Correlation-induced localization

Transcription

1 Correlation-induced localization amplitude in the bulk of the spectrum for all strengths of disorder and all values of the exponent a. Furthermore, in some models [10, 14, 15] a striking duality µ(a) = µ(2 a) in the spatial decay rate µ 2 of the power-law localized wave functions ψ 2 r r 0 µ was discovered [15] that makes the decay rate µ(a) in the interval 0 < a < 2 non-monotonic and symmetric around the point a = 1. In this work we show that the systems [10 15] belong to a new universality class where correlations in the long-range hopping play a significant role enhancing localization. Furthermore, we suggest a new class of models that bridge between the models with uncorrelated hopping and those with fully correlated hopping. These are the models with longranged translation-invariant (TI) hopping integrals in any single realization. In a given realization the hopping integrals H nm = H n m are fully correlated at a given distance n m but they are uncorrelated and sign-alternating at difarxiv: v1 [cond-mat.dis-nn] 2 Oct 2018 P. Nosov, 1, 2, 3 I. M. Khaymovich, 3, 4 and V. E. Kravtsov 5, 6 1 Department of Physics, St. Petersburg State University, St. Petersburg , Russia 2 NRC Kurchatov Institute, Petersburg Nuclear Physics Institute, Gatchina , Russia 3 Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, Dresden, Germany 4 Institute for Physics of Microstructures, Russian Academy of Sciences, Nizhny Novgorod, GSP-105, Russia 5 Abdus Salam International Center for Theoretical Physics - Strada Costiera 11, Trieste, Italy 6 L. D. Landau Institute for Theoretical Physics - Chernogolovka, Russia (Dated: October 4, 2018) A new paradigm of Anderson localization caused by correlations in the long-range hopping along with onsite disorder is considered which requires a revision and a more precise formulation of the basic localizationdelocalization principles. A new class of random Hamiltonians with translation-invariant hopping integrals is suggested and the localization properties of such models are studied both in the coordinate and in the momentum spaces alongside with the corresponding level statistics. Duality of translation-invariant models in the momentum and coordinate space is uncovered and exploited to find a full localization-delocalization phase diagram for such models. In particular an analytical proof of the duality in the wave function decay exponents recently discovered numerically in the models with deterministic translation-invariant power-law hopping is presented. Introduction. The standard picture of the Anderson localization in a three-dimensional single-particle system with short-range hopping [1] is represented by the phase transition between extended ergodic and localized phases at a certain critical disorder strength or energy with a sharp mobility edge separating ergodic and localized states. Exactly at the Anderson localization transition (ALT) non-ergodic (multifractal) extended states have been proved to appear [2, 3]. It is well-known that in low dimensions d = 1, 2 for any tightbinding (or short-range) Hamiltonian with uncorrelated disorder all states are localized. The picture of ALT is restored in low-dimensional systems if disorder is correlated. Indeed, it is commonly believed that correlated disorder in the on-site energies ( diagonal disorder ) tends to delocalize systems. An important example is the Aubry-Andre lattice model [4] with an incommensurate periodic potential, that exhibits ALT. Another possibility to restore ALT is to consider systems with long-range hopping [5 8]. An archetypical example of such nominally one-dimensional systems is suggested in Ref. [9]. In this model the long-ranged hopping terms are completely uncorrelated and Gaussian distributed with a power-law decay of the variance H nm 2 n m 2a with the distance. The parameter that drives the localization transition in this system is the exponent a, with the critical value a c = 1 at ALT. A question, however, arises as to what happens to ALT when the long-range hopping terms are correlated? These correlations in hopping are barely studied. The earlier study of a single-particle system with deterministic (or fully-correlated) power-law decay of long-range hopping [10], which shows analytically localization of most of the states (even for a < 1 where the standard locator expansion breaks down), has been nearly unnoticed until recently. Recently there was a great deal of revival of this intriguing topic. Several works [11 15] reported about other examples of such systems with fully-correlated long-range hopping, confirming the conclusion of the renormalization group (RG) analysis done in Ref. [10]. Neither of these models demonstrates a truly delocalized behavior of wave function Figure 1. Effect of correlations in long-range hopping on the Anderson localization transition (ALT). The increase of correlations in hopping sends the ALT to smaller values of disorder and stretches the critical point into a whole (multi)fractal phase. Threedimensional plots show exemplary spatial distributions of wavefunction intensities in corresponding localized, (multi)fractal, and delocalized phases.

