I present here some results on the mean field theory approach to the statistical mechanics of a D-dimensional array of Josephson junctions in the presence of a magnetic field. The mean field theory equations are obtained by computing the thermodynamical properties. In the high temperature region in the limit D→∞, where the problem is simplified, this limit defines the mean field approximation. Close to the transition point the system behaves very similar to a particular form of spin glasses, i.e., to gauge glasses. We have noticed that in this limit the evaluation of the coefficients of the high temperature expansion may be mapped onto the computation of some matrix elements for the q-deformed harmonic oscillator. The same arguments can be used to predict the thermodynamical properties in the mean field limit. These results can be extended to the low temperature phase using a conjecture on the equivalence of some system without disorder with appropriate random systems.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and
XY spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for
T greater than
Tc, the spin-glass transition temperature. At and below
Tc, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and
XY spins are evaluated using a "many-replica" procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for
T greater than
Tc, the spin-glass transition temperature. At and below
Tc, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
The study of the low temperature phase of spin glass models by means of Monte Carlo simulations is a challenging task, because of the very slow dynamics and the severe finite-size effects they show. By exploiting at the best the capabilities of standard modern CPUs (especially the streaming single instruction, multiple data extensions), we have been able to simulate the four-dimensional Edwards-Anderson model with Gaussian couplings up to sizes L=70 and for times long enough to accurately measure the asymptotic behavior. By quenching systems of different sizes to the critical temperature and to temperatures in the whole low temperature phase, we have been able to identify the regime where finite-size effects are negligible: ξ(t)≲L/7. Our estimates for the dynamical exponent (z≃1/T) and for the replicon exponent (α≃1.0 and T independent), that controls the decay of the spatial correlation in the zero overlap sector, are consistent with the replica symmetry breaking theory, but the latter differs from the theoretically conjectured value.
We investigate the low temperature phase of a vector spin glass in the limit of large component numbers
m of the spins, where this model is expected to be replica symmetric. On a Bethe lattice, a random graph with fixed and finite connectivity, we employ a self-consistent effective-field approach to the cavity method to describe equilibrium states of the system in terms of the configuration of cavity fields, which are reduced mean fields onto each spin. We make use of an iterative algorithm to find such equilibrium states for finite system sizes
N and observe a phase transition, which is indicated by an order parameter describing the preferred direction each spin points into in terms of the configuration of cavity, resp. mean local fields. On single realizations of Bethe lattices the transition is sharp and accompanied by the Generalized-Bose-Einstein condensation, in which the cavity, resp. local fields of the spins condense into a subspace of the available
m-dimensional space for large
m. The dimension of this subspace,
meff(T) depends on the temperature. Is either one or two right at the critical temperature and increases below. Furthermore, for fixed temperature it is expected to scale with the system size
N with an exponent
μm which depends on the connectivity. Averaging over many realizations of disorder we find
meff(Tc)=2 and
meff(T)∼Nμm. Even more, the cavity fields are subject to rotations in the low temperature phase, i.e. there is no freezing in time as generally expected. In case of
meff(Tc)=2 the whole configuration rotates at the critical temperature about the same angle and the same axis of rotation corresponding to a single droplet. In addition, we find replica symmetry breaking below the large-
m limit, i.e. for
meff(T)>m. These results are supported by results of parallel tempering Monte Carlo simulations. Replica symmetry breaking is further investigated by an analytical method in the large connectivity limit, where we compare the free energy in terms of spin-spin-correlations of different replicas calculated with a replica symmetric and a one-step broken replica symmetry ansatz. In the large-
m limit, we find that both solutions coexist for finite connectivities and that replica symmetry holds for full connectivity up to corrections in second order around the saddle point for
m→∞. In the last chapter we calculate the sample-to-sample fluctuations of the free energy of the SK model of the large-
m limit. We derive an exact connection between these fluctuations and bond chaos. Making use of a replica calculation for the {\it large deviations} and a scaling ansatz for the {\it small deviations}, we quantify bond chaos, i.e. calculate the probability of finding a given link overlap for two systems with possibly different sets of bonds. With this, we find the sample-to-sample fluctuations to scale with the number of spins,
N, with an exponent
μ in between
1/5 and
3/10.
Spin glasses are the paradigm of complex systems. These materials present really slow dynamics. However, the nature of the spin glass phase in finite dimensional systems is still controversial. Different theories describing the low temperature phase have been proposed: droplet, replica symmetry breaking and chaotic pairs. We present analytical studies of critical properties of spin glasses, in particular, critical exponents at and below the phase transition, existence of a phase transition in a magnetic field, computation of the lower critical dimension (in presence/absence of a magnetic field). We also introduce some rigorous results based on the concept of metastate. Finally, we report some numerical results regarding the construction of the Aizenman-Wehr metastate, scaling of the correlation functions in the spin glass phase and existence of a phase transition in a field, confronting these results with the predictions of different theories.