Critical properties of the Anderson model in high dimension 

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Interactions and Many Body Localization

The problem of localization becomes more complex if we consider interactionsamong electrons: the fundamental question, which has remained unsolved for years, is if the picture we have so far survives or not once we add interactions. For the oneparticle problem, we know that, if the wave function of the electron at the Fermi surface is localized, we can have finite conductance if the electron is thermally activated above the mobility edge. Mott showed that, coupling the electrons with a phonon bath, electrons can hop between the localized states without thermal activation of the mobility edge (“variable range hopping”) [184]. For long time it was discussed if interactions among electrons could play an analogous role as electron-photon interactions, and thus restore a finite conductivity without thermal activation. The question remained unsolved until the work of Basko, Aleiner and Altshuler [48] in 2006: the authors showed that, if the temperature is finite but small enough, electron-electron interaction alone can not cause finite conductivity. Yet, there exist a critical temperature Tc such that for T > Tc the conductivity is finite. For such temperature the system of interacting electrons undergoes thus a phase transition, called Many Body Localization Transition.
The analysis in [48] consists in taking into account the interactions in perturbation theory, studying the inelastic quasiparticle relaxation, represented by the imaginary part of the sigle-particle self-energy. In a pictorial view, MBL can be thought of as localization in the Fock space of Slater determinants, which plays the role of lattice sites in a disordered one-particle Anderson tight-binding model. The problem of N 1 interacting particles in a finite dimensional lattice is thus interpreted as a one-particle localization problem on a very high dimensional lattice, which for spinless electrons consists in an N-dimensional hyper-cube of 2N sites. This makes the study of singleparticle Anderson Localization in very high dimension (and consequently the problem on the Bethe Lattice) an interesting issue for the comprehension of the problem with interactions. The idea of interpreting a many particle problem as a single-particle one in a very high dimensional space appeared in the context of the study of vibrational degrees of freedom of very big molecules [29], and was then applied in [47] to study the problem of electron-electron lifetime in a quantum dot. The work of Oganesyan and Huse [136], already cited in section (I.6), points out instead the link of MBL with Random Matrix Theory, and offer, at least numerically, a way to study the transition in relation to the level spacing distribution and ergodicity properties. We have to stress indeed that for MBL is particularly significant the description of the transition in terms of breaking of ergodicity (see section (I.6)): it is a transition between a thermal phase, in which we expect all the eigenstates to obey the Eigenstate Thermalization Hypothesis (ETH) [28], according to which each eigenstate is representative of the micro canonical ensemble, and the many body localized phase, in which the ETH stops to hold, and the dynamics conserve some memory of the local initial conditions. The study of the features of this transition has received much attention during the last years, and several questions are open, like the determination of the critical properties, and the presence of an intermediate mixed phase, delocalized but non-ergodic (“bad metal” phase), in analogy with the mixed phase argued for the single-particle problem on the Bethe Lattice (see next paragraph). In the language of MBL the “bad metal” phase is a phase in which thermalization and ergodicity is possible only on certain subregions of the configuration space.

The problem of the intermediate phase

In the work of 1997 [47] cited above, Altshuler and coworkers studied the problem of electron-electron lifetime in a quantum dot by mapping it into the problem of localization in the Fock space. In this study they found a delocalized and a localized regime, and they identified a broad critical region near the mobility edge on the side of the delocalized phase, in which the states are multifractal. As mentioned in the previous sections, multifractal states, which are extended but non-ergodic, i.e. occupy zero fraction of the lattice in the thermodynamic limit, exist in the problem of single24 particle Anderson Transition in three dimension, but only at the critical point. The authors of [47] suggested instead the presence of non-ergodic extended state in a whole region near the critical point. Following up this suggestion, and the analogy between MBL and the problem of one-particle localization on the Bethe Lattice, in [39] a numerical study of the one-particle problem on the Random Regular Graph (RRG) has been performed. A RRG of connectivity k + 1 is a graph selected at random among all the possible graph of connectivity k + 1 [185]. As we will explain more in detail in section (II.2.1), for a Bethe Lattice of finite size most sites belong to the boundary, and we expect therefore that boundary effects play an important role in numerical simulations, significantly affecting the results. The RRG is one of the possibility to construct a structure which is locally a tree-like graph, but which has no boundary: it essentially corresponds to a finite portion of a Bethe Lattice wrapped into itself2. The main result of the analysis of Ref. [39] is that the mobility edge computed from the (numerical) solution of the self-consistent equations [8, 186], and the transition revealed observing the level spacing distribution seem to not coincide. In particular, an intermediate phase which is delocalized but which does not show GOE-like behavior seems to be
present. It is however complicated to establish if the observed behavior is the sign of a real intermediate phase transition, or the consequence of very strong finite size effects: it the latter case the data should be interpreted in terms of a finite-size crossover. The answer to this question is non trivial, and the work of Ref. [39] remained unpublished on purpose. More recently, other works on the Anderson Model on the RRG have investigated the problem of the intermediate phase, focusing in particular on the analysis of the statistics of extended wave functions, finding that these statistics may indeed be multifractal [40, 41]. In [44], combining numerical and semi-analytical calculation, has been then found an evidence for a order transition between ergodic and non-ergodic states within the delocalized phase in RRG. The delocalized non-ergodic phase would imply heterogeneous behavior at the level of transport and diffusion: the particle can travel far away from a given site, but only following specific and disorder dependent paths. In the language of MBL, this means that thermalization is possible only on certain subregions of the Hilbert space. As we will see in section (II.3), the presence of such intermediate phase would be in contradiction with the results obtained by the supersymmetric method for the behavior of the moments of the distribution of wavefunction coefficients. The question of the existence of such phase for the single-particle problem on the Bethe Lattice and its implications for MBL is still a subject of research and discussion. We have investigated the problem of the existence of an intermediate phase of this type in Lévy Matrices, which, as we will explain in section (III.1), present connections with the problem of Localization on Bethe Lattice, and could actually be an interesting system to consider in order to understand the mean field properties of Anderson Localization. The possibility of the existence of an intermediate delocalized non-ergodic phase has been advocated for this kind of matrices by [38], and this is one of the reason which motivated us to study this model. Our results support the idea that the delocalized phase of Lévy Matrices is ergodic, both in the sense of the level statistics and the wavefunction statistics.

