Absence of Mobility Edge in Short-range Uncorrelated Disordered Model: Coexistence of Localized and Extended States

Unlike the well-known Mott's argument that extended and localized states should not coexist at the same energy in a generic random potential, we provide an example of a nearest-neighbor tight-binding disordered model which carries both localized and extended states without forming the mobility edge (ME). Unexpectedly, this example appears to be given by a well-studied $\beta$-ensemble with independently distributed random diagonal potential and inhomogeneous kinetic hopping terms. In order to analytically tackle the problem, we locally map the above model to the 1D Anderson model with matrix-size- and position-dependent hopping and confirm the coexistence of localized and extended states, which is shown to be robust to the perturbations of both potential and kinetic terms due to the separation of the above states in space. In addition, the mapping shows that the extended states are non-ergodic and allows to analytically estimate their fractal dimensions.

A mobility edge (ME) [1], separating localized and extended states in disordered systems, has been established and studied for decades.Known to be present in various semiconductors, amorphous media and even in disordered liquid metals, ME has become a hallmark of the Anderson [2] and many-body [3] localization transitions.It is commonly believed that in any short-range model, with random uncorrelated entries, just below the localization transition ME separates the localized and extended states in the energy spectrum.Therefore, eigenstates with different localization properties cannot coexist at the same energy for the same system parameter values.The argument behind this, given by Nevill F. Mott [1], is straightforward: if extended and localized states coexist at the same energy, any perturbation of the disorder potential immediately hybridizes them, making both extended.In this Letter, we provide an example of onedimensional (1D) disordered short-range model, where in any realization both extended or localized states can emerge at a given energy.Hence, disorder averaging forbids the ME formation.In this work, we identify the necessary conditions to be satisfied to observe such coexistence.
First, the system should avoid level degeneracy or attraction, i.e. it should possess some (residual) level repulsion.Indeed, any resonance in the energy levels, corresponding to localized and extended states, should be suppressed in order to observe their coexistence without hybridization.Among short-range uncorrelated models, the natural ensemble for tunable and controllable level repulsion is the so-called β-ensemble, represented by real symmetric tridiagonal matrices, with independent random elements [4].Such an ensemble is parameterized by the Dyson's index β and has the same joint probability distribution of eigenvalues like in the wellknown Gaussian random-matrix ensembles [5], but for any real β along with β = 1, 2, 4. The limit β → 0 yields uncorrelated eigenvalues as observed in integrable systems [6], whereas β ≥ 1 produces correlated spectra as in chaotic systems [7].
The last, but not the least, the NEE states should be separated in space from the localized ones.The construction of β-ensemble [4] introduces inhomogeneity, where the distributions of hopping matrix elements y n significantly depend on the lattice coordinate n, see Fig. 1(a) and Eq.(2).Consequently the eigenstates of β-ensemble become spatially separated.
In this Letter, we show that by fulfilling all of the above three crucial ingredients, β-ensemble provides an ideal platform for realizing coexistent localized and extended eigenstates.We numerically confirm that O(β −1 ) localized states coexist along with the extended states within the middle of the spectral band without forming any ME.In addition, the NEE phase of β-ensemble is shown to exhibit anomalies in the spectral statistics: nearby eigenvalues remain uncorrelated, while two distant eigenvalues, separated by ∆E > (N β) − 1 2 , can be correlated.Here and further, N is the system size.Such a feature is in sharp distinction from the NEE states observed in a paradigmatic Rosenzweig-Porter ensemble (RPE) [16,19,30].This last aspect unveils the origin of the absence of the ME and can be analytically explained by a local equivalence of β-ensemble to an 1D Anderson model with N -dependent hopping strength.This equivalence demonstrates that β-ensemble separates into nearly independent blocks, where localized and (nonergodic) extended eigenstates appear to be located in spatially separated blocks, but share nearly the same spectral energies.
