Fragile ergodic phases in logarithmically-normal Rosenzweig-Porter model
I. M. Khaymovich, V. E. Kravtsov, B. L. Altshuler, L. B. Ioffe
FFragile extended phases in logarithmically-normalRosenzweig-Porter model.
I. M. Khaymovich a , V. E. Kravtsov b,c , B. L. Altshuler d,e , and L. B. Ioffe f a Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187-Dresden, Germany ; b Abdus Salam International Center for Theoretical Physics - StradaCostiera 11, 34151 Trieste, Italy; c L. D. Landau Institute for Theoretical Physics - Chernogolovka, Russia; d Department of Physics, Columbia University, New York, NY 10027,USA; e Russian Quantum Center, Skolkovo, Moscow Region 143025, Russia; f Google Inc., Venice, CA 90291 USA
In this paper we suggest an extension of the Rosenzweig-Porter (RP)model, the LN-RP model, in which the off-diagonal matrix elementshave a wide, log-normal distribution. We argue that this model ismore suitable to describe a generic many body localization problem.In contrast to RP model, in LN-RP model a new, weakly ergodic phaseappears that is characterized by the broken basis-rotation symme-try which the fully-ergodic phase respects. Therefore, in addition tothe localization and ergodic transitions in LN-RP model there existsalso the transition between the two ergodic phases (FWE transition).We suggest new criteria of stability of the non-ergodic phases whichgive the points of localization and ergodic transitions and prove thatthe Anderson localization transition in LN-RP model is discontinu-ous, in contrast to that in a conventional RP model. We also formu-late the criterion of FWE transition and obtain the full phase diagramof the model. We show that truncation of the log-normal tail shrinksthe region of weakly-ergodic phase and restores the multifractal andthe fully-ergodic phases.
Multifractality | Ergodic phases | Random Matrix Theory | ...
1. Introduction
The structure of many body wave function is important for avariety of problems that range from many body localization(MBL) (see Ref. (1) and a recent review (2)) to quantumcomputation. It was recently realized that in many of theseproblems the wave function is neither localized nor completelyergodic. Instead it is characterized by anomalous dimension, D < P i ψ µ ( i ) ln ψ µ ( i ) = − D ln N , where N is thefull dimension of the Hilbert space and ψ µ ( i ) is the wavefunction coefficient h µ | i i of µ -th state, reminiscent of configu-rational entropy of glasses. These fractal wave functions (seeFig. 1(a)) were reported and intensively discussed in the physi-cal problems of localization on random regular graphs (3–12),the Josephson junction chains (13, 14), the random energymodel (15, 16) and even in the Sachdev-Ye-Kitaev model ofquantum gravity (17–19). In quantum computation similarfractal wave functions appear in the search algorithms basedon the efficient population transfer and it is believed that theappearance of the fractal dimensions is linked with quantumsupremacy (20). Moreover, the wave function correspondingto a generic fault tolerant quantum computation is fractalbecause it is confined to the computational space that is muchsmaller than the full Hilbert space. However, despite the ap-parent importance of this phenomena, its understanding andanalytic description is still in its infancy.Generally, one expects that fractal wave function mightappear in the intermediate regime sandwitched between fullyergodic and fully localized states. However, the only solvablemodel that shows the appearance of such a regime in a certainrange of parameters, the Gaussian Rosenzweig-Porter (GRP) model (21–27), is largely oversimplified. Firstly, such a phasein this model is fractal and not multi -fractal. However, moreimportantly, few mini bands in the local spectrum of thismodel (22, 28) are compact and absolutely continuous inthe energy space, and not multiple and fractal as in realisticmany-body systems (14) (see Fig. 1(a)). This behavior isintimately related to the compactness of the wave functioncoefficients in the configurational space and can be traced backto the property of the moments of the Gaussian distribution h| U | q i = h U i q/ .In this paper we introduce a natural generalization of thismodel and show that it displays a much richer phase diagramand a more realistic behavior. In GRP model every site of thereference space (represented by a matrix index) is connectedto every other site with the transition amplitude distributedaccording to the Gaussian law. Such model occurs as theeffective description of the systems without internal structure,in which transition between resonance sites is due to a smallnumber of hops, such as random energy model (15, 16). Inmore realistic models delocalization of the wave function is dueto a long series of quantum transitions. Each transition has arandom amplitude, so their product is characterized by thelog-normal (LN) distribution, rather than the Gaussian oneas in GRP model. Inspired by this argument in this paper weintroduce and study the generalization of RP model in whichthe transition amplitude between sites has a small typical value,as in RP model, but with much wider, log-normal distributionfunction that we define in Section 2 (see Fig. 1(b)).It appears that the rare large hopping matrix elementsfrom the tail of this distribution alter the phase diagram ofthe system by considerably shrinking the region of multifractalphase as the parameter p that controls the weight in the tail,increases. For large enough p the multifractal phase is totallyreplaced by an ergodic one (see Fig. 1(c)). However, thisergodic phase is fragile. Because it is due to very rare hoppingelements, even a far cutoff of the LN distribution functionrestores the multifractal phase and may even extend it in thephase diagram (Fig. 1(d)).Generally, the mere statement that the eigenfunction fractaldimension D = 1 is not sufficient for complete characteri-zation of the ergodic phase. As was shown in Ref. (29), incertain translational-invariant RP models D = 1 in the refer-ence basis, yet in the Fourier-transformed ’momentum’ basisall eigenvectors are localized. Consequently, the eigenvaluestatistics is Poisson, despite extended character of wave func-tions in the reference basis. On the other hand, the ergodicstates in the GRP model remain ergodic in any basis, like inthe classic Wigner-Dyson (WD) random matrix ensemble.This observation urged us to distinguish between the fully-ergodic (FE) phase where: ( i ) the fraction of populated sitesin an eigenfunction is f = 1, ( ii ) the eigenvalue statistics is PNAS |
July 21, 2020 | vol. XXX | no. XX | a r X i v : . [ c ond - m a t . d i s - nn ] J u l LDoS multifractal weakly-ergodic
Mini-bands fully-ergodic
GRPLN-RP (a) P ( U ) U GaussLN, p = = U typ = - - - (b) fully ergodic weakly ergodic m u l t i f r a c t a l localized γ p ( c ) - ρ typ / ρ av p = / γ ( e ) weakly ergodic fully ergodic m ultifractal γ localized p (d) Fig. 1. (Color online)
Fragile extended phases and mini-bands in LN-RP model. (a) Cartoon of different extended states: fully-ergodic, ( D = 1 , f =1 ), weakly-ergodic ( D = 1 , f < ) and multifractal ( D < ). Sparse spacestructure of wave functions corresponds to sparse structure of mini-bands in thelocal spectrum. A compact mini-band in GRP (red) is compared with multiple fractalmini-bands in LN-RP (blue). (b) Gaussian and tailed log-normal (LN) distributions of U = H nm . With increasing the parameter p in Eq. (1) the weight of the tail at large | U | increases. Gaussian RP ensemble corresponds to p → and RRG is associatedwith p = 1 . (c) Phase diagram of LN-RP N × N random matrix model Eq. (1) in the middle of the spectrum. The parameter γ is an effective disorder. The points (0 , and (1 , in ( p, γ ) plane are the tricritical points. With increasing p theweakly-ergodic (WE) phase proliferates and pushes out both the multifractal (MF) andthe fully-ergodic (FE) phases. For p > the MF phase no longer exists. (d) Phasediagram of RP model in the middle of the spectrum with the LN distribution truncatedso that | U | < N − γ tr , ( γ tr = 0 . ). The WE phase shrinks dramatically and givesthe way to MF and FE phases. (e) Dependence on γ of − ρ typ /ρ av , where ρ typ and ρ av are the typical and the mean Local Density of States (LDoS), obtained byexact diagonalization (blue to red curves) and extrapolated to N → ∞ (black curve).The intersection of curves signals of the transition from MF to WE phase. In the inset:dependence of the order parameter φ = 1 − ρ typ /ρ av on γ . Bright blue point isthe FWE transition between FE ( φ = 0 ) and WE ( φ > ) phases. WD and ( iii ) eigenfunction statistics is invariant under basisrotation ∗ , and the weakly-ergodic (WE) phase (30) where thisinvariance is broken together with the WD eigenvalue statistics,and f < phase transition and not a crossoverbetween them. We will refer to this transition between thefully- and weakly ergodic phases as the FWE transition . Notethat the basis-rotation invariance is an emergent symmetrywhich existence is non-trivial and requires the limit N → ∞ .It is probably this WE phase which is responsible for a so-called “bad metal” phase on the ergodic side of the localizationtransition. In such a phase, both many-body systems (31) andhierarchical structures like RRG (9, 10) have been shown todemonstrate the anomalous sub-diffusive transport.It is quite natural that the ergodic phase in LN-RP modelwhich emerges and proliferates as the weight p of the fat tail ∗ i.e in an ensemble of random Hamiltonians with parameters (e.g. p and γ ) in FE phase, the eigen-function coefficients in any basis are distributed according to the Porter-Thomas distribution. increases, appears to be the weakly-ergodic one. We showin this paper that this phase is separated by a new FWE phase transition from the fully-ergodic phase existent at smallerdisorder (see Fig. 1 (c)-(e)). The weakly-ergodic phase in manyrespects can be considered as the under-developed multifractalphase, as at large but finite system sizes it shows a quasi-multifractal behavior.The analytical theory of the Ergodic (ET), Localization(AT), and FWE transitions developed in this paper is verifiedby extensive numerics based on the Kullback-Leibler diver-gence (32, 33) of certain correlation functions KL1 and KL2of wave function coefficients (34) and on numerical investiga-tion of the typical ( ρ typ ) and the mean ( ρ av ) Local Densityof States (LDoS). The quantity φ = 1 − ρ typ /ρ av is an orderparameter for the FWE transition, with φ = 0 in FE phaseand φ > N ) of KL1 and KL2 marksthe AT and ET transitions, respectively (see Fig. 4).
