Transition between localized and extended states in the hierarchical Anderson model
aa r X i v : . [ c ond - m a t . d i s - nn ] A p r Transition between localized and extended states in the hierarchical Anderson model
F. L. Metz , L. Leuzzi , , G. Parisi , , , V. Sacksteder IV Dip. Fisica, Universit`a
La Sapienza , Piazzale A. Moro 2, I-00185, Rome, Italy IPCF-CNR, UOS Roma
Kerberos , Universit`a
La Sapienza , P. le A. Moro 2, I-00185, Rome, Italy INFN, Piazzale A. Moro 2, 00185, Rome, Italy Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China (Dated: February 26, 2018)We present strong numerical evidence for the existence of a localization-delocalization transitionin the eigenstates of the 1-D Anderson model with long-range hierarchical hopping. Hierarchicalmodels are important because of the well-known mapping between their phases and those of modelswith short range hopping in higher dimensions, and also because the renormalization group canbe applied exactly without the approximations that generally are required in other models. Inthe hierarchical Anderson model we find a finite critical disorder strength W c where the averageinverse participation ratio goes to zero; at small disorder W < W c the model lies in a delocalizedphase. This result is based on numerical calculation of the inverse participation ratio in the infinitevolume limit using an exact renormalization group approach facilitated by the model’s hierarchicalstructure. Our results are consistent with the presence of an Anderson transition in short-rangemodels with D >
I. INTRODUCTION
After more than fifty years the Anderson transition between localized and extended wave-functions of a sin-gle quantum particle moving in a disordered mediumremains the focus of considerable interest. Crucialcontributions to this field have been made by exactlysolvable tight-binding models, such as 1-D models withnearest-neighbour hopping and models on the Bethelattice.
Here we consider another interesting class oftight-binding models with long-range hopping arrangedin a hierarchical block structure and decaying accordingto a power law with exponent α . Hierarchical modelshave a long history in statistical physics starting withDyson , and (as we will explain later) they provide anindirect route to understanding phases and critical be-haviour in D -dimensional systems. We study the hierarchical Anderson model (HAM) in-troduced by Bovier, which combines on-site disorder withhierarchically-structured long range hopping. In the ab-sence of disorder, the spectrum is an infinite set of highlydegenerate flat bands that accumulate at the upper spec-tral edge. The degeneracies are arranged in a geometricseries: one half of the pure HAM’s states lie in the lowestenergy band, one quarter in the next highest energy, etc.Hierarchical models preserve their structure under renor-malization group transformations, which has allowedproof of several rigorous results about the site disorderedHAM’s spectrum, and may promise exact exten-sions of the successful scaling theory of localization. Inparticular, the absolutely continuous part of the spec-trum vanishes and the model presents only spectral lo-calization, provided that the hopping decays sufficientlyquickly with distance, i.e., the hopping decay exponent α > / Unfortunately, much less is known about the size of theHAM’s eigenvectors. The degeneracies of the pure modelpermit different choices of mutually orthogonal sets ofeigenvectors. The most extended set consists of infinitelyextended plane waves, while the least extended set hassizes that are strongly band-dependent, with very local-ized states in the lowest band and infinitely extendedstates in the highest band. In the presence of on-sitedisorder, it recently has been argued that all states arealways localized, based on an analogy with the criti-cality results for random-matrix models, such as ensem-bles of ultrametric and power-law random bandedmatrices. Both models, characterized by an exponent α controlling the power-law decaying random hoppings,exhibit an extended phase for α < α >
1. This would rule out the possibility of a tran-sition in the HAM, which has a well-defined macroscopiclimit only for α >
1. However models with random hop-ping are relatively simple: the scattering length vanishesand only the localization length is important. The HAMbelongs instead to the class of models with deterministic hopping, which are much richer because they have non-trivial physics at both length scales. In particular, the1-D Anderson model with on-site disorder and determin-istic power-law hopping exhibits a localization transitionat its upper spectral edge.