2 2 ferent n m [16]. Physically, such models emerge naturally, as the distance-dependent hopping integrals are frequently very sensitive to various uncontrolled parameters which make H n m a very complicated function of n m thus justifying its statistical treatment in the same way as was originally pointed out by Wigner for nuclear spectra [17]. This extension of the class of random Hamiltonians with long-range hopping allows us to formulate a new picture of ALT which is presented in Fig. 1. In the absence of correlations in the hopping there is a certain value of control parameter (which is labeled in Fig. 1 as Disorder ) above which the states are localized and below which they are extended and ergodic. At the critical value the states exhibit a critical, multifractal statistics. The cartoon examples of the typical pattern of wave function amplitudes of each of the three types are shown as 3D plots in Fig. 1. As the correlation in the hopping increases, the point of ALT is typically shifted to lower disorder. Simultaneously, the critical point is stretched into a whole region bounded by ALT at high disorder and by the new ergodic transition [18] point at low disorder. It is important that the critical lines of both transitions bend to the left, i.e. the states which are localized in the absence of correlations in the hopping remain localized when the correlations are present. However, in the parameter range of former ergodic delocalized states the critical or localized states may appear depending on the particular model. In order to characterize delocalized phases of these models we suggest a method based on the localization-delocalization principles both in the coordinate and in the momentum space. They show that the delocalized phases in TI long-range Hamiltonians are never fully ergodic. By fully ergodic we mean the states which wave function statistics is invariant under basis rotation and which level statistics is Wigner-Dyson. This is contrasted by the weak ergodicity [19, 20] which is defined in a given (e.g. coordinate) basis and implies that the wave function support [20, 21] takes a finite fraction of all the corresponding (e.g.coordinate) space. Weak ergodicity does not imply invariance of wave function statistics under basis rotation. The lack of full ergodicity in TI models exhibits itself in the Poisson level statistics both at large and at small disorder strengths, while the statistics of wavefunctions may be different in the coordinate and in the momentum space and may depend significantly on the correlation strength. In general in TI-models without additional correlations for different n m the wavefunctions which are delocalized in the coordinate space can be (multi)fractal or even localized in the momentum space, while fully correlated hopping matrix elements localize typical wavefunctions also in the coordinate space, see Fig. 1. The same method allows us to analytically prove the duality of localized states suggested in Ref. [15] and to uncover other non-trivial properties of TI-models. Among them are the duality of the wavefunction statistics in the coordinate and momentum spaces in a so-called translationinvariant Rosenzweig-Porter model (TI-RP, a close relative of a RP-model [18, 22]) and universality of TI-RP eigenfunction statistics for all TI-models in the momentum space. Localization criteria for models with long-range hopping. The most generic one-dimensional free-particle Hamiltonian is defined as follows: H mn = ε m δ mn + j m,n, (1) where 1 m, n N are lattice sites, ε m are random onsite energies with zero mean ε m = 0 and the variance ε 2 m = 2 [23] representing uncorrelated diagonal disorder. The (possibly correlated) hopping integrals j m,n = jn,m can be deterministic or random with the variance j n,m 2 depending on the distance m n between coupled sites [16]. Throughout the text we refer to the correlations in the hopping terms j m,n simply as correlations. For simplicity we restrict our consideration to d = 1, unless stated otherwise. The basic localization principle originally suggested by Mott [24] states that the wave functions are localized (extended) when the disorder strength is larger (smaller) than the bandwidth p in the absence of diagonal disorder. The results of this paper and other recent works [10 12, 14, 15, 18, 25], however, show that this principle should be reformulated. It is, in fact, the sufficient condition for weakly ergodic delocalization when is smaller than p : p, weak ergodicity (2) but it does not necessarily mean localization at an opposite sign of inequality. In some models (see, e.g., [25]) weak ergodicity may survive beyond the condition (2), showing that (2) is only the sufficient but not the necessary condition of weak ergodicity. The criterion of localization suggested for systems with long-range hopping by Levitov [5, 6] following Anderson s ideas of locator expansion, is different: δ R > j R, localization. (3) The key point of [5, 6] is that one should compare the mean level spacing δ R 1/R d of a d-dimensional system at a certain length scale 1 m n R N with the width of a resonance governed by the average absolute value of hopping integrals j R within the same scale. Then most eigenstates (except measure zero) are localized if (3) holds for sufficiently large R. Indeed, in order to find the eigenstates one can use the perturbation theory in the small parameter j R /δ R. The inequality (3) means convergence of perturbation series and thus localization. If the criterion (3) is violated, both a multifractal [18] and a weakly ergodic [25] extended phases may emerge. Thus the existence of non-(fully)ergodic extended phases is due to the inequivalence of (2) and (3). More surprisingly, violation of (3) does not exclude localization either, provided that the hopping integrals are correlated. Indeed, the presence of correlations cannot destroy localization if the condition (3) is fulfilled and the perturbation series is convergent. Under this condition the main contribution to the eigenfunction amplitude away from the localization center comes from the first-order perturbation theory which knows nothing about correlations in the hopping matrix elements. The situation changes completely when the

