Non-ergodic delocalized phase in Anderson model on Bethe lattice and regular graph

Abstract

We develop a novel analytical approach to the problem of single particle localization in infinite dimensional spaces such as Bethe lattice and random regular graph models. The key ingredient of the approach is the notion of the inverted order thermodynamic limit (IOTL) in which the coupling to the environment goes to zero before the system size goes to infinity. Using IOTL and Replica Symmetry Breaking (RSB) formalism we derive analytical expressions for the fractal dimension D1 that distinguishes between the extended ergodic, D1 = 1, and extended non-ergodic (multifractal), 0 < D1 < 1 states on the Bethe lattice and random regular graphs with the branching number K. We also employ RSB formalism to derive the analytical expression ln Sfor the typical imaginary part of self-energy Styp in the non-ergodic phase close to the Anderson transition in the conventional thermodynamic limit. We prove the existence of an extended nonergodic phase in a broad range of disorder strength and energy and establish the phase diagrams of the models as a function of disorder and energy. The results of the analytical theory are compared with large-scale population dynamics and with the exact diagonalization of Anderson model on random regular graphs. We discuss the consequences of these results for the many body localization. Contents

Renormalization to localization without a small parameter

“…As mentioned in the first section, the presence of the delocalized states with large energies scaling with the system size (or cutoff value) causes the localization in long-range Euclidean matrices in the similar way as in the models with diagonal disorder and translation-invariant hopping terms [36] due to the leakage of most of the charge spectral weight to large energies (and measure zero of states). The lesson which one should take from this is the following: 11 Note that the case a < d/(d + 1) = 1/2 characterized by the delocalized eigenstates at the very spectral edge is an artefact of finite statistics in our numerical simulations. Indeed, in the renormalization group written for the infinite system with a certain cutoff, see, e.g., the bottom left of Fig.…”

Anderson localization on the Bethe lattice using cages and the Wegner flow

“…The qualitative results appear to be independent of the finite coordination number so long as the graph is tree-like, so one can extrapolate from the cubic case to higher-coordination trees. [49] This, along with replacing the tree with a cage, is done to make the problem numerically tractable, hopefully without sacrificing the most salient features of MBL. The Bethe lattice model has the clear advantage of being loopless, allowing for many analytic approaches to succeed with fewer approximations than the usual toy models of MBL, which have to contend with complicated many-particle configurations.…”

Multifractal dimensions for random matrices, chaotic quantum maps, and many-body systems

“…Multifractal statistics appears at the Anderson localization transition for single-particle lattice systems [17,[65][66][67][68][69][70][71]. In addition, recent examples have reported (multi)fractal phases extend-ing over a whole range of parameters [72][73][74][75][76][77][78][79][80][81][82][83][84][85][86]. Multifractal wavefunctions have been found for some quantum maps [68,70,87,88].…”

“…For local many-body quantum Hamiltonians, the ground states have been found to display multifractal behavior, even in cases for which eigenstates at the center of the many-body spectrum show random-matrix behavior [34,57,60,[89][90][91]. Also, the question of the existence of a multifractal phase in the vicinity of the many-body localization transition as well as its relation to the slow dynamical phases is under active debate [31,57,61,73,74,[92][93][94][95][96]).…”