2. Log-Normal Roseizweig-Porter model
We introduce a modification of the RP random matrix ensem-ble (21, 22) in which the Gaussian distribution of indepen-dent, identically distributed (i..i.d.) off-diagonal real entries H nm = U is replaced by the logarithmically-normal one: P ( U ) = A | U | exp (cid:20) − ln ( | U | /U typ )2 p ln( U − ) (cid:21) , U typ ∼ N − γ/ . [1]It is characterized by two parameters: the disorder-parameter γ which determines the scaling of the typical off-diagonalmatrix element with the matrix size N and the parameter p that controls the weight of the tail.The i.i.d. diagonal entries are supposed to remain Gaussiandistributed, as in the original RP model: h H nn i = 0 , h H nn i = W ∼ N . [2]This LN-RP model is principally different from the Lévy ran-dom matrix models (see, e.g., (35–37) and references therein)exactly because the Gaussian distribution Eq. (2) is not tailed .For numerical purposes we will replace it by the box distribu-tion which is plain in the interval [ − W/ , W/ h| U | q i q ∼ N − γ q / that scale differently with N for differentvalues of q : γ q = γ (1 − pq/ . [3]The limit p → γ q = γ , corresponds to the GRPmodel. It is shown in Ref. (38) that p = 1 is associated withRRG due to the hidden β -symmetry on the local Cayley tree(see Eqs. (6.5)-(6.8) in Ref. (39), Eqs. (D.2), (D.17) in Ref. (4)and Appendix C in Ref. (38)). Finally, the limit p → ∞ corresponds to the Lévy power-law distribution of U (40).For any physically meaningful quantity in the bulk of thespectrum with a bandwidth E BW , only the values | U | < E BW are relevant. For larger values of | U | = | H nm | the states arepushed to the Lifshits tails of the spectrum which we arenot interested in this paper. As in the non-ergodic part ofphase diagram E BW ∼ W is of the order of the spread ofon-site energies, in these regimes the distribution P ( U ) iseffectively cut off for | U | > W . However, for ergodic states E BW is determined by the off-diagonal matrix elements andis divergent with N . In this case the effective cutoff E BW in Eq. (1) should be determined self-consistently (41). | Khaymovich et al. . Criteria of Localization, Ergodic and FWE transi-tions for dense random matrices In this section we consider simple criteria of the disorder-driven † localization, ergodic and FWE transitions for random N × N matrices with the long-range uncorrelated randomhopping h H nm i = 0 and diagonal disorder ∼ O (1). Moregeneral picture and examples of systems are presented inRefs. (29, 42).The first criterion, which is referred to as the Andersonlocalization criterion , states that if the sum : S = N X m =1 h| H n,m |i = N h| U |i < ∞ [4]converges in the limit N → ∞ then the states are Andersonlocalized.Here h .. i stands for the disorder averaging over the dis-tribution, Eq. (1), which is cut off at | U | > W ∼ N . Thereason for such a cutoff is the following. The physical meaningof Eq. (4) is that the number is sites in resonance with a givensite n is finite. The probability that two sites n and m are inresonance is: P n → m = Z W/ − W/ dε n W Z W/ − W/ dε m W Z ∞| ω | P ( H nm ) d ( H nm ) , [5]where for simplicity we consider the box-shaped distribution F ( ε ) of on-site energies. Then integration over ( ε n + ε m ) / ω = ε n − ε m gives: P n → m = Z W − W dU P ( U ) (cid:18) | U | W − U W (cid:19) + Z ∞ W P ( U ) dU. [6]One can easily see that at U typ ∼ N − γ/ (cid:28) O (1) the lastintegral in Eq. (6) can always be neglected. The values of | U | involved in the first integral are bounded from above | U | < W ,which is equivalent to imposing a cutoff at | U | > W on thedistribution P ( U ). As the second term in this integral is atmost 1/2 of the first term, the number of sites in resonancewith the given site, P m P n → m , coincides with Eq. (4) up toa pre-factor of order O (1).The second criterion referred to as the the Mott’s criterion is a sufficient criterion of ergodicity. It states that if the sum S = N X m =1 h| H nm | i E BW = N h U i E BW → ∞ [7]diverges in the limit N → ∞ then the system is in the one ofthe ergodic phases (29).In Eq.(7) the subscript E BW implies that the distribution,Eq.(1) should be truncated at U ∼ E BW , where E BW ∼ W ∼ N is the total spectral bandwidth in the non-ergodic phase.The physical meaning of Eq. (7) is that the mean Breit-Wignerwidth Γ ∼ E − BW N h U i E BW (43) that quantifies the escaperate of a particle created at a given site n , is much largerthan the spread of energy levels W ∼ N due to disorder.Then the fulfillment of the Mott’s criterion implies that thewidth Γ is of the same order as the total spectral bandwidth E BW ∼ √ S and thus there are no mini-bands (which width † The problem of mobility edge and energy-driven transitions in systems with broadly distributedhopping is non-trivial (37) and we leave it for future publications. is Γ) in the local spectrum (see Fig. 1(a)). As the presence ofsuch mini-bands is suggested (14, 28, 44) as a “smoking gun”evidence of the non-ergodic extended (e.g. multifractal) phase,the fulfillment of the Mott’s criterion Eq. (7) immediatelyimplies that the system is in the ergodic extended phase.The multifractal phase realizes provided that in the limit N → ∞ both Eq. (4) and Eq. (7) are not fulfilled: S → ∞ , S < ∞ . [8]Finally, the fully ergodic phase is realized when S , S → ∞ and also: S = (cid:0)P Nm =1 h| H nm | i typ (cid:1) P Nm =1 h| H nm | i E BW = N U h U i E BW → ∞ , [9]is divergent in the N → ∞ limit, where h| H nm | i typ ≡ U =exp h ln | H nm | i ‡ . If only S , S → ∞ but S is not, the weaklyergodic phase is realized.The physics behind the condition Eq.(9) is that the typical escape rate Γ typ ∼ E − BW N U = √ S (43) is much largerthan the disorder strength W ∼ N . The two conditions,Eqs.(7),(9), coincide for a Gaussian distribution of U but aredifferent for the tailed ones, like LN distribution, Eq.(1).
