From Anderson localization on Random Regular Graphs to Many-Body localization
FFrom Anderson localization on Random Regular Graphs toMany-Body localization
K. S. Tikhonov
Skolkovo Institute of Science and Technology, Moscow, 121205, RussiaL. D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, Russia
A. D. Mirlin
Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology, 76021 Karlsruhe, GermanyInstitute for Condensed Matter Theory, Karlsruhe Institute of Technology, 76128 Karlsruhe, GermanyL. D. Landau Institute for Theoretical Physics RAS, 119334 Moscow, RussiaPetersburg Nuclear Physics Institute,188300 St. Petersburg, Russia.
Abstract
The article reviews the physics of Anderson localization on random regular graphs (RRG) and itsconnections to many-body localization (MBL) in disordered interacting systems. Properties ofeigenstate and energy level correlations in delocalized and localized phases, as well at criticality,are discussed. In the many-body part, models with short-range and power-law interactions areconsidered, as well as the quantum-dot model representing the limit of the “most long-range” in-teraction. Central themes—which are common to the RRG and MBL problems—include ergod-icity of the delocalized phase, localized character of the critical point, strong finite-size e ff ects,and fractal scaling of eigenstate correlations in the localized phase. Keywords:
Anderson localization, random regular graphs, many-body localization, ergodicity,critical behavior, eigenfunction and energy level statistics
Contents1 Introduction 22 Anderson localization on Random Regular Graphs 4 d to RRG. . . . . . . . . . . . . 122.3.3 Eigenfunction correlations on RRG: Single wave function . . . . . . . . 142.3.4 Dynamical correlations of eigenfunctions on RRG . . . . . . . . . . . . 162.3.5 Return probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.6 Adjacent eigenstate correlations . . . . . . . . . . . . . . . . . . . . . . 20 Preprint submitted to Annals of Physics February 12, 2021 a r X i v : . [ c ond - m a t . d i s - nn ] F e b .4 Level statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1 Gap ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.2 Level correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Wavefunction correlations: Localized phase . . . . . . . . . . . . . . . . . . . . 26 More than sixty years ago, the celebrated Anderson’s paper [1] marked a discovery of Ander-son localization, which has greatly influenced the development of the condensed matter physicssince then. Transport and localization properties of quantum particles subjected to a random po-tential or other types of disorder have been systematically explored. It was found that transitionsbetween localized and delocalized phases—known as Anderson transitions—show a remarkablyrich physics depending on spatial dimensionality, symmetries, and topologies [2].More recently, the physics of the many-body localization (MBL) in disordered interactingsystems [3, 4] attracted a great deal of research attention. The MBL addresses localization ordelocalization in highly excited states of interacting many-body systems (i.e., states with a finiteenergy density). One can thus consider the MBL as a generalization of Anderson localizationfrom single-particle to many-body setting. We refer the reader to Refs. [5, 6, 7, 8] for recentreviews on various aspects of the MBL.The extension from a non-interacting to an interacting problem strongly complicates the theo-retical investigation—analytical as well as computational. On the analytical side, the approachesto the MBL transition in Refs. [3, 4] (and in the later closely related paper [9]) were based on theanalysis of the corresponding perturbative expansion. Later, it was shown that matrix elementsof Hartree-Fock type (which were discarded in Refs. [3, 4]) essentially enhance delocalizationand parametrically shift the transition point as given by the perturbative analysis, due to the e ff ectof spectral di ff usion [10]. The schemes based on the analysis of the perturbation theory do not2nclude, however, e ff ects related to exponentially rare regions of anomalously high or anoma-lously weak disorder. As was understood in recent years [11, 12], such regions may likely playan important role for the scaling of the MBL transition and the corresponding critical behaviorin the thermodynamic limit of a large system. Based on these ideas, several phenomenologicalrenormalization-group schemes have been proposed [13, 14] that were argued to describe thescaling at the MBL transition.On the computational side, exact diagonalization (ED) studies are restricted to systems with ≈
20 qubit-like binary degrees of freedom (spins, orbitals of fermions or hard-core bosons,Josephson qubits, etc), with the corresponding Hilbert-space size being ∼ ∼ . Whilenumerical simulations on systems of this size do provide a clear evidence of the MBL transition[15, 16], the corresponding finite-size scaling analysis yields exponents that are inconsistent withthe Harris criterion. This is a clear indication of the fact that the system sizes that are accessibleby the ED are way too small for the purpose of observing the ultimate large-system critical be-havior. Indeed, it has been estimated that this requires spin chains of the length L (cid:38)
50 – 100[17, 18, 19]. Quantum dynamics around the MBL transition in systems of this size can be studiedby means of numerical approaches based on matrix product states, and the results are indeed ingeneral agreement with analytical expectations [20, 21, 22, 23]. These simulations can, however,only probe the dynamics at moderately long time scales.Since a controllable analytical treatment of the MBL problem is notoriously di ffi cult (and stillremains a big challenge for future work), simplified models that are amenable to such a treatmentare highly useful. The Anderson localization problem on random regular graphs (RRG) servesas such a toy-model of the MBL problem.An RRG is a finite-size graph that has locally the structure of a tree with a fixed coordina-tion number m + ff er-ence between finite Anderson localization on Bethe lattices and RRG has been demonstrated inRefs. [25, 26]; we will only consider RRG in this review, in view of their connection to the MBLproblem.As a first indication of an analogy between the RRG and MBL models, one can inspect thesize of the Hilbert space. For a spin chain of length L (a paradigmatic MBL-type system), theHilbert-space size scales with the length as 2 L . The RRG model has the same property of anexponential growth of the Hilbert-space volume N with the “linear size” L (the distance betweenthe most distant sites on the RRG lattice), N ∼ m L . Another key property—which is related tothe exponential growth of the lattice—is the suppression of small-scale loops, i.e. the locallytree-like character of the graph. A similarity between the interaction-induced structure in theHilbert space of a many-body problem and the RRG structure has recently triggered a surge ofinterest in Anderson localization on RRG [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39].3t is fair to say that by now we have a rather complete understanding of key properties of thismodel, including the position of the thermodynamic-limit transition point W c , values of criticalexponents and of various observables characterizing properties of eigenstates and energy levels.Of course, the RRG model is distinct from a genuine many-body problem. Specifically, theRRG model discards correlations between matrix elements in the Hilbert space of the many-bodyproblem, thus representing its “simplified version”. Existence of such correlations is clear fromthe fact that the number of independent parameters in a many-body Hamiltonian is much smallerthan the number of non-zero matrix elements. Nevertheless, there are remarkable analogiesbetween the localization transitions in the RRG model and in genuine MBL models, which showup in a variety of key physical properties. In particular, the most salient qualitative properties ofthe Anderson-localization transition on RRG include:(i) the critical point of the Anderson transition has a localized character,(ii) there are strong finite-size e ff ects which manifest themselves in a drift of the apparenttransition point towards stronger disorder with increasing system size,(iii) the Hilbert-space “correlation volume” increases exponentially when the localization tran-sition is approached,(iv) the delocalized phase is ergodic (which means the Wigner-Dyson (WD) level statistics, the1 / N asymptotic scaling of the inverse participation ratio and further associated propertes).These properties of the RRG model have been proven analytically [36] (see also Refs. [40, 41, 42]where a related model of sparse random matrices (SRM) was investigated) and also verified nu-merically [29, 30, 31, 36, 34]. Analytical arguments and numerical simulations for the MBLmodels lead to analogous conclusions. (The analytical arguments in the case of MBL are, how-ever, less rigorous than for RRG, as was pointed out above.) There are thus strong connectionsbetween the RRG and MBL problems. These connections become even closer for models withlong-range interaction (decaying su ffi ciently slowly as a power-law of distance), for which rareregions do not play any essential role, see Ref. [43].In this review article, we first overview recent advances on the Anderson model on RRG(Sec. 2), and then discuss its connections with MBL-type problems (Sec.3). In the many-bodypart of the review, models with short-range and with long-range (power-law) interactions areconsidered (including the quantum-dot model, which is the limiting case of the “most long-range” interaction).
