Localization transition on the Random Regular Graph as an unstable tricritical point in a log-normal Rosenzweig-Porter random matrix ensemble
V. E. Kravtsov, I. M. Khaymovich, B. L. Altshuler, L. B. Ioffe
LLocalization transition on the Random Regular Graph as an unstable tricritical pointin a log-normal Rosenzweig-Porter random matrix ensemble.
V. E. Kravtsov,
1, 2
I. M. Khaymovich, B. L. Altshuler,
4, 5 and L. B. Ioffe
6, 7 Abdus Salam International Center for Theoretical Physics - Strada Costiera 11, 34151 Trieste, Italy L. D. Landau Institute for Theoretical Physics - Chernogolovka, Russia Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187-Dresden, Germany Department of Physics, Columbia University, New York, NY 10027, USA Russian Quantum Center, Skolkovo, Moscow Region 143025, Russia Department of Physics, University of Wisconsin – Madison, Madison, WI 53706 USA Google Inc., Venice, CA 90291 USA
Gaussian Rosenzweig-Porter (GRP) random matrix ensemble is the only one in which the robustmultifractal phase and ergodic transition have a status of a mathematical theorem. Yet, this phasein GRP model is oversimplified: the spectrum of fractal dimensions is degenerate and the mini-bandin the local spectrum is not multifractal. In this paper we suggest an extension of the GRP modelby adopting a logarithmically-normal (LN) distribution of off-diagonal matrix elements. A familyof such LN-RP models is parametrized by a symmetry parameter p and it interpolates between theGRP at p → and Levy ensembles at p → ∞ . A special point p = 1 is shown to be the simplestapproximation to the Anderson localization model on a random regular graph. We study in detailthe phase diagram of LN-RP model and show that p = 1 is a tricritical point where the multifractalphase first collapses. This collapse is shown to be unstable with respect to the truncation of thelog-normal distribution. We suggest a new criteria of stability of the non-ergodic phases and provethat the Anderson transition in LN-RP model is discontinuous at all p > . I. INTRODUCTION
The development of Quantum Computing algorithmsand the problem of Many Body Localization (MBL) [1] ininteracting systems (e.g. in spin chains) ignited an inter-est to the single-particle (Anderson) localization on ran-dom graphs as a proxy for MBL. The analogy comes fromthe mapping in which bit strings of spins- / (describ-ing many-body configurations) correspond to sites on agraph and interaction provides transitions between thesebit strings represented by a link between sites. The bitstrings directly accessible from a given one are uniquelydetermined by the interaction Hamiltonian, so does thestructure and topology of the corresponding graph.Since the seminal work [2] there is a mounting evidencethat a representative class of interacting systems can bemodeled by a graph with a local tree structure but with-out a boundary. The simplest graph of such kind is a ran-dom regular graph (RRG), Fig. 1, which was suggestedin Refs. [3, 4] as a toy model for many-body localization.In particular, it was conjectured in these works [3, 4]that a special Non-Ergodic Extended (NEE) phase is re-alized on RRG which random eigenfunctions are multi-fractal. In view of the above analogy with the systemof interacting qubits this statement, if correct, is of ex-treme importance. Recently such NEE states were sug-gested [5, 6] as mediators for implementing efficient pop-ulation transfer in the Grover’s Quantum Search Algo-rithms with potential application to Machine Learning.While the existence of NEE phase on RRG is still un-der debate [4, 7–13] it is rigorously proven to exist ina much simpler Rosenzweig-Porter (RP) random matrixmodel [14–20]. On the face of it, the RP model seemsto have little to do with RRG. However, in this paper FIG. 1. (Color online)
Random regular graph with thebranching number K + 1 = 3 and N = 8 . we show that the modification of RP model, the logarith-mically normal RP model (LN-RP), offers the simplestapproximation to RRG which accounts for a key differ-ence between an RRG and a finite Cayley tree.This LN-RP model is important on its own, as it in-terpolates between the Gaussian RP model and the Lévyrandom matrix models (see, e.g., [21, 22] and referencestherein) with power-law distribution of the off-diagonalmatrix elements. We introduce a symmetry parameter p that controls this interpolation with p = 0 correspondingto the Gaussian RP model and p = ∞ corresponding tothe Lévy RP model. a r X i v : . [ c ond - m a t . d i s - nn ] M a r A particular case of the Anderson model on RRG cor-responds to p = 1 due to the hidden β -symmetry (seeEqs. (6.5)-(6.8) in Ref. [23], Eqs. (D.2), (D.17) in Ref. [4]and Appendix C) on the local Cayley tree. We showthat p = 1 is the tricritical point such that for p < theLN-RP model supports the NEE phase, while at p ≥ a direct transition from the localized to the Ergodic Ex-tended (EE) phase occurs. However, this EE phase isunstable, as it results from large off-diagonal matrix ele-ments from the tail of the logarithmically-normal distri-bution. Any truncation of this tail (as well as breakingthe β -symmetry) is shown to lead to the reappearance ofthe NEE phase separating localized and EE phases.The analytical theory of the Ergodic (ET) and Local-ization transitions (AT) developed in this paper is verifiedby extensive numerics based on the Kullback-Leibler di-vergence [24, 25] of certain correlation functions of wavefunction coefficients [26]. We also present the theory ofthis new measure of eigenfunction statistics.The rest of the paper is organized as follows. In Sec. IIwe provide the motivation of the model with long-taileddistribution of off-diagonal elements and its relevance tothe Anderson problem on RRG. In Sec. III we describeLN-RP model and introduce the symmetry parameter p .In Sec. IV we present the phase diagram of LN-RP modelbased on the Anderson localization and Mott ergodicitycriteria. Section V shows the numerical data confirmingthe analytical predictions of Sec. IV by making use of thetwo types of Kullback-Leibler divergence. The criticalvalues of both ergodic and localization transitions withthe corresponding critical exponents are extracted fromthe exact diagonalization with finite-size scaling analysis.In Sec. VI we show that for large symmetry parameter p ≥ the system is unstable with respect to the emer-gence of the multifractal phase by pushing the ergodictransition to smaller disorder values. In Sec. VII we de-velop a new theory of stability of non-ergodic states withrespect to their hybridization. The analytical theory ofthe wave function support set fractal dimension is pre-sented in Sec. VIII. Conclusions and discussion of theresults are given in Sec. IX. The Appendices A – F givethe details of the derivation of the above results and in-depth discussion of the new methods and ideas. II. DEFINITIONS AND EMERGENCE OFLONG-RANGE MODELS ON TIGHT-BINDINGHIERARCHICAL GRAPHS
Let us first sketch the mapping of the Anderson local-ization model with nearest neighbor hopping on RRG tothe log-normal Rosenzweig-Porter random matrix modelwith infinite-range hopping. The reader interested in theproperties of the LN-RP model itself can go directly tothe next section.The random K -regular graph (RRG) of N sites is agraph in which any site is connected to K + 1 other sitesin a random way (see Fig. 1). The hopping amplitude p ( r ) rN = d = FIG. 2. (Color online)
Distribution of distances r be-tween two points on RRG of N = 2 = 8192 sitesand branching number K = 2 . The diameter of the graph d = 16 is only by 4 larger than the most probable distance r ∗ = 12 , both being approximately equal to ln N/ ln K in thelimit N → ∞ . V = 1 between nearest neighbor sites is fixed for all links.The Anderson localization model on RRG adds a randompotential ε n fluctuating independently at any site n withthe site-independent distribution F ( ε ) characterized bythe variance (cid:10) ε n (cid:11) ∼ W .RRG is known to have locally a Cayley-tree structurewith the branching number K . However, in contrast toa finite Cayley tree which has a strict hierarchical struc-ture and grows from a special point (the root) onwards,any point of RRG can be considered as a root of a localCayley tree. This is because RRG has loops (of whichoverwhelming majority are long loops with the length ofthe order of the diameter of the graph) which connectone local tree with the other thus making all points ofthe graph statistically equivalent.The local tree structure and the predominance of longloops on RRG lead to the exponential growth of the dis-tribution p N ( r ) ∼ K r of distances r between pairs ofpoints on this graph. This growth lasts nearly up to themaximal distance on RRG (the diameter d ), followed byan abrupt drop to zero, Fig. 2 (see also Fig. 12 in [4]). Themost probable distance, r = r ∗ , differs from the graph di-ameter d only by few extra links and for large graphs of N points they are approximately equal r ∗ (cid:39) d (cid:39) ln N/ ln K .Moreover, due to the exponential growth of p N ( r ) with r , in the thermodynamic limit N → ∞ a finite fraction ofpairs of sites on RRG is at the most probable distance, p N →∞ ( r ∗ ) → f > . This “condensation of large dis-tances” is the crucial point for the analogy between RRGand RP models.Indeed, let’s consider the set of equally spaced siteson RRG with the most abundant distance r ∗ ≈ d − (see Fig. 2). An effective random matrix model involv-ing only such (“marked”) sites which constitute a finitefraction ( ≈ at K = 2 ) of all sites on RRG canbe written using the Anderson impurity model (see Ap-pendix A for more details). Those marked sites interactwith each other through the remaining tree sites similarto the indirect interaction between Anderson impuritiesin a metal. This indirect interaction is long-range , as wellas the RKKY interaction [27] of magnetic impurities me-diated by electrons in a