 2 ) model with random positive, negative and zero nearest-neighbour exchange coefficients is considered at zero temperature. The authors define two 'spin glass' susceptibilities, through average sums over ((s i s j ) 2 ) av and ( mod (s i s j ) mod ) av , where ( ) means averaging over degenerate ground states and ( ) av means averaging over the random bond distribution. These susceptibilities are expanded in powers of the concentrations of the non-zero bonds, for the triangular (seven terms), square (nine terms) and d-dimensional hypercubic (eight terms) lattices. All the series show a singularity, which is associated with the paramagnetic to spin-glass phase transition. The phase boundary between these phases, and the corresponding critical exponents, are estimated.
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.
For many real spin-glass materials, the Edwards-Anderson model with continuous-symmetry spins is more realistic than the rather better understood Ising variant. In principle, the nature of an occurring spin-glass phase in such systems might be inferred from an analysis of the zero-temperature properties. Unfortunately, with few exceptions, the problem of finding ground-state configurations is a non-polynomial problem computationally, such that efficient approximation algorithms are called for. Here, we employ the recently developed genetic embedded matching (GEM) heuristic to investigate the nature of the zero-temperature phase of the bimodal XY spin glass in two dimensions. We analyze bulk properties such as the asymptotic ground-state energy and the phase diagram of disorder strength vs. disorder concentration. The main focus of this work is on an investigation of the excitation spectrum as probed by changing the boundary conditions. Using appropriate finite-size scaling techniques, we consistently determine the stiffness of spin and chiral domain walls and the corresponding fractal dimensions. Results for the overlap distribution do not support the possibility of a multitude of thermodynamic pure states.