2. Anderson localization on Random Regular Graphs
We study non-interacting spinless fermions hopping over RRG with connectivity p = m + H = t (cid:88) (cid:104) i , j (cid:105) (cid:16) ˆ c † i ˆ c j + ˆ c † j ˆ c i (cid:17) + (cid:88) i = (cid:15) i ˆ c † i ˆ c i , (1)where the first sum runs over the nearest-neighbor sites of the RRG. The energies (cid:15) i are inde-pendent random variables distributed uniformly on [ − W / , W / t can be set to be t = .1. Field-theoretical description Statistical properties of observables in this model can be expressed in terms of certain func-tional integrals, either in supersymmetric [44, 40, 36] or in the replicated version [45, 46, 47].These approaches are equivalent for many purposes (see, for example, their comparison in Ref.[48] in a related context). On the other hand, the supersymmetric approach is preferential whenproperties of individual eigenstates or level statistics at the scale of level spacing is studied. Inview of this, we will use the supersymmetric formulation in what follows.The model defined by Eq. (1) has two sources of disorder: randomness in the structure of theunderlying graph and fluctuations of on-site energies (cid:15) i . Various disorder-averaged properties ofa disordered system can be derived in terms of averaged products of Green functions. Such aderivation in the framework of the supersymmetric field theory was performed, with applicationsto the level statistics and to the scaling of the inverse participation ratio (IPR) in the SRM model,in Refs.[40, 41]. In general, averaged products of retarded and advanced Green functions (withenergies E + ω/ E − ω/
2, respectively) can be evaluated as superintegrals of the form[44] (cid:90) (cid:89) k [ d Φ k ] e −L H ( Φ ) U ( Φ ) , (2)where the preexponential factor U ( Φ ) represents the quantity in question and[ d Φ k ] = dS (1) k , dS (2) k , d χ ∗ k , d χ k , dS (1) k , dS (2) k , d χ ∗ k , d χ k , is the supervector integration measure. Here S stay for real commuting and χ for anticommutingvariables. The action L H ( Φ ) is given by L H ( Φ ) = − i (cid:88) i j Φ † i ˆ Λ (cid:26)(cid:20) E + (cid:18) ω + i η (cid:19) ˆ Λ (cid:21) δ i j − H i j (cid:27) Φ j , (3)where η > Λ is a diagonal superma-trix with the first four components (retarded sector) equal to + − e −L H ( Φ ) over the distribution matrix elements of the Hamiltonian. We consideran ensemble of N × N Hamiltonians with the following joint distribution of diagonal H ii ando ff -diagonal H i j = H ji = A i j t i j matrix elements: P ( { H ii } , { A i j } , { t i j } ) = (cid:89) i γ ( H ii ) × (cid:89) i < j (cid:20)(cid:18) − pN (cid:19) δ ( A i j ) + pN δ ( A i j − (cid:21) × (cid:89) i δ (cid:88) j (cid:44) i A i j − p (cid:89) i < j h ( t i j ) . (4)Here A i j is the adjacency matrix, and the delta-function in the last line of Eq. (4) ensures thatthe coordination number of each vertex is p . For the purpose of generality, we have includedin Eq. (4) an arbitrary distribution h ( t ) of non-zero hopping matrix elements. For a RRG model5ith fixed hoppings t =
1, we have h ( t ) = δ ( t − Φ i associated with di ff erent sites. This is done by means of a functional generalizationof the Hubbard-Stratonovich transformation. As a result, we obtain the expression for physicalobservables in terms of an integral over functions g ( Φ ): (cid:104)O(cid:105) = (cid:90) Dg U O ( g ) e − N L ( g ) . (5)The integration (cid:82) Dg runs over functions of a supervector g ( Φ ) with the action N L ( g ) where L ( g ) = m + (cid:90) d Ψ d Ψ (cid:48) g ( Ψ ) C ( Ψ , Ψ (cid:48) ) g ( Ψ (cid:48) ) − ln (cid:90) d Ψ F ( m + g ( Ψ ) , (6)with m = p − F ( s ) g ( Ψ ) = exp (cid:26) i E Ψ † ˆ ΛΨ + i (cid:18) ω + i η (cid:19) Ψ † Ψ (cid:27) ˜ γ ( 12 Ψ † ˆ ΛΨ ) g s ( Ψ ) . (7)Here the function ˜ γ ( z ) is the Fourier transform of the distribution γ ( (cid:15) ) of on-site energies, ˜ γ ( z ) = (cid:82) d (cid:15) e − i (cid:15) z γ ( (cid:15) ). Finally, C ( Ψ , Ψ (cid:48) ) is a kernel of an integral operator inverse to that with the kernel˜ h ( Φ † ˆ ΛΨ ), where ˜ h ( z ) is the Fourier transform of the distribution h ( t ) of hoppings; for an RRGmodel with fixed hoppings t = h ( z ) = e − iz .Since the action is proportional to N , see Eq. (5), in the limit of large N this theory can betreated via the saddle-point approximation. The saddle-point configuration g ( Ψ ) of the action isdetermined by varying Eq. (6) with respect to g , which yields the equation g ( Ψ ) = (cid:82) d Φ ˜ h ( Φ † ˆ ΛΨ ) F ( m ) g ( Φ ) (cid:82) d Φ F ( m + g ( Φ ) . (8)This equation is equivalent to the self-consistency equation for the same model but defined onan infinite Bethe lattice. This property is a manifestation of the fact that, with probability unity,RRG has locally (in the vicinity of any of its sites) a structure of a tree with fixed connectivity p . [More formally, at large N and κ < /
2, a 1 − o (1) portion of RRG nodes have their κ log m N -neighbourhood loopless.]Due to supersymmetry, the denominator in Eq. (8) is thus equal to unity, so that the saddle-point equation reduces to g ( Ψ ) = (cid:90) d Φ ˜ h ( Φ † ˆ ΛΨ ) F ( m ) g ( Φ ) . (9)Equation (9) is identical to the self-consistency equation describing the model on an infiniteBethe lattice, as derived within the supersymmetry formalism in Ref. [44]. For symmetry rea-sons, the saddle-point solution is a function of two invariants g ( Ψ ) = g ( x , y ); x = Ψ † Ψ , y = Ψ † ˆ ΛΨ (10)and the Eq. (9) reduces to a non-linear integral equation for g ( x , y ) (recall that for the modelwith non-random hoppings, as defined in Eq. (1), one has ˜ h ( z ) = e − iz ). For a purely imaginary6 (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) (cid:31) o o o o o o o o o o oo o o oo Figure 1: (a) The disorder-dependence of the largest eigenvalue λ / ( W ) of the operator L / in Eq. (15) in the vicinityof the transition point. The critical disorder is determined by λ / = /
2, which gives W c = . ± .
