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

by G. Biroli (SPhT CEA Saclay)

Tuesday 2 Oct 2012, 16:00 → 18:00 Europe/Rome

Aula Conversi (Dip. di Fisica - Edificio G. Marconi)

Quantum ergodic systems are characterized by eigenenergies statistics which are in the same universality class of Gaussian random matrices and by eigenfunctions that are delocalized. On the contrary, non-ergodic quantum systems, such as integrable models, display Poisson statistics of energy levels and localized wave-functions. Starting from Anderson's pioneering papers, similar properties have also been found for eigenstates of electrons hopping in a disordered environment. All that has lead to the conjecture that delocalization, 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 statistics of the energy levels is Poisson. This new phase, in which the system is non-ergodic but delocalized, may play an important role in several fields, in particular it could be related to the non-ergodic metallic phase conjectured to exist in the context of Many-Body Localization.