Spontaneous Breaking of U(N) symmetry in invariant matrix models
by
Fabio Franchini(TS)
→
Europe/Rome
Description
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, from holographic models to
supersymmetric theories in the localization limit. 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 to a sub-manifold of
the Hilbert space and that the system looses ergodicity.
We also consider models with log-normal weight, such as those emerging
in Chern-Simons and ABJM theories. 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, such as at the Anderson Metal/Insulator
Transition. 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, arguing that this structure is a reflection of the
non-trivial (multi-fractal) eigenvector statistics.
