## Difference between Ergodicity, Level Statistics and Localization Transitions on the Bethe Lattice

### Giulio Biroli, IPhT Saclay

Random Matrix Theory was initially developed to explain the eigen-energy distribution of heavy nuclei. It has become clear by now that its domain of application is much broader and extends to very different fields such as number theory and quantum chaos, just to cite a few. In particular, it has been conjectured—and proved or verified in some special cases—that quantum ergodic (or chaotic) systems are characterized by eigen-energies statistics in the same universality class of random matrices and by eigen-functions that are delocalized over the configuration space. On the contrary, non-ergodic quantum systems, such as integrable models, are expected to display a Poisson statistics of energy levels and localized wave-functions. Starting from Anderson’s pioneering papers, similar properties have also been studied for electrons hopping in a disordered environment. Remarkably, also in this case, similar features of the energy-level statistics have been found. All that has lead to the conjecture that delocalization in configuration space, ergodicity and level statistics are intertwined properties.

In this talk we revisit the old problem of non-interacting electrons hopping on a Bethe lattice with on-site disorder. By using numerical simulations, the cavity method and mapping to directed polymers in random media we unveil the existence of an intermediate phase in which wave-functions are delocalized but the energy-level statistics is Poisson. This new phase, in which the system is non-ergodic but delocalized, may play an important role in several fields from random matrix theory to strongly interacting quantum disordered systems, in particular it could be related to the non-ergodic metallic phase conjectured to exist in the context of Many-Body Localization.