In this work we show that the HAM exhibits alocalization-delocalization transition near its upper spec-tral edge. We perform a thorough numerical study ofthe inverse participation ratio, which is the inverse of theeigenstate volume. Thanks to the HAM’s invariance un-der block renormalization group (RG) transformations,we obtain recurrence equations for calculating the resol-vent matrix. This recursive method allows us to calculatethe IPR in systems large enough to precisely determine t t t t t t t t ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ t t t t t t t p = 1 p = 2 p = 3 FIG. 1: Schematic representation of the hierarchical Andersonmodel, cf. Eq. (1), with L = 2 sites. Lines denote hoppingenergies t p between sites in distinct blocks of size 2 p − . their infinite size behavior. Our results also suggest thatthere is a critical value of α above which all states arelocalized, in analogy with the lower critical dimension D = 2 below which finite-dimensional short-range sys-tems are always localized and above which an Andersontransition was predicted using RG arguments. II. THE HIERARCHICAL ANDERSON MODEL.
The HAM is a 1-D tight binding model with L = 2 N equally spaced sites, and independently distributed ran-dom site potentials ǫ i , i = 1 , . . . , L . The Hamiltonianreads H N = N X i =1 ǫ i | i ih i | (1)+ N X p =1 V p N − p X r =1 1 , p X i = j | ( r − p + i ih ( r − p + j | , where | i i is the canonical site basis. The second line isthe hierarchical hopping matrix introduced by Dyson. It is the heart of the hierarchical Anderson model, andis organized in a tree as illustrated in Fig. 1. The high-est level of the tree has index p = N and the lowestlevel has index p = 1. At each level the system is di-vided into 2 N − p separate blocks, each of which contains2 p sites. The hopping between any two sites within a sin-gle block has energy V p . As seen in Fig. 1, the hoppingbetween sites in two different blocks is determined by lev-els higher in the hierarchy and has energy t p = P Nn = p V n .We study the deterministic HAM, which has hopping en-ergies V p = 2 − α ( p − . This exponential decay in the levelindex p ensures that in large N ≫ O ( L )decays according to a power law t p ∝ O ( L − α ), the sameas 1-D Anderson models with power-law hopping. We study the infinite volume limit of the average den-sity of states (DOS) ρ ( E ) and of the inverse participation ratio P ( E ). The former is defined as ρ ( E ) = lim L →∞ D L L X µ =1 δ ( E − E µ ) E , (2)where h . . . i is the average with respect to the disorder po-tential ǫ i and E µ are the HAM’s eigenvalues. The DOSmeasures the averaged spectrum, but does not containany signal of the eigenstates’ localization or delocaliza-tion. We therefore study the average inverse participa-tion ratio (IPR) of the normalized eigenstates | ψ µ i : P ( E ) = lim L →∞ Lρ ( E ) D L X µ =1 I Lµ δ ( E − E µ ) E , (3)where I Lµ = P Li =1 ( h i | ψ µ i ) is the IPR of an individualeigenstate. Its inverse measures the eigenstate’s volume.The IPR is restricted to the interval 0 ≤ P ( E ) ≤ P ( E ) = 1, and states that are equally distributed acrossall sites satisfy P ( E ) = 1 /L → W = 0 HAM the DOS is a series of flatbands ρ pure ( E ) = P ∞ p =1 − p δ ( E − E pure p − ). Each flat bandis related to a level in the HAM’s hierarchy. The bands’degeneracy decreases repeatedly by factors of two as onemoves to higher energy, thus yielding the factor 2 − p . Thedifference between consecutive energetic levels falls off as E pure p +1 − E pure p ∝ − ( α − p and, hence, these accumulate atthe upper spectral edge E pure ∞ . Near E pure ∞ the integrateddensity of states N ( E pure p ) = P pℓ =1 − ℓ follows a powerlaw similar to that of short-range finite-dimensional sys-tems: N ( E pure p ) = 1 − C (cid:0) E pure ∞ − E pure p (cid:1) d s / . Here d s = 2 / ( α −
