Anderson localization and ergodicity on random regular graphs

Abstract

A numerical study of Anderson transition on random regular graphs (RRG) with diagonal disorder is performed. The problem can be described as a tight-binding model on a lattice with N sites that is locally a tree with constant connectivity. In certain sense, the RRG ensemble can be seen as infinitedimensional (d → ∞) cousin of Anderson model in d dimensions. We focus on the delocalized side of the transition and stress the importance of finite-size effects. We show that the data can be interpreted in terms of the finite-size crossover from small (N Nc) to large (N Nc) system, where Nc is the correlation volume diverging exponentially at the transition. A distinct feature of this crossover is a nonmonotonicity of the spectral and wavefunction statistics, which is related to properties of the critical phase in the studied model and renders the finite-size analysis highly non-trivial. Our results support an analytical prediction that states in the delocalized phase (and at N Nc) are ergodic in the sense that their inverse participation ratio scales as 1/N .Introduction. Anderson localization [1] is a fundamental quantum phenomenon that remains in the focus of current research. A disordered quantum system can be driven (e.g., by increasing disorder) through Anderson transition (AT) between delocalized and localized phase [2]. For some class of models (defined on the Bethe lattice, a tree with constant connectivity) the problem of the AT allows for an exact solution, making it possible to establish the transition point and the corresponding critical behavior [3][4][5][6][7][8]. Recently, the Anderson localization on tree-like graphs has attracted much attention in view of its connections with problems of many-body localization in quantum dots [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and in extended systems with localized single-particle states .

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

“…The key conclusions of Ref. 55 have been corroborated by subsequent numerical studies of the RRG and SRM models 56,57 .…”

“…It is then natural to ask whether this localization is compatible with eigenfunction statistics at the root characteristic for delocalized (ergodic or fractal) regime, as found in Ref. 61.…”

“…Indications of a peculiar character of eigenstates on a Cayley tree were provided by Monthus and Garel 59,60 who studied numerically the statistics of transmisson amplitudes on a Cayley tree in a scattering geometry. Recently, two of the present authors studied, both analytically and numerically, the statistics of eigenfunctions in the root of a Cayley tree 61 . Our results proved that-in line with above expectations of a role of boundary conditions-the statistics on the Cayley tree is qualitatively different from that on RRG.…”

“…53 and 54. These works motivated an intensive numerical research on properties of the delocalized phase in the RRG and SRM models [55][56][57][58] . A detailed numerical investigation of level and eigenfunctions statistics on the delocalized side of the Anderson transition on RRG carried out in Ref.…”

“…A detailed numerical investigation of level and eigenfunctions statistics on the delocalized side of the Anderson transition on RRG carried out in Ref. 55 −1/2 implying an exponential divergence of the correlation volume at the transition point. For N N c the system exhibits a flow towards the Anderson-transition fixed point which has on RRG properties similar to the localized phase.…”