β-ensemble is composed of the matrices H, with the following nonzero elements [4] where N (0, 1) is the normal distribution and χ k is the chi-distribution with a degree of freedom k.H represents a 1D lattice with an open boundary, where a particle can randomly hop to the nearest neighbors under disordered on-site potentials, Fig. 1(a).The relative strengths of onsite potentials { x n } and the hopping amplitudes { y n } make it convenient to re-parameterize β as β = N −γ , leading to the typical behavior of hopping amplitudes, see Appendix A Thus, on average y n increases across the lattice and presents a highly inhomogeneous system.Due to such inhomogeneity, β-ensemble hosts three distinct phases: ergodic (γ ≤ 0), NEE (0 < γ < 1) and localized (γ ≥ 1), separated by second-order phase transitions [29].In the localized phase, all the levels are uncorrelated and Poisson-distributed as the eigenstates are localized with a finite support in the thermodynamic limit (N → ∞), see the left state in Fig. 1(a).Contrarily in the ergodic phase, all energies are correlated irrespective of their distance and the bulk eigenstates are extended over the entire Hilbert space.Different phases in β-ensemble exist due to its inhomogeneous hopping terms, otherwise phase transition is absent in generic 1D systems with uncorrelated short-range hopping [32,33].
The Hilbert-space structure in the NEE phase can be understood from the system-size scaling of the inverse participation ratio (IPR), 4 of the eigenstate at the energy E j having nth component Ψ j (n).It is observed that log I j −D typ 2 log N , where D typ 2 ≈ 1 − γ is a typical fractal dimension, quantifying the eigenstate support set, Fig. 1(b).Hence most of the eigenstates occupy an extensive number, but vanishing fraction of the Hilbert space in the NEE phase.However, the density of IPR shows a peak around I = 1 indicating the presence of strongly localized states, |Ψ loc , along with a finite fraction of extended states with I 1, see Appendix B. We consider a small tolerance value δI 1 and identify |Ψ loc as a state with I > 1−δI.In Fig. 1(b), we show that the ensemble-averaged number of strongly localized states, N loc ∝ N γ coexisting with the finite fraction of NEE states.Now, looking at the joint density of IPR and energy, Fig. 1(c), we unveil the spectral structure of |Ψ loc .In the NEE phase of β-ensemble, |Ψ loc appears only within an energy window (−E G , E G ), centered around midspectrum (E = 0).The ensemble average Does this coexistence form a ME? ME has been observed in the Lévy ensemble [20,34], quasiperiodic lattice [11,[35][36][37][38][39][40][41], 3-D Anderson model [42], quantum random energy model [43] and many-body localization [3].In order to search for ME in the β-ensemble, first, we compute the energy-dependent IPR However, IPR has a fat-tailed distribution in the β-ensemble and may not be a self-averaging quantity [29], requiring more convincing measures.Thus, we extract the energy-dependent fractal dimension D 2 (E) from the system-size scaling of median(I) within small windows across the energy spectrum.Fig. 2 This convincingly shows that the ME is absent in the NEE phase of β-ensemble despite the coexistence of O(N γ ) localized states around E = 0.
In addition to the spectral homogeneity of the localization properties, one can consider energy-level correlations across the spectrum.The ensemble-averaged level-spacing ratio r [44,45] exhibits criticality around γ = 0, implying that the neighboring eigenvalues are typically uncorrelated in the NEE phase [29] in thermodynamic limit, having some residual level repulsion β = N −γ at finite sizes.In order to understand the energy dependence of short-range energy correlation we compute The ensemble-averaged r in the γ-E plane, Fig. 2(c), also shows no energy-dependent crossover from Poisson, r ≈ 0.38, to Wigner-Dyson, r ≈ 0.53, statistics [44,45], for all γ.Thus, short-range energy correlations are also homogeneous over the bulk energy spectrum.
Besides the spectral homogeneity at short energy scales, one needs to characterize long-range two-level correlation.This can be captured by the fluctuations of the nth unfolded energy level, E n [46] around its mean position, n via the power spectrum P (ω) vs frequency ω, Fourier-dual to n.For Poisson (Wigner-Dyson) statistics, the power spectrum of the fluctuations δ n ≡ E n − n decays as ω −2 (ω −1 ) [47][48][49].In β-ensemble, P (ω) shows a heterogeneous behavior in the frequency domain, see Fig. 3: The critical frequency, ω c , corresponds to the unfolded energy scale N γ such that two unfolded energy levels , and uncorrelated otherwise.Therefore, in an energy window shows only Poisson behavior for ∆E < N γ irrespective of E. Such a longrange correlation is unusual and complimentary to the energy correlations typically observed in various models like RPE [16,50,51], deformed Poisson ensemble [21], or driven Aubry-André models [10].Usually the eigenstates hybridize below the Thouless energy, E Th [52], while distant eigenvalues, separated by ∆E > E Th , remain uncorrelated [23,53].