01. (b) N ξ ( M ) = exp (cid:104)− ln Im G (cid:105) M evaluated at pool size M at η = − for W =
17 (filled symbols) and W =
19 (empty symbols). For W =
17 (delocalized phase), N ξ ( M ) evolves from η − (upper dashed line) to N ξ ≡ N ξ ( M → ∞ ) (lower dashed line)which equals the true correlation volume. For W =
19 (localized phase) N ξ ( M ) remains constant of order η − for all poolsizes. From Ref. [37]. frequency ( ω = g ( x , y ) has an important physical interpretation[44]. Specifi-cally, it equals the Fourier-Laplace transform of the joint probability distribution f ( m ) ( u (cid:48) , u (cid:48)(cid:48) ) ofreal and imaginary parts of local Green function, g ( x , y ) = (cid:90) du (cid:48) (cid:90) du (cid:48)(cid:48) f ( m ) ( u (cid:48) , u (cid:48)(cid:48) ) e i ( u (cid:48) y + iu (cid:48)(cid:48) x ) , (11)where G ( m ) A (0 , E ) = (cid:104) | ( E − H − i η ) − | (cid:105) = u (cid:48) + iu (cid:48)(cid:48) computed for a slightly modified lattice,with the site 0 having only m neighbors. When rewritten in terms of f ( m ) ( u (cid:48) , u (cid:48)(cid:48) ), Eq. (9) becomesthe self-consistency equation of Abou-Chacra et al., Ref. [49].A closely related object is the function g ( m + ( Ψ ) which is expressed via g ( Ψ ) as g ( m + ( Ψ ) = (cid:90) d Φ ˜ h ( Φ † ˆ ΛΨ ) F ( m + g ( Φ ) . (12)The function g ( m + ( x , y ) is the Fourier-Laplace transform of the joint probability distribution f ( m + ( u (cid:48) , u (cid:48)(cid:48) ) of real and imaginary parts of local Green function at any site of the undeformedBethe lattice. The self-consistency equation, Eq. (9) can thus be presented in the form G ( m ) d = E − i η − (cid:15) − (cid:80) mi = G ( m ) i , (13)where the symbol d = denotes the equality in distribution (assuming G ( m ) i to be independent copiesof G ( m ) ). The distribution of the local Green function G ( m + on an original lattice can be recov-ered from an auxiliary relation G ( m + d = E − i η − (cid:15) − (cid:80) m + i = G ( m ) i . (14)The distributions of G ( m ) and G ( m + are qualitatively very similar. Below we use a short notation G ≡ G ( m ) . 7 .2. Localization transition and critical behavior Saddle-point evaluation of the functional integral in Eq. (5) provides an interesting perspec-tive on the Anderson localization transition as a spontaneous symmetry breaking phenomenon. Inthe localized phase, the saddle point solution has the symmetry of the equation, g ( x , y ) = g ( y ).To be more precise, g ( x , y ) depends on the variable x only on the scale ∼ η − , due to the termwith η in the action that breaks the symmetry explicitly. Thus, the integral (5) in the localizedphase (as well as at the critical point) is determined by a contribution of a unique saddle point g .In the delocalized phase, dependence on the variable x survives even in the limit of ω, η →
0. Asa result, a manifold of saddle-points emerges, signifying spontaneous symmetry breaking withthe function g ( x , y ) playing the role of an order parameter. In this situation, the integral (5) runsover the manifold of saddle points.The approach to locating the transition point was first established in Ref. [49]; equivalentresults were later obtained within the supersymmetry formalism in Ref. [44]. It amounts to eval-uating the stability of the real solution to the self–consistency equation (obtained by by setting η = W c marks the transition between these two types ofbehavior. The first step in computing W c is thus to find a real ( η =
0) solution P ( G ) to thedistributional Eq. (13). The criterion of its stability can then be expressed in terms of the largesteigenvalue of a certain integral operator, whose kernel can be written in terms of P ( G ). For m = L β can be found in Refs. [49, 44]): L β ( x , y ) = | x | β y (cid:90) d (cid:15) γ ( (cid:15) ) P ( y − − x − (cid:15) ) . (15)The real solution is stable if and only if the largest eigenvalue λ β of the operator L / is smallerthan 1 / m .The largest eigenvalue λ / as a function of disorder is shown in Fig. 1a. Solving the equation m λ / ( W ) = m = W c = . ± . . (16)A very close value of W c was found by a similar method in Ref. [50].Let us now discuss the critical behavior in the delocalized phase. As has been already dis-cussed above, the purely real solution of the self-consistency equation ( η =
0) is unstable tointroduction of finite η in the delocalized phase where a non-trivial distribution P (Re G , Im G )emerges. It can be derived from the function g ( x , y ) which acquires dependence on the variable x on a scale x ∼ N ξ . This scale diverges exponentially when W approaches the critical value W c and has the meaning of the correlation volume which can be related to the correlation length ξ : N ξ ∼ m ξ . (17)In order to evaluate the correlation volume numerically, Eq. (13) was solved in Ref. [37]via the pool method, also known as population dynamics (PD). In this approach, the distribution P ( G ) is represented by a large sample of random variables and Eq. (13) is iterated until conver-gence. The resulting sample of M variables is distributed (at M → ∞ ) according to a desireddistribution. The PD calculation is conducted at a finite (although large) pool size M and a finite(although small) imaginary part of the energy, η . The resulting distribution can be characterized8 igure 2: Critical behavior at the Anderson transition on RRG. (a) Double logarithm of the correlation volume N ξ asa function of τ = − ln(1 − W / W c ) (with W c = . W , ln N ξ ) coordinates. (b) Flowingcorrelation-length exponent ν del ( τ ) = ∂ ln ln N ξ /∂τ . The limiting value, ν del ( τ → ∞ ), gives the critical index of thecorrelation length ν del = /
2, Eq. (21). From Ref. [37]. by “e ff ective correlation volume” N ξ ( η, M ) defined as the PD result for exp (cid:104)− ln Im G (cid:105) , where (cid:104) .. (cid:105) includes averaging over many iterations after the approximate convergence is reached. Thecorrelation volume N ξ is given by the double limit N ξ = lim η → lim M →∞ N ξ ( η, M ) . (18)The role of the pool size, is illustrated by the Fig. 1b, where the dependence of N ξ ( η, M ) on M for W =
17 and fixed η = − . This quantity evolves from η − (characteristic for the localizedphase) at small pool sizes N ξ ( η, M ) to the value N ξ ( η, ∞ ) at large M . At su ffi ciently small η , suchthat η (cid:28) N − ξ , the behavior of N ξ ( η, M ) at large M is essentially independent on η , illustrating thespontaneous symmetry breaking phenomena. This data should be contrasted to the ones for thelocalized side of the transition, exemplified by the results for W =
19, demonstrating the stabilityof the localized phase with respect to η .Apart from studying the specific averages (such as correlation volume N ξ ), we can char-acterize the entire function P ( G ) and determine, in particular, the distribution of the LDOS ρ = (1 /π )Im G . On the delocalized side close to the transition it is expected to be of the fol-lowing form P ( ρ ) ∼ N − / ξ ρ − / , N − ξ < ρ < N ξ ; (19)outside of this range the probability is strongly suppressed. This behavior of P ( ρ ) is essentiallythe same as the one found in the σ model on the Bethe lattice [51, 52]. Equation (19) can bederived via analysis of the symmetry-broken solution near the critical point in both Anderson[44] and σ -model [53, 54]. Main features of this solution are connected to properties of thelargest eigenvalue λ β of the operator (15) via equation m λ β =
1. For W below (and close to) W c the solution is β = / ± i σ , with σ ∼ ( W c − W ) − / . The real part 1 / β translatesinto the exponent 3 / N ξ ∼ exp( π/σ ) which controls the range of validity of the power-law distribution (19).Thus, the critical behavior of the correlation volume readsln N ξ ∼ ( W c − W ) − / . (20)9ccording to Eqs. (20) and (17), the critical index of the correlation length on the delocalizedside of the transition equals : ξ = ln N ξ ln m ∼ ( W c − W ) − ν del ; ν del = . (21)The disorder dependence of the correlation volume N ξ obtained according to Eq. (18) isshown in Fig. 2a. More specifically, we plot ln ln N ξ as a function of τ ≡ − ln(1 − W / W c ). In thisrepresentation, we evaluate ν del ( τ ) = ∂ ln ln N ξ ∂τ , (22)and asymptotic value ν del ( ∞ ) yields the critical index ν del . The τ dependence of the slope ν del ( τ )is shown in Fig. 2b. This figure demonstrates that ν del ( τ ) varies substantially in the range of τ corresponding to W =
14 – 18. At the same time, it does saturate for τ → ∞ (i.e., W → W c ) at ν del = /
2, in a perfect agreement with the analytical prediction Eq. (21). We can write a moreprecise equation, which includes a prefactor and subleading correction:1ln N ξ = c ( W c − W ) / + c ( W c − W ) / , (23)where c = . c = . < W < W c . It is interesting to compare the numerical values for the critical disorder W c , Eq. (16), and for the correlation volume, Eq. (23), with analytical results in the large- m approximation. It turns out that it works quite well already for m = W c ≈ .