1) is the spectral dimension whichcontrols both diffusion and the long-distance physicsof second-order phase transitions such as the Andersontransition. For
W >
0, the integrated DOS of the HAMexhibits a Lifshitz tail at the upper spectral edge, with aLifshitz exponent given by the spectral dimension. Thisis the same behavior as observed in short-range finite-dimensional systems with on-site disorder , where the in-tegrated DOS exhibits a Lifshitz tail controlled by theEuclidean dimension. Overall, as E pure ∞ is approached,the spectral properties of the pure HAM become sim-ilar to short-range finite-dimensional systems. This is,thus, the most promising region for studying localizationtransitions. Much of the interest in hierarchical modelsoriginates in their mapping to short-range models whoseEuclidean dimension is strictly related to the spectral di-mension (see, e.g., Ref. [30] and Refs. therein). III. RENORMALIZATION EQUATIONS FORTHE RESOLVENT
We obtain the DOS and IPR from the diagonal ele-ments of the resolvent matrix G ( N ) ( z ) = ( z − H N ) − ,where z = E − iη , and η is a small positive regularizerthat smooths our numerical results over an interval inthe spectrum with width proportional to η . We usethe following formulas: ρ ( E ) = lim η → + lim L →∞ Lπ L X i =1 D Im G ( N ) i ( z ) E , (4) P ( E ) = lim η → + lim L →∞ ηπLρ ( E ) L X i =1 D | G ( N ) i ( z ) | E . (5)The HAM’s hierarchical structure allowed us to developa block RG approach which recursively calculates the re-solvent for one instance of the disorder. Our calculationhas two phases: a sweep up the hierarchy, and then asweep back down. At each step ℓ of the sweep up weremove the basis states associated with one flat band,and calculate an energy-dependent effective Hamiltonianwhich acts in the reduced basis but exhibits the samepoles found in the original full-basis Hamiltonian. Thiseffective Hamiltonian retains the hierarchical form butits hopping energies { V ( ℓ ) p } and disorder potentials { µ ( ℓ ) i } are renormalised according to µ ( ℓ ) i = 2 µ ( ℓ − i − µ ( ℓ − i µ ( ℓ − i − + µ ( ℓ − i + 2 V ( ℓ − , i = 1 , .., N − ℓ (6) V ( ℓ ) p = 2 V ( ℓ − p +1 , p = 1 , . . . , N − ℓ . (7)Hopping energies and disorder potentials at the begin-ning of the sweep up, V (0) p = V p and µ (0) i = ǫ i − z − P Np =1 V p , are those of the original hierarchical Hamilto-nian. After ℓ = N steps we reach the top of the hierarchyand obtain a single site effective Hamiltonian with disor-der potential µ ( N )1 . The resolvent of this Hamiltonian issimply G ( z ) = − /µ ( N )1 . We use this resolvent to beginthe sweep back down, in which we progressively restorethe original basis and recursively calculate the resolvent’sdiagonal elements in the restored basis: G ( N − ℓ +1)2 i − ( z ) = 2 " µ ( ℓ − i γ ( ℓ − i G ( N − ℓ ) i ( z ) − γ ( ℓ − i , (8) G ( N − ℓ +1)2 i ( z ) = 2 " µ ( ℓ − i − γ ( ℓ − i G ( N − ℓ ) i ( z ) − γ ( ℓ − i , (9)with γ ( ℓ − i = µ ( ℓ − i − + µ ( ℓ − i . This procedure yields thediagonal elements of the resolvent in the original system,and its memory consumption and computational timegrow only linearly with L . The derivation of Eqs. (6-9)is presented in App. A.