metal. Since all the marked sitesare at a distance r ∗ (cid:39) d from each other, the effectivehopping matrix elements between them H eff ,n (cid:54) = m can beall taken independent and identically distributed (i.i.d.).Thus we arrive at an infinite-range random matrix modelof the Rosenzweig-Porter type. The diagonal matrix ele-ments H eff ,nn = ε n have the same statistics F ( ε ) as theon-site energies on RRG.In the same way as RKKY interaction depends onthe details of the Fermi surface in a metal, in our casethe effective hopping matrix elements H eff ,n (cid:54) = m betweenmarked sites encode the hierarchical structure of the treesites in their distribution function. In Appendices B – Cwe show that in contrast to the Gaussian Rosenzweig-Porter (RP) random matrix theory (RMT) [14, 15], thedistributions of diagonal F ( ε ) and off-diagonal P ( U ) matrix elements for a RP model associated with RRGare drastically different. Unlike F ( ε ) which is compact , (cid:104) ε n (cid:105) ∝ (cid:104) ε (cid:105) n , the distribution of P ( U ) of H eff ,n (cid:54) = m hasa fat tail which makes any moment (cid:104)| U | q (cid:105) ∼ N − γ q of U characterized by its own exponent γ q , the averages withlarge enough q being divergent.For not very large branching number K the distribu-tion of P ( U ) associated with RRG in its delocalized phaseis logarithmically normal . The log-normal distribution ofeffective hopping reflects the hierarchical structure of thetree sites and follows from the representation of H eff ,n (cid:54) = m as a product of one-point Green’s functions on a treealong the path between the sites n and m . Due to therandom graph structure the logarithm ln H eff ,n (cid:54) = m is rep-resented by the sum of large number r ∗ of nearly indepen-dent elements and, thus, its distribution can be approxi-mated by Gaussian (see Appendices A and C for detailsof derivation and limits of applicability). This leads tothe log-normal distribution of the hopping term itself.We will see that the Rosenzweig-Porter RMT withthe compact distribution of diagonal elements and log-normal distribution of off-diagonal elements (LN-RP) isvery rich with numerous potential applications which jus-tify its detailed consideration independently of the anal-ogy with the Anderson model on RRG. III. LOG-NORMAL ROSENZWEIG-PORTERRMT
As shown in Appendices B – C, the distribution func-tion P ( U ) of off-diagonal elements of the RP ensembleassociated with RRG is of the “multifractal” form of the large deviation ansatz : P ( U ) ∼ U − exp (cid:20) − ln N G (cid:18) ln U ln N (cid:19)(cid:21) , (1) where G ( x ) is a certain function and the large param-eter ln N is proportional to the diameter of RRG. Thisform is very special, as ln N appears both in front of thefunction G ( x ) and in its argument. It emerges in manydifferent physical problems ranging from distribution ofamplitudes of multifractal wave functions to statistics ofwork in driven systems out-of-equilibrium [28].The simplest choice of G ( x ) is a linear function whichgives rise to a power-law distribution P ( U ) . However,in many relevant cases where the Central Limit Theoremapplies to ln U , the distribution P ( U ) is log-normal whichcorresponds to a parabolic function G ( x ) : P ( U ) = AU exp (cid:34) − ln ( U/U typ )2 p ln( U − ) (cid:35) , U typ ∼ N − γ/ . (2)This distribution is controlled by two parameters: theparameter γ > that governs the typical value of thehopping matrix element in LN-RP model and the sym-metry parameter p .The reason we refer to this parameter as the symmetryparameter is related to the basic symmetry on the Cayleytree (see Appendix C) which gives rise to the dualityrelation: P ( U − ) = U P ( U ) , ⇔ p = 1 . (3)When applied to Eq. (2) this relation requires p = 1 .However, it is useful to keep this parameter free to inter-polate between the LN-RP with long-tail cut in P ( U ) (the case p → which is equivalent to the GaussianRP [14, 15]) and the case p → ∞ when P ( U ) approachesthe Lévy power-law distribution.Another model parameter γ is related to the Lyapunovexponent λ on the disordered Cayley tree. It is defined [4]via the exponential decay of the typical absolute value ofthe Green’s function | G r | typ with the distance r : λ = − lim r →∞ r − ln | G r | typ . (4)By substituting r = r ∗ (cid:39) d = ln N/ ln K and | G r | typ ∼ N − γ/ in Eq. (4) one immediately obtains: γ = 2 λ ln K . (5)As shown in Appendix C, the log-normal distribution of P ( U ) is asymptotically exact for RRG at small disorder.It is also quantitatively accurate in the entire delocalizedphase for not very large branching number K .It is important to note that for the distribution Eq. (2)the scaling of the typical value of U with N differs fromthat of the mean value. The latter exists only for p < and is given by (cid:104) U (cid:105) ∼ N − γ av / , with γ av = γ (1 − p/ , (0 < p < . (6) IV. PHASE DIAGRAM OF LN-RP. COLLAPSEOF THE MULTIFRACTAL PHASE AT THETRICRITICAL POINT AT p = 1 . It was first shown in Ref. [15] that the Rosenzweig-Porter RMT with a Gaussian P ( U ) has three phases:ergodic, γ < γ ET , multifractal, γ ET < γ < γ AT , andlocalized, γ > γ AT and two transitions between them at γ ET = 1 and γ AT = 2 . The same transition points areexpected for the LN-RP in the limit p → . In this sectionwe consider simple “rule of thumb” criteria formulated inRefs. [29, 30] which show how the phase diagram of LN-RP is modified as the symmetry parameter p increases.The physical picture and details of these transition andthe corresponding phases will be considered in Sec. VII.The first criterion, nicknamed as Anderson localizationcriterion , applies to random matrices with uncorrelatedentries and states [29, 30] that if the sum: S = N (cid:88) m =1 (cid:104)| H n,m |(cid:105) W < ∞ (7)converges in the limit N → ∞ then the states are An-derson localized. Here (cid:104) .. (cid:105) W stands for the disorder av-eraging.The physical meaning of this criterion is that the num-ber is sites in resonance with a given site n is finite. In-deed, consider for simplicity the box-shaped distribution F ( ε ) of on-site energies. The probability that two sites n and m are in resonance is: P n → m = W − (cid:90) W/ − W/ dε n (cid:90) W/ − W/ dε m (cid:90) ∞| ε n − ε m | P ( H nm ) d ( H nm ) . (8)Then simple integration over ( ε n + ε m ) / and integrationby parts over ε n − ε m gives: P n → m = (cid:90) W dU P ( U ) (cid:18) UW − U W (cid:19) + (cid:90) ∞ W P ( U ) dU, (9)where U = | H nm | .One can easily see that at U typ ∼ N − γ/ (cid:28) O (1) thelast integral in Eq. (9) is always small and the secondterm in the first integral is at most of the same order asthe first term. Thus with the accuracy up to a constantof order 1 we obtain: P n → m ∼ (cid:104)| H nm |(cid:105) W W , (10)where the subscript W in (cid:104) ... (cid:105) W implies that the distri-bution P ( U ) should be truncated at U max = W . Thenumber of sites in resonance with the given site is thesum (cid:80) m P n → m which coincides with Eq. (7) up to a pre-factor of order O (1) .Importantly, the above derivation (9) gives an elabo-ration to Eq. (7). Indeed, in the case of the long-tailed distribution P ( U ) , one should cut it off at U max = W ∼ O (1) in order to obtain a correct sufficient criterion ofAnderson localization. Note that such a cutoff of P ( U ) is automatically embedded into the RRG/LN-RP corre-spondence (see Appendix B).The second criterion suggested in Refs. [29, 30] andnicknamed in [30] the Mott’s criterion is a sufficient cri-terion of ergodicity. It states that if the sum S = N (cid:88) m =1 (cid:104)| H nm | (cid:105) W → ∞ (11)diverges in the limit N → ∞ then the system is in the(fully) ergodic phase [30].Note that similar to Eq. (7), the averaging in Eq. (11)should be done with the distribution truncated at U max ∼ O (1) . The reason for that is that rare large matrix el-ements | H nm | (cid:29) O (1) split the resonance pair of levelsso much that they are pushed at the Lifshitz tail of thespectrum and do not affect statistics of states in the bodyof spectrum that we are studying [31].The physical meaning of Eq. (11) is that the Breit-Wigner width Γ that quantifies the escape rate of a par-ticle created at a given site n , is much larger than thespread of energy levels W ∼ O (1) due to disorder. Inother words, the fulfillment of the Mott’s criterion impliesthat the width Γ is of the same order as the total spec-tral bandwidth and thus there are no mini-bands (whichwidth is Γ ) in the local spectrum. As the presence ofsuch mini-bands is suggested [32–34] as a “smoking gun”evidence of the non-ergodic extended (e.g. multifractal)phase, the fulfillment of the Mott’s criterion (11) imme-diately implies that the system is in the ergodic extended phase.The above non-ergodic extended (e.g., multifractal)phase realizes provided that in the limit N → ∞ : S → ∞ , S → . (12)The case of a finite S in the limit N → ∞ is moredelicate and may imply merely weak ergodicity [35].The fact that it is the second moment of | H nm | = U which enters Eq. (11) is related to the Fermi Golden Ruledetermining the Breit-Wigner width: Γ = 2 π N (cid:88) m =1 (cid:10) ρ m | H nm | (cid:11) W ≈ π ρ N (cid:88) m =1 (cid:104)| H nm | (cid:105) W , (13)where ρ m and ρ ∼ W − are the density of final states andthe density of on-site energies, respectively. The pertur-bative Eq. (13) is valid as long as Γ (cid:46) W and one canneglect contribution of off-diagonal matrix elements H nm to the density of states. Then the total spectral band-width is limited by W . In the opposite limit Γ (cid:29) W thetotal spectral bandwidth ρ − (cid:39) Γ is dominated by theoff-diagonal matrix elements and should be determinedself-consistently ρ N (cid:88) m =1 (cid:104)| H nm | (cid:105) W ∼ . (14) multifractal localized ergodic p γ FIG. 3. (Color online)
The phase diagram of thelogarithmically-normal Rosenzweig-Porter RMT.