67 andln N ξ (cid:39) π (cid:114)
23 ln / ( W c / / ( W c / e ) (cid:114) W c W c − W , which gives c ≈ . N and the correlation volume N ξ plays a crucialrole for properties of the system one the delocalized side of the transition, i.e., at W smallerthan (but su ffi ciently close to) W c . For N (cid:28) N ξ the system is critical, which means, to thefirst approximation, that it looks localized. In the opposite limit, N (cid:29) N ξ , the system becomesergodic. We will demonstrate below how this crossover from the critical regime, N (cid:28) N ξ , to theasymptotic ergodic regime, N (cid:29) N ξ , manifests itself in key observables.It is worth emphasizing once more that the critical point in the RRG model has the localizedcharacter. This means that, when the system is on the localized side of the transition, W > W c ,there is no qualitative change in the behavior of many observables of interest with increasing N . As an example, the IPR P is of order unity for any N for W > W c —in stark contrast to adramatic change of its behavior for W < W c , from the critical regime N (cid:28) N ξ to the ergodicregime N (cid:29) N ξ . In view of this, we put more emphasis on the discussion of the delocalizedside, W < W c (including, of course, the critical regime), in this review. At the same time,there is also a very interesting physics in the localized phase. In particular, Sec. 2.5 below willaddress dynamical correlation of eigenstates in the localized phase, which are strongly enhancedwhen the system approaches the transition point. We also refer the reader to Ref. [38] where thebehavior of other observables in the localized phase was considered.10 n s u l a t o r good metal Figure 3: System size dependence of the inverse participation ratio (IPR) P in the Anderson model on RRG as givenby ED. (a) ln NP as a function of the system size for various disorder strengths W . Dots: simulation, lines: smoothinterpolation. (b) System size dependence of the fractal exponent µ for various W . From Ref. [29]. The simplest quantity characterizing the wavefunction statistics is ensemble-averaged IPR, P ( W , N ) = (cid:42) N (cid:88) i = | ψ i | (cid:43) . (24)The dependence of NP ( W , N ) on system-size for a set of disorder values as given by ED [29] isshown in Fig. 3a. In the localized phase we have P ∼
1, so that NP ( W , N ) ∝ N at large N , asthe data for W =
25 show. For the delocalized side, analytical calculation in the framework ofthe theory sketched in Sec. 2.1 (see also Sec. 2.3.3 below) yields, for N (cid:29) N ξ , P ∼ N ξ N , N (cid:29) N ξ . (25)In the opposite regime of N (cid:28) N ξ , the system is critical, with the result P ∼ , N (cid:28) N ξ . (26)Thus, in the large- N limit—and more specifically, under the condition N (cid:29) N ξ —the product NP ( W , N ) saturates at an N -independent value C ( W ), C ( W ) = lim N →∞ NP ( N ) , (27)which is manifestation of ergodicity. This behaviour is indeed supported by the data on theFig. 3a, when the system is far from the transition, W (cid:46)
11. The value of C ( W ) as providedby the theory of Sec. 2.1 can be calculated via the PD. As is discussed below, the results arein perfect agreement with those of ED. The value of C ( W ) increases with W approaching W c as C ( W ) ∼ N ξ , i.e., exponentially fast according to Eq. (20). Therefore, when W becomes11 igure 4: Flowing fractal exponent µ as a function of the system size for selected disorders in the range from W =
13 to W =
17. Inset: disorder-dependence of system size corresponding to the minimum of µ ( N ). From Ref. [29]. su ffi ciently close to W c , the condition of ergodic regime, N (cid:29) N ξ , is not fulfilled any more evenfor the largest N accessible via ED, so that the saturation of NP ( N ) is not reached. This isindeed what is observed in Fig. 3a for 11 (cid:46) W < W c .The data of Fig. 3a can be interpreted via the “flowing fractal exponent” µ ( W , N ) = − ∂ ln P ( W , N ) ∂ ln N , (28)shown in Fig. 3b as a function of system size for a set of W . For moderate disorder, W (cid:46)
11, theexponent µ saturates at the ergodic value µ = NP ( N )in Fig. 3a). For stronger disorder (see Fig. 4), 11 (cid:46) W < W c , we observe a non-monotonicbehavior: µ first flows towards its value µ c = µ c = N exceeds the correlation length ξ ( W ) (see inset in Fig. 4), the flow turns towards thedelocalized fixed point with the ergodic value of the exponent, µ = N (cid:29) N ξ ) has been also confirmedby numerical simulations in Refs. [30, 34]. Let us now turn to discussion of wavefunction correlations. It is instructive to recall firstthe corresponding results for Anderson problem on a cubic d -dimensional lattice. Indeed, theRRG model can be viewed, in a certain sense, as a d → ∞ limit of the d -dimensional problem.This limit is, however, very singular. This is clear from comparing the volume as function of thelinear size L : in d dimensions its a power law L d and on RRG its an exponential function m L .We discuss below how one can “guess” the RRG results on the basis of those for d dimensions.After this, we will present the results derived directly for the RRG model, see Sections 2.3.3 and2.3.4. We will see that the “educated guess” based on d dimensional results is largely correct butmisses some important subleading factors.Correlation function of the same wavefunction at di ff erent spatial points is formally defined12s α E ( r , r ) = ∆ (cid:42)(cid:88) k | ψ k ( r ) | | ψ k ( r ) | δ ( E − E k ) (cid:43) . (29)Here ψ k are eigenstates and E k the corresponding energy levels, E is the energy at which thestatistics is studied, ∆ = /ν ( E ) N is the mean level spacing, and ν ( E ) = N − (cid:68) Tr δ ( E − ˆ H ) (cid:69) is thedensity of states. For coinciding points, this correlation function reduces to the IPR, P = (cid:90) d d r α E ( r , r ) = L d α E ( r , r ) . (30)At finite spatial separation | r − r | , a wavefunction in the delocalized phase near the Andersontransition in a d -dimensional system exhibits strong self-correlations up to the correlation length ξ , L d α E ( r , r ) ∼ ( | r − r | / min( L , ξ )) ∆ , (31)for | r − r | < ξ . At the transition point (diverging ξ ), the critical correlations (31) extend over thewhole system. At finite but large ξ (in the vicinity of the transition point) the correlations remaincritical as long as L (cid:28) ξ .The correlation function, characterizing the overlap of di ff erent (close in energy) wavefunc-tions is defined as follows: β E ( r , r , ω ) = ∆ R − ( ω ) (cid:42)(cid:88) k (cid:44) l | ψ k ( r ) ψ l ( r ) | δ (cid:18) E − ω − E k (cid:19) δ (cid:18) E + ω − E l (cid:19)(cid:43) , (32)where ω is the energy di ff erence between the states, and the level correlation function R ( ω ) = ν ( E ) (cid:104) ν ( E − ω/ ν ( E + ω/ (cid:105) (33)is introduced. In finite d Anderson problem, the correlation function β E ( r , r , ω ) exhibits thescaling L d β E ( r , r , ω ) ∼ ( | r − r | / min( L ω , ξ )) ∆ (34)for | r − r | < min( L ω , ξ ). Here L ω ∼ ( ων ) − / d is the length scale associated with the frequency ω (or, equivalently, with the time ∼ ω − ) at criticality. The exponent ∆ determines also the scalingof the di ff usion propagator at criticality [55].For the sake of brevity of notations, we will mainly omit below the subscript E indicating de-pendence of the correlation functions α E and β E on energy around which the statistics is studied.Let us now “translate” Eq. (31) to RRG. First, the factor L d can be interpreted as the systemvolume and thus replaced on the RRG by the number of sites N . Second, the multifractal expo-nent can be replaced by its large- d limit ∆ → − d . Finally, the factors of the type r d , should beunderstood as counting the number of sites in a sphere of a radius r centered at a given site, andreplaced by their RRG counterpart m r . As a result, we come to the following conjecture for thescaling of the same-wavefunction-correlator (31) on RRG: α ( r , r ) ∼ (cid:40) N − m ξ − r , metallic , r < ξ ; N − m − r , critical. (35)13ere r is the the length of the shortest path between the two points r and r on RRG. The cor-relation function of two di ff erent eigenfunctions, Eq. (34), can be obtained in the same manner: β ( r , r , ω ) ∼ (cid:40) N − m ξ − r , metallic , r < ξ ; N − ω − m − r , critical. (36)We turn now to the analytical derivation of the correlation functions on RRG as well as totheir numerical investigation. The correlation functions introduced above can be evaluated for RRG using supersymmet-ric field theory, Sec. 2.1 [36]. Let us start with α (0), which in the delocalized phase (and atsu ffi ciently large N ) can be expressed in terms of the saddle-point solution: α (0) = N g ( m + , xx π ν . (37)The coe ffi cient g ( m + , xx in Eq. (37) has an important physical meaning. Since the function g ( m + ( x , y )is the Fourier-Laplace transform of the distribution of local Green functions on an infinite Bethelattice (see discussion around Eq. (11)), g ( m + , xx is proportional to the average square of the localdensity of states ν ( j ) = − (1 /π )Im G ( j , j ): g ( m + , xx = ( π / (cid:104) ν (cid:105) BL . (38)The subscript “BL” here indicates that the average should be computed using the solution of theself-consistency equation that describes the model on an infinite Bethe lattice. Equation (37) canthus be written in the form α (0) = N (cid:68) ν (cid:69) BL ν . (39)Let us recall a relation between α (0) and the average IPR P P = N α (0) , (40)cf. the analogous formula (30) for a d -dimensional system. Equation (39) yields the ergodic( ∼ / N ) IPR scaling (25). Furthermore, it provides an exact expression for the correspondingprefactor [denoted C ( W ) in Eq. (27)] in terms of the Bethe-lattice correlation function (whichcan be calculated by PD), C ( W ) = (cid:68) ν (cid:69) BL ν . (41)Extending this analysis, one can calculate correlation functions α ( r ) and β ( r , ω ) on RRG.The results are expressed, in the limit of large N , in terms of averaged products of two Greenfunctions on an infinite Bethe lattice. There are two such correlation functions: K ( r ) = (cid:104) G R ( i , i ) G A ( j , j ) (cid:105) BL = (cid:104)