4. Phase diagram
For the log-normal distribution Eq. (1) one easily computesthe moments h| U | q i truncated at U max ∼ N : h U q i = ( N − γq ( − pq ) , if pq < N − γ p , if pq ≥ U typ = N − γ/ the following critical points of the localization ( γ AT ),ergodic ( γ ET ) and FWE ( γ FWE ) transitions: γ AT = ( − p , if p < p, if p ≥ γ ET = ( − p , if p < / p, if p ≥ / γ FWE = 11 + p . [13]The phase diagram at a fixed energy in the middle of spectrumresulting from Eq. (11)-Eq. (13), is presented in Fig. 1(c).The main conclusion we may draw from this phase diagramis the emergence and proliferation of the weakly-ergodic phasethat pushes away both the multifractal (MF) phase and thefully ergodic phase, as the strength of the tail p in the distribu-tion Eq. (1) increases. For p > | U | < N − γ tr , γ tr >
0, eliminates the WE phase andrestores the MF phase, as well as increases the range of thefully-ergodic one (see Fig. 1(d) and Appendix A for details).
Khaymovich et al.
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July 21, 2020 | vol. XXX | no. XX | )b) c) Fig. 2. (Color online)
Hybridization of fractal support sets (a), (b) Two differentfractal support sets, (c) The hybridized fractal support set.
5. Stability of non-ergodic states against hybridization
In this section we consider the stability of non-ergodic (multi-fractal and localized) states against hybridization. It allowsus not only to derive expressions, Eq. (4) and Eq. (7), for theAnderson localization and ergodic transitions in a different waybut also find the fractal dimension D ( p, γ ) of the multifractalsupport set. Furthermore, the new method presented belowis physically transparent and generic enough to be applied toanalysis of the multifractal states in other systems.Let us consider two states ψ µ and ψ ν on different fractalsupport sets as it is shown in Fig. 2(a) and (b). We assumethat both states are multifractal with m ∼ N D sites on afractal support set where the coefficients | ψ ( i ) | ∼ N − D .Here we apply a usual Mott’s argument for hybridizationof states, Fig. 2(c), when the disorder realization, in thiscase the off-diagonal matrix element, changes from H ij to H ij = H ij + δ H ij . The key new element in the theory we areintroducing here is the hopping matrix element V µ,ν betweenthe states and not between the sites as is customary: V µ,ν = X i,j δ H ij ψ µ ( i ) ψ ν ( j ) . [14]Here ψ µ ( i ) is the eigenfunction of the µ -th state of H ij , and δ H ij = H ij − H ij , where H ij is drawn from the same log-normal distribution as H ij .Introducing g ij = − ln δ H ij / ln N and suppressing the in-dices i, j for brevity we conveniently rewrite Eq. (1) as follows § : P ( g ) = const N − pγ ( g − γ ) , ( g ≥ . [15]By the constraint g ≥ | U | ∼ O ( N ) discussed in Sec. 3.The typical number of terms in the sum Eq. (14) in theinterval dg is N D N D P ( g ) ∼ N σ ( g,D ) dg where σ ( g, D ) = 2 D − pγ (cid:16) g − γ (cid:17) . [16]If σ ( g, D ) <
0, the sum, Eq. (14), is dominated by a sin-gle term with the largest | G ij | . For positive σ ( g, D ) > P ( V ≡ | V µ,ν | ) becomes Gaussian. In general, there are bothcontributions P ( V ) = P LN ( V ) + P Gauss ( V ) . [17] ‡ In the ergodic phase the bandwidth E BW is growing with N faster than U that makes the maincontribution to h U i and the average in Eq.(9) can be done using the full log-normal distribu-tion (43). § Here we omit a small deviation from the log-normal distribution for g ij = − ln | H ij − H ij | / ln N > γ/ which is not important in the current setting, see Appendix B for details. The condition of stability of the multifractal phase againsthybridization is derived similar to the Anderson criteria ofstability, Eq. (4), of the localized. The difference is that now wehave to replace the matrix element between the resonant sites U by the matrix element V between the resonant non-ergodicstates and take into account that on each of M = N − D different support sets there are m = N D wave functions whichbelong to the same mini-band and thus are already in resonance with each other. Therefore the total number of independent states-candidates for hybridization with a given state shouldbe smaller than the total number of states M m = N andlarger than the number of support sets M . This number is infact equal to their geometric mean √ NM = M √ m = N − D .With this comment, the criterion of stability of the multi-fractal phase reads in the limit N → ∞ as N − D Z W dV V P ( V ) < ∞ . [18]The contribution of the Gaussian part P Gauss in Eq. (17)to Eq. (18) is: N − D p h V i = N − D − γ eff ( D ) < ∞ , [19]where h V i ≡ N − γ eff , [20]and for stability it must be finite as N → ∞ . The contributionof P LN in Eq. (17) to the stability criterion Eq. (18) is N − D − ∆( D < ∞ , where Z σ ( g,D ) < dg N σ ( g,D ) − g − D ≡ N − ∆( D . [21]Thus the multifractal phase is stable against hybridization ifthe following inequalities are both fulfilled D + γ eff ( D ) ≥ , [22] D + ∆( D ) ≥ . [23]The functions γ eff ( D ) and ∆( D ) are computed in Ap-pendix B and discussed in the next Section.A particular case D = 0 of Eq. (22), Eq. (23) describesthe stability criterion of the localized phase. If the localizedphase is not stable, then hybridization produces an avalancheof multifractal states living on fractal support which dimen-sionality grows until inequalities Eq. (22), Eq. (23) are bothfulfilled for the first time at some 0 < D min1 <
1. If this ispossible in some parameter region then the multifractal stateis stable, otherwise the only stable extended phase is ergodic.
6. Fractal dimension of the NEE support set
In this section we re-consider the phase diagram Fig. 1(c)from the viewpoint of stability criteria given in the previoussection by Eq. (22), Eq. (23) and derive the expression forthe fractal dimension D ( γ ) of the support set of multifractalwave functions.To this end in Fig. 3 we plot γ eff ( α ) γ + α = ( α − √ αp, α < p, p /γ ET ( p ) + α, otherwise , [24] | Khaymovich et al. ( α )/ γ + α p / / γ AT p / α γ eff ( α )/ γ + α / γ D ( γ )/ γ p < / D / γ (a) Δ ( α )/ γ + α α / γ AT / < p < / γ p / / D ( γ )/ γγ eff ( α )/ γ + α D / γ (b) / γ α D / γ = / γ /( p ) p > γ eff ( α )/ γ + αΔ ( α )/ γ + α / γ AT (c) Fig. 3. (Color online)
The functions
Eq. (24) (blue curve) and
Eq. (25) (orange curve) entering inequalities
Eq. (22) , Eq. (23) in different regions of p : (a) p < / ; (b) / ≤ p ≤ ; (c) p > . Intervals of α = D /γ with different functional dependence are shown by dashed vertical lines. The Anderson localization transitioncorresponds to the lower of the blue and orange curves equal to /γ at α = 0 . This transition is always determined by the orange curve representing the log-normal partof the distribution P ( V ) . On the contrary, the stable fractal dimension D ( γ ) = 2 − γ/γ ET ( p ) for γ ≤ γ AT is always determined by the blue curve representing theGaussian part of the distribution P ( V ) . The Anderson transition in all cases but p = 0 is discontinuous , with the minimal stable fractal dimension of the support set being D min1 = D ( γ AT ) = 2 − γ AT /γ ET ( p ) > (shown by a gray dotted arrow). The ergodic transition corresponds to D ( γ ) = 1 and it is continuous . For p ≥ there isno solution D < to the system of inequalities Eq. (22) , Eq. (23) in the region of parameters where the localized phase is unstable. In this case the multifractal phase isabsent. γ KL1 γ p = (a) KL1 γ p = - - KL1 / KL1c x = ( γ - γ c ) ( lnN ) / ν (b) KL1 γ p = KL1 / KL1c - - -
10 0 10 20 300.00.51.01.52.02.5 x = ( γ - γ c ) ( lnN ) / ν (c) KL2 γ γ p = (d) KL2 γ p = KL2 - KL2c - - - x = ( γ - γ c ) ( lnN ) / ν (e) KL2 γ p = KL2 - KL2c -
50 0 50 - - x = ( γ - γ c ) ( lnN ) / ν (f) p ν , ν ν ,theor ν ,theor ν ν (g) Fig. 4. (Color online)
Plots of
KL1 and
KL2 vs. γ for LN-RP model at N = 2 L , with L from to with the step (from red to violet). The logarithmic in N divergenceof KL1 for γ > γ AT ≈ and of KL2 for γ > γ ET ≈ is demonstrated in a wide interval of γ for p = 0 . , as well as insensitivity of KL1 to the ergodic transition.Intersection for
KL2( γ ) curves is sharp at the isolated continuous ergodic transition at γ ET ≈ for p = 0 . and at γ ET ≈ . for p = 0 . , it is smeared out for p = 1 . when the ergodic transition merges with the localization transition. Intersection of curves for KL1 at the Anderson localization transition ( γ AT ≈ . for p = 0 . , γ AT ≈ . for p = 0 . , γ AT ≈ . for p = 1 ) is sharp in all the cases. The insets show the collapse of the curves at the proper choice of γ c and the critical exponents ν and ν for KL1 and
KL2 at the AT and ET, respectively. The plot Fig.4 (g) gives a behavior of ν and ν vs. p conjectured on the basis of results of finite-size scalingpresented in Table 1 and shown on the plot. In the limit p → the critical exponents approach their values ν = ν = 1 for the Gaussian RP model (34). For p ≥ weconjecture the mean-field values ν = ν = 1 / . and∆( α ) γ + α = α − √ αp, p < α < p /γ AT ( p ) − α, α < p, p α + 2 √ αp, α > p , [25]as functions of α = D /γ . Here γ AT ( p ) ≥ γ ET ( p ) ≥ /γ , seeFig. 3. First, we note that the localized phase which formallycorresponds to D = 0, is stable if the lowest of the blueand orange curves in Fig. 3 is higher than 2 /γ at α = 0and it is unstable otherwise. One can see that at α = 0for all values of p the log-normal contribution to Eq. (17)(orange curve) is lower than the Gaussian one (blue curve).This means that the stability of the localized phase is alwaysdetermined by the log-normal part of P ( V ). Moreover, since at α = 0 Eq. (24), Eq. (25) reduce to α + γ eff ( α ) /γ = 1and α + ∆( α ) /γ = 2 /γ AT , respectively, the stability of thelocalized phase implies that γ > γ AT ( p ) ≥ D >
0. Those states are, however, unstable until their supportset reaches the fractal dimension D min > γ decreases below the critical value γ AT ,the stable fractal dimension D ( γ ) increases from D min1 beingalways determined by the intersection of the horizontal line y = 2 /γ > /γ AT ( p ) (red line in Fig. 3) with the blue line.Thus the stable fractal dimension D ( γ ) is always determinedby the Gaussian part of P ( V ) and according to the secondline of Eq. (24) and Fig. 3 is equal to: D ( γ ) = 2 − γ eff = 2 − γγ ET ( p ) , p ≤ . [26]At γ = γ ET the fractal dimension D ( γ ) reaches unity, andat this point a continuous ergodic transition happens. Thus Khaymovich et al.