Theergodic transition (orange) and the Anderson localizationtransition (blue) lines merge at the tricritical point p = 1 (which is associated with RRG) and γ = 4 . This criticalpoint corresponds (see Eq. (5), (6)) to the Lyapunov expo-nent λ = 2 ln K , or to λ av ≡ γ av ln K = γ ln K = ln K which is the known criterion of the Anderson transition ona Cayley tree [4, 36]. For p < the transition from the lo-calized to the ergodic phase goes through the intermediatemultifractal phase; for p ≥ a direct transition happens fromthe localized to the ergodic phase. This means that in the fully ergodic phase the total spec-tral bandwidth is blowing up with increasing Nρ − ∼ [ S ( N )] → ∞ . (15)For the log-normal distribution (2) one easily computesthe moments (7) and (11) truncated at U max ∼ O (1) : (cid:104) U q (cid:105) W = (cid:40) N − γq ( − pq ) , if pq < N − γ p , if pq ≥ (16)and using this equation finds: S = (cid:40) N − γ ( − p ) , if p < N − γ p , if p ≥ , (17) S = (cid:26) N − γ (1 − p ) , if p < / N − γ p , if p ≥ / , (18)leading to the critical points of the localization ( γ AT )and ergodic ( γ ET ) transitions from the conditions (7)and (11) that S or S , respectively, are of order O (1) : γ AT = (cid:26) − p , if p < p, if p ≥ (19) γ ET = (cid:26) − p , if p < / p, if p ≥ / (20) The resulting phase diagram for the log-normalRosenzweig-Porter ensemble, Eq. (2), which is the mainresult of this paper, is presented in Fig. 3.It is remarkable that the point p = 1 which correspondsto the Anderson model on RRG, is the tricritical pointon this phase diagram. At this point in the pure log-normal RP ensemble and in LN-RP ensemble with P ( U ) truncated at U > U max ∼ O (1) , the multifractal phasevanishes. However, as we demonstrate in Sec. VI, thecollapse of the multifractal phase at p ≥ and thus thetricritical point, is unstable with respect to any trunca-tion of P ( U ) with U max (cid:28) O (1) . V. KULLBACK-LEIBLER (KL) MEASURE
The numerical verification of the phase diagram anddetermination of the critical exponents at the Andersonlocalization and ergodic transitions is done in this pa-per using the Kullback-Leibler divergence (KL) [24–26]of certain correlation functions of random eigenstates (formore detailed multifractal analysis of this model see [37]).The Kullback-Leibler correlation functions KL and KL are defined as follows [26]. The first one is definedin terms of wave functions of two neighboring in energy states: KL (cid:88) i | ψ α ( i ) | ln (cid:18) | ψ α ( i ) | | ψ α +1 ( i ) | (cid:19) . (21)The second one is similar but the states ψ and ˜ ψ cor-respond to different (and totally uncorrelated) disorderrealizations: KL (cid:88) i | ψ ( i ) | ln (cid:18) | ψ ( i ) | | ˜ ψ ( i ) | (cid:19) . (22)The idea to define such two measures is the following.In the ergodic phase each of the states has an amplitude | ψ ( i ) | ∼ N − of the same order of magnitude. Thenthe logarithm of their ratio is of order O (1) , and for thenormalized states KL ∼ KL ∼ O (1) . (23)For fully-ergodic states in the Wigner-Dyson limit theeigenfunction coefficients are fully uncorrelated, even forthe neighboring in energy states. Thus there is no differ-ence between KL and KL . Using the Porter-Thomasdistribution one finds: KL KL . (24)Deeply in the localized phase ln | ψ α ( i ) | ∼ −| i − i α | /ξ ,where i α is the position of the localization center. Sincethe positions of localization centers i α are not correlatedeven for the states neighboring in the energy, KL and KL are proportional to L ∼ ln N and divergent in thethermodynamic limit: KL ∼ KL ∝ ln N → ∞ . (25) γ KL1 γ p = KL1 γ p = KL1 γ p = KL1 W3D AM
10 12 14 16 18 202.02.53.03.54.04.5
KL2 γ γ p = KL2 γ p = KL2 γ p = KL2 W3D AM
10 12 14 16 18 20 22 245101520
FIG. 4. (Color online)
Plots of KL and KL vs. γ for LN-RP model at N = 2 L , with L from to with the step (from red to violet) and vs. W for the d Anderson model at N = L , with L = 8 , , , , , , . The logarithmic in N divergence of KL for γ > γ AT ≈ and of KL for γ > γ ET ≈ is demonstrated in a wide interval of γ for p = 0 . , as wellas insensitivity of KL to the ergodic transition. Intersection for KL γ ) curves is sharp at the isolated continuous ergodictransition at γ ET ≈ for p = 0 . and at γ ET ≈ . for p = 0 . , it is smeared out for p = 1 . when the ergodic transitionmerges with the localization transition, and it disappears completely for three-dimensional ( d) Anderson model. Intersectionof curves for KL at the Anderson localization transition ( γ AT ≈ . for p = 0 . , γ AT ≈ . for p = 0 . , γ AT ≈ . for p = 1 ,and W ≈ . for 3D Anderson model) is sharp in all the cases. The properties of KL, Eqs. (24), (25) are fully confirmedby numerics presented in Fig. 4.A qualitative difference between KL and KL is inthe multifractal NEE phase. In this phase the neigh-boring in energy states | ψ α ( i ) | and | ψ α +1 ( i ) | are mostprobably belonging to the same support set and hencethey are strongly overlapping: | ψ α ( i ) | ∼ | ψ α +1 ( i ) | .Furthermore, eigenfunctions on the same fractal supportset can be represented as: ψ α ( i ) = Ψ( i ) φ α ( i ) , where Ψ( i ) is the multifractal envelope on the support set and φ α ( i ) is the fast oscillating function with the Porter-Thomasstatistics [3]. Thus the ratio | ψ α ( i ) | / | ψ α +1 ( i ) | and hence KL has the same statistics as in the ergodic phase. Inother words, KL is not sensitive to the ergodic transi-tion but is very sensitive to the localization one, Fig. 4.In contrast, the eigenfunctions ψ ( i ) and ˜ ψ ( i ) corre-sponding to different realizations of a random Hamilto-nian in KL , overlap very poorly. This is because thefractal support sets which contain a vanishing fraction ofall the sites, do not typically overlap when taken at ran-dom. Therefore the ratio ln( | ψ ( i ) | / | ˜ ψ ( i ) | ) ∼ ln N in KL is divergent in the thermodynamic limit in the multifrac-tal phase, very much like in the localized one. This makes KL very sensitive to the ergodic transition, Fig. 4.A more detailed theory of KL and KL in the mul-tifractal phase is given in Appendix D. The main con-clusion of the analysis done in Appendix D is that thecurves for KL γ, N ) for different N have an intersec-tion point at the critical point γ = γ AT of the Andersonlocalization transition. At the same time, the intersec-tion point for curves for KL γ, N ) coincides with theergodic transition [26], provided that it is continuous andwell separated from the Anderson localization transition.If the localization and ergodic transition merge together and the multifractal state exists only at the transitionpoint (as in 3D Anderson model) then intersection of KL curves is smeared out and may disappear whatso-ever. However, the intersection of KL curves remainssharp in this case too (see Fig. 4).The intersection of finite-size curves for KL and KL helps to locate numerically the critical points γ AT and γ ET . First, we checked that for the well studied d Anderson transition the intersection point of KL curves exactly corresponds to the known critical disor-der W ≈ . , while KL curves show no intersectionwhatsoever (see Fig. 4). The results for γ AT and γ ET forLN-RP model are shown in the Table 6. They coincide(for p > after the extrapolation) with the theoreticalprediction Eq. (19), (20) with the deviation less than .The next step is to analyze the finite-size scaling (FSS)by a collapse of the data for KL and KL at different N in the vicinity of the localization and ergodic transition,respectively. To this end we use the form of FSS derivedin Appendix D: KL (ln N | γ − γ AT | ν ) , (26a) KL − KL c ( N ) = Φ (ln N | γ − γ ET | ν ) . (26b)The input data for the collapse is KL and KL versus γ and W for values of N is shown in Fig. 4. The fittingparameters extracted from the best collapse are ν ( ν )and the critical points γ AT ( γ ET ). The critical valueof KL c ( N ) = KL γ ET , N ) is determined by the bestfitting for γ ET . For the localization transition where thecritical point γ AT is well defined by the intersection in KL , one may look for the best collapse by fitting only ν .This procedure of the finite-size scaling has been testedfor the 3D Anderson model with sizes L = 8 − . The ν = γ c = N = - p = KL1 / KL1c x = ( γ - γ c ) ( lnN ) / ν KL1c lnN - - -
10 0 10 20 300.00.51.01.52.02.5
KL2c lnN
KL2 - KL2c p = x = ( γ - γ c ) ( lnN ) / ν ν = γ c = N = - -
50 0 50 - - KL2 - KL2c
KL2c lnN p = x = ( γ - γ c ) ( lnN ) / ν ν = γ c = N = - - - - FIG. 5. (Color online)
The best collapse of the KL and KL data for LN-RP with p = 1 and p = 0 . . The collapse for KL and KL is done in the vicinity of the localization (for KL ) and ergodic (for KL ) transitions by recursive procedurethat finds γ c and ν by minimizing the mean square deviation of data from a smooth scaling function which is updated atany step of the procedure. (insets) The critical value of KL and KL as a function of ln N . It stays almost a constantfor KL and for KL at p = 0 . when the ergodic transition is continuous and well separated from the localized one but itgrows logarithmically in N at p = 1 when the ergodic and localization transitions merge together. This growth is the reasonof smearing of the intersection of KL curves in Fig. 4. The exponent ν significantly depends on p and is consistent with ν ≈ ν = 0 . at p = 1 and ν = 1 at p = 0 . . results for the scaling collapse of data are presented inAppendix E. Note that in this case there is no inter- p AT g ET g {extr},[theor] {extr},[theor] FIG. 6.
Comparison of analytical predictions (blue),Eq. (19), (20), and numerical data for the transitionpoints γ AT and γ ET for LN-RP model. Numerical data(black) is obtained by exact diagonalization of LN-RP randommatrices with N = 512 − from the intersection pointsin KL and KL , Fig. 4 and from finite-size scaling, Fig. 5.For p > a linear in / ln N extrapolation to N → ∞ ofthe position of the intersection point for two consecutive N isshown in red. p n n – –– –– –– – –– –– FIG. 7. Critical exponents ν and ν in the finite-size scaling,Eq. (26), obtained from the best collapse of KL and KL data, respectively, see Fig. 5. In the [ ... ] are the conjecturedvalues of ν and ν . n n n p FIG. 8. (Color online)
The conjectured dependence ofthe critical exponents ν and ν on the symmetry pa-rameter p for LN-RP at the localization and ergodic tran-sitions, respectively. In the limit p → the critical expo-nents approach their values ν = ν = 1 for the Gaussian RPmodel [26]. For p ≥ we conjecture the mean-field values ν = ν = 1 / . In the interval < p < the critical expo-nents of the Anderson ( ν ) and ergodic ( ν ) transitions aredifferent with ν > ν . section in KL whatsoever (see Fig. 4). Yet, the bestcollapse corresponds to a well-defined W c ≈ which isreasonably close to the value W c = 16 . found from theintersection in KL and known in the literature. Thisencourages us to use the best collapse of KL data todetermine γ ET and ν for LN-RP model where the inter-section of KL curves does exists, albeit smeared out.The results are shown in the Tables 6, 7 while represen-tative samples of the data collapse are shown in Fig. 5.On the basis of these numerical results we conjecture thedependence of the critical exponents ν and ν on thesymmetry parameter which is shown in Fig. 8. γ γ tr γ ET γ AT Ergodic Multifractal Localized p = γ γ tr = γ AT γ ET p FIG. 9. (Color online)
Phase diagram of LN-RP model with U typ ∼ N − γ truncated at U max ∼ N − γ tr ( γ > γ tr > ). (Left panel) Phase diagram in the plane γ − γ tr at a fixed value p = 1 of the symmetry parameter. (Right panel) Phase diagramin the plane γ − p at fixed γ tr = 0 . . At any γ tr > the multifractal NEE phase emerges at p ≥ and fills the gap between theergodic and localized phases. At a small γ tr the Anderson localization transition (blue line) is almost unaffected by truncation,while the ergodic transition (orange line) is pushed to smaller values of γ . Thus the multifractal NEE phase substitutes theergodic one as the truncation parameter γ tr increases demonstrating the fragility of the ergodic phase which existence is dueto atypically large values of the transition matrix elements U . VI. TRUNCATED LN-RP AND FRAGILITY OFERGODIC PHASE.