116 ( Ψ † i , ˆ K Ψ i , )( Ψ † j , ˆ K Ψ j , ) (cid:105) BL ; (42) K ( r ) = (cid:104) G R ( i , j ) G A ( j , i ) (cid:105) BL = (cid:104)
116 ( Ψ † j , ˆ K Ψ i , )( Ψ † i , ˆ K Ψ j , ) (cid:105) BL . (43)14hese correlation functions (for ω = η →
0) have been computed in Ref. [44]. In thelocalized phase, W > W c , the correlation functions K ( r ) and K ( r ) have 1 /η singularity, areequal to each other, and decay with r as K ( r ) = K ( r ) ∼ η m − r e − r /ζ r − / , (44)where ζ is the localization length. In the delocalized phase, W < W c , close to the transition point,the result for the function K ( r ) reads K ( r ) ∼ N ξ m − r r − / (45)and K ( r ) (cid:39) K ( r ) + K ( d )1 , (46)where K ( d )1 = |(cid:104) G R ( j , j ) (cid:105)| is disconnected part of K ( r ).The eigenfunction correlations on RRG are expressed [36], using the theory presented inSec. 2.1, in terms of these correlation functions. In the localized phase, W > W c , U ( g ) in thefunctional integral (5) is simply (cid:104) ( Ψ † , ˆ K Ψ , )( Ψ † r , ˆ K Ψ r , ) (cid:105) BL , which is the correlation function K ( r ), so that α ( r ) = πν N lim η → η K ( r , η ) . (47)Using Eq. (44), we immediately find α ( r ) ∼ N m − r e − r /ζ r − / . (48)This result is extended to the critical point by setting ζ = ∞ , which yields α ( r ) ∼ N m − r r / . (49)In the delocalized phase, the functional integral is reduced to an integral over a manifold ofsaddle points (see discussion in Sec. 2.2). The result for the correlation of the same wavefunctionreads: α ( r ) = π N [ K ( r ) + K ( r ) − Re (cid:104) G R (0) (cid:105) − (cid:104) G R ( r ) (cid:105) ] (50)where the last term in square brackets is relatively small for large N ξ . Using Eqs. (46) and (45),we find: α ( r ) ∼ N ξ N m − r r / , r < ξ. (51)For r > ξ the correlation function α ( r ) is governed by disconnected parts in Eq. (50), yielding α ( r ) (cid:39) d results to d → ∞ . They include, however, an additional factor r − / .These analytical predictions for α ( r ) on RRG can be compared with results of the ED forthe RRG model (1) in the middle of the band [36]. In Fig. 5a we show the connected part α ( c ) ( r ) = α ( r ) − / N evaluated for RRG ( N = , m + =
3) in the delocalized phase. Thechosen values of W are su ffi ciently close to W c , so that the correlation volume is large, N ξ (cid:29) N (cid:29) N ξ ). In this regime, ergodicity of wavefunctions manifests15 r l n N α ( c ) ( r ) W ξ r . . . N m r α ( r ) Figure 5: Eigenfunction self-correlations α ( r ) on RRG. Left: connected part N α ( c ) ( r ) = N α ( r ) − N = inthe delocalized phase, for disorder values W =
8, 10, and 12. Dashed black lines: fit to ln N α ( c ) ( r ) = − r ln m − c ( α )1 ln( r + − c ( α )2 , see Eq. (51). Star symbols: values of α (0) derived from PD, Eq.(39). Inset: correlation length ξ ( W )determined from ln N α ( c ) ( r ) = Right: α ( r ) at the critical point ( W =
18) as found by ED: Nm r α ( r ) as a function of r on double-logarithmic scale. Blue solid line: c / r / , see Eq. (51). From Ref. [36]. itself via system size independence of N α ( r ), see Eq. (51). This is fully confirmed by the EDresults for three di ff erent system sizes, shown in the right panel of Fig. 5. The finite-size e ff ectscan be seen on this plot for r approaching the linear size of the graph ln N / ln m .The value of N α (0) = NP is in full agreement with the analytical prediction (39), (41),where the r.h.s. was evaluated by PD (shown by star symbols). While the agreement is perfectfor W = W =
10, a small deviation for W =
12, which is fully expected since at this valueof disorder the ratio N / N ξ is not so large any more.The dependence of ln[ N α ( c ) ( r )] on r is approximately linear, in agreement with exponentialdecay, predicted by Eq. (51). The fits by ln[ N α c ( r )] = − r ln m − c ( α )1 ln( r + − c ( α )2 , with c ( α )1 as a free parameter, are shown in the left panel of Fig. 5 by dashed lines. The fitted values of c ( α )1 are smaller than 3 / ξ , drifting in the direction of power–lawexponent 3 / ξ → ∞ ). The values of the correlation length ξ definedvia condition ln[ N α ( c ) ( r )] = ξ ( W ) estimated from the analysis of the N –dependence of IPR, see inset to Fig. 4.The r -dependence of α ( r ) at the critical point is also in a very good agreement with theanalytical prediction (49). To demonstrate this, the right panel of Fig. 5 shows the product Nm r α ( r ) as a function of r . At criticality, α ( r ) is predicted to behave as 1 / N and the lines indeedalmost collapse. With increasing N , the finite-size corrections become less relevant and thecurves approach the straight line with the slope 3 /
2, in agreement with Eq. (49).
We now turn to the analysis of the correlation function β E ( r , ω ), defined by Eq. (32). Thiscorrelation function depends on the frequency ω , thus describing dynamical correlations.To evaluate β E ( r , ω ) by using supersymmetric field theory, Sec. 2.1 [36], one first expressesit in terms of Green functions on RRG: α i j ( E ) δ ( ω/ ∆ ) + β i j ( E , ω ) ¯ R ( ω ) = ∆ B i j ( E , ω ) , (52)where i and j are two cites on RRG separated by a distance r ,¯ R ( ω ) = R ( ω ) − δ ( ω/ ∆ ) (53)16s the non-singular part of the two-level correlation function (33) and LDOS correlation functionis introduced as follows B i j ( E , ω ) = (cid:68) ν i ( E − ω/ ν j ( E + ω/ (cid:69) = π Re (cid:20)(cid:28) G R ( i , i , E + ω G A ( j , j , E − ω − G R ( i , i , E + ω G R ( j , j , E − ω (cid:29)(cid:21) . (54)In the delocalized phase at N (cid:29) N ξ , one finds [36] B i j ( E , ω ) = ¯ R WD ( ω ) 12 π Re (cid:104) K ( r , ω ) − (cid:104) G R (0) (cid:105) (cid:105) , (55)where ¯ R WD ( ω ) is the WD level correlation function and K ( r , ω ) = (cid:104) G R ( i , i , E + ω G A ( j , j , E − ω (cid:105) BL , (56)which (for W < W c and N (cid:29) N ξ ) is given by its RMT form, ¯ R WD ( ω ). Thus, one finally gets inthis regime β ( r , ω ) = π N Re (cid:104) K ( r , ω ) − (cid:104) G R (0) (cid:105) (cid:105) . (57)Similarly to Eq. (50), this equation expresses a correlation function of eigenfunctions on RRGwith large N in terms of a correlation function defined on an infinite Bethe lattice (or, equiva-lently, via a self-consistency equation). In this review, we will focus on the correlation function β ( r , ω ) at r =
0, introducing the short notation β (0 , ω ) ≡ β ( ω ). For analytical and numericalstudy of r –dependence of β ( r , ω ), see Ref. [36].To calculate β ( ω ), as given in terms of infinite-Bethe lattice correlation function by Eqs. (57),(56), one can use self-consistency equations for the joint distribution function of two Greenfunctions on di ff erent energies, u = G R ( i , i , E + ω/
2) and v = G A ( i , i , E − ω/ f ( m ) ( u , v ) = (cid:90) d (cid:15) γ ( (cid:15) ) (cid:90) m (cid:89) r = du r dv r f ( m ) ( u r , v r ) × δ (cid:34) u − E + ω + i η − (cid:15) − (cid:80) mr = u r (cid:35) δ (cid:34) v − E − ω − i η − (cid:15) − (cid:80) mr = v r (cid:35) ; (58) f ( m + ( u , v ) = (cid:90) d (cid:15) γ ( (cid:15) ) (cid:90) m + (cid:89) r = du r dv r f ( m ) ( u r , v r ) × δ u − E + ω + i η − (cid:15) − (cid:80) m + r = u r δ v − E − ω − i η − (cid:15) − (cid:80) m + r = v r . (59)Now we discuss application of the general formula (57) to specific regimes. At criticality( W = W c , or, more generally, N ξ (cid:29) N ), performing in Eq. (44) an analytical continuation to realfrequency, η → − i ω/
2, and setting r =
0, one gets K ( r = , ω ) ∼ − i ω . (60)17ccording to Eq. (57), the term (60) does not contribute to β ( ω ). Thus, we need to evaluatecorrections at criticality which are expected to be governed by inverse powers of ln 1 /η : K ( r = (cid:39) c ( K )1 η + c ( K )2 η ln z /η , (61)The subleading factor in a form of a power–law of the logarithm of frequency is a natural counter-part of the subleading r − / factor in the r -dependence, see Eq. (45). Equation (61) was verifiedin Ref.[36] by numerical solution of the self-consistency equation. The numerical value of theexponent z is z (cid:39) /
2. It seems likely that z = / η → − i ω/
