Eigenstate correlations around many-body localization transition
EEigenstate correlations around many-body localization transition
K. S. Tikhonov
1, 2 and A. D. Mirlin
3, 4, 2, 5 Skolkovo Institute of Science and Technology, Moscow, 121205, Russia L. D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia Institute for Quantum Materials and Technologies,Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany Institute for Condensed Matter Theory, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany Petersburg Nuclear Physics Institute,188300 St. Petersburg, Russia.
We explore correlations of eigenstates around the many-body localization (MBL) transition intheir dependence on the energy difference (frequency) ω and disorder W . In addition to the genuinemany-body problem, XXZ spin chain in random field, we consider localization on random regulargraphs (RRG) that serves as a toy model of the MBL transition. Both models show a very similarbehavior. On the localized side of the transition, the eigenstate correlation function β ( ω ) showsa power-law enhancement of correlations with lowering ω ; the corresponding exponent depends on W . The correlation between adjacent-in-energy eigenstates exhibits a maximum at the transitionpoint W c , visualizing the drift of W c with increasing system size towards its thermodynamic-limitvalue. The correlation function β ( ω ) is related, via Fourier transformation, to the Hilbert-spacereturn probability. We discuss measurement of such (and related) eigenstate correlation functionson state-of-the-art quantum computers and simulators. I. INTRODUCTION
The many-body localization (MBL) in disordered in-teracting systems [1, 2] is one of active directions of themodern condensed-matter physics research (see recent re-views [3, 4]). The MBL is of fundamental importanceas it can break ergodicity and suppress low-temperaturetransport in a great variety of complex systems. The pre-diction of the MBL transition [1, 2] has been corroboratedby numerous subsequent analytical and computationalstudies, see, in particular, Refs. 5–20. While the MBLcan be destabilized in the thermodynamic limit (systemsize L → ∞ , with other parameters fixed) due to long-range interactions [21–27], spatial dimensionality d > W c ( L ) depending on the systemsize. On the experimental side, the evidence of the MBLtransition was reported and the associated physics wasstudied in a variety of structures. These include systemsof cold atoms and ions in optical traps[31–39], of spindefects in a solid state[40–43], and of superconductingqubits[44, 45], as well as InO films [46–48].The MBL can be viewed as an extension of the Ander-son localization [49] from single-particle to many-bodysetting. Correspondingly, the MBL transitions are coun-terparts of Anderson localization transitions between lo-calized and delocalized phases [50] of a quantum particlesubjected to a random potential in d > d = 2 for some symmetry classes). A hallmark of Ander-son transitions is the multifractality of eigenstates [50]that implies strong fluctuations of eigenfunction ampli-tudes at criticality and around the transition point, witha non-trivial power-law scaling of the corresponding mo-ments (inverse participation ratios). Furthermore, themultifractality implicates a complex pattern of enhance-ment of correlations between eigenstate amplitudes, both in the coordinate and the energy spaces [50–53]. To un-derstand the physics of the MBL, it is of central impor-tance to explore eigenstate correlations at and aroundthe MBL transitions. This is the main goal of the presentwork. More specifically, we focus on correlations betweeneigenstates in the Hilbert space as a function of energyseparation and disorder—the problem that can be posedvery generally, for any spatial structure of the system. Arelated but different question was addressed in Ref. [54]which considered matrix elements of local operators.To explore the eigenstates correlations, we use twomodels. First, we consider the Anderson model on ran-dom regular graphs (RRG), which has emerged as a toymodel of MBL. Second, we study a genuine many-bodyproblem, the XXZ spin chain in a random field, which hasbecome a paradigmatic model for the MBL transition.The RRG are finite-size graphs that have locally tree-like structure with fixed coordination number but do nothave boundary (i.e., have large-scale loops). The struc-ture of these graphs mimics that of Hamiltonians of in-teracting systems in the many-body Hilbert space. Theidea that single-particle models on a tree (Bethe lattice)can be useful for the analysis of many-body problems wasput forward in Ref. 55 in the context of a quasiparticledecay in a quantum dot. Later work has demonstratedthat one can think about tree-like graphs more generallyas approximately modelling the Hilbert-space structureof a finite many-body system and that the appropriategraphs are then not Bethe lattices but rather RRG. TheRRG model oversimplifies the many-body problem bydiscarding matrix-element correlations in Hilbert spacestates resulting from the fact that the number of inde-pendent parameters in the Hamiltonian is much smallerthan the number of non-zero matrix elements. One im-portant consequence is that, in the localized phase, theinverse participation ratios of eigenstates in the RRGmodel are of order unity [56], while in the MBL mod- a r X i v : . [ c ond - m a t . d i s - nn ] S e p els they exhibit a multifractal scaling with respect to theHilbert-space volume [9, 14, 25, 57]. Despite this, thereare remarkable analogies between the localization transi-tions in the RRG and true MBL models. In particular,in both models (i) the critical point has a localized char-acter, (ii) there are strong finite-size effects with a driftof the apparent transition point towards stronger disor-der, (iii) the “correlation volume” of the Hilbert spacegrows exponentially when the transition is approached,(iv) the delocalized phase is ergodic. For the RRG modelthese results have been analytically proven [56] (see alsoRefs. 58–60 where a related sparse-random-matrix modelwas studied) and numerically verified [56, 61–64]. For theMBL models, analytical arguments are of less rigorouscharacter but still lead to analogous conclusions, in con-sistency with numerical simulations (see, in particular,the MBL papers cited above and references therein). Theconnection between the Anderson model on RRG and theMBL transition is especially close in models with long-range interaction (decaying as a power-law of distance),see Ref. 25.In view of a close similarity between the RRG andMBL problems, and since the RRG model is much moreamenable to the analytical treatment, it is advantageousto perform a numerical study in parallel for both modelswhenever the MBL observable can also be defined in theRRG problem. This is exactly the case for the eigenstatescorrelations studied in the present work. Let us illustrateone of advantages of using RRG as a benchmark model.For the RRG model, we know exactly the position of thethermodynamic-limit transition point W c (as well as val-ues of critical exponents and various other observables)[65]. This allows us to determine the magnitude of finite-size effects in exact-diagonalization computation, whichturn out to be rather strong. As a result, one gets alower bound for finite-size effects in the MBL problem(which can only be stronger due to additional, rare-eventfluctuations related to a smaller number of independentparameter in MBL as comparison to RRG).As an additional motivation for this work, it is worthpointing out that correlations between many-body eigen-states can be measured in quantum computers and sim-ulators. In particular, the eigenstate correlation func-tion β ( ω ) studied in this paper (see Sec. II for precisedefinition) is related via the Fourier transformation tothe probability p ( t ) of return to an initial many-bodystate after time t . Such probabilities can be measuredin state-of-the-art engineered many-body systems. Re-cent examples of experimentally implemented systems onwhich related measurements were performed include one-dimensional (1D) arrays of 53 trapped ions [66] and 51atoms [67] as well as 1D and two-dimensional (2D) ar-rays of superconducting qubits (with up to 21 qubits)[45]. Furthermore, very recently a quantum processorwith 53 superconducting qubits was used to demonstratethe quantum supremacy [68]. The key observable in thisdemonstration is the fidelity defined as a correlation func-tion of two many-body wave functions (corresponding to the idealized and perturbed Hamiltonians, respectively).While it is somewhat different from the quantity we studyin the present work—correlation function of two eigen-functions of the same Hamiltonian with different ener-gies, Ref. [68] makes evident the importance of Hilbert-space correlations between many-body wave functions forquantum-information physics and quantum technologies.The structure of the article is as follows. Section IIdeals with the eigenstate correlations across the local-ization transition in the RRG model. In Section III ananalogous study is carried out for the XXZ spin chain.Section IV contains a summary of our findings as wellas a discussion of their implications and of prospectiveresearch directions. II. RANDOM REGULAR GRAPHS
We study a model of non-interacting spinless quantumparticles hopping over a random regular graph (RRG)with connectivity p = m + 1 (number of sites adjacent toany given site) in a potential disorder, H = (cid:88) (cid:104) i,j (cid:105) (cid:16) c † i c j + c † j c i (cid:17) + (cid:88) i (cid:15) i c † i c i . (1)Here the index i = 1 , . . . , N labels sites of the graph andthe sum in the first term is over the pairs of nearest-neighbor sites of the RRG. The energies (cid:15) i are indepen-dent random variables sampled from a uniform distribu-tion on [ − W/ , W/ (cid:104) . . . (cid:105) goes overthe random structure of the underlying graph and overthe random potential (cid:15) i .An important statistical characteristic of a disorderedsystem is the correlations of different (but relatively closein energy) eigenstates with a given energy separation ω .Formally, we define the corresponding correlation func-tion as follows: β ( ω ) = ∆ R − ( ω ) × (cid:42)(cid:88) k (cid:54) = l | ψ k ( j ) ψ l ( j ) | δ (cid:16) E − ω − E k (cid:17) δ (cid:16) E + ω − E l (cid:17)(cid:43) . (2)Here ψ k are eigenstates and E k the corresponding energylevels, E is the energy at which the statistics is stud-ied, ∆ = 1 /ν ( E ) N is the mean level spacing, ν ( E ) = N − (cid:68) Tr δ ( E − ˆ H ) (cid:69) is the density of states, and R ( ω )the level correlation function R ( ω ) = 1 ν (cid:104) ν ( E − ω/ ν ( E + ω/ (cid:105) . (3)The argument j in Eq. (2) is the lattice site; since all sitesare equivalent, the r.h.s. does not actually depend on j upon ensemble averaging. In the numerical computationsbelow, we average also over j . In the sequel, it will beconvenient to present results for β ( ω ) multiplied by N .For two completely uncorrelated wave functions one has N β ( ω ) = 1.The correlation function β ( ω ) in the delocalized phaseand at the critical point on RRG has been studied in Ref.[56] with the following results: N β ( ω ) ∼ N ξ , ω < ω ξ , ω ln / /ω , ω > ω ξ . (4)In this equation, N ξ (which depends on W ) stands forthe correlation volume and ω ξ ∼ /N ξ for the associatedlevel spacing. The correlation volume N ξ exhibits onRRG the following critical behavior when the disorder W approaches from the delocalized side the critical point W c [65]: ln N ξ ∼ ( W c − W ) − / . (5)More specifically, for the “minimal”, p = 3, RRG modeland in the center of the band ( E = 0), the critical disor-der is W c = 18 .