The main result, Fig. 3, of Sec. IV confirmed numer-ically in Sec. V is the collapse of the multifractal phaseat p ≥ and existence of the tricritical point in LN-RP which is associated, via the qualitative arguments ofSec. II, with the localization transition on RRG.In this section we show that the ergodic phase thatemerges at the localization transition in this tricriticalpoint (and for all p ≥ ) is unstable with respect to adeformation of LN-RP model such that P ( U ) is cut fromabove at: U max ∼ N − γ tr / (cid:28) O (1) , ( γ tr > . (27)As the result of this truncation the multifractal phasere-appears by substituting a part of the ergodic phase ina non-truncated LN-RP model (see Fig. 9) [38]. To thisend we use the expression that generalizes Eq. (16): (cid:90) N − γ tr / dU U q P ( U ) ∼∼ N − qγ ( − pq ) , γ tr < γ (1 − pq ) N − pγ (cid:20) ( γ − γ tr)24 + pq γγ tr (cid:21) , γ tr > γ (1 − pq ) (28)and apply the same criteria Eq. (7), (11) to find thecritical points of the localization and ergodic transitions.Then we obtain for the critical point γ AT of the Andersonlocalization transition: γ AT = 2 p − ( p − γ tr + (cid:113) (2 p − ( p − γ tr ) − γ tr , (29)if γ tr > γ AT (1 − p ) . In the opposite case truncation doesnot affect γ AT . For the critical point γ ET of the ergodic transition inthe same way we find for γ tr > γ ET (1 − p ) : γ ET = 2 p − (2 p − γ tr + (cid:113) (2 p − (2 p − γ tr ) − γ tr . (30)The results of Eq. (29), (30) are plotted in Fig. 9.One can see that at any positive non-zero γ tr the mul-tifractal NEE phase emerges at p ≥ in between of thelocalized and ergodic ones. At small γ tr the line of lo-calization transition is almost insensitive to truncation,while the line of ergodic transition is pushed to smallervalues of γ corresponding to larger typical transition ma-trix elements U (smaller effective disorder). This provesthe fact that the ergodic phase in LN-RP with p ≥ isvery fragile and exists only due to atypically large transi-tion matrix elements. It is substituted by the multifrac-tal NEE phase as soon as such matrix elements are madeimprobable by truncation.We believe that this scenario of the multifractal phaseemergence at p ≥ is quite generic and happens for thewide class of perturbations of the LN-RP model [39]. Inthe case of RRG corresponding to the tricritical point, p = 1 , of the non-truncated LN-RP model, the effectof the local Cayley tree structure in the exact mappingof the Anderson model on RRG onto LN-RP model isunexplored in detail and might, in principle, lead to aneffective truncation of the above type. In any scenario thetricritical nature of p = 1 point in LN-RP makes this case(and the corresponding case of RRG) significantly morecomplicated and different from the conventional Ander-son localization transition in finite dimensions. This isthe reason, in our opinion, of the long-lasting debateson the existence of NEE phase in the Anderson model onRRG (see the debates in [4, 7–11] and references therein). a)b) c) FIG. 10. (Color online)
Hybridization of fractal supportsets (a), (b) Two different fractal support sets, (c) The hy-bridized fractal support set.
VII. STABILITY OF NON-ERGODIC STATESAGAINST HYBRIDIZATION
In this section we consider the stability of non-ergodic(multifractal and localized) states against hybridization.It allows us not only to derive expressions, Eqs. (7)and (11), for the lines of the Anderson localization andergodic transitions in a different way but also find inSec. VIII the fractal dimension D ( p, γ ) of the multifrac-tal support set. Furthermore, the new method presentedbelow is physically transparent and generic enough to beapplied to analysis of the multifractal NEE states in othersystems.Let us consider two states ψ µ ( i ) and ψ ν ( i ) on differentfractal support sets as it is shown in Fig. 10(a) and (b).We assume that both states are multifractal with m ∼ N D sites on a fractal support set where | ψ ( i ) | ∼ N − D .The key new element in the theory we are introducinghere is the transmission matrix element V µ,ν between the states and not between the sites as we did in the previoussections V µ,ν = (cid:88) i,j G ij ψ µ ( i ) ψ ν ( j ) , (31)where G ij is the two-point Green’s function.Introducing g ij = − ln G ij / ln N and suppressing theindices i, j for brevity we conveniently rewrite Eq. (C9)as follows: P ( g ) = const N − pγ ( g − γ ) , ( g ≥ . (32)By the constraint g ≥ we implemented a cutoff at G max ∼ O (1) discussed in Section IV and Appendix B.The typical number of terms in the sum Eq. (31) with g in the interval dg is N D N D P ( g ) ∼ N σ ( g,D ) dg where σ ( g, D ) = 2 D − pγ (cid:16) g − γ (cid:17) . (33)If σ ( g, D ) < (region I in Fig. 16 of Appendix F), thesum, Eq. (31), is dominated by a single term with the largest G ij . For positive σ ( g, D ) > (region II in Fig. 16of Appendix F), many terms contribute to this sum andthe distribution P ( V ≡ | V µ,ν | ) becomes Gaussian. Ingeneral, there are both contributions given by P ( V ) = (cid:90) g ∈ I dg N σ ( g,D ) δ ( V − N − D − g )+ (cid:90) g ∈ II dg P ( g ) (cid:42) δ V − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) ij G ij ψ µ ( i ) ψ ν ( j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:43) . (34)The condition of stability of the multifractal phaseagainst hybridization is derived similar to the Andersoncriteria of stability, Eq. (7), of the localized phase donein Sec. VII. The difference is that now we have to re-place the matrix element between the resonant sites U bythe matrix element V between the resonant non-ergodicstates and take into account that on each of M = N − D different support sets there are m = N D wave functionswhich belong to the same mini-band and thus are alreadyin resonance with each other. Therefore the total numberof independent states-candidates for hybridization with agiven state should be smaller than the total number ofstates M m = N and larger than the number of supportsets M . This number is in fact equal to their geometricmean √ N M = M √ m = N − D .With this comment, the criterion of stability of themultifractal phase reads in the limit N → ∞ as N − D (cid:90) W dV V P ( V ) < ∞ . (35)The contribution of the Gaussian part of P ( V ) toEq. (35) is: N − D (cid:112) (cid:104) V (cid:105) = N − D − γ eff ( D ) < ∞ , (36)where (cid:104) V (cid:105) ≡ N − γ eff . (37)The contribution of the first (log-normal) term inEq. (34) to the stability criterion is: N − D (cid:90) g ∈ I dg N σ ( g,D ) − g − D ≡ N − D − ∆( D ) < ∞ . (38)Thus the multifractal phase is stable against hybridiza-tion if the following inequalities are both fulfilled D + 12 γ eff ( D ) ≥ , (39) D + ∆( D ) ≥ . (40)The function γ eff ( D ) and ∆( D ) are computed in Ap-pendix F.A particular case D = 0 of Eqs. (39), (40) describesthe stability criterion of the localized phase. If the0 Δ ( α )/ γ + α p / / γ AT p / α γ eff ( α )/ γ + α / γ D ( γ )/ γ p < / D / γ Δ ( α )/ γ + α α / γ AT / < p < / γ p / / D ( γ )/ γγ eff ( α )/ γ + α D / γ / γ α D / γ = / γ /( p ) p > γ eff ( α )/ γ + α Δ ( α )/ γ + α / γ AT FIG. 11. (Color online)
The functions (41) (blue curve) and (42) (orange curve) entering inequalities Eqs. (39), (40)in different regions of p : (left) p < / ; (middle) / ≤ p ≤ ; (right) p > . Intervals of α = D /γ with different functionaldependence are shown by dashed vertical lines. The Anderson localization transition corresponds to the lower of the blue andorange curves equal to /γ at α = 0 . This transition is always determined by the orange curve representing the log-normal partof the distribution P ( V ) . On the contrary, the stable fractal dimension D ( γ ) = 2 − γ/γ ET ( p ) for γ ≤ γ AT is always determinedby the blue curve representing the Gaussian part of the distribution P ( V ) . The Anderson transition in all cases but p = 0 is discontinuous , with the minimal stable fractal dimension of the support set being D min1 = D ( γ AT ) = 2 − γ AT /γ ET ( p ) > (shown by a gray dotted arrow). The ergodic transition corresponds to D ( γ ) = 1 and it is continuous . For p ≥ there is nosolution D < to the system of inequalities Eqs. (39), (40) in the region of parameters where the localized phase is unstable.In this case the multifractal phase is absent. localized phase is not stable, then hybridization pro-duces an avalanche of multifractal states living on frac-tal support which dimensionality grows until inequalitiesEqs. (39), (40) are both first fulfilled for some < D < .If this is possible in some parameter region then the mul-tifractal state is stable. If the only solution to the systemof inequalities Eq. (39), (40) corresponds to D ≥ thenthe only stable extended phase is ergodic. VIII. FRACTAL DIMENSION OF THE NEESUPPORT SET