3 3 perturbation theory diverges. In this case all orders in perturbation theory contribute to the eigenfunction amplitude on an equal footing and correlations come into play. As recent examples [11 15] show, the effect of correlations when (3) is violated may be localization of states which were extended in the absence of correlations. These examples, in which hopping is deterministic, prove that (3) is a sufficient but not a necessary condition of localization. Translation-invariant models. A natural generalization of latter models is a family of translation-invariant (TI) models of the form (1), with the hopping term j m,n = j m n [26] depending only on the directed distance m n between coupled sites [16]. An equivalent dual form of the Hamiltonian H mn = ε m δ m,n + j m n in the momentum basis is H pq = Ẽpδ p,q + J p q, with new on-site energies Ẽ p = Ẽ p = and new hopping integrals J p = J p = 1 N N 1 m=0 N 1 m=0 j m e 2πi pm N, (4) ε m e 2πi pm N, (5) exchanging their roles after Fourier transforming (FT). It allows one to generalize the Levitov s localization principle (3) to the momentum space, with p q P δ P > J P, δ P = Ẽp Ẽq P, JP = J p q P. (6) This criterion implies that at small enough disorder there exist states localized in the momentum space (p-localized states) obeying, thus, the Poisson level statistics. At the same time in the coordinate space these states must be extended due to the dual criterion (2). As the level statistics is basisinvariant, we come to the conclusion that in TI models a coexistence of Poisson levels statistics and the delocalized (but not fully ergodic!) character of eigenstates is possible. The dual counterpart of the weak ergodicity criterion (2) for TI-models differs from (2) only by the opposite sign of the inequality: p, p = max Ẽp p,p Ẽp. (7) Equations (2, 3, 6, 7) form the basis of our qualitative analysis throughout the manuscript. Below we consider two families of random matrix models as examples and show the effect of correlations on their localization properties applying two dual pairs of these localization-delocalization principles. Rosenzweig-Porter family. We start by considering a family of infinitely long-range models with the localization transition. They are known as the Rosenzweig-Porter (RP) ensemble [18, 22, 27 40] and the Yuzbashyan-Shastry (YS) model [11 15]. The latter model is exactly (and trivially) integrable [11, 12]. The Hamiltonian of the RP-ensemble takes the form (1) with (fully uncorrelated) hopping matrix elements j mn with Figure 2. Localization-delocalization phase diagrams for (left) RP and (right) PLBM families of ensembles. Additionally to coordinate-space diagrams (above horizontal lines) and levelstatistic diagrams (in the middle) for TI-models the momentumspace diagrams are shown below the lines. The types of spectral statistics (Wigner-Dyson, Poisson and hybrid) is indicated for each phase. The increase of correlations in hopping (from bottom to top) affects the delocalized phase of all models, first, making TI-systems non-ergodic and then (YS and BM) localizing wavefunctions in the coordinate space. The eigenfunction statistics in the coordinatespace of TI ensembles is the same as that for their non-ti counterpart. Notice also the same sequence of phases in the momentum space of all TI ensembles. zero mean and the variance j mn 2 = 2 N γ scaling with the matrix size N. The diagonal elements are characterized by ε 2 m = 2. In the YS model the hopping terms in (1) form a rank-1 matrix j mn = N γ/2 g m g n with deterministic or random g m N 0. Further we focus on the translation invariant case with deterministic g m = 1. There are two transitions in RP-model: the ALT occurring at γ = 2 and governed by the criterion (3) and the ergodic transition at γ = 1 [18] governed by (2) [41]. The YS model demonstrates localization at all γ > 2 confirmed by (3)and the critical behavior (equivalent to γ eff = 2 for the RP-model) at γ < 2 [14, 15, 42], see left panel of Fig. 2. The level statistics of RP-ensemble [18, 27 31, 34] is of Wigner-Dyson form for γ < 1 and Poisson for γ > 2. For 1 < γ < 2 it shows the Wigner-Dyson-like level repulsion at small energies E < E T h and the Poisson statistics at E > E T h, with E T h N 1 γ. Further we refer to this level statistics as the hybrid one. Low-energy level repulsion is well-represented by a socalled ratio- or r-statistics, see Fig. 3: r = ( min r n, 1 ), r n = E n E n 1, (8) r n E n+1 E n showing the value r for Gaussian orthogonal ensemble (GOE), r for Gaussian unitary ensemble (GUE), and r = 2 ln for Poisson level statistics [43]. We would like to stress once again that despite the r-statistics is widely used to locate the localization transition, it is not capable to distinguish between the WD level statistics of fully ergodic phases and the hybrid statistics. In order to distinguish between them one should study, e.g. the level number variance (δn) 2 at a large average number n E T h /δ N 2 γ of levels in the studied energy interval

4 4 which for the hybrid statistics should show the quasi-poisson behavior (δn) 2 n. A relevant measure of eigenfunction statistics is the distribution of amplitudes P ( ψ E (m) 2 ) encoded in the spectrum of fractal dimensions [8] f(α) = 1 α+ lim N ln[p ( ψ E(m) 2 = N α )]/ ln N. (9) As shown in [18] for RP f(α) takes a simple linear form { 1 + (α γ)/2, max(0, 2 γ) < α < γ f(α) =,, rest α (10) for γ 1 with an additional point f(0) = 0 for γ > 2. The f(α) in the ergodic phase, γ < 1, coincides with the one at γ = 1 and is represented by the only point f(1) = 1, see Fig. 4. The YS-model is characterized by the same properties as RP-ensemble with γ replaced by γ eff = max(2, γ) [42]. A translation-invariant relative of RP ensemble, namely TI- RP, is described by a variant of (1) H mn = ε m δ m,n + j m n, jm n 2 = 2 N γ. (11) with independent identically distributed (i.i.d.) Gaussian random (GR) hopping integrals with zero mean and the variance. Because of translation invariance j n,m = j n m, the TI-RP model possesses the duality of properties in the coordinate and the momentum spaces [44]. Indeed, FT of i.i.d. real {ε n } or complex {j n = j n} GR numbers are i.i.d. complex { J p = J p} or real {Ẽp} GR numbers with the dual variances [45]: Ẽ2 p N jn 2 N 1 γ, (12) Jp 2 N 1 ε 2 n (13) To avoid complications related to the correlations (degeneracy) {Ẽp = Ẽ p} of FT of real symmetric GR {j n = j n = j n} here and further we consider the class of Gaussian unitary ensembles. For discussion of orthogonal class of ensembles see [45]. Thus the ratio J p 2 / Ẽ 2 p N γ p where: γ p = 2 γ, (14) is a parameter dual to γ in