17 and the scaling of ln N ξ reads (with asubleading term included) [65]1 / ln N ξ = c ( W c − W ) / + c ( W c − W ) / , (6)where c = 0 . c = 0 . < W < W c .Let us briefly comment on the physical significance ofEq. (4). The first line of this equation describes eigen-states correlations in the “metallic” regime. The factor N ξ in this formula implies that the correlations get en-hanced when the system approaches the transition point.This is related to strong spatial fluctuations (multifrac-tality) of eigenstates near criticality. Independence ofthis formula of ω demonstrates that eigenstates separatedby a sufficiently small energy ω < ω ξ exhibit essentiallythe same “multifractal pattern” (despite its randomness).The second line of Eq. (4) describes critical correlations,which is why it does not depend on N ξ . The distanceto the critical point enters only via the range of valid-ity, ω > ω ξ . Exactly at critical point, W = W c , wehave ω ξ = 0, so that the second line of Eq. (4) holds inthe whole range of frequencies (limited only by the levelspacing ∆ ∼ /N ).The goal of this section is to extend the study of theRRG correlation function β ( ω ) to the localized phaseand thus to obtain a full description of eigenstates cor-relations around the localization transition on RRG. Webegin by presenting qualitative arguments concerning ex-pected behavior of β ( ω ) on the localized side of the transi-tion. First, in the limit of very strong disorder, W → ∞ ,individual eigenstates are essentially localized on differ-ent sites and do not overlap. We thus expect N β ( ω ) → W → ∞ . Combining this with Eq. (4),we conclude that, for a small ω , the correlation func-tion N β ( ω ) should be a non-monotonic function of W that shows a maximum in the vicinity of the critical point W = W c .Second, for a given W > W c , two localized states willbe typically located in remote regions of the system andoverlap very weakly in view of the exponential decay ofthe localized wave functions. However, there is a certainprobability that two such states turn out to be in res-onance, which then strongly enhances the overlap. Theprobability of a resonance is enhanced for small energyseparation ω , so that N β ( ω ) is expected to decay with ω in the localized phase. For the single-particle problemin d dimensions, this decay was studied in Ref. [53], withthe result N β ( ω ) ∼ ξ d − d ln d − ( δ ξ /ω ) , ω < δ ξ , (7)where ξ is the localization length, δ ξ ∼ ξ − d the levelspacing in the localization volume, and d the multifrac-tal exponent. It was pointed out in Ref. [53] that thelogarithmic enhancement of correlations with lowering ω in Ref. (7) is closely related to the Mott’s behavior lawfor the ac conductivity.What kind of behavior of N β ( ω ) can one expect onthis basis in the localized phase on RRG? The RRGmodel can be in a certain sense viewed as a d → ∞ limitof the d -dimensional Anderson model; this limit is, how-ever, highly singular [56]. Equation (7) suggests that theenhancement of correlations for small ω on RRG shouldbe faster than a power of ln ω . It is even more difficultto guess what the dependence on disorder [encoded inthe localization length ξ and the corresponding spacing δ ξ in Eq.(7)] transforms into when the RRG model isconsidered. As we show below, the eigenstate correlationfunction N β ( ω ) has a power-law dependence on ω inthe localized phase of the RRG model, with an exponentthat is a function of disorder. We will also see that sucha behavior holds also for a genuine MBL problem.We have computed β ( ω ) numerically by exact diago-nalization of the RRG model with the connectivity p = 3,focussing on the vicinity of the band center, E = 0. Weconsider system sizes N in the range from 2 to 2 . Foreach realization of disorder, we average over N/
32 statesnear the band center. In addition, we average over dis-order realizations; their number ranges from 50000 forsmaller systems to 50 for the largest systems.In Fig. 1 we show the results of exact-diagonalizationstudy for N = 32768 and several disorder values rang-ing from the delocalized phase ( W = 10 , ,
14) throughthe critical point ( W = 18) to the localized phase W =24 , ,
42. (Note that the difference between W = 18and the exact critical value W c = 18 .