In this section we re-consider the phase diagram of non-truncated LN-RP, γ tr ≤ , from the viewpoint of stabilitycriteria given in the previous section by Eqs. (39), (40)and derive the expression for the fractal dimension D ( γ ) of the support set of multifractal wave functions.To this end in Fig. 11 we plot γ eff ( α ) γ + α = (cid:26) α − √ αp, α < p, p /γ ET ( p ) + α, otherwise , (41)and α ) γ + α = α − √ αp, p < α < p /γ AT ( p ) − α, α < p, p α + 2 √ αp, α > p , (42)calculated in Appendix F as functions of α = D /γ .Here γ AT ( p ) ≥ and γ ET ( p ) ≥ are given by Eqs. (19)and (20), respectively.According to the stability criteria Eqs. (39), (40) thefunctions Eqs. (41), (42) should be compared to /γ , seeFig. 11. First, we note that the localized phase whichformally corresponds to D = 0 , is stable if the lowestof the blue and orange curves in Fig. 11 is higher than /γ at α = 0 and it is unstable otherwise. One can see that at α = 0 for all values of p the log-normal con-tribution (orange curve) is lower than the Gaussian one(blue curve). This means that the stability of the local-ized phase is always determined by the log-normal partof P ( V ) . Moreover, since at α = 0 Eqs. (41), (42) re-duce to α + γ eff ( α ) /γ = 1 and α + 2∆( α ) /γ = 2 /γ AT ,respectively, the stability of the localized phase impliesthat γ > γ AT ( p ) ≥ in agreement with (19).If the localized phase is unstable then different local-ized states hybridize and form a multifractal state with D > . Those states are, however, unstable until theirsupport set reaches the fractal dimension D min > where Eqs. (39), (40) are first both fulfilled.As the parameter γ decreases below the critical value γ AT , the fractal dimension D ( γ ) increases from D min1 being always determined by the intersection of the hori-zontal line y = 2 /γ > /γ AT ( p ) (red line in Fig. 11) withthe blue line. Thus the stable fractal dimension D ( γ ) is always determined by the Gaussian part of P ( V ) andaccording to the second line of Eq. (41) and Fig. 11 isequal to: D ( γ ) = 2 − γγ ET ( p ) , p ≤ . (43)At γ = γ ET the fractal dimension D ( γ ) reaches unity,and at this point a continuous ergodic transition hap-pens. Thus the critical point of ergodic transition coin-cides with that determined by Eq. (20).Note that while D ( γ ) is linear in γ , as for the Gaus-sian RP model [15], other fractal dimensions D q ( q > )are not necessarily equal to D ( γ ) as it was the case inRef. [15]. The calculation of D q with q > goes beyondthe scope of this paper and will be studied elsewhere [37].Note that, unlike the ergodic transition, the Andersontransition is discontinuous : the stable fractal dimension D ( γ ) is separated by a finite gap D min = D ( γ AT ) from1the localized state D = 0 : D min = (cid:40) − γ AT ( p ) γ ET ( p ) , < p < , p ≥ (44)This gap is shown by the gray dotted arrow in Fig. 11.The right panel of Fig. 11 demonstrates that for p ≥ the minimal fractal dimension D min = 1 , so that themultifractal phase is no longer possible in LN-RP modelwith γ tr ≤ . However, as it is shown in Sec. VI, itappears if γ tr > . IX. CONCLUSION AND DISCUSSION
In this paper we introduce a log-normal Rosenzweig-Porter (LN-RP) random matrix ensemble characterizedby a long-tailed distribution of off-diagonal matrix ele-ments with the variance controlled by the symmetry pa-rameter p . We calculate the phase diagram of LN-RPusing the recently suggested Anderson localization andMott ergodicity criteria for random matrices. An alter-native approach based on the analysis of stability withrespect to hybridization of multifractal wave functionsdeveloped in this paper gives results identical to thoseobtained from the above criteria and consistent with nu-merical calculations. It also helps to compute the dimen-sion D of the eigenfunction fractal support set and showthat the Anderson localization transition is discontinuouswith D min1 > at all p > .This LN-RP model has many potential applicationsand we use it to develop an alternative approach to the lo-calization problem on random regular graph. It is basedon the partition of all sites on RRG into two groups:(i) the “marked sites” remote from each other at the mostabundant distance of the order of the graph diameter and(ii) the “tree sites” at much smaller distance from eachother. This partition is only meaningful for the graphsin which the marked sites take a finite fraction of all siteslike in the graphs with a local tree structure. Then westudy an effective random matrix model involving onlythe marked sites and show that it is a special case p = 1 of the log-normal Rosenzweig-Porter random matrix en-semble introduced in this paper. An important result ofthis paper is that in this p = 1 LN-RP model arising fromthe above partition, there is a direct transition from thelocalized to the ergodic phase similar to the one obtainedin Refs. [7, 8]. However, the point p = 1 appears to bevery special: it is a tricritical point of the LN-RP modelwhich is unstable to deformations of this model. In par-ticular, it is unstable to truncation of the far tail in thelog-normal distribution considered in Sec. VI and sensi-tive to modification of the loop statistics in RRG [40]leading to the different (possibly non-convex) distribu-tion of off-diagonal matrix elements in the correspondingLN-RP model.We would like to emphasize that our mapping of thelocalization problem on RRG to LN-RP random matrix ensemble is an approximation which is justified only qual-itatively. Note that the approach adopted in Ref. [7, 8]is not free of approximations too. While the local treestructure of RRG is treated exactly in the framework ofthe supersymmetric sigma-model, the final solution restson the “self-consistency” condition (Eqs. (24) and (30) inRef.[8]): g ( Q ) = (cid:90) D Q (cid:48) [ g ( Q (cid:48) )] K e − ST r [ − g ( Q − Q (cid:48) ) + η Λ Q (cid:48) ] . (45)This condition is an approximation , as the correspondingequations do not take into account a detailed topologyof the graph (statistics of loop lengths, etc.) but only(i) the local tree structure (encoded in the non-linearterm [ g ( Q (cid:48) )] K ), and (ii) the statistical homogeneity ofthe graph as a whole (encoded in the fact that only the zero spatial mode component Q of the supersymmetric Q ( i ) -field enters in Eq. (45)).In a sense, this approximation is in many respects sim-ilar to our mapping onto LN-RP model. Indeed, the zero spatial mode component Q is known to describethe Wigner-Dyson random matrix ensembles [41]. TheGaussian Rosenzweig-Porter model is a Wigner-Dysonrandom matrix ensemble with parametrically enhancedfluctuations of diagonal matrix elements. The break-down of basis-rotation invariance by the special diagonal(like in the Rosenzweig-Porter ensemble) is described inEq. (45) by the ’gradient’ term − g ( Q − Q (cid:48) ) which be-comes non-zero (in contrast to the Wigner-Dyson case)due to the presence of the “external” Q supermatrix inthe non-linear integral equation (45).Therefore, one may conjecture that our mapping ontothe p = 1 LN-RP model is equivalent to the self-consistentapproximation of Ref. [8] and earlier works by Mirlin andFyodorov (see Refs. [42, 43] and references therein). Anadditional support to this conjecture comes from the factthat at p = 1 the critical exponent ν = ν ≡ ν reachesits mean-field value ν = 1 / (see Table 7 and Fig. 8).Whether or not both approximations give correct predic-tions for the phases on RRG is, in our opinion, still anopen issue. ACKNOWLEDGMENTS
V.E.K. and I.M.K are grateful for support and hospi-tality to GGI of INFN and University of Florence (Italy)where this work was initiated. V.E.K and B.L.A. ac-knowledge the support and hospitality of Russian Quan-tum Center during the work on this paper and G. V.Shlyapnikov for illuminating discussions there. V.E.Kgratefully acknowledges support from the Simons Cen-ter for Geometry and Physics, Stony Brook Universityat which part of the research for this paper was per-formed. This research was supported by the DFG projectKH 425/1-1 (I. M. K.), by the Russian Foundation for Ba-sic Research Grant No. 17-52-12044 (I. M. K.), and by2Google Quantum Research Award “Ergodicity breakingin Quantum Many-Body Systems” (V. E. K.).
Appendix A: RRG-to-LN-RP correspondence.
As mentioned in Sec. II, the local tree structure andthe predominance of long loops on RRG lead to “conden-sation of large distances” when most of the pairs of sitesare located at a certain distance of the order of the graphdiameter d (cid:39) ln N/ ln K . This leads to the set of equallyspaced sites on RRG with the most abundant distance r ∗ ≈ d − . Those marked sites interact with each otherthrough the remaining tree sites similar to the indirect in-teraction between Anderson impurities in a metal. Thisindirect interaction is long-range and the effective hop-ping matrix elements ( H eff ) nm of such a model can befound using the Anderson impurity model.Indeed, let us consider the marked sites n and m asAnderson impurities imbedded into the Cayley tree whichsites are connected by a hopping V . Those impurities(which are not directly connected) are supposed to beconnected with the neighboring tree sites by the samehopping V . Then the impurity Green’s function G =( E − H eff ) − can be expressed through an exact Green’sfunction G on a Cayley tree as follows: G nm = g n V G m (cid:48) n (cid:48) g m + g n δ nm , (A1)where n (cid:48) and m (cid:48) are the sites on a tree neighboring tothe marked sites (’impurities’) m and n , respectively, and (cid:98) g nm = g n δ nm = ( E − ε n ) − δ nm is the bare impurityGreen’s function. Thus, inverting Eq. (A1) and assuming V || (cid:98) g || || (cid:98) G || (cid:28) we obtain: G − = (cid:98) g − (1 + V (cid:98) g (cid:98) G ) − ≈ (cid:98) g − − V (cid:98) G. (A2)Substituting here G − = ( E − (cid:98) H eff ) and (cid:98) g − = E − (cid:98) ε oneobtains: (cid:98) H eff ( E ) = (cid:98) ε + V (cid:98) G ( E ) . (A3)where (cid:98) G ( E ) is an exact Green’s function on a Cayley treeand (cid:98) ε = diag { ε n } . Appendix B: Full and cavity Green’s functions on aCayley tree
Note that (cid:98) G ( E ) with components G E ( i, j ) ≡ G ( i, j ) in Eq. (A3) is the full two-point Green’s function on aCayley tree. It is convenient to express it in terms of theproduct Π( i, j ) of the cavity Green’s functions G p → p − ,where p − is an immediate descendant of p along the pathfrom i to j (see Fig. 12): Π( i, j ) = (cid:89) i
