Abstract
 
 
Matrix Models have a strong history of success in describing a variety of situations, from nuclei spectra to conduction in mesoscopic systems, from strongly interacting systems to various aspects of mathematical physics. Traditionally, the requirement of base invariance has lead to a factorization of the eigenvalue and eigenvector distribution and, in turn, to the conclusion that invariant models describe extended systems. Moreover, Wigner-Dyson statistics for the eigenvalues is a hallmark of eigenvector delocalization. We show that deviations of the eigenvalue statistics from the Wigner-Dyson universality reflects itself on the eigenvector distribution and that a gap in the eigenvalue density breaks  the U(N) symmetry to a smaller one. This spontaneous symmetry breaking means that egeinvectors become localized and that the system looses ergodicity.
We also consider models with log-normal weight. Their eigenvalue distribution is intermediate between Wigner-Dyson and Poissonian, which candidates these models for describing a system intermediate between the extended and localized phase. We show that they have a much richer energy landscape than expected, with their partition functions decomposable in a large number of equilibrium configurations, growing exponentially with the matrix rank. We will discuss the meaning of this energy landscape and its implication, commenting on the perspectives for localization problems.
 
- F. Franchini; "On the Spontaneous Breaking of U(N) symmetry in invariant Matrix Models"; arXiv:1412.6523.
- F. Franchini; "Toward an invariant matrix model for the Anderson Transition"; arXiv:1503.03341.