17 is immaterialfor system sizes amenable for exact diagonalization.) Inthe delocalized phase, W = 10, 12, and 14, the behavior(4) is clearly observed: the power-law (1 /ω ) behavior athigh frequencies and a saturation at lower frequencies.The saturation frequency ω ξ becomes smaller when thedisorder increases, i.e., the system approaches the criticalpoint. At criticality, W = 18, the power-law (approxi-mately 1 /ω ) behavior is indeed observed in the whole − − − − ω N β ( ω ) FIG. 1. Eigenstate correlation function β ( ω ) for RRG withdisorder strengths W = 10 , , , , , ,
42 (from cyan tomagenta). The first three values are on the delocalized sideof the transition, W = 18 is essentially the critical point, andthe three largest values are on the localized side. The systemsize is N = 32768. range of frequencies. Remarkably, the power-law behav-ior of β ( ω ) survives in the localized phase W = 24, 30,and 42, where it is characterized by a disorder–dependentexponent µ ( W ). The numerical results thus unambigu-ously suggest the power-law scaling of eigenstate corre-lations in the localized phase: N β ( ω ) ∼ ω − µ ( W ) . (8)At criticality, µ ( W = W c ) = 1, while in the localizedphase, W > W c , the exponent µ ( W ) is less than unityand gradually decreases towards zero as W grows. InFig. 2, we highlight the correlation function β ( ω ) in thelocalized phase. The right panel shows the numericallydetermined exponent µ ( W ) obtained from the data pre-sented in the left panel.It is worth emphasizing that the exponent µ ( W ) is,strictly speaking, defined in the limit of large system size N and low frequency ω . For finite N there are correc-tions to an apparent position of the critical point (thatare discussed in more detail below), which also influencethe numerically determined values of µ ( W ). Also, theremight be in principle a subleading (e.g. logarithmic) fac-tor in ω -dependence in Eq. (8). We know that such a log-arithmic factor does exist at criticality, see second line ofEq. (4). Its emergence at criticality has a clear physicalreason, as it provides convergence of the time-dependentcorrection to return probability, see Ref. [56]. On the lo-calized side, such a convergence is already guaranteed bythe power-law dependence (8) with µ ( W ) <
1, so thatwe do not have any arguments in favor of such a factor.Also, the corresponding lines in the left panel of Fig. 2are rather straight (up to fluctuations), without any clearindication of such a factor. When fitting the numericaldata to extract the exponent µ , we thus assume a purepower-law dependence (8), without any subleading pref- actors.We turn now to the analytical approach to eigenstatecorrelations. In Ref. [56], we have shown how variousobservables characterizing the RRG model can be ex-pressed in terms of a solution of a saddle-point equationfor the effective action of the problem. This equation isequivalent to a self-consistency equation for the distribu-tion of local Green functions on an infinite Bethe lattice(BL). This is a non-linear integral equation, and a fullanalysis of its solution is by no means an easy task. Inthe localized phase, the solution is singular in the low-frequency limit. Leading contribution to the correlationfunction that is needed for our purposes determines prop-erties of individual eigenstates (e.g., the participation ra-tio). To obtain the correlation function of different eigen-states, one needs a subleading term (see Ref. [56]), whichmakes the analysis much more difficult. We first discussthe numerical solution of the self-consistency equation;below we discuss an analytical solution in the limit ofstrong disorder W . To solve numerically this equation,one can use the population-dynamics approach, see alsoRefs. [63, 70]. In particular, we calculated in this way inRef. [56] the correlation function β ( ω ) as well its Fouriertransform, the return probability p ( t ), in the delocalizedphase and demonstrated perfect agreeement with exact-diagonalization results. Here we demonstrate that thepopulation-dynamics approach to the solution of the self-consistency equation can also be used for computing β ( ω )on the localized side, even though it turns out to requiremuch more efforts.Determining β ( ω ) in this way amounts to evaluation ofthe correlation function of local densities of states ρ ( (cid:15) ; j )on an infinite Bethe lattice K ( ω ) = (cid:104) ρ ( E + ω/ j ) ρ ( E − ω/ j ) (cid:105) BL (cid:104) ν ( E ) (cid:105) , (9)where ρ ( (cid:15) ; j ) = − π Im G R ( j, j, (cid:15) ) , (10)and G R ( j, j, (cid:15) ) is the retarded Green function at energy (cid:15) with equal spatial arguments j . Calculation of β ( ω ) bypopulation-dynamics approach (see Supplementary Ma-terial [69] for more details) requires introducing a finiteimaginary part of frequency ω → ω + iη , evaluation of K ( ω ) for a complex frequency and then considering thelimit of η → +0, N β ( ω ) = lim η → +0 K ( ω ) . (11)This is rather non-trivial in the localized phase, as takingthe limit requires considering very small values of η . InFig. S1 of the Supplementary Material [69], we show the η -dependence of K ( ω ) for W = 24 and two values of ω asobtained by population dynamics. It is seen that K ( ω )does have a finite limit at η → η dependence takes place at very small η . The resultingvalues of N β ( ω ) are shown by dots in the left panel of − − − − ω N β ( ω )
20 25 30 35 40 W . . . . µ ( W ) FIG. 2. Eigenstate correlations in the localized phase on RRG. The system size is N = 32768. Left: β ( ω ) for disorders W = 18 (essentially the critical point), 24, 30, and 42 (from cyan to magenta). The straight lines on the log-log scale implya power-law dependence of β ( ω ) on ω in the localized phase, with a disorder-dependent exponent µ ( W ). Dots show resultsfrom the population dynamics for W = 24 (see Supplementary Material [69] for details); they have the same (blue) color asthe corresponding line of exact-diagonalization data. Right:
Exponent µ ( W ) characterizing the frequency scaling of β ( ω ), seeEq. (8). the Fig. 2 and are in good agreement with results of theexact diagonalization.In order to shed light on the physical origin of thepower-law scaling of the correlation function β ( ω ) in thelocalized phase on RRG, we perform its analysis by as-suming a limit of strong disorder W . In this limit, almostevery single-particle state is localized within a small lo-calization length ζ around a certain lattice site (localiza-tion center). Typically, two localized states are located adistance of order of system size L = ln N/ ln m apart andhave an exponentially small overlap ∝ e − L/ζ . However,there are rare resonant events: two states located farapart may form a resonant pair and strongly hybridise,so that the resulting states will have amplitudes of orderunity at both localization centers. Such a resonant pairwill thus realize the strongest possible overlap and there-fore give a maximal possible contribution to the correla-tion function β ( ω ). Even though such resonance eventsare rare, they determine the average value β ( ω ) in thecase of d -dimensional system with d >
1, see Ref. [53].Specifically, the factor ln d − ( δ ξ /ω ) in Eq. (7) representsresonant enhancement of eigenstate correlations. Clearly,its role increases with increasing d . More generally, it isknown that the role of resonances is particularly impor-tant in the RRG and MBL models, in view of the effec-tively infinite-dimensional character of the correspond-ing Hilbert space. It is thus natural to expect that thepower-law scaling of β ( ω ) on RRG can be understood inthe framework of the resonance mechanism. This turnsout to be indeed the case.To shed light on the origin of the power-law scaling, itis instructive to make the following simplistic estimate.It is known that, upon averaging, localized eigenstates ona tree-like graph decay in the following way [56, 71–73]: (cid:104)| ψ ( r ) |(cid:105) ∼ m − r exp {− r/ζ ( W ) } , (12) where r is the distance from the “localization center” ofthe state and ζ ( W ) is the localization length that divergesat the transition point as ζ ( W ) ∼ ( W − W c ) − and di-minishes slowly at strong disorder, ζ ( W ) ∼ / ln( W/W c ).While Eq. (12) yields the average, let us assume thatall eigenstates decay in this way; this will be sufficientto understand the W -dependent power-law in eigenstatecorrelations. Consider two such eigenstates separated bya distance R . The corresponding overlap matrix elementis then M ∼ m − R exp {− R/ζ ( W ) } . The optimal condi-tion of the Mott-like resonance for two eigenstates withthe energy difference ω is M ∼ ω . Under this condition,two considered eigenstates (let’s call them ψ k and ψ l ) getstrongly hybridized, so that (cid:88) j | ψ k ( j ) ψ l ( j ) | ∼ . (13)Expressing the distance between the eigenstates centersthrough the frequency, we find R ( ω ) (cid:39) ln(1 /ω )ln m + ζ − ( W ) . (14)The total number of states whose centers are separatedby distance R from that of the state ψ k is N R ( ω ) ∼ m R ( ω ) ∼ ω − µ ( W ) , (15)where µ ( W ) = ζ ( W ) ln mζ ( W ) ln m + 1 . (16)The formation of the resonance