2, is performed in Eq. (61) andthe result is substituted into Eq. (57), the first term in Eq. (61) drops and the following scaling of β ( ω ) at criticality is found: β ( ω ) ∼ N ω ln z + /ω . (62)The applicability of the critical scaling (62) is limited, on the side of small ω , by the finite size N of the system. In the limit ω → β ( ω → ∼ / N . (63)This can be shown by using the fact that β ( ω → /α (0) = / N (cid:29) N ξ , see below. It follows that β ( ω → /α (0) ∼ N (cid:28) N ξ . UsingEq. (49), we find Eq. (63). The critical scaling (62) matches the low-frequency value (63) at thescale ω N ∼ N ln z + N , (64)which is parametrically (by a logarithmic factor) smaller than the mean level spacing ∆ ∼ / N .We consider now the behavior of β ( ω ) in the delocalized phase, W < W c and N (cid:29) N ξ . In thesmall-frequency limit, a comparison of Eqs. (57) and (50) yields β ( ω → = α (0) ∼ N ξ N . (65)The factor 1 / K (0 , ω ) and hence β ( ω ) remain nearly frequency independent for not too high frequen-cies before the system enters the critical regime, see Eq. (62). The crossover frequency ω ξ isdetermined by matching Eqs. (62) and (65); it is found as ω ξ ∼ N − ξ (up to a logarithmic factor).Summarizing, the results for the dynamical correlation function β ( ω ) in the delocalized phaseread β ( ω ) ∼ N ξ N , ω < ω ξ , N ω ln z + /ω , ω > ω ξ . (66)These results largely confirm the expectation, Eq. (36), based on the d → ∞ extrapolation (up toan additional logarithmic frequency–dependent factor).18 − − − − − − ω N β ( ω ) t − − − p ( t ) t l n / p ( t ) Figure 6: Correlation function β ( ω ) of di ff erent eigenfunctions at the same spatial point and return probability p ( t ) onRRG. Left: N β ( ω ) for W < W c . Solid lines: ED results ( N = ), cyan dashed line: result of Eq. (57) with K (0 , ω )determined from numerical solution of Eqs. (59), (58), black dashed line: 1 /ω scaling. Black dots: α (0) / Right: returnprobability p ( t ) (ED result) for W = p ( t ) at initial time crosses overto finite-size saturation at p ∞ = N α (0) ∼ N ξ / N (the dashed, dotted, and solid lines correspond to various system sizes( N = , 2 , and 2 ), respectively). Inset: comparison of p ( t ) (solid, W =
8) to, classical di ff usion over the infiniteBethe lattice, p ( t ) = ap ( Dt ) (dashed). From Ref. [36]. The numerical results for the correlation function β ( ω ) are shown in Fig. 6. In the left panel,the results for the delocalized phase are presented. Solid lines ( W = , ,
12) are obtained byED. This plot also shows (by dashed line) results for W =
13 as obtained from Eq. (57) and (56),with finite- ω correlations on an infinite Bethe lattice derived from the self-consistency equations(59), (58). These two types of numerical results are in a very good agreement between eachother as well as with the analytical prediction Eq. (66). Both the critical behavior (1 /ω , up tocorrections that are di ffi cult to observe in this plot) and the low-frequency saturation are evident.As an additional check, this figure also includes the numerically obtained values of α (0) / β ( ω → /α (0) = /
3, see Eq. (65), is perfectly fulfilled. It is possible to extractthe correlation length ξ from either the value of β ( ω →
0) or from the crossover scale in thefrequency–dependence. Both ways give values of ξ ( W ) close to those shown in the inset ofFig. 5. In this section, we consider spreading of a state, localized at t = j andevaluate probability p ( t ) to find it at the same site at a later time t >
0. Formally, it is defined asfollows: p ( t ) = (cid:42) N (cid:88) j | (cid:104) j | e − i ˆ Ht | j (cid:105) | (cid:43) , (67)where averaging over the initial site j is performed. It is straightforward to express the Fouriertransform of return probability p ( ω ) in terms of eigenstates correlations functions, defined inEqs. (29) and (32): p ( ω ) = N δ ( ω ) (cid:90) dE ν ( E ) α E (0) + N (cid:90) dE ν ( E ) R E ( ω ) β E ( ω ) . (68)Thus, return probability encodes both α E (0) (or, equivalently, IPR) via its t → ∞ limit, and β E ( ω )via its time–dependence. Below we assume that the sum in Eq. (67) is projected to the states inthe vicinity of a certain energy E and omit the subscript E in the notation for p E ( t ).19n the critical regime, we find by a straightforward calculation [36], from Eq. (68): p ( t ) (cid:39) p ∞ + c ( p ) ln z t , t → ∞ , (69)with a numerical constant c ( p ) .In the metallic regime ( W < W c and N (cid:29) N ξ ), the return probability p ( t ) can be describedby a classical random walk over the tree. Such a random walk is described by p ( t ) = p ( Dt ) ∼ Dt ) / e − Dt , (70)which gives the probability for a particle to be found at the starting point after time t [56, 57].The di ff usion coe ffi cient can be expressed in terms of a certain integral involving the solutionof the self-consistency equation [44] (see also a similar computation for σ model [54]); thecorresponding asymptotics at N ξ (cid:29) D ∼ N − ξ ln N ξ , (71)where N ξ scales according to Eq. (21). In a finite system the decay, described by Eq. (70)saturates at a value, given by the first term in Eq. (68): p ∞ ∼ N α (0) ∼ N ξ N , (72)with 1 / N scaling of p ∞ being another manifestation of ergodicity of the delocalized phase on theRRG.The right panel of Fig.6 presents results for return probability p ( t ) as obtained by ED of theRRG model (projected to the middle of the band). At long times, this decay is limited by thesystem size N , and p ( t ) saturates at the value p ∞ . While for W =
8, the di ff usive exponentialdecay is apparent from the plot, the curve for disorder W =
12 shows a di ff erent type of be-havior: p ( t ) exhibits a nearly flat part up to t ∼ − . This is sign of a critical behavior which,for longer times, crosses over to an exponentially fast decay representative for the delocalizedregime. However, it does not have much time to develop, since the saturation dictated by thesystem size sets in.In the inset to the right panel of Fig. 6, the return probability is compared to the solution of aclassical di ff usion problem on the Bethe lattice [56, 57]. The correspondence is very good, untilfinite-size e ff ects become important at t (cid:38) Let us now turn to another correlator that is closely related to β ( ω )—a correlation functionof adjacent-in-energy eigenstates: β nn = ∆ (cid:42)(cid:88) k δ ( E k − E ) | ψ k ( j ) ψ k + ( j ) | (cid:43) . (73)Here the subscript “nn” stands for “nearest neighbor” (in energy space). We follow the analysisof β nn on RRG that was performed in Ref. [39].20
10 15 20 25 30 W N β nn ( W ) W c W . . . . . . µ nn ( W ) W c Figure 7: Correlation of adjacent wavefunctions on RRG (ED results) for N = , , , , (from cyan tomagenta). Left:
Correlation function β nn ( W ). Dashed line: the asymptotic behaviour of N β nn for W < W c accordingto Eqs. (75) and (23). Vertical dotted corresponds to W = W c . Right:
Exponent µ nn characterizing the N scaling ofadjacent-state correlations, see Eq. (76). Dashed line shows the analytically expected N → ∞ behaviour, see Eq. (74)for the delocalized phase and Eq. (75) for the localized phase (this part of the dashed line is a guide for an eye.) FromRef. [39]. Clearly, β nn (cid:39) β ( ω ∼ ∆ ), where ∆ is the level spacing. Thus, in the delocalized phase and inthe large- N limit (the condition is N (cid:29) N ξ ) one has N β nn = N ξ / . (74)The coe ffi cient 1 / P (cid:39) N ξ / N at N (cid:29) N ξ . The behavior in the localized phase is discussed in detail inSec. 2.5 below; the result is, according to Eq. (100), N β nn ∼ N µ ( W ) . (75)To characterize the evolution of β nn with the system size N , it is useful to define a disorder-and size-dependent exponent (cf. a similar procedure in Sec. 2.3.1 where ED data for IPR areanalyzed in a similar way): µ nn ( W , N ) = ∂ ln (cid:16) N β nn (cid:17) ∂ ln N . (76)On the delocalized side, N β nn is independent on N at large 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 → ∞ . Onthe localized side, Eq. (75) yields µ nn ( W > W c , N ) → µ ( W ) in the large- N limit.In Fig. 7, ED data for N β nn ( W , N ) (left panel) and the corresponding results for µ nn ( W , N )(right panel) are shown. As expected, for W < W c the µ nn ( N ) curves gradually drift downwards,towards zero, with increasing N . Closer to W c , this drift is in fact non-monotonic (first upward,then downward); the reason for this was discussed in Sec. 2.3.1. Still closer to the critical point,for 15 (cid:46) W < W c only upward drift is observed. This is related to finite–size limitation andthe upward is expected to be superseded by a downward drift at su ffi ciently large N to give µ nn ( W ) → N → ∞ limit. On the localized side, W > W c , numerical simulations yielda nearly N -independent µ nn ( W , N ), in consistency with the expected limiting behavior µ nn ( W > W c , N ) → µ ( W ). 21 WD r P Figure 8: Mean adjacent gap ratio r . Left: r ( W ) at various N . Inset: drift of the crossing point W ∗ with linear system sizeln N . Right: r (ln N ), at various W . From Ref. [29]. Eigenenergies E k of disordered tight-binding models are correlated random variables whosestatistics on d -dimensional lattices has been investigated intensively for several decades. Twomost popular means to characterize the multivariate distribution function of energy levels P ( { E i } )are the statistics of P ( ω ) of spacings between adjacent levels and the two-level correlation func-tion R ( ω ). The energy levels have qualitatively distinct statistical properties in the localized and delo-calized phase of the model. This transition in the level statistics, which becomes a crossover for afinite system size, has been studied in detail in finite- d models [58, 59, 60, 61, 62, 63]. FollowingRefs. [64, 27], we use the ensemble-averaged ratio r = (cid:104) r n (cid:105) of two consecutive spacings, r n = min( δ n , δ n + ) / max( δ n , δ n + ) , (77)which changes from r P = .