i lie on the same3path passing through the root of the tree, one formallyreplaces in Eq.(B4) l by i and sets Π( i, i ) = 1 .Eq.(B4) is a product of the cavity Green’s functionsalong the path from i to j , with the single exceptionthat at the common descendant l of i and j the cavityGreen’s function is replaced by the full one. This is theminimal modification of the product which makes theGreen’s function G ( i, j ) (as well as G ( l, l ) ) singular atthe eigen-energy E = E n .However, this modification is crucial not only for thecorrect spectral properties. It drastically restricts fluc-tuations of | G ( i, j ) | . Indeed, from the recursion relationEq.(B2) it follows that if at some step G j → i is anoma-lously large than at the next step G i → k will be anoma-lously small, so that the product G j → i G i → k is of orderone. At the same time anomalously small G j → i does notresult in an anomalously large G i → k . For this to happen all other terms in r.h.s. of Eq.(B2) should be anomalouslysmall which is much less probable. This means that aproduct Π( i, j ) along a long path is typically small, asfor large number of terms in the product the probabil-ity to have a small G j → i increases. However, there arerare events when Π( i, j ) is anomalously large. This hap-pens only when the last term in the product G p → p − with p − = i (which cannot be compensated) is anomalouslylarge. These rare events are responsible for the symmet-ric distribution of y = | G ( i, j ) | − discussed in AppendixC.Now let us consider Eq.(B4). One can easily see thatthe anomalously large end-terms G l − → l and G l + → l in Π( i, l ) and Π( j, l ) cancel out by the corresponding termsin the denominator of Eq.(B5), if we neglect very improb-able evens when they both are large. With this restrictionone may write: r − ln | G ( i, j ) | = (cid:26) r − ln | Π( i, j ) | , if ln | Π( i, j ) | < ≈ , otherwise , (B6)where r = | i − j | (cid:29) .Note that the above analysis applies also to one-dimensional Anderson model which formally correspondsto K = 1 . Then Eq.(B6) is consistent with the exactresult [45] that T = | G ( i, j ) | ≤ , where T is the trans-mission coefficient through a chain of the length L .This intrinsic cutoff at large values | i − j | = d of | G ( i, j ) | affects the Anderson localization (7) and theMott ergodicity (11) principles for the corresponding LN-RP random matrix model and leads to the phase diagramwith the tricritical point at p = 1 . Appendix C: ’Multifractal’ distribution of | Π( i, j ) | Now we consider generic properties of the distributionof the product Π r ≡ | Π( i, j ) | of cavity Green’s functions G p → p − on a Cayley tree at large distances r ≡ | i − j | .As it is shown in Ref. [4], in the limit of a long path r (cid:29) the distribution function F ( y ) of y = Π − r has a special symmetry: F ( y ) = F (1 /y ) . (C1)Correspondingly, the distribution function P (Π r ) obeysthe symmetry: P (1 / Π r ) = Π r P (Π r ) . (C2)In order to proceed further we make use of theexpression for F ( y ) in terms of its moments I n = (cid:82) F ( y ) y − m dy : F ( y ) = 2 y (cid:90) B dm πi y m ( I m ) r , (C3)where the integration is performed over the Bromwich m = c + iz contour which goes parallel to the imaginaryaxis ( z ∈ [ −∞ , + ∞ ] ) on the positive side of the real one( c > ). Eq. (C3) is nothing but a Mellin transform whichallows to restore the distribution function, given that the(analytically continued) moments I m are known.The moments I m at m ∈ [0 , obey the following sym-metry which reflects the symmetry Eq. (C1) [4]: I m = I − m , (C4)with a minimum at m = 1 / and: I = I = 1 , ∂ m I m | m =1 = − ∂ m I m | m =0 . (C5)This symmetry is another representation of the basic β -symmetry on a Cayley tree established in the seminalwork [23].Computing the Mellin transform in the saddle-pointapproximation one finds with the exponential accuracy: ln( y F ( y )) = r (ln I m − m∂ m ln I m ) m = m ∗ , (C6)where m ∗ is found from the stationarity condition:
12 ( ∂ m ln I m ) m = m ∗ ( y ) = − ln yr . (C7)Eq. (C7) implies that m ∗ is a function of the argument ln( y ) /r . Then it follows from Eq. (C6) that: F ( y ) ∼ y − exp (cid:20) − r G (cid:18) ln yr (cid:19)(cid:21) , ( r (cid:29) , (C8)where G ( x ) some function of ln( y ) /r .The form Eq. (C8) is very special. A large parameter r appears both in front of G ( x ) and in its argument ina reciprocal way. This form is know as the large devia-tion , or multifractal ansatz . It appears in many differentproblems of statistical mechanics (see e.g. Ref. [28] andreferences therein) and is a non-trivial generalization ofCentral Limit Theorem when the logarithm of the fluc-tuating quantity is a sum of many terms with specialcorrelations between them.4The simplest choice of the function G ( x ) is a lin-ear function which corresponds to a power-law distribu-tion. A parabolic function G ( x ) appears when ln y isthe sum of nearly uncorrelated terms which leads to thelogarithmically-normal distribution P (Π r ) : P (Π r ) = A ( r )Π r exp (cid:34) − ln (Π r / Π typ )2 p ln(Π − ) (cid:35) , A ( r ) = 1 √ π pλ r . (C9)where Π typ ∼ e − λ r , with the Lyapunov exponent λ , isthe typical value of Π r , and p is the symmetry parameter.The symmetry Eq. (C2) corresponds to p = 1 . Any p (cid:54) =1 modifies the power of Π r in r.h.s. of the symmetryrelation, Eq. (C2).Note also that the product Π r Π r has the log-normaldistribution Eq.(C9) with r = r + r , if both Π r and Π r are distributed log-normally as in Eq.(C9) with r = r and r , respectively. This implies that the distributionof the product | Π( i, j ) | depends only on the distance be-tween the points i and j , no matter whether i and j areon the same path passing through the root of a tree orthey are on two different branches (as in Fig. 12). Theproperty P (Π r Π r ) = P (Π r + r ) is required of any sen-sible distribution of the product of local quantities on aCayley tree.The distribution of the off-diagonal matrix elements U = | G ( i, j ) | | i − j | = d of the corresponding RP RMT canbe found from P (Π r ) by setting r ≈ d = ln N/ ln K , U = Π r , and employing Eq.(B6) which imposes a cutoffat U max ∼ N .Here an important comment is at place. Eq.(C9) with p = 1 and r = L applies also to a one-dimensional An-derson model at weak disorder and localization length ξ << L much smaller than the system size L . Indeed,denoting ln Π − = L/ (2 ξ ) ≡ x/ one reduces Eq.(C9) tothe distribution function of √ T of Ref.[45], where T (cid:28) is the transmission coefficient. The comparison with theexact result of Ref.[45] could give a more precise limit ofapplicability of the log-normal distribution than Eq.(B6).It shows that the log-normal distribution of T = | G ( i, j ) | valid for ln(1 /G ) (cid:38) x , is modified for (cid:46) ln(1 /G ) (cid:28) x (where G = | G ( i, j ) | for brevity) by an additional pre-exponential factor x − ln(1 /G ) which is not essential un-less in the vicinity of G = 1 . Thus in 1D case the essentialcut-off of the log-normal distribution, indeed, happens at G = 1 .One can expect that on the Cayley tree the additionalfactor which replaces x − ln(1 /G ) is a function of (cid:96) − =ln(1 /G ) / ln N . Such a factor is of order O (1) for any G ∼ N − γ tr , γ tr ∼ and thus cannot lead to efficienttruncation of the log-normal distribution.One can show that the distribution Eq. (C9) with p = 1 is asymptotically exact on an infinite Cayley tree in thelimit of small disorder. In particular, for a granular Cay-ley tree described by the non-linear sigma model (NL σ M) ln I m m W = - ( - m ) - - - - FIG. 13. (Color online) Plots of ln I m from Eq. (C12) for W = 20 and the parabolic dependence − . m (1 − m ) . the moments I m are given by [7]: I m = (cid:114) π g (cid:2) K m +1 / ( g ) g sinh g (C10) + K m − / ( g ) ( g cosh g − m sinh g )] , where g is the dimensionless conductance (the coefficientin front of ( ∇ Q ) in the NL σ M). In the limit of largeinter-grain conductance g (cid:29) one obtains: ln( I m ) ≈ − (2 g ) − m (1 − m ) , (C11)which according to Eq. (C3) implies the log-normal distri-bution of y and Π r . The asymptotic expression Eq. (C11)reproduces Eq. (C10) very accurately down to g ∼ . .The same is true for an ordinary Cayley tree with asingle orbital per site. In this case the ’two-brick’ ap-proximation (Eq. (90) in Ref. [4]) gives for I m : I m = sinh (cid:2) (2 m −
1) ln (cid:0) W (cid:1)(cid:3) (2 m −
1) sinh (cid:2) ln (cid:0) W (cid:1)(cid:3) . (C12)One can show that ln I m from Eq. (C12) is approachingEq. (C11) with (2 g ) − → ( W − / for W → andremains an almost perfect parabola in a broad interval < W (cid:46) (see Fig. 13). We conclude therefore that thelog-normal distribution of G r is quantitatively accurate inthe whole range of disorder strengths up to the Andersontransition point W c ∼ K ln K if the branching number K (cid:46) ∼ O (1) .However, for large ln( W/ (cid:29) I m ≈ ( | m − | −
1) ln( W/ (C13)is linear in m everywhere except for a small interval ofthe width ∼ / ln( W/ in the vicinity of the minimumat m = 1 / in which ln I m can be approximated by aparabola (see Fig. 14). In this case the saddle pointEq. (C7) does not have a solution for | ln y | > r ln( W/ (see Fig. 14), and the distribution F ( y ) is truncated. For | ln y | < r ln( W/ we obtain: F ( y ) ∼ C exp (cid:20) − ln y Σ (cid:21) , ( | ln y | < r ln( W/ , (C14)5 - - ∂ m ln I m m - / r ln I m m - - - - - FIG. 14. (Color online) Plots of ln I m (main panel) and itsderivative (inset) from Eq. (C12) for ln( W/
2) = 13 . For | ln y | > r ln( W/ the saddle-point Eq. (C7) does not have asolution. where C = e − r ln( W/ and Σ = (2 / r ln ( W/ .Then from the results of Appendix B it follows that P ( U ) is truncated from below at U min ∼ N − ln( W/ K andfrom above at U max ∼ . Between these limits it can beapproximated by: P ( U ) ∼ AU exp (cid:20) − ln ( U )(2 / d ( N ) ln ( W/ (cid:21) (cid:18) U typ U (cid:19) , (C15)where U typ ∼ N − ln( W/ K and d ( N ) = ln N/ ln K .One can see that the probability to find U largerthan the typical one is considerably smaller than theone resulting from the forward scattering approximation(FSA): P FSA ∼ A (cid:48) U exp (cid:20) − ln ( U/U typ ) d ( N ) ln ( W/ (cid:21) . (C16)Furthermore, because of the resonances (neglected in theFSA but captured by Eq. (C12)) the transition matrixelements U are never described by the FSA, no matterhow large is ln( W/ . Appendix D: Kullback-Leibler measures in themultifractal phase