IV. RESULTS
Fig. 2 compares the DOS and IPR calculated withour renormalization method (solid lines) and η = 0 . −2 −1 −2 0 2 4 EP(E) −3 −2 −1 r (E) W=1.8W=1.4W=1.0W=0.6 −1 E E E E E ..... FIG. 2: Comparison of the HAM’s average DOS and IPR:numerical diagonalization (filled circles) vs. RG method (solidlines), with hopping decay exponent α = 7 /
4. The energiesof the pure model’s flat bands are marked with cyan verticallines in the upper pane. The inset shows that when the IPRis small the RG method is sensitive to the spectral line width η ; the dashed and solid lines were obtained with η = 0 .
01 and η = 0 .
005 respectively. a system of size L = 2 . The potential ǫ i is generatedfrom a Gaussian distribution with zero mean and stan-dard deviation W . Diagonalization results are averagedover N ǫ = 10 disorder realizations and renormalisationresults over N ǫ = 2 × realizations. Fig. 2 showsexcellent agreement between the two methods.The only important discrepancy is found in the IPRat small disorder W = 0 .
6, where the DOS falls precipi-tously. The observed discrepancy is explained by Fig. 2’sinset, which compares results with two values of the regu-larization parameter: η = 0 .
01 and 0 . η → + is reachedonly at η ≪ . α = 7 / W = 0 . , . , . , .
8. At small disor- −3 −2 −1 EP(E)
W=2.0W=1.8W=1.6W=1.4W=1.2W=1.0W=0.810 −4 −3 −2 −1 r (E) FIG. 3: Average DOS and IPR near the upper spectral edgein a large L = 2 system at several disorder strengths and α = 3 / η = 5 × − and N ǫ = 30. der W = 0 . P ( E ) show that theeigenstates are bigger in the band centers and smaller atthe band edges. When the disorder is increased the bandsprogressively blur together and P ( E ) steadily increasesas the eigenstates become ever more localized. Fig. 3shows the same behavior at α = 3 / L = 2 . Reaching such large sizes allows us to exploresmaller η values and obtain detailed results about manybands near the upper spectral edge. Indeed, in orderto obtain statistically significant results the spectral linewidth η must considerably exceed the mean level spacing[ N ρ ( E )] − .In general the IPR exhibits several local minima cor-responding to large states near the centers of the HAM’sbands, and the global minimum lies near HAM’s upperspectral edge. In order to verify the existence of extendedstates at finite W we focus on the asymptotic value ofthe global minimum of the IPR, P min ( W ), in the L → ∞ limit and for infinitesimal η → + . The main graphin Fig. 4 summarizes our calculation of P min ( W ) for aparticular hopping decay α = 3 / W = 0 . L = 2 system atthree different values of η . Statistical errors at smaller η are larger because of η ’s proximity to the level spacing.App. B includes a detailed discussion of these errors inthe limit η → + . In particular, we have checked that for L ≥ the IPR curves at fixed η do not change with L ,which signals that they accurately represent the infinitevolume limit.The IPR depends on η , and as η → + the globalminimum deepens and shifts toward higher energy. Thiseffect is not significant at larger disorder W > .
0, but at −4 −3 −2 −1 EP(E) h=10 -3 h=10 -4 h=10 -5 −5 −4 −3 h −4 −3 −2 P( h ) W=1.0 , E=5.480W=0.95 , E=5.465W=0.9 , E=5.445W=0.8 , E=5.405
FIG. 4: The η → + limit. The three curves show the averageIPR at three values of η and α = 3 / W = 0 . L = 2 , and N ǫ = 10. The global minimum decreases and shifts to higherenergy. The inset shows the IPR versus η at the energy ofthe global minimum. Again α = 3 /
2, but now L = 2 and N ǫ = 100. The solid lines are linear fits to P ( η ) = P min + bη . α W c π A χ / ndf ndf3 / . .
96 67 / . . . .