requires that one of thesestates is separated by an energy difference ∼ ω from thestate ψ k . Thus, the probability p ω of the resonance in thefrequency interval [ ω, ω ] involving the given state ψ k isequal to a ratio of the frequency ω to the level spacing ∼ N − R ( ω ) , p ω ∼ ωN R ( ω ) ∼ ω − µ ( W ) . (17)Using the definition (2), we get N ωβ ( ω ) ∼ N (cid:90) ωω dω (cid:48) β ( ω (cid:48) )= (cid:88) l : ω< | E k − E l | < ω (cid:42)(cid:88) j | ψ k ( j ) ψ l ( j ) | (cid:43) ∼ ω − µ ( W ) . (18)In the second line of Eq. (18) the state k is fixed; thesummation goes over states l with the energy difference inthe [ ω, ω ] interval. In the last line, we used Eq. (17) forthe probability of a resonance in this interval and Eq. (13)for the resonant overlap. Comparing the starting andthe final expressions in Eq. (18), we finally come to theresult, Eq. (8), for the scaling of N β ( ω ) with frequency ω , where the exponent µ ( ω ) is given by Eq. (16).Inspecting Eq. (16) for the exponent µ ( W ), we findthe following asymptotic behavior. When the disorder W approaches the critical point (from the localized side),Eq. (16) yields µ ( W ) → , W → W c + 0 . (19)This matches the critical scaling β ( ω ) ∝ /ω (up to alogarithmic correction), see second line of Eq. (4). In theopposite limit of large W , we get, by using the asymptoticbehavior of the localization length, µ ( W ) ∼ W/W c ) , W (cid:29) W c . (20)Thus, µ ( W ) decays to zero at W → ∞ but this decay islogarithmically slow. These results are in good agreementwith the numerical observations presented above.As we have already mentioned, Eq. (12) describes theaverage decay of a wavefunction. At the same time, wave-functions fluctuate strongly; in particular, decay of thetypical wavefunction amplitude is described by a differ-ent localization length [73]. In the Supplementary Ma-terial [69] we present a more accurate version of theabove resonance-counting analysis, which takes into ac-count strong fluctuations of eigenstates around the av-erage (12). It confirms the power-law scaling (8) andyields qualitatively the same results for the behavior ofthe exponent µ ( W ).It is useful to introduce a correlator that is closelyrelated to β ( ω )—a correlation function of adjacent-in-energy eigenstates: β nn = ∆ (cid:42)(cid:88) k δ ( E k − E ) | ψ k ( j ) ψ k +1 ( j ) | (cid:43) . (21) Here the subscript “nn” stands for “nearest neighbor” (inenergy space). Clearly, β nn (cid:39) β ( ω ∼ ∆), where ∆ is thelevel spacing. Thus, in the delocalized phase we and inthe large- N limit (the condition is N (cid:29) N ξ ) we have N β nn = N ξ / N ξ . The value 1 / P = (cid:42)(cid:88) j | ψ k ( j ) | (cid:43) , (23)is P (cid:39) N ξ /N at N (cid:29) N ξ [56]. In the localized phase,we have, according to Eq. (8), N β nn ∼ N µ ( W ) . (24)In the left panel of Fig. 3, we show results of nu-merical simulations for the correlator N β nn for severalsystem sizes. For W < W c , the lines clearly approach alimiting ( N → ∞ ) curve, in agreement with Eq. (22).According to Eq. (22), this limiting curve is determinedby the disorder dependence of the correlation volume, N ξ ( W ). Indeed, we observe a perfect agreement withthe asymptotic behavior of N ξ ( W ) given by Eq. (6)(shown by dashed line). For any given N , the curve N β nn ( W ) deviates from the limiting curve upon increas-ing W , since the condition N (cid:29) N ξ ceases to be satis-fied, shows a maxmum at certan size-dependent disorder W peak ( N ), and then decays. The non-monotonic behav-ior of N β nn ( W ) is a general feature of a system thatundergoes a localization transition, see qualitative dis-cussion above. The position of the maximum W peak ( N )can be viewed as a size-dependent apparent critical pointwhich drifts to larger W with growing system size. Inthe limit of N → ∞ the drift stops at the limiting value W peak ( N → ∞ ) = W c (cid:39) . W peak ( N ) for several system sizes N . The drift towards W c is evident but it is rather slow. Looking at theseslowly drifting values, one could naively think, that theyare close to the actual value of W c . This is not true,however: these values are still rather far from the truecritical point. (One indication of this is absense of aclear trend to saturation.) For the RRG model, we havea luxury of knowing the true critical point with a highprecision, W c = 18 .
17, which is obtained by a very dif-ferent approach—investigation of stability of the solutionof the saddle-point equation corresponding to the local-ized phase [65]. Therefore, the RRG model, as a toymodel of MBL, is very useful for benchmarking exact-diagonalization studies of MBL problems. We see fromTable I that if only exact-diagonalization data would beavailable, it would be very hard to determine the positionof the N → ∞ critical point with a reasonable accuracy. W N β nn ( W ) W c W . . . . . . µ nn ( W ) W c FIG. 3. Correlation of adjacent wavefunctions on RRG. System sizes are N = 4096 , , , , Left:
Correlation function β nn ( W ). Dashed line is the expected asymptotic behaviour of N β nn on the delocalizedside, see Eqs. (24) and (6). Vertical dotted line marks the critical point of the localization transition, W c = 18 . Right:
Exponent µ nn characterizing the N scaling of adjacent-state correlations, see Eq. (27). Dashed line shows theoretically expected N → ∞ behaviour, see Eq. (22) for the delocalized phase and Eq. (24) for the localized phase (this part of the dashed line isschematic.) An even more difficult task for the exact-diagonalizationnumerics is to find the true (asymptotic) value of the crit-ical exponent ν of the correlation length ξ = ln N ξ / ln m .Asymptotically, the drift of the peak can be characterizedby the critical exponent via the scaling relationln ln N = − ν ln [ W c − W peak ( N )] . (25)While this equation is valid in the limit of N → ∞ , it isconvenient to introduce an apparent finite-size exponentvia 1 ν ( N ) = − ∂ ln [ W c − W peak ( N )] ∂ ln ln N . (26)The last line of the Table I presents values of ν ( N ) ob-tained by numerical differentiation according to Eq. (26).We see a strong variation of ν ( N ) towards the trueasymptotic value ν = 1 / ν ( N ) approaches its asymp-totic value 1/2 from above, and that values of N muchlarger than those amenable to exact diagonalization areneeded to obtain numerically 1/2 with a good accuracy,was demonstrated in detail in Ref. [65]. Our find-ings are in full agreement with these previous results. Itshould be stressed that when calculating ν ( N ) in TableI, we used the high-precision value of the critical disor-der, W c = 18 .
17. For the MBL problems, W c is foundfrom numerical simulations with a much lower precision(see the discussion above), which further increases uncer-tainty of numerical determination of the critical exponent ν . To characterize the evolution of β nn with the systemsize, we define a disorder- and size-dependent exponent: µ nn ( W, N ) = ∂ ln (cid:0) N β nn (cid:1) ∂ ln N . (27) log N
12 13 14 15 16 ∞ W peak ( N ) 13 .
70 13 .
78 13 .
89 14 .
06 14 .
28 18 . ν ( N ) 4 .
31 3 .
52 2 .
22 1 .
42 0 .
96 1 / W peak ( N ) of the maximum of N β nn ( W )curves that can be viewed as an N -dependent apparent criticalpoint. Upon increasing N , it shows a slow drift towards thelimiting ( N → ∞ ) value W c = 18 .
17. The lower line of thetable shows the “flowing ( N -dependent) critical exponent”extracted from W peak ( N ) according to Eq. (26). It evolves tothe asymptotic ( N → ∞ ) value ν = 1 / On the delocalized side, N β nn is independent on N atlarge N , which implies that µ nn ( W < W c , N ) → N (cid:29) N ξ ( W ). At the critical point, W = W c , we have µ nn ( W c ) → N → ∞ . On the localized side, Eq. (24)yields µ nn ( W > W c , N ) → µ ( W ) in the large- N limit. Inthe right panel of Fig. 3, we show numerical results for µ nn ( W, N ). As expected, for
W < W c the µ nn ( N ) curvesgradually drift downwards, towards zero, with increasin N . Closer to W c , this drift is in fact non-monotonic(first upward, then downward), see Ref. [61] for a discus-sion of the physical origin of such behavior on RRG. For15 (cid:46) W < W c we observe only upward drift; one needsmuch larger N to see that it will be eventually supersededby a downward drift with the ultimate large- N limit µ nn ( W ) →
0. On the localized side,
W > W c , we find anearly N -indepedent µ nn ( W, N ), in consistency with theexpected limiting behavior µ nn ( W > W c , N ) → µ ( W )and in agreement with numerical data for µ ( W ) (whichare, of course, also subjected to finite-size corrections) inthe right panel of Fig. 2. − − − − ω N β ( ω ) FIG. 4. Eigenstate correlation function β ( ω ) for spin chainof length L = 16. The disorder strengths are W =1 . , . , , , , ,
10 (from cyan to magenta). The first threevalues are deeply in the ergodic phase. The next two valuesare also on the delocalized side of the transition but corre-spond to the critical regime for relatively small systems avail-able for exact diagonalization. The largest two values are onthe localized side of the transition. This figure is a spin-chaincounterpart of Fig. 2 for the RRG model.