In this section we give a more detailed quantitativedescription of KL . In order to do this we employ theansatz for the wavefunction moments: M q = (cid:42)(cid:88) i | ψ ( i ) | q (cid:43) = N − D q ( q − f q ( L/ξ q ) , (D1)where D q is the fractal dimension and f q ( x ) is thecrossover scaling function: f q ( L/ξ q → ∞ ) → const . multifractal phaseconst . N ( q − D q − , ergodic phaseconst . N ( q − D q localized phase (D2) Note that for the graphs with the local tree structure thelength scale L ∝ ln N , so that the scaling function is ingeneral a function of two arguments ln N/ξ q and N/e ξ q representing the length - and volume scaling [12, 13]. Onthe finite-dimensional lattices N ∝ L d , and the volumescaling can be represented as the length scaling in themodified scaling function. In this case a single argument L/ξ q is sufficient.When L ∝ ln N the volume scaling is the leading onefor L (cid:29) ξ q , and it is this scaling that provides the asymp-totic behavior Eq. (D2). The length scaling is importantin the crossover region L (cid:46) ξ q . Below for brevity we willuse the short-hand notation L/ξ q in all the cases.There are two trivial cases: M = N and M = 1 (which follows from the normalization of wave function).As a consequence we have D = 1 and f ( x ) = f ( x ) ≡ . (D3)Next using the statistical independence of ψ and ˜ ψ inEq. (22) and normalization of wave functions we repre-sent KL (cid:42)(cid:88) i | ψ ( i ) | ln | ψ ( i ) | (cid:43) − N − (cid:42)(cid:88) i ln | ψ ( i ) | (cid:43) . (D4)Now we express both terms in Eq. (D4) in terms of M q using the identity: ln | ψ α ( i ) | = lim (cid:15) → (cid:15) − ( | ψ α ( i ) | (cid:15) − (D5)The first term is equal to: (cid:42)(cid:88) i lim (cid:15) →∞ | ψ ( i ) | (cid:15) ) − | ψ ( i ) | (cid:15) (cid:43) = lim (cid:15) →∞ (cid:20) (cid:15) ( M (cid:15) − (cid:21) . (D6)The second term can be expressed as: − N (cid:42)(cid:88) i lim (cid:15) →∞ | ψ ( i ) | (cid:15) − (cid:15) (cid:43) = − lim (cid:15) → (cid:20) (cid:15) (cid:0) N − M (cid:15) − (cid:1)(cid:21) . (D7)Now expanding M (cid:15) and M (cid:15) in the vicinity of q = 0 , and defining f (cid:15) ( x ) = 1 + (cid:15) ϕ ( x ) + O ( (cid:15) ); (D8) f (cid:15) ( x ) = 1 − (cid:15) ϕ ( x ) + O ( (cid:15) ) , (D9)we obtain: KL KL c ( N ) + ϕ ( L/ξ ) + ϕ ( L/ξ ) , (D10)where KL c is logarithmically divergent: KL c = ln N (1 − ∂ (cid:15) D (cid:15) | (cid:15) =0 − D ) + const . (D11) = ln N ( α − D ) + const . Here we used the identity for α describing the typicalvalue of the wave function amplitude | ψ | typ = N − α : α = dτ (cid:15) d(cid:15) | (cid:15) =0 = ∂ (cid:15) [ D (cid:15) ( (cid:15) − | (cid:15) =0 . (D12)6 KL1 - KL1c x = ( W - W c ) L / ν ν = W c = L = -
3D AM - -
20 0 20 40 60 - KL2 - KL2c x = ( W - W c ) L / ν
3D AM ν = W c = KL2c lnN L = - - -
50 0 50 100 - - FIG. 15. (Color online) Collapse of KL and KL data in the vicinity of the localization transition in d Anderson model.The transition point in the KL collapse and the corresponding critical exponent ν were found by the best collapse with theminimal χ deviation from the scaling function which was updated at any step of iterative collapse process. Such processconverges and gives the optimal values of γ c and ν , as well as the scaling function (parameterized by a 6-order polynomial),despite there is no intersection in the KL curves for different N (see Fig. 4). Note that, generally speaking, the characteristiclengths ξ ∼ | γ − γ c | − ν (0) and ξ ∼ | γ − γ c | − ν (1) mayhave different critical exponents ν (0) and ν (1) . If this isthe case, the smallest one will dominate the finite-sizecorrections near the critical point: KL − KL c ( N ) = Φ ( L | γ − γ c | ν ) , ν = min { ν (0) , ν (1) } . (D13)Eq. (D13) is used in this paper for the numerical char-acterization of the phases. Deeply in the multifractalphase and at L (cid:29) the scaling function Φ ( x ) accordingto Eq. (D2) is a constant. Then KL c ( N ) and KL areboth logarithmically divergent, as α > and D < inEq. (D11) in the multifractal phase.The scaling function Φ ( x ) is also a constant deeplyin the ergodic phase but in this case α = D = 1 andthe logarithmic divergence of KL c is gone. As the result KL is independent of N deeply in the ergodic phase.At the continuous ergodic transition α = D = 1 , andthe critical value KL c ( N ) of KL is independent of N .This results in crossing at γ = γ ET of all the curves for KL at different values of N which helps to identify the ergodic transition [26].However, if the ergodic transition coincides with theAnderson localization transition and is discontinuous ,(i.e. α and D are not equal to 1 at the transition),the critical value KL c ( N ) is no longer N -independent.In this case the crossing is smeared out and can disap-pear whatsoever. Nonetheless, by subtracting KL c from KL one can still identify the transition from the con-dition of the best collapse by choosing an optimal γ c inEq. (D13). However, it is safer to use KL in this case.The derivation of finite size scaling (FSS) for KL pro-ceeds in the same way by plugging the identity Eq. (D5) into: KL (cid:42)(cid:88) i | ψ α ( i ) | ln | ψ α ( i ) | (cid:43) − (cid:42)(cid:88) i | ψ α | ln | ψ α +1 ( i ) | (cid:43) . (D14)and employing the ansatz: (cid:42)(cid:88) i | ψ E ( i ) | q | ψ E + ω ( i ) | q (cid:43) ∼ N β N αω × F q ,q ( L/ξ q , L/ξ q ) , (D15)where N ω = 1 / ( ρω ) and ρ is the mean DoS.Applying for large ω ∼ ρ − ( N ω (cid:39) ) the “decouplingrule”: (cid:42)(cid:88) i | ψ E ( i ) | q | ψ E + ω ( i ) | q (cid:43) ∼ (cid:88) i (cid:10) | ψ E ( i ) | q (cid:11) (cid:10) | ψ E + ω ( i ) | q (cid:11) , (D16)and for small ω ∼ δ ( N ω (cid:39) N ) the “fusion rule”: (cid:42)(cid:88) i | ψ E ( i ) | q | ψ E + ω ( i ) | q (cid:43) ∼ (cid:42)(cid:88) i | ψ E ( i ) | q +2 q (cid:43) , (D17)one easily finds: β = − D q (1 − q ) + D q (1 − q ) , (D18) α + β = − D q + q (1 − q − q ) . ψ α and ψ α +1 we obtain fromEq. (D1): (cid:42)(cid:88) i | ψ α ( i ) | q | ψ α +1 ( i ) | q (cid:43) ∼ F q ,q ( L/ξ q + q ) × N − D q q ( q + q − . (D19)Substituting Eq. (D19) in Eqs. (D5), (D14) we observecancelation of the leading logarithmic in N terms in KL deeply in the multifractal phase: KL c = const . (D20)We obtain: KL ( L | γ − γ c | ν ) . (D21)where ν = ν (1) ≥ ν and the crossover scaling function Φ ( x ) is: Φ ( x ) = ∂ (cid:15) f (cid:15) ( x ) − ∂ (cid:15) f ,(cid:15) ( x ) | (cid:15) =0 . (D22)As it is seen from Eq. (D21), KL is independent of N at the Anderson transition point γ = γ AT . Thus allcurves for KL at different values of N intersect at γ = γ AT . This gives us a powerful instrument to identify theAnderson localization transition point. Appendix E: Finite-size scaling collapse for KL and KL for 3D Anderson model In Fig. 15 we present the result for the data collapse for KL and KL in the vicinity of the localization transi-tion in the d Anderson model. This result demonstratesthat our iteration procedure is convergent and gives agood approximation for the critical point from the col-lapse of KL data which do not show any intersectionof KL vs. W curves at the critical point. From thiscollapse we found the critical exponents: ν = 1 . ± . , ν = 1 . ± . . (E1)Note that from the results of Appendix D it follows thatquite generally at the same critical point: ν ≤ ν , (E2)since ν is given by the minimal of the two values ν (0) and ν (1) (see Eq. (D13)) corresponding to the wave functionmoments Eq. (D1) with q = 0 and q = 1 , respectively. Atthe same time, ν = ν (1) . Our result Eq. (E1) satisfies theinequality Eq. (E2). On the theory side it follows fromnowhere that there is only one single critical exponent ν of any FSS in a situation where there is a continuousmultitude of multifractal dimensions. In our opinion, itis more natural to assume that the exponent ν is specificto the quantity which FSS is studied, as it is shown inthe Appendix D for KL and KL . However, our samples are too small and our FSS anal-ysis is too simplistic (e.g. it does not take into accountirrelevant scaling exponents) to claim that ν and ν arereally different.Note that for different ergodic and Anderson localiza-tion transitions the inequality (E2) is not valid in generaland thus, cannot be applied to the LN-RP at p < , whilefor p ≥ it is saturated, see Table 7 and Fig. 8. Appendix F: Analysis of stability
In this section we calculate the contributions to P ( V ) from the log-normal and Gaussian parts to Eq. (34).One can easily compute the variance of the Gaussianpart of P ( V ) leaving in it only the bi-diagonal terms with i = i (cid:48) and j = j (cid:48) : (cid:104) V (cid:105) = (cid:90) g ∈ II dg N − pγ ( g − γ ) − g (F1) ∼ max g ∈ II (cid:110) N − pγ ( g − γ ) − g (cid:111) ≡ N − γ eff . The maximum in Eq. (F1) at g belonging to region II inFig. 16 can be reached (i) inside the region II at g = g ∗ ,(ii) at the border of this region at g = g ∗ , and (iii) at thecut-off of P ( g ) at g ∗ = 0 (see Fig. 16 and Fig. 17(left)).The expression for γ eff ( D takes the form: γ eff ( D ) = γ (1 − p ) , pγ < D < , p < D + γ − √ D γp, D < min (cid:16) pγ , γ p (cid:17) γ p , γ p < D < , p ≥ . (F2)Next we compute the function ∆( D ) = − max g ∈ I { σ ( g, D ) − g − D } . (F3)in Eq. (38).The details of the calculation which is similar tocalculation of γ eff ( D ) in Eq. (F1) are illustrated inFig. 17(right). The resulting expression for ∆( D i ) is: g / 2 g D I III *2 1 g D p g g= - ( , ) g D s *2' 1 g D p g g= + FIG. 16. (Color online) Regions of g contributing to the log-normal (I) and Gaussian (II) parts of the distribution function P ( U µ,ν ) .