the momentum space. A non-trivial property [42] (confirmed also by numerics, see Figs. 2 and 4) is that the eigenfunction statistics in the coordinate space of RP and TI-RP ensembles are the same for γ > 0 [45]. For γ > 2 in both ensembles the states are localized in the coordinate (r)-space, as it follows from Eq. (3). For 1 < γ < 2 they are multifractal and characterized by the linear spectrum of fractal dimensions (10), and for γ < 1 they are weakly ergodic, as follows from (2). Because of the symmetry (14) the character of eigenfunction statistics in the momentum (p-) space of TI-RP is inverted with respect to the point γ = 1 (see Fig. 2). The level statistics of TI-RP is symmetric with respect to the self-dual point γ = γ p = 1, see Fig. 3. It shows the hybrid behavior (the same as for RP in the interval 1 < γ < 2) in the Figure 3. r-statistics (average level-spacing ratio) for (a) RP and (b) PLBM families of models numerically calculated for Random Matrix Ensembles of unitary symmetry for the system size N = 2 14 and N r = 10 3 disorder realizations. In both cases deterministic models (YS and BM) show only localized or critical behavior, while TI-models (TI-RP and TI-PLRBM) demonstrate delocalized phase in a finite range of parameters returning to Poisson statistics both at small and large hopping integrals, corresponding to localization in real and momentum space. Upper (lower) horizontal lines shows the r-values for Wigner-Dyson (Poisson) statistics. Right (left) vertical line shows the Anderson localization transition in real (momentum) space for TI-models. entire interval 0 < γ < 2 and the Poisson behavior outside of it. In contrast to RP model, the Wigner-Dyson level statistics in TI-RP model do not occur. The key principles to understand this behavior of level statistics are (i) the level statistics is basis-invariant, (ii) the Wigner-Dyson (WD) statistics holds only if the eigenfunction statistics is basis-invariant and (iii) if there is a basis in which the states are localized, the level statistics must be Poisson. One of the consequence of these principles for TI-RP is the absence of WD level statistics, because nowhere (except the dual critical point γ = 1) the eigenfunction statistics is the same in the r - and p - spaces. The other consequence is the hybrid level statistics in the interval 0 < γ < 2 where eigenstates are not localized in both r - and p - spaces. The spectrum of fractal dimensions f p (α p ) in the momentum space is given then by (10) with γ replaced by γ p, (14), see the inset of Fig. 4(a). Power-law banded matrix family. The next family we consider is the one of the power-law random banded matrices (PLRBM) [8, 9] of the form of (1), with j mn = 0 and jmn 2 = [1 + ( m n /b) 2a ] 1, and its fully correlated counterpart with deterministic power-law decaying hopping integrals [10, 14, 15, 46 49] j m,n = j 0 (1 δ m,n )/ m n a, to which we refer further as the Burin-Maksimov (BM) model. PLRBM shows ALT at a = 1, with ergodic states for a < 1 and localized states for a > 1 governed by the criterion (3). The parameter b matters only at the transition point a = 1 and determines the strength of multifractality [8, 9]. By contrast, the BM-model demonstrates the power-law localization for most of the states (except measure zero) not only at a > 1, but also at a < 1 [15] with an intriguing symmetry of the exponents in the power-law decay of wave functions. The level statistics of PLRBM is of the Wigner-Dyson form for a < 1 and Poisson for a > 1 [8, 9]. In contrast, for the BM-model they are always Poisson, except for an integrable point a = 0 coinciding with the YS-model with γ = 0, where the statistics is critical, see right panel in Fig. 3.

5 5 Figure 4. Spectrum of fractal dimensions f(α) for the Rosenzweig-Porter family of models (RP, TI-RP, YS) for (a) γ = 0.5, (b) 1.5, (c) 2.5 numerically extrapolated from system sizes N = with N r = 10 3 disorder realizations in each. Dashed lines show analytical predictions (10) for f(α). (inset) Spectrum of fractal dimensions in the momentum space f p(α p) with analytical predictions (10) (dashed lines) and γ p = 2 γ for TI-RP, demonstrating the difference between RP and TI-RP ensembles in their delocalized phases. Figure 5. Average of ln( ψ(n) 2 ) for power-law banded matrix family of models (PLRBM, TI-PLRBM, BM) for different exponents in the hopping power-law decay (a) a = 0.25, (b) 0.75, (c) 1.75 numerically calculated for the system size N = 2 14 and N r = 10 3 disorder realizations. All models are power-law localized at the tails for a > 1, while for a < 1 only BM shows localization with effective exponent a eff mentioned in all panels. Dashed lines show analytical prediction (15) of this power-law decay. (inset) spectrum of fractal dimensions in the momentum space f p(α p) with analytical predictions (10) (dashed lines) and γ p = 2(1 a) given by (17), demonstrating difference between PLRBM and TI-PLRBM ensembles in their delocalized phases a < 1. Both power-law models have a built-in spatial structure. Therefore the eigenstate statistics can be characterized in more detail comparing to RP-family. Indeed, considering the typical decay of the wave function intensity ψ E (n) 2 with the distance n n 0 [16] from its maximal value ψ E (n 0 ) 2, one finds at large distances: ψ E (n) 2 typ exp [ ln ψ E (n) 2 ] n n 0 µ, (15) with µ = 2a for a > 1 both in PLRBM and in the BMmodel by the perturbation theory. At a < 1 the random model shows µ = 0, while the deterministic one gives µ = 2a eff = 2 (2 a), as shown numerically in [15], see Fig. 5. A TI relative of the PLRBM model, namely TI-PLRBM, is described by H mn = ε m δ m,n +j m n, with i.i.d. GR hopping integrals with zero mean and the variance: jm n 2 = (1 δ n,m )/ m n 2a. (16) In the momentum space both BM and TI-PLRBM ensemble are characterized by i.i.d. GR hopping integrals J p q with Jp 2 2 /N and a certain p-dependent realization of the on-site disorder Ẽp. For TI-PLRBM the typical value Ẽ(typ) p of Ẽp is N (1 2a)/2 when the sum in (4) is dominated by m N at a < 1/2 and it is of the order of 1 when this sum is dominated by m 1 at a > 1/2 [45]. Then one obtains Jp 2 /[E p (typ) ] 2 N γ(eff) p, with an effective γ p parameter for this model being: γ (eff) p = max{1, 2(1 a)}. (17) Eq. (17) and the results obtained for the TI-RP model show that for 0 < a < 1/2 (1 < γ p (eff) < 2), (see inset of Fig. 5(a)), the eigenfunction statistics in TI-PLRBM