Scaling Theory of the Anderson Transition in Random Graphs: Ergodicity and Universality

Abstract

We study the Anderson transition on a generic model of random graphs with a tunable branching parameter 1 < K ≤ 2, through large scale numerical simulations and finite-size scaling analysis. We find that a single transition separates a localized phase from an unusual delocalized phase which is ergodic at large scales but strongly non-ergodic at smaller scales. In the critical regime, multifractal wavefunctions are located on few branches of the graph. Different scaling laws apply on both sides of the transition: a scaling with the linear size of the system on the localized side, and an unusual volumic scaling on the delocalized side. The critical scalings and exponents are independent of the branching parameter, which strongly supports the universality of our results.Ergodicity properties of quantum states are crucial to assess transport properties and thermalization processes. They are at the heart of the eigenstate thermalization hypothesis which has attracted enormous attention lately [1]. A paramount example of non-ergodicity is Anderson localization where the interplay between disorder and interference leads to exponentially localized states [2]. In 3D, a critical value of disorder separates a localized from an ergodic delocalized phase. At the critical point eigenfunctions are multifractal, another non trivial example of non-ergodicity [3,4]. Recently, those questions have been particularly highlighted in the problem of many-body localization [5][6][7][8][9]. Because Fock space has locally a tree-like structure, the problem of Anderson localization on different types of graphs [10][11][12][13][14][15] has attracted a renewed activity [16][17][18][19][20][21][22][23][24][25]. In particular, the existence of a delocalized phase with non-ergodic (multifractal) eigenfunctions lying on an algebraically vanishing fraction of the system sites is debated [19-21, 23, 25].The problem of non-ergodicity also arises in another context corresponding to glassy physics [26]. For directed polymers on the Bethe lattice [27], a glass transition leads to a phase where a few branches are explored among the exponential number available. As there is a mapping to directed polymer models in the Anderson-localized phase [10,[28][29][30], it has been recently proposed that this type of non-ergodicity (where the volume occupied by the states scales logarithmically with system volume) could also be relevant in the delocalized phase [18]. Note however that it has been envisioned that this picture could be valid only up to a finite but very large length scale [31].In this letter, we study the Anderson transition (AT) in a family of random graphs [32][33][34], where a tunable parameter p allows us to interpolate continuously between the 1D Anderson model and the random regular graph model of infinite dimensionality. Our main tool is the single parameter scaling theory of localization [35]. It has been used as a crucial tool to interpret the numerical simulations of Anderson localization in finite dimensions [3,[36][37][38] and to achieve...

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

“…Correspondingly, no evidence of non-ergodic extended phase was reported in the earlier works 34,35,39,40 that predicted asymmetric critical behavior. The conclusion that D = 1 in the delocalized regime was reached in recent mostly numerical studies 44,50 that we discuss in detail below. However, very recently two papers 41 reported the existence of the non-ergodic extended, multifractal phase found in the framework of the non-linear sigma-model on a finite Cayley tree.…”

“…We start with the very recent work 44 . This work studied the model in the regimes were non-ergodic phase is expected to be narrow or disappear, so such conclusion is to be expected.…”

“…In the NEE phase both of them are given by the large-deviation ansatz P (ln Z) = A exp ln M f ln Z ln M , cf. (44). This is a very general type of distribution function characterized by a function f (α) and a large parameter M → ∞.…”

Renormalization to localization without a small parameter

“…Moreover, the insets show that the above collapse works relatively well until the maximum of |q ε (R)| 2 , but not only in the bulk of the spectrum. Finally, our theory predicts that, for a < d, |q ε (R)| 2 as a function of ε must have a sharp maximum at ε * max ∼ R d−a 0 with the magnitude of the order of R d 0 , see (24) and the corresponding discussion. By combining these two estimates we get the one describing the energy dependence of the maximal magnitude with increasing cutoff R…”

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

“…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.…”