Many-body localization and new critical phenomena in regular random graphs and constrained Erdős-Renyi networks

V. Avetisov, A. Gorsky, S. Nechaev, O. Valba

Published 2016-11-25Version 1

We consider from the localization perspective the new critical behavior discovered recently for the regular random graphs (RRG) and constrained Erd\H{o}s-Renyi networks (CERN). The diagonal disorder for standard models, we replace by the fugacity $\mu$ of triads in the RRG and CERN. At some critical value of $\mu$ the network decays into the maximally possible number of almost full graphs, and the adjacency matrix acquires the two-gapped structure. We find that the eigenvalue statistics corresponds to delocalized states in the central zone, and to the localized states in the side one. The mobility edge lies between zones. We apply these findings to the many-body localization assuming the approximation of the hierarchical structure of the Fock space (for some interacting many-body system) by the RGG and by CERN with some vertex degree. We allow the 3-cycles in the Fock space and identify particles in the many-body system above the phase transition with clusters in the RRG. We discuss the controversial issue of the additional phase transition between ergodic and non-ergodic regions in the delocalized phase in the Fock space and find the strong "`memory-dependence"' of the states in the delocalized phase.