21 5TABLE I: Values of the parameters and the χ of the powerlaw fit to the IPR data shown in Fig. 5. W c is the criticaldisorder where the delocalization transition occurs, and π isa critical exponent. smaller disorder it forces us to use considerable care withthe η → + extrapolation. The inset in Fig. 4 displaysour extrapolation to the limit η → + at four weak dis-order strengths. At each W we find the energy E min ( W )of the local minimum at the lowest value of the spectralwidth parameter employed, η = 10 − , and then graphthe IPR at that energy as a function of η . Concerninguncertainty in E min ( W ), we have checked that it affectsour results only slightly, and in any case can only causean unduly careful overestimate of the IPR. The fittingcurves in Fig. 4 show that the IPR depends linearly on η via P ( η ) = P min + bη . This allows us to determine veryaccurately the asymptotic global minimum of the IPR.The straight lines in Fig. 5’s log-log plot are the cen-tral result of our work: strong numerical evidence thatthe minimum IPR P min ( W ) converges to zero accordingto a power law P min ( W ) = A ( W − W c ) π , similar tothe power law observed in finite dimensional short-rangesystems. At smaller disorder W ≤ W c the HAMexhibits a delocalized phase. Table I reports the bestfit parameters for two values of the hopping decay expo-nent α = 3 / , /
4. In both cases our data excludes thepossibility that W c = 0. −5 −4 −3 −2 −1 −1 W−W c P min (W) a=3/2a=7/4 FIG. 5: Power law behaviour of the minimum IPR P min ( W )near the critical disorder strength W c where it convergesto zero. Solid lines represent the power-law fit P min ( W ) = A ( W − W c ) π , with parameters from table I. V. CONCLUSIONS.
We have analyzed the DOS and the IPR of the hierar-chical Anderson model by means of a RG-based calcula-tion of the resolvent matrix, finding strong evidence fora localization-delocalization transition at finite disorderat α = 3 / /
4. Since it has been proven rigor-ously that the absolutely continuous part of the spec-trum vanishes for α > / our results indicate thatspectral localization may not imply the existence of ex-ponentially localized eigenvectors. A study of the spatialdecay of the resolvent elements should clarify this pointand we expect our work to stimulate further research inthis direction. Our results also indicate that the HAMdiffers from the 1-D tight-binding model with power-lawhopping, where all states are localized for α ≥ / Since the HAM’s spectral dimension can be mapped tothe spatial dimension of Anderson models with short-range hopping, we expect that an Anderson transitionexists in the regime 1 < α <
2, with α ≃ Lastly we mention that ourRG method can be used to compute off-diagonal elementsof the resolvent, allowing determination of other relevantquantities such as the longest localization length.
Acknowledgments
FLM is greatly indebted to Lucas Nicolao, Jacopo Roc-chi, Pierfrancesco Urbani and Izaak Neri for many in-teresting and useful discussions. VS thanks Tomi Oht-suki and Koji Kobayashi for discussions and hospitality.The research leading to these results has received fund-ing from the European Research Council (ERC) grant agreement No. 247328 (CriPheRaSy project), from thePeople Programme (Marie Curie Actions) of the Euro-pean Union’s Seventh Framework Programme FP7/2007-2013/ under REA grant agreement No. 290038 (NE-TADIS project) and from the Italian MIUR under theBasic Research Investigation Fund FIRB2008 program,grant No. RBFR08M3P4, and under the PRIN2010 pro-gram, grant code 2010HXAW77-008.