is p-fractal, the entire region a > 1/2 corresponds to p-delocalized but not fully ergodic eigenfunction statistics (like in the critical point γ = 1 of ergodic transition in RP), and for a < 0 (γ p (eff) > 2) the states are localized in the p-space. As in TI-RP, the eigenfunction statistics in TI-PLRBM in the r-space is the same as for PLRBM for a > 0 [42, 45], see Figs. 2 and 5. Notice also (see Fig. 2) that the statistics of eigenstates in the p-space of TI-PLRBM can be mapped onto that of TI-RP if one replaces a γ/2. This reflects a general property of TI ensembles: the phase diagram in the p-space of all of them is similar to the one of TI-RP. The reason for that is that the hopping matrix elements J p have the same statistics which stems from the uncorrelated diagonal on-site energies ε i. The level statistics in TI-PLRBM (see Fig. 2) can be easily identified using the three principles formulated above and then checked numerically, Fig. 3. It is Poisson at a < 0

6 6 and a > 1 and a hybrid one at 0 < a < 1/2 and at 1/2 < a < 1. As mentioned above, the latter interval in TI-PLRBM corresponds to γ p (eff) = 1. Therefore the behavior of (δn) 2 = χ n (with level compressibility 0 < χ 1) should be quasi-poisson, as in the point γ = 1 of ergodic transition in RP ensemble [18]. In the interval 0 < a < 1/2 the hybrid character of level statistics follows from the lack of basis invariance of the eigenfunction statistics (see Fig. 2). At the same time, the eigenfunction statistics (in the r-space) of PLRBM in this interval has been found to be Porter-Thomas [25]. Thus TI-PLRBM, which has the same eigenfunction statistics in the r-space as PLRBM [42, 45], is an example of a RM ensemble where the Porter-Thomas eigenfunction statistics can coexist with a hybrid level statistics. Another example of this sort is an invariant RMT described by the PDF of RM Hamiltonian Ĥ given by P (Ĥ) exp[ T r V (Ĥ)] with weak confinement V (x) ln 2 (x) discussed in Refs. [50 52]. In this case correlations of hopping matrix elements result from a strongly non-gaussian form of P (Ĥ). Duality of the exponent µ in BM model. Within the same method one can explain the duality µ(a) = µ(2 a), (18) of the power-law decay in (15) in the BM-model discovered in [15]. Indeed, according to (4) the spectrum of the disorder-free BM model H 0 mn = j 0 / n m a behaves as [45, 46, 49] Ẽ p /(2j 0 ) ζ a + A a ( N p ) 1 a, for p N, (19) ( ) 2 2q Ẽ p /(2j 0 ) Ẽmin + B a, for q N (20) N for a 0, 1, with q = N/2 p, the Riemann zeta-function ζ a, and dimensionless constants A a, B a, and Ẽmin given in [45]. According to (5) and (6) all states with P 2 a < p 2 a 2N 3/2 a j 0 a 1 / are p-localized (extended in real space) and disorder cannot delocalize them [53] (see similar arguments in [49]). The typical states live close to the bottom of the spectrum E p (typ) N 0, (20), with the level spacing δẽp N 2 [47]. Then as follows from the RG analysis (analogous to [47, 54]) these typical states, P 2 a > p 2 a are always localized in the coordinate space [45]. In addition, in the momentum space analogously to TI-PLRBM, (17), one obtains for BM J p 2 /[E p (typ) ] 2 N γ(eff) p, with an effective parameter γ p (eff) = 1, as J p 2 2 /N. For the YS-model, a = 0, the spectrum of the disorder-free hopping model, Eqs. (19, 20), simplifies to Ẽ0 j 0 N 1 γ/2 and Ẽ p 0 = 0 and the effective parameter γ p (eff) is infinitely large. In both models it shows that the eigenfunction statistics in the momentum space is weakly ergodic for all states, except measure zero [55]. One also might suggest in the spirit of [14] that localization properties of typical states can be uncovered by truncating the Hilbert space to the fast oscillation sector P > p. However, as recently pointed out in [56], this naive procedure gives an effective model of the form of the PLRBM ensemble with the critical exponent, a eff = 1, and strongly correlated random hopping terms that do not explain the localization phenomenon at a < 1 and a surprising duality (18). Instead, the effective decay rate a eff = 2 a for a < 1 which explains the duality (18), can be found within a more accurate method, using hopping matrix inversion. The starting point is an identity for an eigenstate ψ E (m): ψ E (m) = N M m l (E + E 0 ε l )ψ E (l), (21) l=1 where E is the eigenfunction energy, ( E 0 ) N 0 is a constant taken to be below the bottom of the spectrum Ẽmin < 0, and M = (E 0 + ĵ) 1 is the inverted hopping matrix ĵ shifted by E 0. The elements of the matrix M for a < 1 are given by: M n = C 1 e κ n + C 2 (1 δ n,0 ) j 0 n a eff, a eff = (2 a). (22) where dimensionless constants C 1, C 2, and κ can be found in [45]. The second term in (22) is nothing but the long-range part of inverse Fourier transform of (E 0 + Ẽp) 1 1/Ẽp, where Ẽp is given by (4), (19). This result is crucially based on the fact that at a < 1 the spectrum Ẽp is unbounded from above in the limit N, while E min in (20) (and E 0 ) remains finite. The first term in (22) is given by the shortrange part of inverse Fourier transform of 1/Ẽp given by (20). Because a eff > 1 at a < 1, the sum in r.h.s. of (21) converges if ψ E (l) behaves as in (15) with { 2 aeff = 2 (2 a), a < 1 µ = 2a, a > 1. (23) Then one concludes that the l.h.s. of (21), ψ E (m), for a < 1 behaves at large distances m in the same way as M m, thus making assumption (23) self-consistent. Note that this derivation is not valid for TI models with uncorrelated and sign-random hopping integrals. The reason is that due to the random-sign nature of j n in the r-space, Ẽ p for a < 1 is unbounded both from above and from below in the limit N. Thus, (E 0 + ĵ) 1 is singular for any N-independent shift E 0 N 0 which leads to a delocalized behavior of ψ E (m) [45]. Possibly the same arguments apply to the models with sign-alternating non-random hopping integrals [57, 58]. Conclusion and discussions. We demonstrate that the correlations in long-range hopping integrals may change drastically the localization-delocalization phase diagram of many models turning extended phase into the (multi)fractal or even localized one. We show that the well-known localization principles (2) and (3) are not equivalent and are in fact the sufficient (but not the necessary) conditions for weakly ergodic delocalization and localization, respectively. This conclusion is illustrated by the localization-delocalization phase diagram Fig. 2 which is the main result of this paper.