1/ 2 p < D p g *1 g *2 g min of ( ) at p<1/2 is reached here eff a g a
1/ 2 p > D p g *2 g * g = min of ( ) at p>1/2 is reached here eff a g a p < p g *1 g *2 g D *2' g p > p g * g = D *2' g p g FIG. 17. (Color online) (Left panel) Different possible positions g ∗ , g ∗ or g ∗ = 0 that maximize Eq. (F1) in region II dependingon p , γ and D . The configuration of maximum realized in each sector of parameters is shown by an ikon in the correspondingsector. (Right panel) Different possible positions g ∗ , g ∗ or g ∗ = 0 that maximize Eq. (F3) in region I. The configuration ofmaximum realized in each sector of parameters is shown by an ikon in the corresponding sector. For D > γ/ p the maximumin Eq. (F3) is reached at the edge of the right segment of region I, g = g ∗ (cid:48) (not to be confused with the edge of the left segment g = g ∗ , see Fig. 16 ). It leads to a higher branch of the orange curve ∆( α ) + α/ in Fig. 11 (not shown in Fig. 11) which isseparated by a gap from the blue curve in Fig. 11 and thus is irrelevant for our analysis. ∆( D ) = γ (cid:0) − p (cid:1) − D , < D < γp , p < D + γ − √ D γp, γp < D < γ p , p < γ p − D , < D < γ p , p ≥ . (F4) [1] D.M. Basko, I. L. Aleiner, and Boris L. Altshuler,“Metal-insulator transition in a weakly interacting many-electron system with localized single-particle states,”Ann. Phys. (N. Y). , 1126–1205 (2006).[2] Boris L. Altshuler, Yuval Gefen, Alex Kamenev, andLeonid S. Levitov, “Quasiparticle Lifetime in a FiniteSystem: A Nonperturbative Approach,” Phys. Rev. Lett. , 2803–2806 (1997).[3] Andrea De Luca, BL Altshuler, VE Kravtsov, andA Scardicchio, “Anderson localization on the Bethe lat-tice: Nonergodicity of extended states,” Phys. Rev. Lett. , 046806 (2014).[4] V.E.Kravtsov, B.L.Altshuler, and L.B.Ioffe, “Non-ergodic delocalized phase in Anderson model on Bethelattice and regular graph,” Annals of Physics , 148–191 (2018).[5] Vadim N. Smelyanskiy, Kostyantyn Kechedzhi, SergioBoixo, Sergei V. Isakov, Hartmut Neven, and Boris Alt-shuler, “Nonergodic delocalized states for efficient popu-lation transfer within a narrow band of the energy land-scape,” Phys. Rev. X , 011017 (2020).[6] K. Kechedzhi, V. N. Smelyanskiy, J. R McClean, V. SDenchev, M. Mohseni, S. V. Isakov, S. Boixo, B. L.Altshuler, and H. Neven, “Efficient population transfervia non-ergodic extended states in quantum spin glass,”(2018), arXiv:1807.04792.[7] K. S. Tikhonov and A. D. Mirlin, “Fractality of wavefunctions on a Cayley tree: Difference between tree andlocally treelike graph without boundary,” Phys. Rev. B , 184203 (2016).[8] K. S. Tikhonov and A. D. Mirlin, “Statistics of eigen-states near the localization transition on random regulargraphs,” Phys. Rev. B , 024202 (2019). [9] Giorgio Parisi, Saverio Pascazio, Francesca Pietracap-rina, Valentina Ros, and Antonello Scardicchio, “An-derson transition on the Bethe lattice: an approach withreal energies,” Journal of Physics A: Mathematical andTheoretical , 014003 (2019).[10] S. Bera, G. De Tomasi, I. M. Khaymovich, andA. Scardicchio, “Return probability for the Andersonmodel on the random regular graph,” Phys. Rev. B ,134205 (2018).[11] Giuseppe De Tomasi, Soumya Bera, Antonello Scardic-chio, and Ivan M. Khaymovich, “Sub-diffusion in theAnderson model on random regular graph,” (2019), ac-cepted for publication in PRB(R), arXiv:1908.11388.[12] I. García-Mata, O. Giraud, B. Georgeot, J. Martin,R. Dubertrand, and G. Lemarié, “Scaling theory of theanderson transition in random graphs: Ergodicity anduniversality,” Phys. Rev. Lett. , 166801 (2017).[13] I. García-Mata, J. Martin, R. Dubertrand, O. Giraud,B. Georgeot, and G. Lemarié, “Two critical localizationlengths in the anderson transition on random graphs,”Phys. Rev. Research , 012020 (2020).[14] N. Rosenzweig and C. E. Porter, “"Repulsion of EnergyLevels" in Complex Atomic Spectra,” Phys. Rev. B ,1698 (1960).[15] V. E. Kravtsov, I. M. Khaymovich, E. Cuevas, andM. Amini, “A random matrix model with localization andergodic transitions,” New J. Phys. , 122002 (2015).[16] Per von Soosten and Simone Warzel, “Non-ergodic de-localization in the Rosenzweig–Porter model,” Letters inMathematical Physics , 1–18 (2018).[17] D. Facoetti, P. Vivo, and G. Biroli, “From non-ergodiceigenvectors to local resolvent statistics and back: A ran-dom matrix perspective,” Europhys. Lett. , 47003 (2016).[18] K. Truong and A. Ossipov, “Eigenvectors under a genericperturbation: Non-perturbative results from the randommatrix approach,” Europhys. Lett. , 37002 (2016).[19] M. Amini, “Spread of wave packets in disordered hierar-chical lattices,” Europhys. Lett. , 30003 (2017).[20] C. Monthus, “Statistical properties of the green func-tion in finite size for anderson localization models withmultifractal eigenvectors,” J. Phys. A: Math. Theor. ,295101 (2017).[21] P. Cizeau and J. P. Bouchaud, “Theory of lévy matrices,”Phys. Rev. E , 1810–1822 (1994).[22] E. Tarquini, G. Biroli, and M. Tarzia, “Level statisticsand localization transitions of lévy matrices,” Phys. Rev.Lett. , 010601 (2016).[23] R Abou-Chacra, P.W. Anderson, and D.J. Thouless, “ Aselfconsistent theory of localization,” J. Phys. C. , 1734(1973).[24] Solomon Kullback and Richard A Leibler, “On informa-tion and sufficiency,” The annals of mathematical statis-tics , 79–86 (1951).[25] Solomon Kullback, Information Theory and Statistics (John Riley and Sons, 1959).[26] M Pino, J Tabanera, and P Serna, “From ergodic tonon-ergodic chaos in Rosenzweig–Porter model,” Journalof Physics A: Mathematical and Theoretical , 475101(2019).[27] M. A. Ruderman and C. Kittel, “Indirect Exchange Cou-pling of Nuclear Magnetic Moments by Conduction Elec-trons,” Phys. Rev. , 99 (1954).[28] I. M. Khaymovich, J. V. Koski, O.-P. Saira, V. E.Kravtsov, and J. P. Pekola, “Multifractality of randomeigenfunctions and generalization of Jarzynski equality,”Nature Comms. , 7010 (2015).[29] E. Bogomolny and M. Sieber, “Power-law random bandedmatrices and ultrametric matrices: Eigenvector distribu-tion in the intermediate regime,” Phys. Rev. E , 042116(2018).[30] P. A. Nosov, I. M. Khaymovich, and V. E. Kravtsov,“Correlation-induced localization,” Physical Review B , 104203 (2019).[31] Note that in [30] this criterion has been modified in orderto exclude measure zero of modes with atypically largehopping energies.[32] M. Pino, V. E. Kravtsov, B. L. Altshuler, and L. B. Ioffe,“Multifractal metal in a disordered Josephson junctionsarray,” Phys. Rev. B , 214205 (2017).[33] Giuseppe de Tomasi, Moshen Amini, Soumya Bera,Ivan M. Khaymovich, and Vladimir E. Kravtsov, “Sur-vival probability in Generalized Rosenzweig-Porter ran- dom matrix ensemble,” SciPost Phys. , 014 (2019).[34] P. A. Nosov and I. M. Khaymovich, “Robustness of delo-calization to the inclusion of soft constraints in long-rangerandom models,” Phys. Rev. B , 224208 (2019).[35] Here we call wave-function “weakly ergodic” if it occupiesa finite fraction of the total Hilbert space. Such statesplay an important role in several recent works [10, 11,29, 30, 34, 46–48].[36] Michael Aizenman and Simone Warzel, “Extended Statesin a Lifshitz Tail Regime for Random Schrödinger Oper-ators on Trees,” Phys. Rev. Lett. , 136804 (2011).[37] I. M. Khaymovich and V. E. Kravtsov, (2020), (unpub-lished).[38] Note that the truncation at U max (cid:38) O (1) , γ tr ≤ , doesnot alter the phase diagram in Fig. 3.[39] One possible perturbation of the Anderson model onRRG with respect to its structure considered in [40] ex-plicitly shows the above mentioned emergence of the mul-tifractal phase.[40] V Avetisov, A Gorsky, S Nechaev, and O Valba,“Localization and non-ergodicity in clustered randomnetworks,” Journal of Complex Networks (2019),10.1093/comnet/cnz026, cnz026.[41] K.B. Efetov, Supersymmetry in disorder and chaos (Cambridge University Press, 1996).[42] A. D. Mirlin and Y. V. Fyodorov, “Localization transi-tion in the Anderson model on the Bethe lattice: spon-taneous symmetry breaking and correlation functions,”Nucl. Phys. B , 507 (1991).[43] A. D. Mirlin and Y. V. Fyodorov, “Statistical propertiesof one-point Green functions in disordered systems andcritical behavior near the Anderson transition,” J. Phys.France , 655 (1994).[44] M. Warzel S. Aizenman, “Resonant delocalization forrandom Schrödinger operators on tree graphs,” Journalof the European Mathematical Society , 1167–1222(2013).[45] V. I. Mel’nikov, “Fluctuation of resistance of finie disor-dered system,” Sov. Phys. Solid State , 782–786 (1981).[46] I. M. Khaymovich, M. Haque, and P. A. McClarty,“Eigenstate thermalization, random matrix theory, andbehemoths,” Phys. Rev. Lett. , 070601 (2019).[47] Arnd Bäcker, Masudul Haque, and Ivan M Khaymovich,“Multifractal dimensions for random matrices, chaoticquantum maps, and many-body systems,” Phys. Rev. E100