Appendix A: Derivation of the renormalizationequations
We discuss here how to derive the RG equations (6-9)that are used in the main text. This calculation can beperformed using only linear algebra, but we find it moreconvenient to use Gaussian integrals. We first rewrite { G ( N ) k ( z ) } k =1 ,...,L , the diagonal elements of the resolvent G ( N ) ( z ) = ( z − H N ) − , as the Gaussian integrals G ( N ) k ( z ) = ı R d φ φ k exp (cid:2) L ( N ) ( φ ,..., N ) (cid:3)R d φ exp (cid:2) L ( N ) ( φ ,..., N ) (cid:3) , (A1) L ( N ) ( φ ,..., N ) = ı N X j =1 µ j φ j + W ( N ) (cid:0) φ ,..., N ; V ,...,N (cid:1) , where d φ = Q N j =1 dφ j and µ j = ǫ j − z − P Np =1 V p . Thefunction W ( N ) encodes the hierarchical hoppings: W ( N ) (cid:0) φ ,..., N ; V ,...,N (cid:1) = ı N X p =1 V p N − p X r =1 p X j =1 φ ( r − p + j . We have introduced the simplified notation x ,..., A ≡ x , . . . , x A . The function L ( N ) ( φ ,..., N ) has the sameform as the HAM’s Hamiltonian and therefore preservesits formal structure under a RG transformation: a localterm incorporating the random potential and a non-localhierarchical hopping term. We make a change of integra-tion variables ψ ± j = 1 √ φ j − ± φ j ) , j = 1 , . . . , N − , which transforms the hierarchical term as follows: W ( N ) (cid:0) φ ,..., N ; V ,...,N (cid:1) = ıV N − X j =1 ( ψ + j ) + W ( N − (cid:16) ψ +1 ,..., N − ; V ′ ,...,N − (cid:17) , where V ′ p = 2 V p . This transformation allows us to explic-itly calculate the integrals over { ψ − j } j =1 ,..., N − in Eq.(A1), halving the number of degrees of freedom. Afterperforming the transformation and integration we obtainan equation which relates { G ( N ) i ( z ) } i =1 ,...,L for the orig-inal model with L sites to { G ( N − i ( z ) } i =1 ,...,L/ for amodel with L/ { G ( N ) i ( z ) } i =1 ,..., N into two sectors (onefor the even sites and another for the odd sites), we obtainthe following expressions G ( N )2 k − ( z ) = ı R d ψ + d ψ − ( ψ + k + ψ − k ) e H ( N − ( ψ ± ) R d ψ + d ψ − e H ( N − ( ψ ± ) , (A2) G ( N )2 k ( z ) = ı R d ψ + d ψ − ( ψ + k − ψ − k ) e H ( N − ( ψ ± ) R d ψ + d ψ − e H ( N − ( ψ ± ) . (A3)In the above expressions we have changed integrationvariables to d ψ ± = Q N − j =1 dψ ± j and we have defined H ( N − (cid:0) ψ ± (cid:1) = ı N − X j =1 σ j ( ψ + j ) + ı N − X j =1 ∆ j ( ψ − j ) + ı N − X j =1 C j ψ + j ψ − j + W ( N − (cid:16) ψ +1 ,.., N − ; V ′ ,..,N − (cid:17) , where the following quantities are complex valued: σ j = 12 ( µ j − + µ j ) + 2 V , ∆ j = 12 ( µ j − + µ j ) ,C j = 12 ( µ j − − µ j ) . Since Eqs. (A2) and (A3) involve only simple Gaussianintegrals with respect to ψ − ,..., N − , these variables can beintegrated out one by one. We map the resulting expres-sion to Eq. (A1) for a system with 2 N − sites, renormal-ized disorder µ ′ ,..., N − which obeys Eq. (8) in the maintext, and renormalised hopping potential V ′ ,...,N − . Weobtain Eqs. (6-9) at the first RG step ℓ = 1 of the orig-inal