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We show that non-interacting disordered electrons on a Bethe lattice displaya new intermediate phase which is delocalized but non-ergodic, i.e. it ischaracterized by Poisson instead of GOE statistics. The physical signature ofthis phase is a very heterogenous transport that proceeds over a few disorderdependent paths only. We show that the transition to the usual ergodicdelocalized phase, which takes place for a disorder strength smaller than theone leading to the localization transition, is related to the freezing-glasstransition of directed polymers in random media. The numerical study of leveland eigenstate statistics, and of the singular properties of the probabilitydistribution of the local density of states all support the existence of thisnew intermediate phase. Our results suggest that the localization transitionmay change nature in high dimensional systems.

Difference between level statistics, ergodicity and localization transitions on the Bethe lattice