Multifractality of wave functions on a Cayley tree: From root to leaves

Abstract

We explore the evolution of wave-function statistics on a finite Bethe lattice (Cayley tree) from the central site ("root") to the boundary ("leaves"). We show that the eigenfunction moments Pq = N |ψ| 2q (i) exhibit a multifractal scaling Pq ∝ N −τq with the volume (number of sites) N at N → ∞. The multifractality spectrum τq depends on the strength of disorder and on the parameter s characterizing the position of the observation point i on the lattice. Specifically, s = r/R, where r is the distance from the observation point to the root, and R is the "radius" of the lattice. We demonstrate that the exponents τq depend linearly on s and determine the evolution of the spectrum with increasing disorder, from delocalized to the localized phase. Analytical results are obtained for the n-orbital model with n 1 that can be mapped onto a supersymmetric σ model. These results are supported by numerical simulations (exact diagonalization) of the conventional (n = 1) Anderson tight-binding model.

Anderson transition on the Bethe lattice: an approach with real energies

“…In several recent works [42,41,44] it has been suggested that the delocalized phase of the Anderson model on the RRG with N sites has two distinct phases: one in which the participation ratios of the eigenfunctions is ∝ N and one in which it is ∝ N α for some α < 1. This was questioned in several other papers [48,49,50] (however a genuine multifractal region was shown to exist for the finite Cayley tree with N vertices [54,65,66], and to control the RRG properties below a given critical volume [67]); to the best of our understanding the question is whether this phase exists on the RRG or it is washed away when going from the Cayley tree to the RRG. This question can not be addressed within the framework discussed in this work.…”

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

“…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]).…”

Multifractality in random networks with power-law decaying bond strengths

“…Our approach may be used to investigate the multifractality of eigenfunctions in other random network models. Indeed, similar studies have been already performed to explore the multifractality of eigenfunctions of the Anderson model on Cayley trees (AMCT) [13,53,54] and random graphs [55]. It is relevant to stress that there are three important differences between the network model studied here and the AMCT studied in Refs.…”

“…[13,53]: (i) Cayley trees have a fixed degree (the AMCT in [13,53] is characterized by k = 3), while due to the randomnetwork nature of the dPBRM model the degree is defined as an average quantity here. (ii) The dPBRM model represents networks with randomly-weighted bond strengths between vertices, while the AMCT in [13,53] is defined as a network with constant bond strengths. (iii) The dPBRM model possesses an infinite line of critical points characterized by the parameter b ∈ (0, ∞) (that we did not examine here since we fixed b = 1 in Eq.…”