model. Performing these steps recursively leads tothe recurrence equations (6) and (7) shown in the maintext: G ( N − ℓ +1)2 i − ( z ) = 2 " µ ( ℓ − i γ ( ℓ − i G ( N − ℓ ) i ( z ) − γ ( ℓ − i ,G ( N − ℓ +1)2 i ( z ) = 2 " µ ( ℓ − i − γ ( ℓ − i G ( N − ℓ ) i ( z ) − γ ( ℓ − i , where γ ( ℓ ) i = µ ( ℓ )2 i − + µ ( ℓ )2 i . Appendix B: Performing the η → + limitnumerically The regularization parameter η in the resolvent giveseach eigenvalue a line width proportional to η . This canbe understood by analysing our equation for the DOS ρ ( E ) = lim η → + lim L →∞ Lπ L X i =1 D Im G ( N ) i ( z ) E . (B1) −3 −2 EP(E)
L=2 L=2 L=2 L=2 L=2 FIG. 6: Size effects on the IPR at α = 1 . W = 0 . η = 10 − and N ǫ = 10. −3 −2 −1 −7 −6 −5 −4 −3 h a=3/2a=7/4 FIG. 7: η dependence of the ratios η/ ∆ L, N ,η ( E ) (open cir-cles) and σ L, N ,η ( E ) /ρ L, N ,η ( E ) (filled circles). ∆ L, N ,η ( E )is the approximate mean level spacing and σ L, N ,η ( E ) isthe standard deviation around ρ L, N ,η ( E ). Results were ob-tained using Eqs. (6-9) in the main text with N = 28 and N ǫ = 100. We set ( W = 0 . , E = 5 . α = 3 /
2, and( W = 0 . , E = 3 . α = 7 /
4. These values were used toproduce the left-most (smallest disorder) data point in Fig. 5of the main text.
The right hand side of this equation is the limit η → + of a sum of Lorentzian functions with width η andcentered at E . The Lorentzians quantify the distances of H N ’s eigenvalues from the energy E . As η approachesthe mean level spacing from above our observables willdisplay larger and larger fluctuations, since our averageswill include smaller and smaller numbers of eigenstates.If η is smaller than the level spacing then one obtainsresults which have no physical meaning. Accurate resultsfor very small η are obtained only if the system size L andthe number of samples N ǫ are large enough. Fig. 6 showshow this issue influences the IPR. We fix the number ofsamples N ǫ and spectral line width η and vary the systemsize. Convergence is obtained at L ≥ .If we define ρ L, N ,η ( E ) as the average DOS of a finitethough very large system, we can estimate the mean levelspacing ∆ L, N ,η ( E ) around E as∆ L, N ,η ( E ) ∼ L N ǫ ρ L, N ,η ( E ) . We estimate the error at small η by calculating σ L, N ,η ( E ) /ρ L, N ,η ( E ) and η/ ∆ L, N ,η , where σ L, N ,η ( E ) isthe standard deviation of ρ L, N ,η ( E ). Typical results aredisplayed in Fig. 7. When we decrease η → + we findmonotonic growth in σ L, N ,η ( E ) /ρ L, N ,η ( E ) and mono-tonic decay in η/ ∆ L, N ,η . For small enough η we reach aregime where σ L, N ,η ( E ) /ρ L, N ,η ( E ) = O (1), η/ ∆ L, N ,η = O (1), and ρ L, N ,η ( E ) exhibits large fluctuations. We con- clude that the limit η → + is achieved, for practicalpurposes, when ∆ L, N ,η ≪ η ≪
1, i.e. in very large sys-tems. Therefore we establish a sensible lower cutoff on η by imposing a maximum value of the average DOS’srelative error σ L, N ,η ( E ) /ρ L, N ,η ( E ).The results for the minimum IPR displayed in Fig. 5of the main text were obtained by choosing η = 10 − as the lower cutoff when α = 3 / η ∈ [10 − , − ]when α = 7 /
