A random matrix model with localization and ergodic transitions

Abstract

Motivated by the problem of many-body localization and the recent numerical results for the level and eigenfunction statistics on the random regular graphs, a generalization of the Rosenzweig-Porter random matrix model is suggested that possesses two transitions. One of them is the Anderson localization transition from the localized to the extended states. The other one is the ergodic transition from the extended non-ergodic (multifractal) states to the extended ergodic states. We confirm the existence of both transitions by computing the two-level spectral correlation function, the spectrum of multifractality f ( ) a and the wave function overlap which consistently demonstrate these two transitions.

Many-body-localization transition: strong multifractality spectrum for matrix elements of local operators

“…Note that this linear spectrum for the Localized phase has already been found for the MBL case in [113] and for an Anderson Localization matrix model in [53].…”

“…36 (as found for Anderson localization on trees [51,112] or as in some matrix model [53]). One general constraint is however that the minimal exponent has to be strictly positive…”

“…[45,46]. This delocalized non-ergodic scenario is usually explained within the point of view that the MBL transition is somewhat similar to an Anderson Localization transition in the Hilbert space of 'infinite dimensionality' as a consequence of the exponential growth of the size of the Hilbert space with the volume [47][48][49][50][51][52][53]. Note however that the existence of a non-ergodic delocalized phase remains very controversial even for Random-Regular-Graphs that are locally tree-like without boundaries, as shown by the two very recent studies with opposite conclusions [54,55].…”

Integrable matrix theory: Level statistics

“…Despite this fact, integrable matrices do exhibit a parameter-dependent localization property [43] in which almost all eigenstates of the matrix H(x) = xT + V are localized in the basis of V for all values of x. The stability of this property when a random matrix perturbation is added to H(x), including the possibility of a multifractal phase accompanying the localized and delocalized regimes [40], is the subject of future study.…”

“…(11) and Refs. 24, 25. It is interesting to note that many-body localized [36] (MBL) systems are also expected to display Poisson level statistics [37,38], and there exist random matrix ensembles which model localization and its statistical signatures [39,40]. Such ensembles are basis-dependent, which is natural because localization is a basis-dependent property.…”