7 7 We suggest a natural extension of the class of models with correlated long-range hopping integrals by introducing the translation-invariant random matrix models, where hopping integrals are fully correlated along the diagonals but the correlations between the diagonals are absent. We identify phases with different character of localization/delocalization in these models both in the coordinate and in the momentum spaces together with the spectral statistics. The results are summarized in Fig. 2. It is shown that at moderately small disorder strengths the delocalized phases in TI-models are never fully ergodic, and the eigenfunction statistics are different in the momentum and the coordinate spaces. The eigenfunction statistics in the momentum space of different TI models are shown to be similar and represented by a new universal model, TI-RP, showing the fractal and localized phases in the momentum space. In the coordinate space the eigenfunction statistics in all TI models is the same as for the corresponding non-ti counterparts for hopping integrals decreasing either with the system size (γ > 0) or with the distance (a > 0). We formulated the principles to identify the level statistics in the considered models as belonging to the Wigner-Dyson, Poisson or the new hybrid class. Within the same method, for the random matrix models with deterministic power-law decaying hopping integrals j n m n m a we confirm that both for a > 1 (see [46 49]) and a < 1 [15] the typical states are localized with the power-law tails ψ n (m) n m a eff at a 0 and uncover an origin of the duality a eff = max(a, 2 a). It is also worth noticing that our arguments are not restricted only to the one-dimensional case, d = 1. Recent work [58] has shown the presence of localized states in the case of isotropic deterministic power-law decaying hopping in three-dimensional cubic lattices for a < d = 3, which is wellunderstood in terms of our formalism. However, the certain form of correlations, such as sign-alternation (or anisotropy) of hopping terms, with translational invariance [58] kept intact, may delocalize the system in the coordinate space. The sign-alternating (or anisotropic) hopping acts as a pseudo disorder since the level-spacing structure does not exhibit well-separated Hilbert subspaces (cf. [14]). Such models are expected to behave similar to TI ensembles, showing p-fractal and p-localized phases at small disorder instead of localization in real space. Another intriguing direction of research is the interplay between correlations in the hopping integrals and in the on-site energies. As recently shown the correlated on-site disorder (quasi-periodic potential [4]) may destroy localization and produce a whole bunch of (multi)fractal phases depending on the power a [59] in the BM-model with deterministic power-law hopping integrals. In addition, the BM-model (for a > 1) with long-range interactions has been considered recently [60] in the problem of entanglement dynamics and many-body localization. One of the promising directions would be to consider effect of longrange interactions with a < 1 on Many Body Localization. We are grateful to D.N. Aristov, A.L. Burin, R. Moessner, and A.S. Ovchinnikov for stimulating discussions. P. N. appreciates warm hospitality of the Max-Planck Institute for the Physics of Complex Systems, Dresden, Germany, extended to him during his visits when this work was done. P. N. acknowledges funding by the RFBR, Grant No , and the Foundation for the Advancement to Theoretical Physics and Mathematics BASIS Grant No I. M. 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9 9 APPENDIX This document provides supplemental material discussing some technical details of analytic considerations and numerical results. In particular, the Supplemental Information contains the following notes: In the main text, we have introduced several translation-invariant (TI) matrix ensembles and made use of scaling criteria in momentum space. In S1 we consider properties of Discrete Fourier Transform (DFT) of Gaussian random variables which lead to the duality for TI-Rosenzweig-Porter (TI-RP) model. In S2 we provide some details of numerical calculations and describe the algorithm of extracting the spectrum of fractal dimensions f(α). The S3 is devoted to the comparison of the wave function distributions for TI and non-ti uncorrelated models in the weakly ergodic delocalized phase, a, γ < 1, with the fully ergodic random-matrix theory (RMT) prediction. In the main text we have used scaling arguments for Burin-Maksimov (BM) [10] ensemble both in coordinate and momentum spaces, which require calculation of the disorder-free spectrum and level-spacing structure. In S4 we present its detailed derivation and the estimate for the number of decoupled delocalized states. In S5 we make use of the Renormalization Group (RG) approach introduced in [47, 54] for BM. We apply this approach to the bottom of the band, where most of the states live, and claim that all these states are localized even for a < 1 in full agreement with scaling arguments and predictions of our effective real-space