Anderson Localization on the Bethe Lattice: Nonergodicity of Extended States

Abstract

Statistical analysis of the eigenfunctions of the Anderson tight-binding model with on-site disorder on regular random graphs strongly suggests that the extended states are multifractal at any finite disorder. The spectrum of fractal dimensions f (α) defined in Eq.(3), remains positive for α noticeably far from 1 even when the disorder is several times weaker than the one which leads to the Anderson localization, i.e. the ergodicity can be reached only in the absence of disorder. The oneparticle multifractality on the Bethe lattice signals on a possible inapplicability of the equipartition law to a generic many-body quantum system as long as it remains isolated.Introduction.-Anderson localization (AL) [1,2], in its broad sense, is one of the central paradigms of quantum theory. Diffusion, which is a generic asymptotic behavior of classical random walks [3], is inhibited in quantum case and under certain conditions it ceases to exist [2]. This concerns quantum transport of noninteracting particles subject to quenched disorder as well as transport and relaxation in many-body systems. In the latter case the many-body localization (MBL) [4] can be thought of as localization in the Fock space of Slater determinants, which play the role of lattice sites in a disordered tight-binding model. In contrast to a d-dimensional lattice, the structure of Fock space is hierarchical [5]: a twobody interaction couples a one-particle excitation with three one-particle excitations, which in turn are coupled with five-particle excitations, etc. This structure resembles a random regular graph (RRG) -a finite size Bethe lattice (BL) without boundary. Interest to the problem of single particle AL on the BL [6,7] has recently revived [8][9][10][11][12] largely in connection with MBL. It is a good approximation to consider hierarchical lattices as trees where any pair of sites is connected by only one path and loops are absent. Accordingly the sites in resonance with each other are much sparser than in ordinary d > 1-dimensional lattices. As a result even the extended wave functions can occupy zero fraction of the BL, i.e. be nonergodic. The nonergodic extended states on 3D lattices where loops are abundant are commonly believed [13][14][15][16] to exist but only at the critical point of the AL transition.In this paper we analyze the eigenstates of the Anderson model on RRG with connectivity K + 1 (K is commonly used to refer to the branching of the corresponding BL) and N sites:where ψ(i) (i = 1, ..., N ) can be characterized by the moments[13] (I 1 = 1 for the normalization). One can define the ergodicity as the convergence in the limit N → ∞ of the real space averaged |ψ(i)| 2q (equal to I q /N ) to its ensemble average value |ψ(i)| 2q = I q /N . This happens when the fluctuations of |ψ(i)| 2 are relatively weak andThe latter condition turns out to be both necessary and sufficient for the convergence of I q to I q (see Supplementary Materials for the discussion). Deviations of τ (q) from q − 1 are signatures of the nonergodic state. I...

Low-frequency conductivity in many-body localized systems

“…It is not presently clear whether or not the stretched-exponential effective interactions that occur at putative critical points within the MBL phase [9,14,21,22] substantially modify the above story. Also, our analysis of near-transition behavior assumes that the delocalized phase is thermal, and thus may not apply to hypothesized transitions between an MBL phase and a nonthermal delocalized phase [47][48][49].…”

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

“…[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].…”

Early Breakdown of Area-Law Entanglement at the Many-Body Delocalization Transition

“…These effects cause an overestimate of the stability of the MBL phase, which actually becomes unstable to thermalization earlier only at much longer length scales. Note that an interesting alternate possibility is the existence of an intermediate phase between the ETH and MBL phase, with neither thermal nor area-law entanglement [64][65][66], which we do not pursue further. This onset of thermalization at high order is what one would expect from a long lengthscale delocalization mechanism.…”