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July 21, 2020 | vol. XXX | no. XX | lnN KL2 γ p = ET AT ( a ) ∂ lnN KL2 γ p = / ET AT ( b ) Fig. 5. (Color online)
Derivative of
KL2 w.r.t. ln N vs. γ for LN-RP model extrapolated from pairs of sizes N = 512 − (red solid lines) for (a) p = 0 . and (b) p = 0 . with the theoretical predictions Eq. (32) and
Eq. (33) (grey dashedlines). The jump is related to the jump in D , Eq. (27) , for all p > . At p → theAT is continuous and instead of the jump in the function d KL2 /d ln N vs. γ thereis only a jump in its γ -derivative at the AT. At p = 1 / , instead, the jump manifestsitself in the dramatic increase of slope near γ = γ AT . the critical point of ergodic transition coincides with thatdetermined by Eq. (12).Note that, unlike the ergodic transition, the Andersontransition is discontinuous : the stable fractal dimension D ( γ )is separated by a finite gap D min = D ( γ AT ) from the localizedstate D = 0: D min = ( − γ AT ( p ) γ ET ( p ) , < p < , p ≥ p ≥ D min = 1, so that the multifractal phase is nolonger possible in LN-RP model Eq. (1). However, it is restoredif the LN distribution is truncated at | U | ∼ N − γ tr with γ tr >
7. Kullback-Leibler (KL) measure
The numerical verification of Eq. (11), Eq. (12) and determina-tion of the critical exponents at the Anderson localization andergodic transitions is done in this paper using the Kullback-Leibler divergence (KL) (32–34, 45) ¶ .The Kullback-Leibler correlation functions KL1 and KL2are defined as follows (34, 45). The first one is defined interms of wave functions of two neighboring in energy states ψ µ ( i ) and ψ µ +1 ( i ) at the same disorder realization:KL1 = *X i | ψ µ ( i ) | ln (cid:18) | ψ µ ( i ) | | ψ µ +1 ( i ) | (cid:19)+ . [28]The second one is similar but the states ψ and ˜ ψ correspondto different (and totally uncorrelated) disorder realizations:KL2 = *X i | ψ ( i ) | ln (cid:18) | ψ ( i ) | | ˜ ψ ( i ) | (cid:19)+ . [29]The idea to define such two measures is the following. In theergodic phases each of the states has an amplitude | ψ ( i ) | ∼ N − of the same order of magnitude. Then the logarithm oftheir ratio is of order O (1), and for the normalized statesKL1 ∼ KL2 ∼ O (1) . [30] ¶ For more detailed multifractal analysis of this model see (46)
For fully-ergodic states the eigenfunction coefficients are fullyuncorrelated, even for the neighboring in energy states. Thusthere is no difference between KL1 and KL2. Using the Porter-Thomas distribution one finds:KL1 = KL2 = 2 . [31]For weakly-ergodic states KL2 is still O (1) but is larger thanthe Porter-Thomas value due to the fact that there are ’pop-ulation holes’ where N | ψ ( i ) | is N -independent but small,Fig. 1(a).Deeply in the localized phase ln | ψ µ ( i ) | ∼ −| i − i µ | /ξ ,where i µ is the position of the localization center. Since thepositions of localization centers i µ are not correlated even forthe states neighboring in the energy, the logarithm of the ratioof the two wave function coefficients in Eq. (28), Eq. (29) isdivergent in the thermodynamic limit. For Anderson localizedstates on finite-dimensional lattices this divergence is linear inthe system size L . However, localization on graphs such asRRG and RP models is not a conventional localization (3, 22).In this case there is a power-law in 1 /N background with themost probable (typical) value of | ψ ( i ) | ∼ N − α far fromthe localization center and therefore:KL1 ∼ KL2 = α ln N → ∞ [32]with α = ( γ AT / γ − γ AT ) + 2 for LN-RP model.A qualitative difference between KL1 and KL2 is in themultifractal phase. In this phase the neighboring in en-ergy states | ψ µ ( i ) | and | ψ µ +1 ( i ) | are most probably belong-ing to the same support set ‖ and hence they are stronglyoverlapping: | ψ µ ( i ) | ∼ | ψ µ +1 ( i ) | . Furthermore, eigenfunc-tions on the same fractal support set can be representedas: ψ µ ( i ) = Ψ( i ) φ µ ( i ), where Ψ( i ) is the multifractal en-velope on the support set and φ µ ( i ) is the fast oscillatingfunction with the Porter-Thomas statistics (3). Thus the ratio | ψ µ ( i ) | / | ψ µ +1 ( i ) | and hence KL1 in MF phase has the samestatistics as in the ergodic one. We conclude that KL1 is notsensitive to the ergodic transition but is very sensitive to the localization one, Fig. 4.In contrast, the eigenfunctions ψ ( i ) and ˜ ψ ( i ) in KL2 cor-responding to different realizations of a random Hamiltonian,overlap very poorly in MF phase. This is because the fractalsupport sets which contain a vanishing fraction of all the sites,do not typically overlap when taken at random. ThereforeKL2 = ( α − D ) ln N = 2(1 − D ) ln N [33]is divergent in the thermodynamic limit in the multifractal phase of RP models, with ( α − D ) = 2( γ/γ ET − >
0, Eq. (26), very much like in the localized one. This makesKL2 very sensitive to the ergodic transition. The propertiesof KL1 and KL2, Eq. (31), Eq. (32), are fully confirmedby numerics presented in Fig. 4. The jump in the slope α ( γ AT + 0) − α ( γ AT −
0) + D min = 2 D min at the Andersontransition, γ = γ AT , originates from the jump in D , Eq. (27).Numerically it is clearly seen in the derivative of KL2 overln N versus γ shown in Fig. 5. We also show in Fig. 6 thatKL2 is sensitive to the FWE transition and can be operativein identifying it. ‖ In the many-body systems undergoing MBL transition it is not the case as the breakdown of theergodicity of the many-body wave function is accompanied by the transformation of the level statis-tics from Wigner-Dyson to Poisson and thus, neighboring in energy wave functions live far awayfrom each other, see the results for KL in (45). | Khaymovich et al. γ AT { ext } , [ th ] γ ET { ext } , [ th ] ν ν .