III. SPIN CHAIN
In this section, we apply a similar methodology tostudy the model of the S = Heisenberg chain in arandom magnetic field, governed by the Hamiltonian H = (cid:88) i ∈ [1 ,L ] S i · S i +1 − h i S zi , (28)with h i drawn from a uniform distribution [ − W, W ] andwith periodic boundary conditions, S L +1 ≡ S . (Notethat the total magnetization S z = (cid:80) i S zi is conserved.)This model has become one of paradigmatic models forthe investigation of the MBL physics [74, 75]. Numer-ically, systems of sizes up to L = 22 [9] and L = 24[57] were investigated via exact diagonalization, whichyielded estimates W c = 3 . . W c ≈
15, while the actualvalue is W c = 18 .
17, i.e., about 20% higher. In gen-uine interacting MBL models (like the spin-chain modelconsidered in this section), the finite-size effects are ex-pected to be still stronger due to effects of rare spatialregions. This has been supported by an analysis basedon the time-dependent variational principle with matrix product states which was used to study the dynamics(relaxation of spin imbalance) in Ref. [16] in much largersystems, up to L = 100. This study has demonstrated astrong drift of apparent (size-dependent) W c with systemsize L , suggesting the critical value W c ≈ . β ( ω ), Eq. (2), andthe correlation function of closest-in-energy wave func-tions, β nn , Eq. (21). Let us emphasize that these quanti-ties now characterize exact many-bony eigenstates ψ k ( j ).Here the index k labels eigenstates and the argument j runs over basis states of the Hilbert space which areeigenstates of S zi for all i (and thus are eigenstates of theHamiltonian in the extreme-localization limit W = ∞ ).Before presenting our results for the frequency-dependent wave function correlations, we briefly recallthe existing knowledge about the average inverse partic-ipation ratio which characterizes statistical properties ofan individual many-body eigenstate, Eq. (23). In the de-localized phase, it was found [57] that eigenfunctions inthe model of Eq. (28) are ergodic in the sense that P ∝ /N. (29)Here N is the volume of the many-body space, i.e. thedimensionality of the subspace of the full Hilbert spacethat is allowed by conservation laws. In the model of Eq.(28) and in the zero magnetization sector S z = 0, whichwe consider below (we limit ourselves to even L only),one has N = L ![( L/ (cid:39) L (cid:114) πL . (30)This ergodic behavior of the inverse participation ratioin the spin-chain model is fully analogous to that in thedelocalized phase of the RRG model (see Sec. I). At thesame time, on the localized side of the transition, there issome difference in the scaling of the inverse participationratio in the RRG model and in the genuine many-bodyproblem (like a spin chain). While P ∼ P ∼ N − τ ( W ) (31)in the MBL phase of spin-chain models [14, 25, 57, 77],with a disorder-dependent exponent τ ( W ). It was shown[25] that in the strong disorder regime (large W ), theexponent τ ( W ) scales as τ ( W ) ∝ /W (32)with disorder. Further, at criticality one also finds thefractal scaling (31), with the exponent τ ( W c ) that isequal to the limiting value of τ ( W ) at W → W c + 0.(This is another manifestation of the fact that the criticalpoint in the MBL problem has properties of the localized − − − − ω N β ( ω ) W . . . . µ ( W ) FIG. 5. Dynamical eigenstate correlations for the spin chain of length L = 16. Left:
Correlation function β ( ω ) for disorders W = 5 , , ,
10 (from cyan to magenta).
Right:
Exponent µ ( W ) characterizing the power-law frequency scaling β ( ω ) ∝ ω − µ ( W ) .This figure is a spin-chain counterpart of Fig. 2 for the RRG model. phase.) The non-trivial scaling in Eq. (31) originatesfrom a finite density (in real space) of local resonancesthat are not able to establish a global delocalization butlead to an exponential increase of the support of themany-body wavefunction in the spin configuration ba-sis of the Hamiltonian Eq. (28). A detailed analysis ofthe model (28) revealed [57] that the scaling (32) is validwith a good accuracy up to the critical point and yielded τ ( W c ) ≈ . β ( ω ) in the MBL phaseof the spin chain. Consider a given basis state | j (cid:105) , i.e., aneigenstate of all S zi . For strong disorder (deeply in theMBL phase), | j (cid:105) is close to an exact eigenstate ψ k . Moreprecisely, there exists a small density ∼ /W of lowest-order resonant processes (flips of pairs s r , s r +1 of adja-cent spins) which “dress” the state | j (cid:105) , leading to fractal-ity of the inverse participation ratio discussed above. Toestimate the number of higher-order resonant processes,we consider n –th order of the perturbation theory in in-teraction. It is important that involved spins should forma connected cluster (of maximal length 2 n ) in order toguarantee that this tentative resonant process does notdecouple into independent pieces. This is clear alreadyin case of n = 2: consider a process involving spin flipsin two remote pairs s , s and s r , s r +1 such that r > , r, r +1 in two distinct orders see Ref. [14] and ref-erences therein. As a result, the number of processes thatcan actually lead to resonance in the n –th order of per-turbation theory scales as N n ; L ∼ Lρ ( n ) with ρ ( n ) inde-pendent on the system length L . In other words, ρ ( n ) isthe spatial density of n -th order processes which may po-tentially lead to resonances. Crucially, ρ ( n ) ∼ m n grows exponentially with n , in analogy with the RRG problem[14], and m is independent on n . In a conventional spinchain, we thus have the branching factor m = O (1). (Onecan have a parametrically large m in a chain of coupled“spin quantum dots” with large number of spins per dot[14].)The density ρ ( n ) was considered in the context of acconductivity in Ref. [78], where it was denoted e s ( γ ) n and s ( γ ) was termed “configuration entropy per flipped spinof the possibly resonant clusters”. The argument γ wasintroduced to emphasize that s is actually a fluctuatingquantity. Our effective branching number m thus cor-responds to e s of Ref. [78]; the fluctuations of m arediscarded in our simplified argument.The number of “potentially resonant” processes for agiven initial state scales therefore as N n ; L ∼ Lm n , (33)with m of order unity, which is the same behavior ason RRG, up to an overall factor L . This behavior isresponsible for the MBL transition in a spin chain takingplace at a disorder W c of order unity (i.e., independenton L . We can now repeat, with minor modifications, thesimplified analysis performed for RRG in Sec.II, whichyields [cf. Eq. (18)] N ωβ ( ω ) ∼ (cid:88) l : ω< | E k − E l | < ω (cid:42)(cid:88) j | ψ k ( j ) ψ l ( j ) | (cid:43) ∼ ( N ∆) p ω P res , (34)where ∆ is the many-body level spacing, p ω is the numberof resonances (for a given state k ) in the band [ ω, ω ], and P res is the resonant overlap, P res = (cid:88) j | ψ k ( j ) | | ψ l ( j ) | . N ∆, p ω , and P res in Eq.(34):(i) The typical energy of an eigenstate of all S zi of L spins is ∼ √ L , hence N ∆ ∼ √ L .(ii) The average number of processes that represent po-tential resonances in the band [ ω, ω ] can be esti-mated, in analogy with Eq. (17), as p ω ∼ ωN R ( ω ); L ∼ ωLm R ( ω ) = Lω − µ ( W ) (35)with R ( ω ) as in Eq. (14) and µ ( W ) as in Eq. (16).(iii) The resonant overlap scales in the same way as theinverse participation ratio (31), i.e., P res ∼ N − τ ( W ) . (36)In this respect, the spin-chain problem differs fromthe RRG model, for which P res ∼
1, see Eq.(13).Combining the above estimates and using L ≈ log N ,we finally get N β ( ω ) ∼ ω − µ ( W ) (log N ) / N τ ( W ) . (37)This derivation should be viewed as substantially over-simplified since fluctuations were not fully taken account.Nevertheless, this treatment is sufficient to understandthe emergence of power–law dynamical scaling (revealedby numerical results below) with continuously varyingexponent in the MBL phase.The result (37) is largely the same as Eq. (8) for RRG;the most important factor is the power-law frequency de-pendence ω − µ ( W ) . The only difference is in the additional N -dependent factor. We note, however, that the expo-nent τ ( W ) is parametrically small ( ∼ /W ) in the MBLphase, remaining numerically quite small at the criticalpoint. Also, for realistic N , the factor N τ ( W ) in the de-nominator is essentially compensated by the logarithmicfactor in the numerator. So, in practice, the differencebetween the results for RRG and for the spin chain isrelatively minor.We now present the numerical results for the dynam-ical eigenstate correlations in the spin-chain model (28).To evaluate the correlation function β ( ω ) we computed,via exact diagonalization, eigenstates of Eq. (28) in the S zi basis. We studied systems of sizes L in the range12 – 18 and averaged over 5 · (for smallest systems) –5 · (for largest systems) realizations of disorder. Foreach disorder realization, we determined the middle ofthe band E by the condition ( E − E min ) / ( E max − E min ) =0 .