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Dyson Brownian motion model

As we have seen in section (I.6), thanks to the invariance under rotations, RMT provides an explicit expression for the joint probability distribution P(x1, . . . , xN) of the eigenvalues of N × N matrices belonging to the three classical Gaussian ensembles [106]: P(x1, . . . , xN) = CN e−W, (II.4.1) W = 12 XN j=1 x2j − X i<j log |xi − xj | , (II.4.2).
where = 1, 2 or 4 respectively for the GOE, GUE and GSE case (see section (I.6)). This expression is identical to the probability density of the position of N unit charges on an infinite straight line −1 < x < 1 subjected to the potential energy (I.6.2). If the parameter is identified with the inverse temperature = (KBT)−1, the computation of averages over the distribution (II.4.1) is equivalent to the computation of thermodynamic quantities. With this picture, referred to as the Coulomb gas model, the problem of averaging over Gaussian ensembles is mapped into a statistical mechanics model. Dyson extended this idea, in such a way that the Coulomb gas model acquires meaning not only on a thermodynamic point of view, but also as a dynamical system out of equilibrium: to do that, the variables xj must be interpreted as positions of particles in Brownian motion [209, 210, 211]. Each particle has therefore no inertia and is subjected to a fluctuating force fj and to a frictional force proportional to the velocity: the motion of the particles is thus described by the following system of Langevin equations:  dxj dt = − dW0.

The recursion equation for the resolvent

In section (II.2) we have presented the cavity approximation, which is based on the assumption that the Gaussian probability measure (II.2.2), used to write the resolvent in the form of a Gaussian integral over auxiliary fields i, factorizes over the sites of cavity graph: this assumption is exact on the Bethe Lattice (see section (II.2.1)), thanks to the particular tree-like structure in which loops are absent. We can show that such assumption is justified also in the case of Lévy Matrices, using the fully-connected structure of the model and the generalized central limit theorem. The starting point is the expression of the resolvent (II.2.1) of a system of size N in  terms of a Gaussian integral over auxiliary fields i. Following the authors of Ref. [38] we can imagine to add a row k and its symmetric columns to the matrix H: this, in terms of the tight binding representation of the matrix is equivalent to add a site to  the system. For the resolvent of the system with (N + 1) sites, if we integrate over all fields except k, we obtain G (N+1) kk = i R dk 2 k exp h0.

Table of contents :

Introduction
I The Anderson Localization Transition: introduction 
I.1 Disorder and Localization
I.2 Characteristics of the transition and localized states
I.3 Anderson Localization in one and two dimensions and weak localization
I.4 Scaling theory and field theory formulation
I.5 Anderson transition on the Bethe Lattice: mean field
I.6 Anderson Localization and Random Matrix Theory
I.7 Brief review on numerical results
I.8 Experiments on Localization
I.9 Open problems
I.9.1 Interactions and Many Body Localization
I.9.2 The problem of the intermediate phase
II Overview on analytical techniques and known results 
II.1 Definitions
II.2 Cavity equations
II.2.1 On the Bethe Lattice
II.3 Supersymmetric method
II.4 Dyson Brownian motion model
IIILocalization Transitions of Lévy Matrices 
III.1 Introduction and motivations
III.2 The recursion equation for the resolvent
III.3 The density of states
III.4 Computation of the mobility edge
III.4.1 The mapping to directed polymers in random media
III.5 Numerical check of the phase diagram
III.6 The problem of the intermediate phase: previous results
III.7 The Supersymmetric method applied to Lévy Matrices
III.7.1 Equation on R() with the supersymmetric method
III.8 The Dyson Brownian motion argument
III.9 Numerical results for μ 2 (1, 2)
III.10 Numerical results for μ 2 (0, 1)
III.10.1 Numerical results for Q(G)
III.10.2 Wavefunction statistics and multifractal spectrum
III.11 Summary of the results
IV Critical properties of the Anderson model in high dimension 
IV.1 Numerical results in d = 3, . . . , 6
IV.1.1 Transport properties
IV.1.2 Statistics of level spacings and of wave-functions coefficients .
IV.2 Strong Disorder RG
IV.3 Summary of the results
Conclusion and perspectives
A Transfer Matrix, conductance and localization length 
A.1 Transfer Matrix and conductance
A.2 Transfer Matrix and localization length
B Critical wave functions: multifractality 
C The IPR in terms of the Green function 
D The DoS of Sparse RM model with the supersymmetric method 
E Meaning of the order parameter function 
F The generalized central limit theorem 
G Computation of the mobility edge of Lévy Matrices 
List of Figures
Bibliography 

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