01 2 .
00 [2] 1 .
00 [1] 1 . ± .
03 0 . ± . / .
80 [2 .
67] 2 .
06 [2] 0 . ± .
07 1 . ± . / .
43 [3 .
2] 2 .
96 [3] 0 . ± .
07 0 . ± .
121 4 . { . } , [4] 3 . { . } , [4] 0 . ± .
07 0 . ± . / . { . } , [5] 4 . { . } , [5] 0 . ± .
08 0 . ± . / . { . } , [6] 5 . { . } , [6] 0 . ± .
10 0 . ± . Table 1. Comparison of analytical predictions(blue),
Eq. (11) , Eq. (12) , and numerical data for the transi-tion points γ AT and γ ET and the corresponding critical exponents ν and ν for LN-RP model. Numerical data (black) is obtained by ex-act diagonalization of LN-RP random matrices with N = 512 − from the intersection points in KL1 and
KL2 and from finite-sizescaling by the best collapse of the curves, Fig. 4. For p > a linearin / ln N extrapolation to N → ∞ of the position of the intersectionpoint for two consecutive N is shown in red. A more detailed theory of KL1 and KL2 in the multifractalphase is given in Appendix C. The main conclusion of this anal-ysis is that the curves for KL1( γ, N ) for different N have anintersection point at the critical point γ = γ AT of the Ander-son localization transition. At the same time, the intersectionpoint for curves for KL2( γ, N ) coincides with the ergodic tran-sition (34), provided that it is continuous and well separatedfrom the Anderson localization transition. If the localizationand ergodic transition merge together and the multifractalstate exists only at the transition point, then intersection ofKL2 curves is smeared out and may disappear whatsoever (asin 3D Anderson model). However, the intersection of KL1curves remains sharp in this case too (see Fig. 4).The intersection of finite-size curves for KL1 and KL2 helpsto locate numerically the critical points γ AT and γ ET . Moreprecise determination of the critical points and the correspond-ing critical exponents ν and ν is done by the finite-size scaling(FSS) data collapse (see insets in Fig. 4 and Appendix D). Theresults are shown in the Table 1. On the basis of these numer-ical results we conclude that our expressions Eq. (11), Eq. (12)for the Anderson and ergodic transition points are accurateand conjecture on the p -dependence of the critical exponents ν and ν of AT and ET obtained from KL1 and KL2. (seeFig. 4(g)).
8. Numerical location of the FWE transition
For numerical verification of Eq. (13) for FWE transitionpoint we make use of the ratio of the typical ρ typ and mean ρ av averageln ρ typ = h ln ρ ( x, E + iη ) i , ρ av = h ρ ( x, E + iη ) i , [34]of local density of states (LDOS) ρ ( x, E + iη ) = Im X µ | ψ µ ( x ) | / ( E + iη − E µ ) . [35]As is shown in Ref. (4), at small bare level width η (cid:28) E BW /N ,where E BW = max(Γ , W ) is the total spectrum bandwidth,this ratio ρ typ /ρ av ∼ η N D /E BW grows linearly with η butthen saturates at ρ typ /ρ av ∼ N − D . In the ergodic phase D = 1 and the plateau in ρ typ /ρ av tends to a finite limit as ρ typ / ρ av p = - ρ typ / ρ av γ η - γ = (a) KL2 γ p = (b) Fig. 6. (Color online) (a): The ratio of the typical and average LDoS as a functionof γ for p = 1 at different values of N = 512 − (purple through red) andextrapolated to N = ∞ (black). Intersection of dashed lines gives the position ofFE-WE transition point γ FWE ≈ . (shown by a bright blue point) as predictedby Eq. (13) . Inset: dependence on the level width η . The main plot is done for η shown by an arrow at the plateau of η -dependence. (b): The Zoom of Fig. 4(f) KL2 vs. γ for p=1 for the same values of N and their extrapolation to N → ∞ .Intersection of dashed lines gives the same position of FWE transition γ FWE ≈ . as on Fig.6(a). N → ∞ . This behavior is well seen in the inset of Fig. 6. Weused the properly defined ∗∗ plateau value of φ = 1 − ρ typ /ρ av as the order parameter for the FWE transition. For γ <γ FWE this parameter φ = 0, signaling of the fully-ergodic phase. For γ > γ FWE the order parameter is non-zero. Thisbehavior is shown in Fig. 6 (see also an inset in Fig. 1(e)and figures in Appendix F), where the black curve represents φ = φ ∞ ( γ ) extrapolated to N = ∞ from the finite N values φ N ( γ ) obtained by exact diagonalization. In spite of imperfectextrapolation that does not allow to get a true non-analyticityat γ = γ FWE , the dashed gray lines of continuation of theblack curve intersect exactly at γ = 1 / γ FWE at p = 1. A similar intersection at γ ≈ / γ plot in Fig. 6.They all suggest thatthe FWE transition does exist and is described by Eq. (13).
9. Conclusion
In this paper we introduce a log-normal Rosenzweig-Porter(LN-RP) random matrix ensemble characterized by a long-tailed distribution of off-diagonal matrix elements. We obtainanalytically the phase diagram of LN-RP using the Andersonlocalization and Mott ergodicity criteria for random matricescomplemented by the new criterion for the transition betweenthe fully- and weakly-ergodic phases. This phase diagram isconfirmed it by extensive numerics.An alternative approach to localization and ergodic tran-sitions based on the analysis of stability with respect to hy-bridization of multifractal wave functions developed in thispaper gives results identical to those obtained from the abovecriteria. Using this approach we computed analytically thedimension D of the eigenfunction fractal support set andshowed that the Anderson localization transition in our modelis discontinuous with the minimal fractal dimension D min1 > ACKNOWLEDGMENTS.
V.E.K. and I.M.K are grateful for sup-port and hospitality to GGI of INFN and University of Florence(Italy) where this work was initiated. V.E.K and B.L.A. acknowl-edge the support and hospitality of Russian Quantum Center duringthe work on this paper and G. V. Shlyapnikov for illuminating dis-cussions there. V.E.K gratefully acknowledges support from theSimons Center for Geometry and Physics, Stony Brook Universityat which part of the research for this paper was performed. This re-search was supported by the DFG project KH 425/1-1 (I. M. K.), by ∗∗ at the maximum of the second derivative of this ratio vs. η , see Appendix E for details Khaymovich et al.
PNAS |
July 21, 2020 | vol. XXX | no. XX | he Russian Foundation for Basic Research Grant No. 17-52-12044(I. M. K.), and by Google Quantum Research Award “Ergodicitybreaking in Quantum Many-Body Systems” (V. E. K.). U max (cid:38) O (1) , γ tr ≤ , does not alter ergodic and localiza-tion transitions in the phase diagram in Fig. 1. A. Truncated LN-RP and fragility of ergodic phase.