5, where E min and E max are the lowest and the largesteigenstate energy, and considered 1 /
32 fraction of allstates around the middle of the band. The correlationfunction β ( ω ) for various strengths of disorder, from de-localized to the MBL phase, is shown in Fig. 4, whichis a direct spin-chain counterpart of Fig. 2 for the RRG model. The observed behavior is fully analogous to thatfound in Fig. 2. For sufficiently weak disorder, W = 1 . W = 3 and 4, the tendency towards saturation isalso achieved but we are still far from reaching the fullsaturation. This is an indication of the fact that thesetwo values are also on the delocalized side of the MBLtransition (in the thermodynamic limit) but the systemsizes are too small to observe ergodicity. In other words,these values correspond to the critical regime for systemsizes that can be studied via exact diagonalization. Forstrong disorder, W = 6 and 10, the data exhibit a power-law behavior in the full range of frequencies, which is ahallmark of the MBL phase.In Fig. 5 we show the results for strong disorder,from W = 5 (approximately the critical point) till W =10 (deeply in the MBL phase), cf. the analogous figure2 for the RRG model. In the left panel, we see oncemore that β ( ω ) in the MBL phase shows a power-lawfrequency scaling, β ( ω ) ∝ ω − µ ( W ) , with the disorder-dependent exponent (slope on the log-log scale). In theright panel, the corresponding exponent µ ( W ) is plottedas a function of disorder. This figure is again similar tothe right panel of Fig. 2 although numerical values of theexponent µ ( W ) are somewhat smaller than in the RRGmodel.A related quantity—the adjacent-state correlationfunction β nn defined by Eq. (21)—is shown in Fig. 6.The results are very similar to their RRG analog, Fig. 3.The left panel of Fig. 6 displays the correlation function β nn ( W ) in a broad range of disorder strengths for severalvalues of the system size L . Like in the case of RRG, W < W c , the curves gradually approach, with increas-ing L , a limiting curve, thus demonstrating ergodicityof the delocalized phase. For system sizes available forexact diagonalization the ergodic (large- L ) behavior isreached for W (cid:46)
2. In full analogy with the RRG model,the β nn ( W ) curves exhibit a maximum near W ≈ L → ∞ ) value of W c .The right panel shows the flowing exponent µ nn definedby Eq. (27). In general, the curves are rather similar tothose for RRG in the right panel of Fig. 3. However, it isworth noticing a difference in the maximum value of µ nn for the largest system size. While for the RRG model thismaximum value is equal to unity with a good accuracy, inthe spin-chain case the maximum value is approximately0.75. One reason for this is finite-size effects which areconsiderably stronger for the spin chain than for RRG.In fact, there is also a deeper reason for this difference,which should remain also in the limit N → ∞ . Indeed,let us consider a system at criticality ( W = W c ) in thelarge- N limit. Recall that the inverse participation ratio P at the critical point shows “fractal” scaling (31). Theoverlap of two adjacent states at criticality is expected1 L
12 14 16 18 W peak ( L ) 2 .
66 2 .
84 3 .
01 3 . W peak ( L ) of the maximum of µ nn ( W )curves that can be viewed as an L -dependent apparent criticalpoint. Upon increasing L , it shows a slow drift towards thelimiting ( L → ∞ ) value. to follow the same power-law scaling, N β nn ∼ N − τ ( W c ) [cf. Eq. (36)], which implies (at N → ∞ ) µ nn ( W c ) = 1 − τ ( W c ) , (38)yielding µ nn ( W c ) ≈ .
8. The same result is obtainedfrom Eq. (37) if one extends it from the MBL phase tothe critical point and sets µ ( W c ) = 1 (as on RRG). Moregenerally, Eq. (37) suggests a relation between the expo-nents in the MBL phase, µ nn ( W ) = µ ( W ) − τ ( W ) . (39)At large W the exponent τ ( W ) is small ( ∼ /W ), so that τ ( W ) (cid:28) µ ( W ) and thus µ nn ( W ) ≈ µ ( W ) . (40)This remains valid with reasonable accuracy up to thecritical point since τ ( W c ) is quite small. It is also worthnoting that logarithmic corrections to scaling, like thelogarithmic factor in Eq. (37), and further strong finite-size effects substantially influence numerical values of ex-ponents characterizing the MBL phase as obtained bymeans of exact diagonalization.The expected extrapolation of µ nn ( W ) to the thermo-dynamic limit L → ∞ is shown by a dashed line in theright panel of the Fig. 6. Similarly to the β nn peak,the position of the peak in µ nn provides a finite-size es-timate for the position of the transition and drifts, withincreasing L , towards W c , see Table II. The drift is ap-proximately linear with number of spins L ; these systemsizes are clearly too small to allow for a reliable estimateof the thermodynamic-limit critical disorder W c As in theRRG model, a considerable part of the delocalized phasegives rise to a broad critical regime, 2 . (cid:46) W (cid:46)
5, forsystem sizes available for exact diagonalization.
IV. SUMMARY
In this paper, we have studied dynamical eigenstatecorrelations across the MBL transition. This was donefor two models: (i) the RRG model that serves as atoy-model of the MBL transition and (ii) a spin chainrepresenting a genuine many-body problem. The re-sults for both models were found to be very similar.The main observables that we have considered are thefrequency-dependent eigenstate correlation function β ( ω )and the adjacent-state correlation function β nn . For both of them, we explored dependences on disorder W and onthe system size. We have introduced the exponent µ ( W )controlling the scaling of β ( ω ) with frequency ω and therunning exponent µ nn ( W, N ) characterizing the scaling of N β nn with N . Our key findings are briefly summarizedbelow.(i) For W < W c our results confirm the ergodicity ofthe delocalized phase. In particular, the correlationfunction N β nn shows at large N the ergodic 1 /N scaling, in analogy with the inverse participationratio P . Equivalently, the exponent µ nn ( W, N )tends to zero at N → ∞ .(ii) Dynamical eigenstate correlations in the localizedphase W > W c are characterized, in the large- N limit, by fractal scaling, N β ( ω ) ∼ ω − µ ( W ) and N β nn ∼ N µ nn ( W ) , with disorder-dependent ex-ponents µ ( W ) and µ nn ( W ). The source of thepower-law enhancement of correlations with low-ering ω is Mott-like resonances between distantlocalized states. For finite N (as in the exact-diagonalization numerics), the exponents are sub-jected to finite-size corrections. Our analytical ar-guments (for N → ∞ ) suggest that for RRG model µ ( W ) = µ nn ( W ), while for the spin-chain problemthere is a small difference between these exponentsdue to fractal scaling of the inverse participationratio in the MBL phase. Since the critical pointhas a localized character, the value µ ( W c ) is equalto the limit of µ ( W ) at W → W c + 0, and similarlyfor µ nn ( W c ). On RRG we find (again on the basisof analytical arguments that assume the large- N limit) µ nn ( W c ) = µ ( W c ) = 1, while for the spinchain µ ( W c ) = 1 and µ nn ( W c ) = 1 − τ ( W c ) ≈ . τ ( W ) is the exponent characterizing the frac-tality of the inverse participation ratio in the lo-calized phase. With increasing disorder, the ex-ponents µ ( W ) and µ nn ( W ) decay rather slowly, µ ( W ) ≈ µ nn ( W ) ∼ / [ln( W/W c )].(iii) The correlation function β nn and the correspond-ing exponent µ nn exhibit, as functions of disorder W , a maximum that serves as an indication of theMBL transition. With increasing N , the positionsof this maxima drift towards W c . This drifts is,however, rather slow, so that the position of themaximum remains quite far from the actual W c forall system sizes amenable to exact diagonalization.This is a manifestation of strong finite-size effectsin the MBL problems [16, 76, 79, 80], which makeextremely difficult a reliable determination of thecritical point of the MBL transition and of the as-sociated critical behavior on the basis of exact di-agonalization. A closely related observation is arather broad critical regime on the delocalized sideof W c , where the system sizes that can be treatedby exact diagonalization are too small in order toreach (even approximately) the ergodic behavior.2 W N β nn ( W ) W c W . . . . . . µ nn ( W ) W c µ nn,max = 1 − τ ( W c ) FIG. 6. Correlation of adjacent wavefunctions for spin chain. System sizes are n = 12 , , ,
18 (from cyan to magenta).