Above numerical results unambiguously show that the coexistence of localized and extended states fails to form any ME.This can be analytically understood from a local equivalence of the β-ensemble to an Anderson model.Indeed, in β-ensemble, the hopping amplitudes over the first O(N γ ) sites are much smaller in magnitude in comparison to the typical on-site potentials x typ n ∼ O(1).Hence the sites 1, 2, • • • , n N γ can be considered to be effectively disconnected from the rest of the lattice and hosts |Ψ loc .Corresponding DOS follows the normal distribution, while both short-and long-range energy correlations are given by the Poisson statistics [31].
The structure of the eigenstates at the remaining sites can be understood as follows.
The hopping amplitudes for lattice sites n > N γ , given by Eq. ( 2), are self-averaging and homogeneous with small relative fluctuations |y n+δn − y n | y n for δn n.Consequently, one can partition the entire lattice into the spatial blocks n ∈ ∆ l , where hopping is approximately the same and Here log N , and 0 ≤ l < and the corresponding mean level spacing is The above equivalence explains the properties of the NEE phase of β-ensemble as illustrated below.
Secondly, the eigenstates in the last largest block are least localized with a localization length ξ ∼ N 1−γ .This block contains a finite fraction of all sites, N l ∼ O(N ) and then defines the typical fractal dimension D typ 2 = 1 − γ in the NEE phase of β-ensemble [29].As both the number of eigenstates, with localization length 1 ≤ ξ l ≤ N 1−γ , and the bandwidth, Eq. ( 7), increase with l, the distribution of any localization measure exhibits a fat tail [31].Thus, within (−E G , E G ), where all the bands overlap, all the localization lengths are possible.This structure of spatial-separated states with different ξ l explains the absence of ME and the coexistense of localized and extended states in β-ensemble.
Thirdly, the Anderson equivalence explains the anomalous long-range energy correlations in the NEE phase of β-ensemble.The eigenvalues from all blocks constitute the global DOS, hence the bandwidth β is given by that of the largest block at l max 1−γ δζ with a bandwidth N 1−γ 2 .Thus, global mean level spacing is given by δ Contrarily, the smallest level spacing, locally in a sub-block is δ min = δ lmax ∼ N − 1−γ 2 .As δ min > δ, neighboring eigenvalues come from different sub-blocks and are, thus, uncorrelated, while the correlated ones have at least the energy difference δ min .The unfolding procedure rescales δ → 1, setting a critical dimensionless energy δ min /δ = N γ , in agreement with numerics.Any two unfolded energy levels 2 ), they may belong to the same sub-block and be correlated.This explains the origin of the anomalous behavior in the power spectrum of the energy fluctuations in the β-ensemble, Eq. ( 5).
Finally, we numerically confirm that the eigenstates of β-ensemble in the NEE phase are exponentially decaying using the metric defined in [31,56].Therefore we can order the eigenstates according to their localization centers instead of energy.To understand the correlation among such spatially ordered eigenstates, we look at the covariance matrix The covariance matrix gives a rather complete idea about the Hilbert space structure.By plotting the thresholdfiltered covariance matrix Mij = Θ(M ij − δM ), with the Heaviside step function Θ(x) and a threshold δM < 2 π , we unveil the eigenstate spatial correlation structure.Mij = 1 implies that ith and jth states have a high degree of overlap, i.e. they are hybridizing and vice-aversa.In the ergodic phase M is a dense matrix, while it is sparse in the localized regime.In β-ensemble, see Fig. 4, M shows banded structure with the spatial band, increasing with the indices i, j, confirming the analytical picture of the block ∆ l .We have also shown the energy levels of the spatially ordered eigenstates.This further shows in the β-ensemble the localized and the NEE states can appear at nearly same energy eventually leading to coexistence in the thermodynamic limit.
To sum up, in this letter, we provide the set of main principles on how to avoid the mobilityedge emergence in short-range disordered models and illustrate them in a well-known example of β-ensemble.With various spectral and localization measures, we uncover the structure of the coexistence of localized and extended states in such model and confirm these results analytically by the spatially local mapping to the 1D Anderson model with system-size-dependent hopping.The general principles, formulated in this work and verified on β-ensemble, allow one to realize the coexistence of the localized and extended states in the same energy interval without fine-tuning which is robust against perturbations and disorder realizations.Such systems can be used for quantum memory and faulttolerant quantum calculations, where the localized states, decoupled from the extended modes of the bath, are free from decoherence.As an outlook, it would be of particular interest to find many-body realizations of the above concepts.We observe that all the densities match nicely with P Poisson (r) = 2 (1 + r) 2 .In the inset of