The phase diagram shown in Fig. 1 of the main text confirmednumerically by calculations of the KL-divergence and by the ratioof typical and mean local density of states (LDOS) demonstratesthe collapse of the multifractal phase at p ≥ p = 0 and p = 1.In this section we show that the weakly-ergodic (WE) phase thatemerges at p > p ≥ P ( U ) is cutfrom above at: U max ∼ N − γ tr / (cid:28) O (1) , ( γ tr > . [36]As the result of this truncation the multifractal phase re-appears bysubstituting a part of the ergodic phase in a non-truncated LN-RPmodel (see Fig. 1(d)) (50). To this end we use the expression that | Khaymovich et al. eneralizes Eq. (10): Z min( N − γ tr / ,W )0 dU U q P ( U ) ∼∼ N − qγ ( − pq ) , γ (1 − pq ) > γ tr , N − pγ h ( γ − γ tr)24 + pq γγ tr i , γ tr > γ (1 − pq ) , N − γ p , γ tr , γ (1 − pq ) < γ AT of the Andersonlocalization transition is affected as follows γ AT = 2 p − ( p − γ tr + p (2 p − ( p − γ tr ) − γ tr , [38]only if γ tr > γ AT (1 − p ) ,
0. In the opposite case truncation doesnot affect γ AT .For the critical point γ ET of the ergodic transition in the sameway we find the effect only for γ tr > γ ET (1 − p ) , γ ET = 2 p − (2 p − γ tr + p (2 p − (2 p − γ tr ) − γ tr . [39]The criterion for the fully-weakly ergodic (FWE) transition doesnot have any truncation of (cid:10) U (cid:11) at U ∼ W , thus it is affected bythe truncation at all γ tr > γ FWE (1 − p ) (even negative ones if p > / γ tr > γ FWE (1 − p )at γ FWE = 2 p + (2 p − γ tr + p (2 p + (2 p − γ tr ) + (8 p − γ tr p − . [40]Note that Eq. (38) and Eq. (39) give real solutions for γ tr <γ AT (0) = 2 and γ tr < γ ET (0) = 1, respectively, and both thesesolutions increase with the tail weight p . At the same time FWEtransition replaces ET one for all γ tr > γ FWE ( γ tr = 1) = γ ET ( γ tr = 1) = 1 for all p . Similar thing happens for γ tr >
2, whenFWE transition replaces ALT as well, with γ AT ( γ tr = 2) = 2, butin this case γ FWE ( γ tr = 2) = 2 only for p → any positive non-zero γ tr the multifractalNEE phase emerges at p ≥ γ tr (cid:28) p = 1 ( p > − γ tr / (4 p )) γ AT ’ p − p − γ tr − γ tr p + O (cid:0) γ tr (cid:1) , [41]while the line of ergodic transition is pushed to smaller values of γ at 2 p > − γ tr / (4 p ) γ AT ’ p − p − γ tr − γ tr p + O (cid:0) γ tr (cid:1) , [42]corresponding to larger typical transition matrix elements U (smallereffective disorder). Thus, the width of the MF phase increaseslinearly with γ tr (cid:28) γ AT − γ ET = 2 pγ tr + O (cid:0) γ tr (cid:1) . [43]This proves the fact that the weakly ergodic phase in LN-RP with p ≥ γ tr (cid:28) p > − γ tr / (4 p ) as γ ET − γ FWE = 8 p p − p − − p − γ tr ] + O (cid:0) γ tr (cid:1) , [44]showing linear decrease with γ tr and giving a reasonable approxi-mation of the value of γ tr ’ γ FWE = 4 p + 2(2 p − γ tr p − γ tr p + O (cid:0) γ tr (cid:1) . [45] B. Analysis of stability
In this section we calculate the contributions to P ( V ) from thelog-normal P LN ( V ) and Gaussian P Gauss ( V ) parts to Eq. (17).One can easily compute the variance of the Gaussian part of P Gauss ( V ) leaving in it only the bi-diagonal terms with i = i and j = j : h V i = Z g ∈ II dg N − pγ ( g − γ ) − g [46] ∼ max g ∈ II n N − pγ ( g − γ ) − g o ≡ N − γ eff . The maximum in Eq. (46) at g belonging to region II in Fig. 7 canbe reached (i) inside the region II at g = g ∗ , (ii) at the border ofthis region at g = g ∗ , and (iii) at the cut-off of P ( g ) at g ∗ = 0 (seeFig. 7 and Fig. 8(left)).The expression for γ eff ( D takes the form: γ eff ( D ) = γ (1 − p ) , pγ < D < , p < D + γ − √ D γp, D < min (cid:0) pγ , γ p (cid:1) γ p , γ p < D < , p ≥ . [47]Next we compute the function∆( D ) = − g ∈ I { σ ( g, D ) − g − D } . [48]in Eq. (21).The details of the calculation which is similar to calculation of γ eff ( D ) in Eq. (46) are illustrated in Fig. 8(right). The resultingexpression for ∆( D i ) is:∆( D ) = γ (cid:0) − p (cid:1) − D , < D < γp , p < D + γ − √ D γp, γp < D < γ p , p < γ p − D , < D < γ p , p ≥ . [49]In the end of this section we consider the question of the distri-bution of g ij = − ln | H ij − H ij | / ln N with log-normal distributed H = N − g and H = N − g . Applying the usual logarithmicapproximation ln | H − H | ≈ ln max {| H | , | H |} , we approximate g ij = min( g , g ) [50]and, thus, the distribution P ( g ) is given by P ( g ) = P ( g = g ) Z ∞ g P ( g ) dg ’ ( P ( g = g ) g < γ/ P ( g = g ) g > γ/ g > γ/ g ∗ of ∆( D ) + D for D > γ/ (8 p ) whichnever contributes to the phase diagram. C. Kullback-Leibler measures in the multifractal phase
In this section we give a more detailed quantitative description ofKL1 and KL2 measures.We begin by considering the simpler correlation function, KL2.For that we employ the ansatz for the wavefunction moments: M q = *X i | ψ µ ( i ) | q + = N − D q ( q − f q ( L/ξ q ) , [52] g / 2 g D I III *2 1 g D p g g= - ( , ) g D s *2' 1 g D p g g= + Fig. 7. (Color online) Regions of g contributing to the log-normal (I) and Gaussian (II)parts of the distribution function P ( U µ,ν ) .Khaymovich et al. PNAS |
July 21, 2020 | vol. XXX | no. XX | / 2 p < D p g *1 g *2 g min of ( ) at p<1/2 is reached here eff a g a
1/ 2 p > D p g *2 g * g = min of ( ) at p>1/2 is reached here eff a g a (a) p < p g *1 g *2 g D *2' g p > p g * g = D *2' g p g (b) Fig. 8. (Color online) (a) Different possible positions g ∗ , g ∗ or g ∗ = 0 that maximize Eq. (46) in region II depending on p , γ and D . The configuration of maximum realizedin each sector of parameters is shown by an ikon in the corresponding sector. (b) Different possible positions g ∗ , g ∗ or g ∗ = 0 that maximize Eq. (48) in region I. Theconfiguration of maximum realized in each sector of parameters is shown by an ikon in the corresponding sector. For D > γ/ p the maximum in Eq. (48) is reached at theedge of the right segment of region I, g = g ∗ (not to be confused with the edge of the left segment g = g ∗ , see Fig. 7 ). It leads to a higher branch of the orange curve ∆( α ) /γ + α in Fig. 3 (not shown in Fig. 3) which is separated by a gap from the blue curve in Fig. 3 and thus is irrelevant for our analysis. where D q is the fractal dimension in the corresponding phase and f q ( x ) is the crossover scaling function: f q ( L/ξ q → ∞ ) → const . multifractal phaseconst . N ( q − D q − , ergodic phaseconst . N ( q − D q localized phase [53]that tends to a constant as L → ∞ .Note that graphs with the local tree structure and for LN-RPmatrices the length scale L ∝ ln N , so that the scaling function is ingeneral a function of two arguments ln N/ξ q and N/e ξ q representingthe length - and volume scaling (11, 12). On the finite-dimensionallattices N ∝ L d , and the volume scaling can be represented as thelength scaling in the modified scaling function. In this case a singleargument L/ξ q is sufficient.When L ∝ ln N the volume scaling is the leading one for L (cid:29) ξ q ,and it is this scaling that provides the asymptotic behavior Eq. (53).The length scaling is important in the crossover region L (cid:46) ξ q .Below for brevity we will use the short-hand notation L/ξ q in allthe cases.There are two trivial cases: M = N and M = 1 (which followsfrom the normalization of wave function). As a consequence wehave D = 1 and f ( x ) = f ( x ) ≡ . [54]Next using the statistical independence of ψ and ˜ ψ in Eq. (29)and normalization of wave functions we representKL2 = *X i | ψ ( i ) | ln | ψ ( i ) | + − N − *X i ln | ψ ( i ) | + . [55]Now we express both terms in Eq. (55) in terms of M q using theidentity: ln | ψ α ( i ) | = lim (cid:15) → (cid:15) − ( | ψ α ( i ) | (cid:15) −