Left:
Correlation function β nn ( W ). Vertical dashed line marks an approximate value of the critical point of the MBL transition, W c (cid:39)
5, as obtained from the quantum dynamics in large chains [16].
Right:
Exponent µ nn characterizing the N scaling ofadjacent-state correlations, see Eq. (27). Dashed line shows theoretically expected N → ∞ behaviour, see Eq. (22) for thedelocalized phase and Eq. (24) for the localized phase (this part of the dashed line is schematic.) This figure is a spin-chaincounterpart of Fig. 3 for the RRG model. Let us now discuss a possible way to experimentallymeasure the frequency-dependent eigenstate correlations.In this connection, consider the return probability p ( t ) toa many-body state ψ (0): p ( t ) = |(cid:104) ψ ( t ) | ψ (0) (cid:105)| , (41)where ψ ( t ) = e − iHt ψ (0) follows the dynamics determinedby the Hamiltonian H . Let us choose one of the basisstates as the initial state, | ψ (0) (cid:105) = | j (cid:105) . Expanding ψ (0)and ψ ( t ) in terms of the eigenstates ψ k , we get p ( t ) = (cid:88) kl e − i ( E k − E l ) t | ψ k ( j ) | | ψ l ( j ) | . (42)In the limit t → ∞ , the return probability is determinedby the diagonal ( k = l ) terms in Eq. (42), which yields p ( t → ∞ ) ≡ p ∞ = (cid:88) k | ψ k ( j ) | = P ( j )2 . (43)Here P ( j )2 is the inverse participation ratio (23), with aslight difference that the summation goes over k ratherthan over j (i.e. it characterizes the expansion of a basisstate over exact eigenstates). This difference is not es-sential (and disappears completely upon averaging). Fo-cussing on the MBL phase and the critical point, we havethus p ∞ ∼ N − τ ( W ) . (44)The dynamical part of the return probability p ( t ) isgiven by non-diagonal terms in Eq. (42): p ( t ) − p ∞ = (cid:88) k (cid:54) = l e − i ( E k − E l ) t | ψ k ( j ) | | ψ l ( j ) | = (cid:90) dE ν ( E ) (cid:90) dω e − iωt N β ( ω ) . (45) where β ( ω ) is the eigenstate correlation function (2).(Note that β ( ω ) implicitly depends on E .) The many-body density of states ν ( E ) is sharply peaked near themiddle of the band, so that the integral in Eq. (45) isgoverned by the vicinity of the corresponding value of E . Using Eq. (8) or Eq.(37), we get a power-law tem-poral decay of the many-body return probability in thelocalized phase and at the critical point: p ( t ) − p ∞ ∼ t − µ ( W ) . (46)Here we have discarded the N -dependent factor inEq. (37) that is not that important in practice in viewof smallness of the exponent τ ( W ) and of large compen-sation between the logarithmic and power-law factor forrealistic N , see comment after Eq. (37).As has been already pointed out in the Introduction,Sec. I, the return probability p ( t ) can be efficiently mea-sured in engineered many-body systems (quantum simu-lators or quantum processors), such as arrays of trappedions, atoms, and supeconducting qubits [45, 66–68]. Thestate-of-the-art devices contain ≈
50 elements [“qubits”analogous to spins in the Hamiltonian (28)]; it is expectedthat this number will grow up to ≈
100 in near future.Clearly, the full quantum-state tomography is impossiblein such devices, in view of the huge size of the many-bodyHilbert space, N ∼ – 2 ≈ – 10 . At the sametime, the measurement of the many-body return proba-bility p ( t ) [i.e., of the Fourier transform of the dynamicaleigenstate correlation function β ( ω )] is absolutely feasi-ble. This is done by measuring the evolved state ψ ( t )in the non-interacting basis j (i.e. measuring all S zi ). Ifthe measurement is performed, e.g., ∼ times (as inRefs. [66, 67]), one can determine p ( t ) as long as it is (cid:38) − . For a rather slow, power-law decay (46), this al-lows one to determine p ( t ) up to very long times t > ∼ , implying a possibility to proceed up to p ( t )as small as ∼ − . Experimental investigation of theeigenstate correlations across the MBL transition is thusa very promising avenue for future research.Finally, it is worth mentioning connections betweenour results for the eigenstate correlation function β ( ω )and the behavior of other dynamical observables. In par-ticular, Ref. [54] studied frequency dependence of matrixelements of local (in real space) operator S iz , where i is agiven site of the lattice. In our notations, this means thefollowing correlation function (cid:104)| (cid:0) S iz (cid:1) kl | (cid:105) ≡ (cid:104)|(cid:104) ψ k | S iz | ψ l (cid:105)| (cid:105) , (47)considered as a function of the frequency ω = E k − E l .Here ψ k and ψ l are exact many-body eigenstates, and E k and E l the corresponding energies, as in Eq. (2). The ω → t Fourier transform of Eq. (47) can be viewed as areturn probability in real space, which is in general verydifferent from the return probability in the many-body space p ( t ) given by the Fourier transform of the correla-tion function β ( ω ) studied in the present work. At thesame time, there is a remarkable similarity in the behav-ior of both correlation functions in the localized phase(and at criticality): they both show a power-law depen-dence on frequency, with a continuously changing expo-nent. A related power-law scaling of the ac conductivity, σ ( ω ) ∼ ω α , with 1 < α < β ( ω ) studied in this work and its Fourier transform p ( t )] remains an interesting goal for future research. V. ACKNOWLEDGMENTS
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01 (red) and 0 .
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In this Supplemental Material, we provide some additional information to the analysis performed in the main partof the paper.