386 to r WD = . r for a set of N is shown in Fig. 8, left panel. As expected, theincrease of W produces a crossover from the GOE to the Poisson value. This crossover becomessharper for larger N , remaining rather broad even for N = . This implies that the criticalregime is broad up to the largest N studied.The curves in Fig. 8, left panel apparently demonstrate a crossing point near W =
15. Acloser inspection shows that with N increasing from 2 to 2 , the apparent crossing point driftsfrom W ∗ (cid:39)
14 to W ∗ (cid:39)
16. At the same time, the value of r at the “moving crossing point”gradually decreases towards the Poisson value. This is a manifestation of the localized nature ofthe Anderson transition critical point on tree-like graphs. Close to the lower critical dimension d =
2, the critical point corresponds to weak disorder, and the critical level statistics is close to22 ω ( χ ω − N / − ) N ( − RMT ξ N ξ N diffusive critical − − − ω − χ ( ω ) Figure 9: Level statistics on RRG: Relative level number variance χ ( ω ). Left:
Schematic representation of analyticalpredictions for χ ( ω ) for W < W c in the vicinity of the transition, N (cid:29) N ξ (cid:29)
1, see Eqs. (82), (87) and Eq. (90) for RMT,di ff usive and critical results, correspondingly. Right:
Blue line: ED results for χ ( ω ) ( N = , W = ff usive contribution (87). From Ref. [36]. the WD one (multifractality is weak). With increasing d the critical point moves towards strongdisorder, so that the level statistics approaches the Poisson form (multifractality takes its strongestpossible form) in the limit d → ∞ . As we discussed in Sec. 2.3, the latter limit corresponds totree-like models. Therefore, in contrast to finite– d models, no intermediate true crossing pointfor curves r ( W ) is expected: the crossing point should necessarily drift towards the Poisson limitwith increasing N . This is exactly what we observe in Fig. 8, left panel.A complementary view on the same data is provided on Fig. 8, right panel, where a set ofcurves r ( N ) corresponding to di ff erent W is shown. For moderate disorder ( W < W c ), depen-dence r ( N ) is non-monotonic. This behavior is a simple consequence of the gradual drift of theapparent crossing point which is explained above. Exactly at critical disorder, W = W c , thesystem develops (with increasing N ) the properties, specific to the critical point and, r decreasesdown to r P . On the delocalized side ( W < W c ) in the vicinity of the transition, it behaves as acritical system as long as its linear size log N is smaller than the correlation length ξ ( W ). Thus,for N < N ξ ( W ) ∼ ξ ( W ) , spectral statistics (as well as other observables) develops as at criticality( r decreases with growing N ). As soon as N reaches N ξ ( W ), the system “feels” that it is actuallyin the delocalized phase, and r drifts towards its large- N limit r WD . Thus, N ∼ N ξ marks thepoint of the minimum of r ( N ) curve. There exists a close correspondence between the two-level correlation function R ( ω ) and thevariance Σ ( ω ) of the number of levels I ( ω ) within a band of the width ω , Σ ( ω ) = (cid:68) I ( ω ) (cid:69) − (cid:104) I ( ω ) (cid:105) , (78)which is a convenient characteristics of the rigidity of the spectrum at ω (cid:29) ∆ . In particular, forthe Poisson and GOE statistics the level number variance reads Σ ( ω ) = ω/ ∆ , Poisson , (79) Σ ( ω ) (cid:39) π ln 2 πω ∆ , GOE; ω (cid:29) ∆ . (80)Numerically, it is advantageous to evaluate χ ( ω ) = Σ ( ω ) / (cid:104) I ( ω ) (cid:105) . (81)23or the Poisson statistics it is equal to unity; in the WD case it becomes χ ( ω ) (cid:39) π ∆ ω ln 2 πω ∆ , GOE; ω (cid:29) ∆ . (82)A review of the behavior of R ( ω ) and Σ ( ω ) in a metallic sample of spatial dimensionality d < R ( ω ) = ∆ (cid:88) i j B i j ( E , ω ) , (83)where B i j ( E , ω ) is the correlation function of local densities of states defined by Eq. (54). Usingthe result Eq. (55), the connected part R ( c ) ( ω ) of the level correlation function can be presentedin the form R ( c ) ( ω ) = R ( c )WD ( ω ) + R ( c )di ff ( ω ) , (84)where R ( c )di ff ( ω ) = ∆ π ν (cid:88) r ( m + m r − Re K ( c )1 ( r , ω ) (85)and K ( c )1 ( r , ω ) = K ( r , ω ) −|(cid:104) G R (0) (cid:105)| is the connected part of the Bethe-lattice correlation function(56). For ω (cid:28) N − ξ the correlation function K ( c )1 ( r , ω ) is essentially independent of ω and thuscan be replaced by K ( c )1 ( r , ≡ K ( c )1 ( r ) (cid:39) K ( r ). Using Eq. (45), we see that the sum over r inEq. (85) converges at r ∼
1, yielding R ( c )di ff ( ω ) ∼ N ξ N , ω < N − ξ . (86)The corresponding behavior of the relative number variance χ ( ω ) is χ ( ω ) ∼ N ξ ω. (87)The Eq. (84) suggests that the RMT level statistics describes the level correlations in a ratherwide frequency range when the system is su ffi ciently large, N (cid:29) N ξ . The frequency scale atwhich the level correlation function loses its universal character is determined by comparison ofthe two contributions in Eq. (84). The crossover scale reads ω c ∼ NN ξ ) / . (88)In d < E Th .In the case of RRG, the situations is di ff erent. Indeed, the Thouless energy, is defined as inverseof the time t Th required for particle to di ff usely reach all points of the system. This time scalesonly logarithmically with the volume N of RRG: t Th ∼ D ln N / ln m , (89)where D is the di ff usion coe ffi cient, see Eq. (71).Let us now discuss the spectral statistics right at the transition. As Eq. (64) suggests, thecritical eigenstates are fully correlated only up to the scale ω N , which is smaller that the level24 − − − − − ω . . . . . . χ ( ω ) . . . . . ln ln 1 /ω l n / ( − χ ( ω )) Figure 10:
Left:
Evolution of level number variance χ ( ω ) on the disorder. Solid lines: ED result ( N = ) for delocalizedphase ( W =
8, 10, and 12) and for the critical point ( W = N → ∞ at fixed ω , note that RMT contribution is discardedin this limit. Green dots: W =
10, cyan dots: W = . Right: ln(1 − χ ( ω )) − vs ln ln(1 /ω ) at the critical point. Thestraight line: Eq. (91) with µ (cid:48) =
1. From Ref. [36]. spacing ∆ by a logarithmically large factor ln / N (cid:29)
1. This implies that the level repulsionis significant only for small frequencies ω (cid:46) ω N . Therefore, the level statistics of the RRGmodel at criticality approaches the Poisson statistics with increasing N and the critical levelcompressibility is Poissonian χ ∗ = , (90)in contrast to a finite- d system, for which the critical statistics is intermediate between WD andPoisson forms [67] and 0 < χ ∗ <
1. The approach of χ to the critical value (90) is expected to belogarithmically slow in frequency due to logarithmic dependence of ω N / ∆ on N .The analytical results for the relative level number variance χ ( ω ) for a delocalized RRGsystem close to the metal-insulator transition, N (cid:29) N ξ (cid:29)
1, are summarized in the left panel ofFig. 9.ED numerical results for the spectral correlations are shown in the Figs. 9, right panel and inFig. 10. On the first plot, χ ( ω ) is shown on the log-log scale. The universal RMT and di ff usiveregimes are clearly seen, cf. the left panel of this figure. On the same plot, we show the analyticalprediction (84) which matches numerical result nicely.On the Fig. 10, left panel we show the evolution of χ ( ω ) curve with disorder increasingtowards W c . In the metallic domain ( W < W c ), both RMT and di ff usive regions are observed.The point of the minimum of χ marks the crossover scale, which drifts towards smaller ω withincreasing disorder (i.e., increasing N ξ ). The largest value of χ ( ω ), reached at the right borderof the di ff usive regime, drifts towards the Poisson value, Eq. (90) and the critical regime startsto develop. In order to numerically observe the critical regime, the simulations were performedalso directly at the critical disorder ( W =