1 FIG. 1 . 2 ≈ 1
FIG. 1.(a) Schematic of β-ensemble, given by 1D lattice in Eq. (1).The hopping increases along the lattice (thickness, color of links), yields localized (left) and extended (right) states, coexisting at the same energy, but living in spatially different system parts.(b) Phase diagram of β-ensemble, with three distinct phases along with typical fractal dimension in the NEE phase, D typ 2 ≈ 1 − γ, and the number N loc ∼ N γ of strongly localized states, |Ψ loc .(c) Joint density P I, E of IPR, I and the energy, E = E/ β , rescaled to the bandwidth β for γ = 0.7.The colorbar indicates the values of joint density in log N scale, −E G , E G is the rescaled energy band for coexistent states.Numerical results are for N = 8192 and 128 realizations.
takes a wide range of values from O(1) to N −D2 , convincingly demonstrating the coexistence of localized and extended states.

1 FIG. 2 .
FIG. 2. (a) IPR, (b) fractal dimension D2, (c) mean levelspacing ratio r for N = 8192 in the γ-E plane.Solid black lines indicate −E G , E G , the energy band of localized states.
where ρ (E) is the global density of states (DOS).For a given energy I(E) → 0 ( O(1)) for extended (localized) states.Hence, an existence of a ME would have implied I(E) exhibiting an energy-dependent crossover from 0 to O(1) within (−E G , E G ). Fig.2(a), showing IPR in the γ-E plane of the energy E = E/ β , rescaled by the global energy bandwidth β = 2 E 2 = √ 4 + 2N 1−γ , provides no evidence of ME at any value of γ.

≈ 2 . 2 n x 2 n=
δζ .The number of sites in ∆ l is N l ≈ N γ+ζ l and the hopping amplitudes can be written as y typ l Thus within ∆ l , hopping amplitudes are effectively constant and the model is equivalent to the 1D Anderson model with hopping strength N ζ l 2 and on-site potential O(1).As a result, within each block, the eigenstates exponentially decay, Ψ(j) ∼ exp − |j−j loc | ξ , see Appendix D, with the localization length ξ l ∼ y N ζ l [54] and localization centers j loc randomly distributed in ∆ l .Furthermore, as each sub-block of length ξ l ∼ N ζ l has N ζ l eigenstates, thus, each block ∆ l can accommodate N l /ξ l ≈ N γ sub-blocks.The eigenstates within each sub-block hybridize, but not across sub-blocks.The local Anderson equivalence implies that each sub-block has Gaussian DOS with a bandwidth[55] FIG. 4.(left) Threshold-filtered covariance matrix M , Eq. (8), for N = 1024, γ = 0.5, and δM = 0.5 The red dashed lines denote spatial blocks of the form ∆ l (Eq.(6)) with δζ = 0.1 and l = 0, 1, . . ., 5, while the cyan lines show N γ spatial sub-blocks in the last block.(right) Energies of spatially ordered eigenstates.The blue dashed vertical lines denote −E G , E G , i.e. energy bound of localized states.

1 FIG
FIG. I. (a) Density of IPR for various γ and system size N = 8192.(b) system size scaling exponent of N loc , the ensemble averaged number of localized states, (i.e.N loc ∝ N α loc ) as a function of γ.Inset shows the scaling exponent of the coefficient of variation of N loc , i.e.CV(N loc ) ∝ N α CV .The error-bars denote 99.9% confidence interval obtained from linear fitting in log-log scale.

E/ϵ β 1 FIG
FIG. III.Joint density of IPR and energy for various γ and N = 8192 obtained from 128 disordered samples.The colorbar indicates values of P I, E β in log N scale where β ≡ √ 4 + 2N 1−γ is the global bandwidth.
Fig. II(b), we show the ensemble averaged r as a function of γ for various N and δI.We observe that irrespective of N, γ or δI, r ≈ 0.38 which is expected of a Poisson ensemble.Thus short-range correlation is absent in the energy levels of localized states.Complimentary to the short-range measures like ratio of level spacing, the long-range spectral correlations are captured by P δ ω , the power-spectrum (squared modulus of Fourier transform) of δ n statistics.As E loc /σ follows a normal distribution, the unfolding becomes trivial where the unfolded energy E loc = spectrum of noise in such unfolded energy levels exhibits a power-law behavior w.r.t.ω as shown in Fig. II(c), i.e.P δ ω ∝ ω αω .The inset of Fig. II(c) shows that irrespective of N, γ or δI, the exponent α ω ≈ −2 similar to Poisson ensemble.Thus we can conclude that the energy levels of localized states are uncorrelated at any distance.In Fig. III, we show the joint density of IPR and energy for various values of γ.Such plots show the coexistence localized states in the NEE phase as well as helps us identify (−E G , E G ), the energy bound of localized states.Fig. IV(b) shows that the ensemble average E G does not scale with system size for any value of γ irrespective of δI.In Fig. IV(a), we show that the long-range energy correlations are homogeneous over the energy spectrum in the ergodic and localized phase.

1 FIG
FIG. IV.(a) ensemble averaged power spectrum of noise vs. dimensionless frequency, ω for various γ where ∆E = 1024 and N = 32768.For each γ, solid lines denote PE (ω) at three randomly chosen E and the dashed curves denote PE (ω) averaged over E. Inset shows exponent α vs. unfolded energy where PE (ω) ∝ ω −α .(b) system size scaling of EG ∝ N α G for various δI where (−EG, EG) is the energy bound of localized state (has I > 1 − δI).