4. This ensures that the relative er-ror σ L, N ,η ( E ) /ρ L, N ,η ( E ) is restricted to the interval[10 − , − ], as can be seen in Fig. 7. P. W. Anderson, Phys. Rev. , 1492 (1958). B. Kramer and A. MacKinnon, Rep. Prog. Phys. , 1469(1993). F. Evers and A. D. Mirlin, Rev. Mod. Phys. , 1355(2008). E. N. Economou,
Green’s functions in quantum physics (Springer, Heidelberg, 2006). R. Abou-Chacra, P. W. Anderson, and D. J. Thouless, J.Phys. C: Solid St. Phys. , 1734 (1973). R. Abou-Chacra and D. J. Thouless, Journal of Physics C:Solid State Physics , 65 (1974). F. J. Dyson, Commun. Math. Phys. , 91 (1969). S. Molchanov, Proc. Lukacs Symposiu p. 179 (1996). A. Bovier, J. Stat. Phys. p. 745 (1990). G. Baker, Phys. Rev. B , 2622 (1972). Y. Meurice, J. Phys. A: Math. Theor. , R39 (2007). E. Kritchevski, Proc. Am. Math. Soc. , 1431 (2007). E. Kritchevski, ”Hierarchical Anderson model” in ”Proba-bility and mathematical physics: a volume in honor of S.Molchanov” , vol. 42 (Amer. Math. Soc., 2007). E. Kritchevski, Ann. Henri Poincare , 685 (2008). S. Kuttruf and P. M¨oller, Ann. Henri Poincare , 525(2012). E. Abrahams, P. W. Anderson, D. C. Licciardello, andT. V. Ramakrishnan, Phys. Rev. Lett. , 673 (1979), URL http://link.aps.org/doi/10.1103/PhysRevLett.42.673 . C. Monthus and T. Garel, JSTAT p. P05005 (2011). Y. V. Fyodorov, A. Ossipov, and A. Rodriguez, JSTAT p.L12001 (2009). E. Bogomolny and O. Giraud, Phys. Rev. Lett. ,044101 (2011). A. D. Mirlin, Y. V. Fyodorov, F.-M. Dittes, J. Quezada,and T. H. Seligman, Phys. Rev. E , 3221 (1996). C. Yeung and Y. Oono, Europhys. Lett. , 1061 (1987). A. Rodriguez, V. A. Malyshev, and F. Dominguez-Adame, J. Phys. A: Math. Gen. , L161 (2000). A. Rodriguez et al. , Phys. Rev. Lett. , 027404 (2003). A. V. Malyshev, V. A. Malyshev, and F. Dominguez-Adame, Phys. Rev. B , 172202 (2004). F. A. B. F. de Moura, A. V. Malyshev, M. L. Lyra, V. A.Malyshev, and F. Dominguez-Adame, Phys. Rev. B ,174203 (2005). F. Wegner, Z. Physik B , 209 (1980). F. L. Metz, I. Neri, and D. Boll´e, Phys. Rev. E , 031135(2010). F. Slanina, Eur. Phys. J. B , 361 (2012). W. Kirsch and F. Martinelli, Commun. Math. Phys. ,27 (1983). M. Ibanez Berganza and L. Leuzzi, arXiv:1211.3991v2(2012). M. Janssen, Phys. Rep. , 1 (1998), ISSN 0370-1573. Y. Song, W. A. Atkinson, and R. Wortis, Phys. Rev. B ,045105 (2007). E. N. Economou and M. H. Cohen, Phys. Rev. B , 2931(1972). Y. V. Fyodorov and A. D. Mirlin, Phys. Rev. Lett. ,2049 (1991). J. Bauer, T.-M. Chang, and J. L. Skinner, Phys. Rev. B , 8121 (1990). T.-M. Chang, J. Bauer, and J. L. Skinner, J. Chem. Phys. , 8973 (1990). D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Phys. Rev.Lett. , 88 (1990). F. C. Lavarda, M. C. dos Santos, D. S. Galv˜ao, andB. Laks, Phys. Rev. Lett. , 1267 (1994). F. A. B. F. de Moura and M. L. Lyra, Phys. Rev. Lett. , 3735 (1998).40