theory. Next, in S6 we present derivation of the effective real-space Hamiltonian for BM and discuss its relations to so-called cooperative shielding [14] and spectrum truncation [56]. Additionally, we show how the duality a 2 a can be violated in this model. Finally, in S7 we calculate the effective parameter TI-RP-parameter γ p in the momentum space for TI-power-law random banded matrices (TI-PLRBM). Due to the property Appendix S1: Discrete Fourier transform of Gaussian random variables N 1 m=0 X m X m = N 1 p=0 X p X p (S1.1) of the discrete Fourier transform (DFT) X p = X p = 1 N 1 X N 1/2 m e 2πi pm N m=0 (S1.2) for real (complex) Gaussian statistically independent random variables X m = ε m / N (X m = j m N) with zero mean and fixed variance X i = 0, Xi 2 = σ 2, P (X 0,..., X N 1 ) = N 1 i=0 e X2 i /2σ2 2πσ 2, (S1.3) the real-valued components of their DFT Re X p, Im X p, p = 0, N/2 are also independent Gaussian random variables with the same variance σ 2 for real components X p = X p and two times smaller one σ 2 /2 for each real-valued components of complex elements X p. Note that Im X 0 = 0 for all N and Im X N/2 = 0 for even N giving in total always N real random numbers. As a result of this property, the case of the Gaussian unitary ensemble (GUE) provides only the hermitian condition H mn = Hnm on the matrix elements of the TI-RP Hamiltonian H mn = ε m δ mn + j m n, keeping the duality γ p = 2 γ of the TI-RP ensemble and avoiding approximate degeneracies in its spectrum at small γ. However, in the other case of the Gaussian orthogonal ensemble (GOE) the symmetric constriction H mn = H nm on the realvalued matrix elements correlates the TI-hopping integrals j n = j n and produces the degeneracy Ẽp = Ẽ p in the spectrum Ẽ p = Ẽ p = N 1 m=0 j me 2πi pm N of the hopping problem H mn (0) = j m n. This results in the ratio-statistics going to zero at γ < 0 and lifts the TI-RP symmetry. Therefore in the main text we focus on GUE case.

10 10 Figure S1. Extrapolation of the spectrum of fractal dimensions for TI-RP ensemble in the system size N in (a, b) coordinate and (c, d) momentum spaces. (a, c) The finite size spectrum of fractal dimensions f(α, N) (f p(α p, N)) in the coordinate (momentum) space versus α (α p) for different N; (b, d) f(α, N) (f p(α p, N)) in the coordinate (momentum) space versus 1/ ln N for different values of α (α p) (symbols) with the linear fitting (dashed lines). Appendix S2: Details of numerical calculations In order to calculate the spectrum of fractal dimensions f(α) in all considered models we first collect the empirical distribution P ( ψ E (m) 2, N) of wavefunction intensities ψ E (m) 2 over all N sites 1 m N averaged over the half of the states in the middle of the spectrum of a finite system and over N r = 10 3 disorder realizations in the form of a histogram. The extracted finite-size spectrum of fractal dimensions f(α, N) = ln ( N 1 α P (N α ) ln N ) / ln N has been extrapolated to the limit N over several system sizes with the linear fit f(α, N) = f(α) + c α / ln N, see Fig. S1(b). In order to eliminate the effect of zeros of wave functions ψ which dominate the distribution function P ( ψ E (m) 2 ) at small ψ E (m) 2 N 1 and extract the distribution function of a smooth envelope of ψ E (m) 2 we use the so-called rectification approach suggested in [19] and used in [18]. Indeed, we represent ψ E (m) = ψe env (m) η, where η is GOE random oscillations with the unit average square which are supposed to be statistically independent of ψe env(m). Then the distribution of ln ψ E(m) 2 is a convolution of the distribution ln ψe env (m) and the known distribution of ln η. Making a numerical de-convolution one obtains the distribution P env ( ψe env(m) 2 ) in which the effect of zeros of η is eliminated. One can see that f(α) rectified and extrapolated as explained above has a linear in α part which exactly coincides with the analytical predictions dashed lines in Fig. S1(a). The analysis of the spectrum of fractal dimensions f p (α p ) in the momentum space is completely analogous, see Fig. S1(c, d). Appendix S3: Comparison of eigenstate distributions in a weakly ergodic phase with RMT predictions In this note we consider the eigenstate distributions of considered models in more details, focusing on their deviations from fully-ergodic RMT predictions. The standard RMT predicts the Gaussian distribution of each real-valued component ψ R of the eigenvector (see, e.g., [61, 62]): P (ψ R ) = exp [ βnψ 2 R /2] (2π/(βN)) 1/2, (S3.1) with the matrix size N and the ensemble parameter β taking the values β = 1, 2, 4 for GOE, GUE, and GSE (the Gaussian symplectic ensemble), respectively. Due to the duality of TI-models in the coordinate and momentum spaces in the main text we focus on the GUE case, β = 2, thus, the RMT distribution of the renormalized wave-function intensity y = N ψ 2 in this case takes a simple exponential form, the analogue of the Porter-Thomas distribution for GOE: P (N ψ 2 = y) = e y. (S3.2) Some recent works (see, e.g., [25]) predict the deviation of eigenstate statistics of the PLRBM ensemble from Eq. (S3.2) at a > 1/2 even in the thermodynamic limit N. Another motivation of the detailed study of the distribution P (N ψ 2 = y)

