Both the static and the dynamical behaviour occurring in mean field spin glass models models can be interpreted as consequences of the complex (free) energy landscape that spin glasses have, with many minima, valleys and saddles. Traditionally, much attention has been devoted in the past to the analysis of absolute minima, i.e. equilibrium states. More recently, we have understood that also metastable states, i.e. higher energy minima, can play an important role. In fact, for some spin glass models high energy metastable states can trap the asymptotic dynamics and determine the behaviour of the system at long times.
In general, disordered systems are characterized by the presence of an exponentially large number of metastable states, and there is a finite entropy, called Complexity or Configurational Entropy, related to them. Whether these states are relevant or not, depends on their individual stability and on the robustness of their overall structure. In our recent research, we discovered that two classes of models exist where the structure of states is radically different. We developed a field-theory able to describe such a structure, and showed that robust structures with stable states are described by a supersymmetric theory, while fragile structures of marginal states do break the supersymmetry. These two classes correspond to different dynamical behaviours and display different stability properties towards external perturbations.
Piu’ recentemente, abbiamo mostrato quale siano le conseguenze di cio’ per una tecnica molto utilizzata in vari campi della meccanica statistica quale il metodo della cavita’.
At a formal level, the features of the free energy landscape and the properties of its minima can also be investigated using the Cavity Method, a mathematical approach which allows to write self-consistent equations for the local magnetizations of the system. The stable solutions of these equations correspond to minima of the free energy and identify ground states and metastable states of the system. The Cavity Method is a very powerful approach, which can be generalized and applied to several different instances of complex disordered models, such as neural networks and optimization problems. Our results on super symmetry breaking also have consequences in the Cavity approach. For some models, the standard Cavity Method must be generalized to appropriately deal with sub-optimal marginal solutions.
Cavity Method for Supersymmetry-breaking Spin-Glasses A. Cavagna, I. Giardina and G. Parisi Phys. Rev. B 71, 024422 (2005).
Off-equilibrium confined dynamics in a system with level-crossing states. B. Capone, T. Castellani, I. Giardina, F Ricci-Tersenghi, Phys. Rev. B 74, 144301 (2006)
Metastable states in glassy systems, I. Giardina. In “Les Houches – Session LXXXV: Complex Systems”, J.-P Bouchaud, M. Mezard and J. Dalibard eds., Elsevier, Amsterdam, 2007.