Beyond the Mean Field Approximation for Spin Glasses

We investigate the replica symmetry broken (RSB) phase of spin glass (SG) models in a random field defined on Bethe lattices at zero temperature. From the properties of the RSB solution we deduce a closed equation for the extreme values of the cavity fields. This equation turns out not to depend on the parameters defining the RSB, and it predicts that the spontaneous RSB does not take place homogeneously on the whole system. Indeed, there exist spins having the same effective...

We review recent numerical progress in the study of finite dimensional strongly disordered magnetic systems like spin glasses and random field systems. In particular we report in some details results for the critical properties and the non-equilibrium dynamics of Ising spin glasses. Furthermore we present an overview over recent investigations on the random field Ising model and finally of quantum spin glasses.

We use the Bethe approximation to calculate the critical temperature for the transition from a paramagnetic to a glassy phase in spin-glass models on real-world graphs. Our criterion is based on the marginal stability of the minimum of the Bethe free energy. For uniform degree random graphs (equivalent to the Viana-Bray model) our numerical results, obtained by averaging single problem instances, are in agreement with the known critical temperature obtained by use of the repl...

A scheme to provide various mean-field-type approximation algorithms is presented by employing the Bethe free energy formalism to a family of replicated systems in conjunction with analytical continuation with respect to the number of replicas. In the scheme, survey propagation (SP), which is an efficient algorithm developed recently for analyzing the microscopic properties of glassy states for a fixed sample of disordered systems, can be reproduced by assuming the simplest r...

Infinite-range spin-glass models with Levy-distributed interactions show a freezing transition similar to disordered spin systems on finite connectivity random graphs. It is shown that despite diverging moments of the local field distribution this transition can be analyzed within the replica approach by working at imaginary temperature and using a variant of the replica method developed for diluted systems and optimization problems. The replica-symmetric self-consistent equa...

We discuss the mean-field theory of spin-glass models with frustrated long-range random spin exchange. We analyze the reasons for breakdown of the simple mean-field theory of Sherrington and Kirkpatrick. We relate the replica-symmetry breaking to ergodicity breaking and use the concept of real replicas to restore thermodynamic homogeneity of the equilibrium free energy in a replicated phase space. Embedded replications of the spin variables result in a set of hierarchical fre...

We introduce a mean field spin glass model with gaussian distribuited spins and pairwise interactions, whose couplings are drawn randomly from a normal gaussian distribution too. We completely control the main thermodynamical properties of the model (free energy, phase diagram, fluctuations theory) in the whole phase space. In particular we prove that in thermodynamic limit the free energy equals its replica symmetric expression.

An extensive list of results for the ground state properties of spin glasses on random graphs is presented. These results provide a timely benchmark for currently developing theoretical techniques based on replica symmetry breaking that are being tested on mean-field models at low connectivity. Comparison with existing replica results for such models verifies the strength of those techniques. Yet, we find that spin glasses on fixed-connectivity graphs (Bethe lattices) exhibit...

A longstanding open question in the theory of disordered systems is whether short-range models, such as the random field Ising model or the Edwards-Anderson model, can indeed have the famous properties that characterize mean-field spin glasses at nonzero temperature. This article shows that this is at least partially possible in the case of the random field Ising model. Consider the Ising model on a discrete $d$-dimensional cube under free boundary condition, subjected to a v...

In this thesis, we review and examine the replica method from several viewpoints. The replica method is a mathematical technique to calculate general moments of stochastic variables. This method provides a systematic way to evaluate physical quantities and becomes one of the most important tools in the theory of spin glasses and in the related discipline including information processing tasks. In spite of the effectiveness of the replica method, it is known that several pro...