11 11 Figure S2. Comparison of eigenstate probability distributions P (N ψ 2 = y) in the (weakly) ergodic phase of (a-d) RP and (e-h) TI-RP models (solid lines for different system sizes N) with the RMT prediction P (y) = e y (black dashed line). Figure S3. Comparison of eigenstate probability distributions P (N ψ 2 = y) in the (weakly) ergodic phase of (a-d) PLRBM and (e-h) TI-PLRBM models (solid lines for different system sizes N) with the RMT prediction P (y) = e y (black dashed line). is the statement from the main text that the eigenfunction statistics in the coordinate space of RP (PLRBM) and TI-RP (TI- PLRBM) ensembles are the same for γ > 0 (a > 0), see [42] for more analytical details. To verify both these statements we plot the P (N ψ 2 = y) for several system sizes N = 2 9,..., 2 14 for all four ensembles (color solid lines in Figs. S2 and S3) together with the RMT prediction (S3.2) (black dashed lines). From Fig. S2(a-d) one can see that for all γ < 1 the RP-ensembles shows exponential distribution at least in the thermodynamic limit N (see the flow of the distributions with the system size). In the critical point γ = 1 of the ergodic transition the distribution does not flow towards (S3.2) as at that point the spectral statistics is quasi-poisson with the finite compressibility κ at large energies [18]. The TI-RP ensemble, Fig. S2(e-h), shows the same behavior as its non-ti counterpart for γ > 0 (in the thermodynamic limit), while the point γ = 0 explicitly shows deviations from (S3.2). The deviations at γ = 0 are expected as this point is the Anderson localization transition in the momentum space. On the other hand, the convergence of the eigenstate statistics to the RMT prediction in TI-RP at 0 < γ < 1 provides a non-trivial example of the phase, where the RMT eigenfunction statistics can coexist with a hybrid level statistics. Figure S3(a-d) confirm the results of the recent paper [25] for the PLRBM ensemble, showing the convergence of eigenstate statistics to the RMT prediction at a < 1/2 and deviation from it at 1/2 < a < 1. The TI-counterpart of PLRBM, Fig. S3(e-h), tends to (S3.2) at 0 < a < 1/2 in the large N limit and deviates from it at a > 1/2. The finite size scaling behaviors of PLBRM and TI-PLRBM are slightly different, which still does not contradict to the statement [42]. As in TI-RP, the eigenstate statistics in TI-PLRBM deviates from RMT prediction at a = 0 due to the Anderson localization transition in the momentum space.

12 12 Appendix S4: Derivation of the disorder-free spectrum and number of decoupled delocalized states Here we consider the continuous approximation N of the DFT of hopping terms j n = (1 δ n,0 )/ n a in BM given partially in [46, 49] Ẽ p /(2j 0 ) = n 0 n <N/2 e 2πipn/N 2 n a = Re N/2 n=1 e 2πipn/N n a = Re [ ] ( 1) p e 2πip/N Φ(e 2πip/N, a, 1 + N/2) + Li a (e 2πip/N ), with Lerch transcendent Φ(z, s, b) = n=0 zn /(n + b) s and polylogarithm Li m (z) = n=1 zn /m n functions. The expansion of the polylogarithm gives the main result. Indeed, Ẽ 0 /(2j 0 ) = N/2 n=1 where H n is the Harmonic number and ζ a is the Riemann zeta function, (S4.1) 1 n a = H n N 1 a 1 a + ζ a + O(N a ), a > 0, a 1, (S4.2) Ẽ p /(2j 0 ) ζ a + A a ( p N ) a 1, 0 < p N, (S4.3) with A a = (2π) a 1 Γ 1 a sin πa 2, a 2m + 1, m N, (S4.4) Ẽ p = 2j 0 Re q = N/2 p N, and N/2 n=1 ( 1) n e 2πiqn/N n a 2j 0 N/2 n=1 ( 1) n n a [ 1 n2 2 ( ) ] 2 2πq ( q ) 2 N Ẽmin + B a, (S4.5) N Ẽ min = 2j 0 (2 1 a 1)ζ a < 0 for a > 2; B a = 8π 2 j 0 (1 2 3 a )ζ a 2 2π 2 j 0 a > 0. (S4.6) Now we estimate number N dec of decoupled delocalized states both for BM [15, 46 49] and the Yuzbashyan-Shastry (YS) [11, 12] models. In the integrable case of YS-model, a = 0, with the scaling of the ratio of the hopping amplitude j 0 to the disorder strength : j 0 / N γ/2, the only state, namely the zero-momentum state with the energy Ẽ0 N 1 γ/2, is decoupled from other N 1 degenerate states at γ < 2 [14]. The extensive behavior of the level spacing with N, demonstrated in YS model, survives in BM for all a < 1, but in this case the number of decoupled states is extensive N dec 1. Indeed, according to Levitov s arguments [5, 6] we compare the energy differences (a 0, 1) ( ) 1 a N ( δe p = Ẽp+δp Ẽp 2j δp ) a 1 ( ) 2 a N 1 2π p p 2j δp 0 a 1, δp p (S4.7) p N with sum of absolute values of hoppings p+δp p =p J p δp N, and get that for all the states with p 2 a < p 2 a N 3/2 a 2j 0 a 1 (S4.8) they are localized in p-space (extended in real space) and the disorder N 0 cannot delocalize them, meaning that N dec p for a < 3/2 and N dec = 0 for a > 3/2. Similar arguments are given in [49] for a > 1. Note, however, that in the case of 1 < a < 3/2, the effect of the decoupled delocalized states is small as all their energies are not increasing with N and there is the critical disorder strength of order of the bare bandwidth c p = Ẽ0 Ẽmin (S4.9) above which all states p < p become also localized (see, e.g., [47, 49]). For j 0 / N 0 and a > 3/2 all states are localized for any disorder strength.