1) [56]The first term is equal to: *X i lim (cid:15) →∞ | ψ ( i ) | (cid:15) ) − | ψ ( i ) | (cid:15) + = lim (cid:15) →∞ h (cid:15) ( M (cid:15) − i . [57]The second term can be expressed as: − N *X i lim (cid:15) →∞ | ψ ( i ) | (cid:15) − (cid:15) + = − lim (cid:15) → h (cid:15) (cid:0) N − M (cid:15) − (cid:1)i . [58]Now expanding M (cid:15) and M (cid:15) in the vicinity of q = 0 , f (cid:15) ( x ) = 1 + (cid:15) ϕ ( x ) + O ( (cid:15) ); [59] f (cid:15) ( x ) = 1 − (cid:15) ϕ ( x ) + O ( (cid:15) ) , [60]we obtain: KL2 = KL2 c ( N ) + ϕ ( L/ξ ) + ϕ ( L/ξ ) , [61] where KL2 c is logarithmically divergent, as in Eq. (33):KL2 c = ln N (1 − ∂ (cid:15) D (cid:15) | (cid:15) =0 − D ) + const . = ln N ( α − D ) + const . [62]Here we used the identity α = dτ (cid:15) d(cid:15) | (cid:15) =0 = ∂ (cid:15) [ D (cid:15) ( (cid:15) − | (cid:15) =0 . [63]for α describing the typical value of the wave function amplitude: | ψ ( i ) | typ = N − α [64]Note that, generally speaking, the characteristic lengths ξ ∼| γ − γ c | − ν (0) and ξ ∼ | γ − γ c | − ν (1) in φ and φ may have differentcritical exponents ν (0) and ν (1) . If this is the case, the smallest onewill dominate the finite-size corrections near the critical point:KL2 − KL2 c ( N ) = Φ ( L | γ − γ c | ν ) , ν = min { ν (0) , ν (1) } . [65]Eq. (65) is employed in this paper for the numerical charac-terization of the phases by finite size scaling (FSS). One can seefrom Eq. (62) that KL2 is logarithmically divergent in the mul-tifractal phase, as α > D < ϕ ( x ) and ϕ ( x ) tend to a finite N -independent limit. It is alsologarithmically divergent in the localized phase, as in Eq. (32),where one can formally set D = 0 in Eq. (62):KL2 c = α ln N. [66]However, in the ergodic phase the logarithmic divergence of KL2is gone, since in this case α = D = 1 in Eq. (62). One can easilyshow using the Porter-Thomas distribution: P PT ( x = N | ψ ( i ) | ) = e − x/ √ π x [67]that KL2 = 2 in the fully-ergodic phase.At the continuous ergodic transition, where the correlation length ξ = ∞ and α = D = 1, the critical value KL2 c ( N ) of KL2 isindependent of N . This results in crossing at γ = γ ET of all thecurves for KL2 at different values of N which helps to identify the ergodic transition (34).However, if the ergodic transition coincides with the Andersonlocalization transition and is discontinuous , (i.e. α and D arenot equal to 1 at the transition), the critical value KL2 c ( N ) is nolonger N -independent. In this case the crossing is smeared out andcan disappear whatsoever. Nonetheless, by subtracting KL2 c fromKL2 one can still locate the transition point from the best collapseof KL2 vs. γ curves by choosing an optimal γ c and ν in Eq. (65).However, it is safer to use KL1 in this case.The derivation of finite size scaling (FSS) for KL1 proceeds inthe same way by plugging the identity Eq. (56) into:KL1 = *X i | ψ α ( i ) | ln | ψ α ( i ) | + − *X i | ψ α | ln | ψ α +1 ( i ) | + . [68] | Khaymovich et al. ν = γ c = N = - p = KL1 / KL1c x = ( γ - γ c ) ( lnN ) / ν KL1c lnN - - -
10 0 10 20 300.00.51.01.52.02.5 (a)
KL2 - KL2c
KL2c lnN p = x = ( γ - γ c ) ( lnN ) / ν ν = γ c = N = - - - - (b) KL2c lnN
KL2 - KL2c p = x = ( γ - γ c ) ( lnN ) / ν ν = γ c = N = - -
50 0 50 - - (c) Fig. 9. (Color online)
The best collapse of the KL and KL data for LN-RP with p = 1 and p = 0 . . The collapse for KL and KL is done in the vicinity of thelocalization (for KL ) and ergodic (for KL ) transitions by recursive procedure that finds γ c and ν by minimizing the mean square deviation of data from a smooth scalingfunction which is updated at any step of the procedure. (insets) The critical value of KL and KL as a function of ln N . It stays almost a constant for KL and for KL at p = 0 . when the ergodic transition is continuous and well separated from the localized one but it grows logarithmically in N at p = 1 when the ergodic and localizationtransitions merge together. This growth is the reason of smearing of the intersection of KL curves in Fig. 4. The exponent ν significantly depends on p and is consistent with ν ≈ ν = 0 . at p = 1 and ν = 1 at p = 0 . . and employing the ansatz: *X i | ψ E ( i ) | q | ψ E + ω ( i ) | q + ∼ N β N αω × F q ,q ( L/ξ q , L/ξ q ) , [69]where N ω = 1 / ( ρω ) and ρ is the mean DoS.Applying for large ω ∼ ρ − ( N ω ’
1) the “decoupling rule”: *X i | ψ E ( i ) | q | ψ E + ω ( i ) | q + ∼ X i (cid:10) | ψ E ( i ) | q (cid:11) (cid:10) | ψ E + ω ( i ) | q (cid:11) , [70]and for small ω ∼ δ ( N ω ’ N ) the “fusion rule”: *X i | ψ E ( i ) | q | ψ E + ω ( i ) | q + ∼ *X i | ψ E ( i ) | q +2 q + , [71]one easily finds: β = − D q (1 − q ) + D q (1 − q ) , [72] α + β = − D q + q (1 − q − q ) . Due to the “fusion rule” for ψ α and ψ α +1 we obtain from Eq. (52): *X i | ψ α ( i ) | q | ψ α +1 ( i ) | q + ∼ F q ,q ( L/ξ q + q ) × N − D q q ( q + q − . [73]Substituting Eq. (73) in Eq. (56), Eq. (68) we observe cancelationof the leading logarithmic in N terms in KL1 in the multifractalphase: KL1 c = const . [74]We obtain: KL1 = Φ ( L | γ − γ c | ν ) . [75]where ν = ν (1) ≥ ν and the crossover scaling function Φ ( x ) is:Φ ( x ) = ∂ (cid:15) f (cid:15) ( x ) − ∂ (cid:15) f ,(cid:15) ( x ) | (cid:15) =0 . [76]As it is seen from Eq. (75), KL1 is independent of N at the Andersontransition point γ = γ AT . Thus all curves for KL1 at different valuesof N intersect at γ = γ AT . This gives us a powerful instrument toidentify the Anderson localization transition point.Note that the coefficient in front of ln N in KL2 may help todetect discontinuity of the Anderson transition. Indeed, one can usethe Mirlin-Fyodorov symmetry of fractal dimensions to establishthe relation, see Eq. (33): α = 2 − D , ⇒ α − D = 2(1 − D ) . [77]This tells us immediately that for continuous Anderson transitionwhich is characterized by vanishing D both just below and justabove the transition, the coefficient in front of ln N in KL2 is equal to 2. In particular, we conclude that α on the localized side of thetransition is equal to 2. It appears that in LN-RP this value α = 2 , ( γ = γ AT + 0) . [78]in the localized phase just above the transition remains equal to2 also in the case where the transition is discontinuous . This isin contrast to the corresponding coefficient 2(1 − D ) in front ofln N in KL2 just below the transition which is smaller than 2 if thetransition is discontinuous. Such a jump in the coefficient in frontof ln N in KL2 is a signature of the discontinuity of the transitionwhich is the most easily detectable numerically, see Fig. 5. D. Finite-size scaling collapse for KL and KL . The next step is to analyze the finite-size scaling (FSS) by a collapseof the data for KL1 and KL2 at different N in the vicinity of thelocalization and ergodic transition, respectively. To this end we usethe form of FSS derived in IS C.KL1 = Φ (ln N | γ − γ AT | ν ) , [79a]KL2 − KL2 c (N) = Φ (ln N | γ − γ ET | ν ) . [79b]The input data for the collapse is KL1 and KL2 versus γ for 7 valuesof N is shown in Fig. 4. The fitting parameters extracted from thebest collapse are ν ( ν ) and the critical points γ AT ( γ ET ). Thecritical value of KL2 c ( N ) = KL2( γ ET , N ) is determined by the bestfitting for γ ET . For the localization transition where the criticalpoint γ AT is well defined by the intersection in KL1, one may lookfor the best collapse by fitting only ν .The plots of Fig. 9 demonstrate the quality of the collapse forseveral representative cases. In the insets of the figures we showthe ln N - dependence of the critical values of KL1, KL2 whichwere obtained numerically from KL1( γ = γ AT , N ) and KL2( γ = γ ET , N ), respectively, with γ AT and γ ET found from the bestcollapse. It is demonstrated that the critical value of KL1 is almost N -independent, as well as the critical value of KL2 at p = 1 / p = 1 when ET and ATmerge together the critical value of KL2 increases linearly withln N , signaling of the critical multifractal state at the Andersontransition point, very similar to the case of 3D Anderson transition.This ln N -dependence of KL2 c is the reason of smearing out of theintersection point in KL2 shown in Fig. 4. E. Ratio of typical and mean LDOS
In this section we consider in more details the technical issue withthe determination of the order parameter for the FWE transition φ ( η ) = 1 − ρ typ ρ av , [80]being the ratio of the typical, ρ typ , and the mean, ρ av , LDOS givenby the expressionsln ρ typ = h ln ρ ( x, E + iη ) i , ρ av = h ρ ( x, E + iη ) i , [81] Khaymovich et al.