POPULATION DYNAMICS FOR CORRELATION OF EIGENSTATES IN THE LOCALIZED PHASE
As an alternative (to exact diagonalization) way to calculate β ( ω ) in the RRG model, one can use the field-theoretical approach [56]. The correlation function β ( ω ) is given by Eqs. (9), (10), (11). To determine K ( ω ), onehas to solve numerically the saddle-point equation to the effective action characterizing the problem. This equationis equivalent to the self-consistency equation for the joint distribution function of Green functions at two energies, u = G R ( i, i, E + ω/
2) and v = G A ( i, i, E − ω/ G R,A ( j, j, E ) = (cid:104) j | ( E − H ± iη ) − | j (cid:105) . f ( m ) ( u, v ) = (cid:90) d(cid:15) γ ( (cid:15) ) (cid:90) (cid:32) m (cid:89) r =1 du r dv r f ( m ) ( u r , v r ) (cid:33) × δ (cid:20) u − E + ω + iη − (cid:15) − (cid:80) mr =1 u r (cid:21) δ (cid:20) v − E − ω − iη − (cid:15) − (cid:80) mr =1 v r (cid:21) ; (48) f ( m +1) ( u, v ) = (cid:90) d(cid:15) γ ( (cid:15) ) (cid:90) (cid:32) m +1 (cid:89) r =1 du r dv r f ( m ) ( u r , v r ) (cid:33) × δ (cid:34) u − E + ω + iη − (cid:15) − (cid:80) m +1 r =1 u r (cid:35) δ (cid:34) v − E − ω − iη − (cid:15) − (cid:80) m +1 r =1 v r (cid:35) . (49)Equation (48) is a self-consistency equation for the function f ( m ) ( u, v ), while Eq. (49) expresses the required distri-bution function f ( m +1) ( u, v ) through f ( m ) ( u, v ). The correlation function K ( ω ) is given by K ( ω ) = − π (cid:104) Im u Im v (cid:105) , (50)where the averaging is performed with the function f ( m +1) ( u, v ).Equations (48), (49) have been used to study the level number variance on RRG in Refs. [63, 81] and to exploreeigenstate correlations in the delocalized phase of the RRG model in Ref. [56].We have solved the self-consistency equations (48), (49) by using the pool size M = 2 and the broadening η from η = 10 − down to η = 10 − . The results for the local-DOS correlation function K ( ω ) are shown in Fig. S1. Thevalues for the correlation function β ( ω ) derived from the data in this figure are shown in Fig. 2 of the main texttogether with exact-diagonalization results. OVERLAP OF EIGENFUNCTIONS IN THE LOCALIZED PHASE OF RRG VIA RESONANCECOUNTING
In this section, we describe an analytical approach to evaluation of the frequency-dependent eigenfunction correlationfunction β ( ω ), see Eq. (2), in the localized phase of the Anderson model (1) on a RRG. These correlations are expectedto be controlled by Mott-type resonances between distant localized states. In the main text of the paper, a simplifiedderivation of Eq. (8) is presented that discards strong fluctuations of eigenstates on tree-like graphs. Here we presenta more accurate analysis that takes into account these fluctuations.To calculate the probability of resonances [and, in this way, to evaluate the correlation function β ( ω )], we make useof a resonance counting approach in the spirit of Ref. [55], which is expected to be valid in the strong–disorder regime W (cid:29)
1. The starting point are eigenstates at W → ∞ that are localized at individual sites of the graph. Let uspick up a certain site | (cid:105) with local energy equal to (cid:15) (generation 0) and consider a tree formed by the graph aroundthis node. Sites that are separated by distance n from the site | (cid:105) form the generation n . The number of sites in thegeneration n grows as m n . (We discard the loops, which is certainly justified as along as n is smaller than the linearsize L (cid:39) log m N of the graph.) For a given site | j (cid:105) of the generation n + 1, let us compute the probability that thissite is in resonance with the site | (cid:105) with resonance splitting ∼ ω . The resonance conditions for the local energies andthe effective matrix element M j reads | (cid:15) − (cid:15) j | (cid:46) M j . We focus on resonances with | (cid:15) − (cid:15) j | ∼ M j , since they areobviously much more likely than those with | (cid:15) − (cid:15) j | (cid:28) M j and thus will determine the disorder-averaged correlationfunction. Therefore, the conditions for the resonance with splitting of order ω are as follows: | (cid:15) j − (cid:15) | ∈ [ ω, ω ] (51)and | M j | ∈ [ ω, ω ] . (52)The matrix element M j appears in the n + 1–th order of perturbation theory over the hopping matrix element V (inthe main text of the paper we set V = 1 but here we restore it for clarity): M j = V n (cid:89) i =1 V(cid:15) − (cid:15) i ≡ V A n , (53)7where (cid:15) , ...(cid:15) n are local energies encountered on the (unique) path from | (cid:105) to | j (cid:105) . The resonance probability equals P n ( ω ) = P (1) n ( ω ) P (2) n ( ω ) , where probabilities P (1) n ( ω ) and P (2) n ( ω ) correspond to conditions (51) and (52), respectively. For the first of them,we clearly have P (1) n ( ω ) ∼ ωW . (54)To find P (2) n ( ω ), we use Eq. (9) of Ref. [55] for the distribution of | A n | : P ( | A n | ) ∼ [ln | A n | ( W/V ) n ] n − ( W/V ) n ( n − | A n | , (55)valid for | A n | > ( V /W ) n . This inequality limits applicability of our analysis to ω > V ( V /ω ) n ≡ ω min . This limitationis however not essential, since for W > W c , one has ω min (cid:28) ∆ n , where ∆ n is the level spacing of a RRG with a linearsize n . As a result, we find P (2) n ( ω ) = (cid:90) ω/Vω/V d | A n |P ( | A n | ) ∼ ωV P (cid:16) | A n | ∼ ωV (cid:17) ∼ Vω (cid:18) VW (cid:19) n ln n − [( ω/V )( W/V ) n ]( n − . (56)Hence, the resonance probability equals P n ( ω ) ∼ (cid:18) VW (cid:19) n +1 ln n − [( ω/V )( W/V ) n ]( n − . (57)The average number of resonances that the initial state | (cid:105) encounters in the frequency interval [ ω, ω ] at distance n + 1 is thus N n ( ω ) ∼ m n P n ( ω ) ∼ (cid:20) meVW (cid:18) ln WV + 1 n ln ωV (cid:19)(cid:21) n , (58)where we have discarded a prefactor which does not scale with n . The total number of such resonances at all distancesreads N restot ( ω ) = (cid:88) n N n ( ω ) = (cid:88) n exp (cid:20) − nζ + n ln(1 − y/n ) (cid:21) (59)where ζ is the localization length given by 1 ζ = − ln (cid:18) emVW ln WV (cid:19) (60)and y = ln( V /ω )ln(
W/V ) . (61)In the relevant range of frequencies ω min < ω < V we have 0 < y < n . The sum in Eq. (59) is dominated by thevicinity of the stationary point n ∗ = y/x ∗ ( ζ ) where x ∗ ( ζ ) is determined by equation f ( x ∗ ) = ζ − , with f ( x ) = x − x + ln(1 − x ) . (62)Discarding the prefactor, we find the leading behavior N restot ( ω ) ∼ ( ω/V ) z (63)with z = 1[1 − x ∗ ( ζ )] ln( cW/V ) . (64)8In Eq. (64), we have restored a numerical constant c (of order unity) under the argument of the logarithm. Thisconstant was discarded in the approximation used above, so that its evaluation requires a more accurate analysis.The total number of states in the interval [ ω, ω ] on the RRG of volume N ∼ L m equals N tot ( ω ) ∼ ω ∆ ∼ ωNW . (65)Combining Eqs. (63) and (65), we find that the probability that the fraction of the basis states | j (cid:105) on RRG in therange [ ω, ω ] that are in resonance with the state | (cid:105) equals p res ( ω ) = N restot ( ω ) N tot ( ω ) ∼ (cid:16) ωV (cid:17) − z N . (66)If such a resonance takes place, the states | (cid:105) and | j (cid:105) get strongly hybridized, and for two resulting states | ψ (cid:105) and | ψ (cid:105) we get the overlap (cid:88) j (cid:48) | ψ ( j (cid:48) ) | | ψ ( j (cid:48) ) | ∼ , (67)where the sum goes over the RRG sites j (cid:48) . We expect that the correlation function β ( ω ) is determined by suchresonances, which implies N β ( ω ) ∼ p res ( ω ) ∼ (cid:16) ωV (cid:17) − z N , (68)and finally N β ( ω ) ∼ ( ω/V ) − µ (69)with µ = 1 − z . (70)Thus, we have derived Eq. (8) of the main text, which is the main goal of this section of the Supplemental Material.This computation can be trusted only as long as n ∗ < L where L stays for the linear size of the graph. One maycheck that this inequality implies that our approximation fails for W → W c (when the localization length ζ is large).In the opposite limit, W (cid:29) W c , when ζ (cid:28)
1, the approximation is controllable. To determine the asymptotic behaviorof the exponent z , Eq. (64), we inspect Eq. (62) for small ζ and find11 − x ∗ = ζ − + ln( ζ − ) + 1 + O (cid:18) ζ − ) (cid:19) . (71)Substituting this in Eq. (64), we obtain z ≈ ln( W/V ) − ln m ln( cW/V ) ≈ − ln( cm )ln W/V . (72)Therefore, according to Eq. (70), we have in the limit of strong disorder µ ∼ W/V ) , (73)with proportionality constant of order unity (its exact value is beyond our accuracy). Note that this is the samelogarithmic behavior that was found in the simplified approximation used in the main text, see Eq. (20).As we have pointed out above, this analysis is insufficient to determine the behavior of the exponent µ ( W ) in thelimit W → W c + 0. We know, however, from the matching with the behavior at not too low frequencies on thedelocalized side of the transition, see second line of Eq. (4), that µ ( W cc