Motivated by the problem of many-body localization and the recent numerical results for the level and eigenfunction statistics on the random regular graphs, a generalization of the Rosenzweig-Porter random matrix model is suggested that possesses two transitions. One of them is the Anderson localization transition from the localized to the extended states. The other one is the ergodic transition from the extended non-ergodic (multifractal) states to the extended ergodic states. We confirm the existence of both transitions by computing the two-level spectral correlation function, the spectrum of multifractality f ( ) a and the wave function overlap which consistently demonstrate these two transitions.

We develop a semi-quantitative theory of electron pairing and resulting superconductivity in bulk "poor conductors" in which Fermi energy E F is located in the region of localized states not so far from the Anderson mobility edge E c . We assume attractive interaction between electrons near the Fermi surface. We review the existing theories and experimental data and argue that a large class of disordered films is described by this model.Our theoretical analysis is based on analytical treatment of pairing correlations, described in the basis of the exact single-particle eigenstates of the 3D Anderson model, which we combine with numerical data on eigenfunction correlations. Fractal nature of critical wavefunction's correlations is shown to be crucial for the physics of these systems.We identify three distinct phases: 'critical' superconductive state formed at E F = E c , superconducting state with a strong pseudogap, realized due to pairing of weakly localized electrons and insulating state realized at E F still deeper inside localized band. The 'critical' superconducting phase is characterized by the enhancement of the transition temperature with respect to BCS result, by the inhomogeneous spatial distribution of superconductive order parameter and local density of states. The major new feature of the pseudo-gaped state is the presence of two independent energy scales: superconducting gap ∆, that is due to many-body correlations and a new "pseudogap" energy scale ∆ P which characterizes typical binding energy of localized electron pairs and leads to the insulating behavior of the resistivity as a function of temperature above superconductive T c . Two gap nature of the pseudogapped superconductor is shown to lead to specific features seen in scanning tunneling spectroscopy and point-contact Andreev spectroscopy. We predict that pseudogaped superconducting state demonstrates anomalous behavior of the optical spectral weight. The insulating state is realized due to presence of local pairing gap but without superconducting correlations; it is characterized by a hard insulating gap in the density of single electrons and by purely activated low-temperature resistivity ln R(T ) ∼ 1/T .Based on these results we propose a new "pseudospin" scenario of superconductorinsulator transition and argue that it is realized in a particular class of disordered superconducting films. We conclude by the discussion of the experimental predictions of the theory and the theoretical issues that remain unsolved.

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...

We develop a theory of a pseudogap state appearing near the superconductor-insulator (SI) transition in strongly disordered metals with an attractive interaction. We show that such an interaction combined with the fractal nature of the single-particle wave functions near the mobility edge leads to an anomalously large single-particle gap in the superconducting state near SI transition that persists and even increases in the insulating state long after the superconductivity is destroyed. We give analytic expressions for the value of the pseudogap in terms of the inverse participation ratio of the corresponding localization problem.

Three recently suggested random matrix ensembles (RME) are linked together to represent a class of RME with multifractal eigenfunction statistics. The generic form of the two-level correlation function for the case of weak and extremely strong multifractality is suggested.PACS numbers: 72.15.Rn, 05.45+b Random matrix ensembles turn out to be a natural and convenient language to formulate generic statistical properties of energy levels and transmission matrix elements in complex quantum systems. Gaussian random matrix ensembles, first introduced by Wigner and Dyson 1,2 for describing the spectrum of complex nuclei, became very popular in solid state physics as one of the main theoretical tools to study mesoscopic fluctuations 3 in small disordered electronic systems. The success of the random matrix theory (RMT) 2 in mesoscopic physics is due to its extension to the problem of electronic transport based on the Landauer-Büttiker formula 4 and the statistical theory of transmission eigenvalues 5 . Another field where the RMT is exploited very intensively is the problem of the semiclassical approximation in quantum systems whose classical counterpart is chaotic 6 . It turns out 6 that the energy level statistics in true chaotic systems is described by the RMT, in contrast to that in the integrable systems where in most cases it is close to the Poisson statistics.Apparently the nature of the energy level statistics is related to the structure of eigenfunctions, and more precisely, to the overlapping between different eigenfunctions. This is well illustrated by spectral statistics in a system of non-interacting electrons in a random potential which exhibits the Anderson metal-insulator transition with increasing disorder. At small disorder the electron wave functions are extended and essentially structureless. They overlap very well with each other resulting in energy level repulsion characteristic of the Wigner-Dyson statistics. On the other hand in the localized phase electrons are typically localized at different points of the sample, and in the thermodynamic limit where the system size L → ∞ they "do not talk to each other". In this case there is no correlation between eigenvalues, and the energy levels follow the Poisson statistics.The energy level statistics in the critical region near the Anderson transition turns out to be universal and different from both Wigner-Dyson statistics and the Poisson statistics 7,8 . A remarkable feature of the critical level statistics is that the level number variance Σ 2 (N ) = (δN ) 2 = χN is asymptotically linear in the mean number of levelsN ≫ 1 in the energy window. Such a quasiPoisson behavior was first predicted in Ref. [9]. Later the existence of the linear term in Σ 2 (N ) was questioned 8 , since for this term to appear the normalization sum rule should be violated. It has been shown recently 10 that the new qualitative feature responsible for the violation of the sum rule and the existence of the finite "level compressibility" χ is the multifractality of critical w...

A new paradigm of Anderson localization caused by correlations in the long-range hopping along with uncorrelated on-site disorder is considered which requires a more precise formulation of the basic localizationdelocalization principles. A new class of random Hamiltonians with translation-invariant hopping integrals is suggested and the localization properties of such models are established both in the coordinate and in the momentum spaces alongside with the corresponding level statistics. Duality of translation-invariant models in the momentum and coordinate space is uncovered and exploited to find a full localization-delocalization phase diagram for such models. The crucial role of the spectral properties of hopping matrix is established and a new matrix inversion trick is suggested to generate a one-parameter family of equivalent localization/delocalization problems. Optimization over the free parameter in such a transformation together with the localization/delocalization principles allows to establish exact bounds for the localized and ergodic states in long-range hopping models. When applied to the random matrix models with deterministic power-law hopping this transformation allows to confirm localization of states at all values of the exponent in power-law hopping and to prove analytically the symmetry of the exponent in the power-law localized wave functions.

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

We consider the correlation of two single-particle probability densities $|\Psi_{E}({\bf r})|^{2}$ at coinciding points ${\bf r}$ as a function of the energy separation $\omega=|E-E'|$ for disordered tight-binding lattice models (the Anderson models) and certain random matrix ensembles. We focus on the models in the parameter range where they are close but not exactly at the Anderson localization transition. We show that even far away from the critical point the eigenfunction correlation show the remnant of multifractality which is characteristic of the critical states. By a combination of the numerical results on the Anderson model and analytical and numerical results for the relevant random matrix theories we were able to identify the Gaussian random matrix ensembles that describe the multifractal features in the metal and insulator phases. In particular those random matrix ensembles describe new phenomena of eigenfunction correlation we discovered from simulations on the Anderson model. These are the eigenfunction mutual avoiding at large energy separations and the logarithmic enhancement of eigenfunction correlations at small energy separations in the two-dimensional (2D) and the three-dimensional (3D) Anderson insulator. For both phenomena a simple and general physical picture is suggested.Comment: 16 pages, 18 figure