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July 21, 2020 | vol. XXX | no. XX || vol. XXX | no. XX | -7 -6 -5 -4 -3 -2 -1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 I m G t y p ImG typ
ImG av -7 -6 -5 -4 -3 -2 -1 η d l n I m G t y p / d l n η γ = 1.5, p = 0.01 N=512N=1024N=2048N=4096N=8192N=16384 -7 -6 -5 -4 -3 -2 -1 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 I m G t y p ImG typ
ImG av -7 -6 -5 -4 -3 -2 -1 η d l n I m G t y p / d l n η γ = 2.0, p = 1.0 N=512N=1024N=2048N=4096N=8192N=16384 -7 -6 -5 -4 -3 -2 -1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 I m G t y p ImG typ
ImG av -7 -6 -5 -4 -3 -2 -1 η d l n I m G t y p / d l n η γ = 3.2, p = 1.0 N=512N=1024N=2048N=4096N=8192N=16384 -7 -6 -5 -4 -3 -2 -1 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 I m G t y p ImG typ
ImG av -7 -6 -5 -4 -3 -2 -1 η d l n I m G t y p / d l n η γ = 3.8, p = 1.0 N=512N=1024N=2048N=4096N=8192N=16384
Fig. 10. [upper panels] Plots of the typical ρ typ (solid lines) and mean ρ av (dashed lines) LDOS and [lower panels] of the second derivative of ρ typ /ρ av w.r.t. η forthe log-normal RP model at (left panel) p = 0 . in the multifractal phase γ ET < γ = 1 . < γ AT , (other panels) p = 1 . in the weakly ergodic phase γ FWE < γ = 2 . , . , . < γ ET = γ AT . The positions of the maxima of the second derivative are shown by crosses of the corresponding color for all system sizes in the range from N = 512 (dark blue) to N = 16384 (red). Notice a plateau developing in ρ typ for intermediate values of η with increasing the system size. The maximum of the secondderivative is always inside the plateau region or on its right end. The plateau gradually shrinks with increasing γ and disappears in the localized phase. with the LDOS before averaging written as ρ ( x, E + iη ) = X µ | ψ µ ( x ) | η/π ( E − E µ ) + η . [82]The averaging in Eq. (81) is taken over the disorder realizations,over all coordinates x (which are statistically equivalent in LN-RP)and over 100 energy values in the middle half of the spectrum.As mentioned in the main text the ratio ρ typ /ρ av develops theplateau ∼ N − D in some range of bare level width parameter η (cid:29) δ large compared to the typical level spacing δ . However, atany finite sizes this plateau has a finite slope, especially for the WEphase where N D = fN with a N -independent constant f < N -independent φ ( η (cid:29) δ ) ∝ − f = O (1) [83]which is zero in the FE phase, f = 1, and finite in the WE one, f < p = / - ρ typ / ρ av γ ( a ) - ρ typ / ρ av p = / γ ( b ) - ρ typ / ρ av γ p = ( c ) - ρ typ / ρ av γ p = ( d ) Fig. 11. Plots for φ = 1 − ρ typ /ρ av as a function of γ at p = 0 . , , , for N = 512 , , , , (from blue to red) and extrapolated to N = ∞ (black). The gray dashed lines represent cubic polynomial fits to the pointsof extrapolation away from the transition. The intersection of each of these lines with φ = 0 gives the numerical estimate of γ FWE (shown by bright blue point) which iscompared in Table 2 with the predicted by
Eq. (13) in the main text.
In order to find the FWE transition accurately we develop theprocedure of the automatic selection of η in the middle of theunderdeveloped plateau. For this purpose we take the secondderivative of the ratio ρ typ /ρ av w.r.t. η after the smoothening it with the 5-degree spline and find the point of maximum of thisderivative lying in between of two local minima (see the lower panelsin Fig. 10). Figure 10 shows several examples for p = 0 .
01 and p = 1 where the positions of the maxima of the second derivativeare shown by crosses of the corresponding color for all system sizes N . F. Location of FWE transition
The behavior of the order parameter Eq. (80) helps to locate theFWE transition. In Fig. 11 we present the plots for φ = 1 − ρ typ /ρ av for different values of p as a function of γ calculated numerically byexact diagonalization for several values of N and then extrapolatedto N = ∞ . p .
50 0 .
75 1 .
00 2 .
00 3 . γ theorFWE .
66 0 .
57 0 . .
33 0 . γ numFWE . ± .
04 0 . ± .
04 0 . ± .
04 0 . ± .
07 0 . ± . Table 2. Comparison of analytical predictions (red),
Eq. (13) , andnumerical data (blue) for the FWE transition points. Numerical datais obtained by exact diagonalization of LN-RP random matrices with N = 512 − followed by extrapolation to N → ∞ of the orderparameter, Eq. (80) , given by the ratio of the typical ρ typ and themean ρ av local DOS. KL2 γ p = / ( a ) KL2 γ p = ( b ) Fig. 12. (Color online)
Plots for
KL2 as a function of γ at p = 0 . (a) and (b) for N = 512 , , , , (from blue to red) and extrapolated to N = ∞ (black). The gray dashed lines represent cubic polynomial fits to the pointsof extrapolation away from the transition. The intersection of each of these lines withthe RMT value KL2 = 2 gives the numerical estimate of γ FWE (shown by brightblue point).
While this extrapolation is reliable away from the transition, itis not able to give the true singularity at the transition which would | Khaymovich et al. .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 γ β = l n Γ / l n N p = γ σ E = › ( E − › E fi ) fi / σ ,E = › | E − › E fi | fi Nδ = N › E n +1 − E n fi Nδ typ = N exp › ln | E n +1 − E n | fi N [1 − (1 − p ) γ ] / N p pγ − γ/ p = 1 .0 γ p = Fig. 13. (Color online)
Scaling of the spectral bandwidth E BW with N in different regions of p : (a) p = 1 / ; (b) p = 1 ; (c) p = 1 . extracted numerically from thefitting to E BW = cN β of the eigenvalue standard deviation σ E = h ( E − h E i ) i / , the averaged absolute deviation of E from its mean σ ,E = h| E − h E i|i , themean δ = h E n +1 − E n i and typical δ typ = exp h ln ( E n +1 − E n ) i global level spacings multiplied by N for the system sizes N = 512 − . All measures arecalculated over the % of the states in the middle of the spectrum. The dashed lines represent the analytical prediction, Eq.(91), in the ergodic phases, γ < γ ET , while inthe non-ergodic ones E BW = W ∼ N . require an extrapolation from much larger matrix sizes. There-fore for numerical location of the transition γ we used the cubicpolynomial fit to the points of extrapolation sufficiently remotefrom the transition (represented by gray dashed lines in Fig.11).Intersection of these lines with the dashed line φ = 0 gives thenumerical estimate of γ FWE . Almost the same values of γ FWE canbe obtained by studying KL2 statistics. Some of the plots for KL2vs γ are presented in Fig.12.The results of this analysis are summarized and compared withthe prediction of Eq.(13) in Table 2. One can see that Eq. (13) iswell reproduced by our numerics. G. Mean and typical Breit-Wigner width of the mini-band
According to the definition Eq. (7) and the Fermi Golden rule theBreit-Wigner width is given by the following sumΓ n ∼ ρ av N X m =1 | Hmn |
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