18, black curve). The gradual approach of χ ( ω ) to itscritical value Eq. (90) can indeed be described by χ ( ω ) = − c ( χ ) ln µ (cid:48) (1 /ω ) , (91)with µ (cid:48) (cid:39)
1, as shown in the Fig. 10, right panel (it is probable that µ (cid:48) = W =
10) and cyan W = . ω > ω c . Forlower frequencies, the results of ED cross over to the RMT behavior, while the data of Ref. [31]continue to follow the di ff usive behavior, Eq. (87) (the analysis of Ref. [31] was performed inthe limit N → ∞ at fixed ω which does not allow to include the RMT part). In Sec. 2.3.4, dynamical correlations of eigenstates β ( ω ) on RRG were studied in the ergodicphase and in the critical regime. In this subsection, we consider, following Ref. [36], the RRGcorrelation function β ( ω ) in the localized phase. In combination with Sec. 2.3.4, this yields a fulldescription of eigenstates correlations around the localization transition on RRG.Let us start with a qualitative discussion. For W > W c , individual eigenstates are localizedon di ff erent sites with exponentially decaying wave functions and typically overlap very weakly.However, with certain probability two such states form a resonance, which strongly enhances theoverlap. The probability of a resonance is enhanced for small energy di ff erence ω , hence N β ( ω )in the localized phase should decay with increase of ω . For Anderson localization problem in d dimensions, this decay was characterized in Ref. [68], with the result N β ( ω ) ∼ ξ d − d ln d − ( δ ξ /ω ) , ω < δ ξ , (92)where ξ is the localization length, δ ξ ∼ ξ − d is the level spacing in the localization volume, and d the multifractal exponent. It was pointed out in Ref. [68] that the logarithmic enhancement ofcorrelations with lowering ω in Eq. (92) is closely related to the Mott’s law for the ac conductiv-ity. It is not straightforward to translate the result in Eq. (92) to the the RRG model. Equation(92) suggests that the enhancement of correlations for small ω on RRG should be faster thana power of ln ω . As we show below, in the localized phase of the RRG model the function N β ( ω ) has a power–law dependence on ω , with an exponent that is a slowly decaying functionof disorder.Let us first consider the limit of strong disorder W . In this case, almost every single-particlestate is localized within a small localization length ζ around a certain lattice site. Typically, twolocalized states are separated by a distance of the order of system size L = ln N / ln m and havean overlap ∝ e − L /ζ . However, via rare resonant events, two states located far apart may form aresonant pair and strongly hybridise. Such a pair gives a maximal possible contribution to thecorrelation function β ( ω ). Even though such events are rare, they determine the average value β ( ω ) in the case of d -dimensional system with d >
1, see Ref. [68]. This resonant enhancementis reflected by the factor ln d − ( δ ξ /ω ) in Eq. (92). Its role is clearly increasing with increasing d . As discussed below, the power-law scaling of β ( ω ) on RRG is a direct consequence of thisresonance mechanism.Localized eigenstates on a tree-like graph decay as follows [53, 44, 36, 38], see Eq. (48): (cid:104)| ψ ( r ) |(cid:105) ∼ m − r exp {− r /ζ ( W ) } , (93)where r is the distance from the “localization center” of the state and ζ ( W ) is the localizationlength. It behaves as ζ ( W ) ∼ ( W − W c ) − in the vicinity of the transition and ζ ( W ) ∼ / ln( W / W c )for W (cid:29) W c . While Eq. (93) is valid on average, let us assume for a moment that all eigenstatesdecay in this way; this will be su ffi cient to understand the W -dependent power-law in eigenstatecorrelations. 26 − − − − ω N β ( ω ) Figure 11: Correlation of di ff erent eigenstates β ( ω ) for RRG (ED, N = ) and, from cyan to magenta: W = , , W =
18 for the critical point, and W = , ,
42 on the localized side. From Ref. [39].
Consider two eigenstates ψ k and ψ l localized at sites separated by the distance R . The cor-responding overlap matrix element is M ∼ m − R exp {− R /ζ ( W ) } . The optimal condition of theMott-like resonance for two eigenstates with the energy di ff erence ω is M ∼ ω . Under thiscondition, ψ k and ψ l get strongly hybridized: (cid:88) j | ψ k ( j ) ψ l ( j ) | ∼ . (94)The resonant condition can be written as follows R ( ω ) (cid:39) ln(1 /ω )ln m + ζ − ( W ) . (95)The total number of states in a sphere of radius R centered at the state ψ k equals N R ( ω ) ∼ m R ( ω ) ∼ ω − µ ( W ) , (96)where µ ( W ) = ζ ( W ) ln m ζ ( W ) ln m + . (97)The resonant condition implies that energy of one of these states is separated by ∼ ω fromenergy of the state ψ k . Thus, the probability p ω of the resonance in the frequency interval [ ω, ω ]involving the given state ψ k is equal to p ω ∼ ω N R ( ω ) ∼ ω − µ ( W ) . (98)Using the definition (32), we find N ωβ ( ω ) ∼ N (cid:90) ωω d ω (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 ) . (99)In this consideration, the state k is fixed; we have used Eq. (98) for the probability of a resonancein this interval and Eq. (94) for the resonant overlap. As a result, we arrive at the following result: N β ( ω ) ∼ ω − µ ( W ) , (100)27 − − − − ω N β ( ω )
20 25 30 35 40 W . . . . µ ( W ) Figure 12: Correlation of di ff erent eigenstates in the localized phase of RRG (ED, N = ). Left:
Disorder W = β ( ω ) in thelocalized phase, with exponent µ ( W ) depending on disorder. Dots: PD results for W = Right:
Exponent µ ( W )characterizing the frequency dependence of β ( ω ), see Eq. (100). From Ref. [39]. where the exponent µ ( W ) is given by Eq. (97).Let us evaluate asymptotic behavior of the exponent µ ( W ). In the vicinity of the critical point(on the localized side), Eq. (97) gives µ ( W ) → , W → W c + . (101)This matches (up to a logarithmic correction) the critical behavior β ( ω ) ∝ /ω , see second lineof Eq. (66). In the opposite limit of W (cid:29) W c , we find µ ( W ) ∼ W / W c ) , W (cid:29) W c . (102)Thus, µ ( W ) decays to zero at W → ∞ but this decay is logarithmically slow.In our reasoning above, we relied on Eq. (93) which describes the average decay of a wave-function. At the same time, wavefunctions fluctuate strongly; in particular, decay of the typicalwavefunction amplitude is described by a di ff erent localization length [38]. An account of strongfluctuations of eigenstates around the average does not change the main conclusion about thepower-law scaling, Eq. (100) and yields qualitatively the same results for µ ( W ) [39].Figures 11 and 12 show results of numerical evaluation of β ( ω ) by ED of the RRG modelwith the connectivity p =
3, in the vicinity of the band center, E =
0, and for system sizes N inthe range from 2 to 2 . Figure 11 displays the results for β ( ω ) for N = and several disor-der values covering the while phase diagram. In the delocalized phase, W =
10, 12, and 14, thepower-law (1 /ω ) behavior at high frequencies and a saturation at lower frequencies are observed,agreement with Eq. (66); see also the left panel of Fig. 6. The saturation frequency ω ξ decreaseswith approach to the critical point, so that at criticality, W =
18, the power-law (approximately1 /ω ) behavior is observed in the whole 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 adisorder–dependent exponent µ ( W ). These numerical results support the power-law scaling, Eq.(100). The exponent µ ( W ) satisfies Eq. (101) at criticality and gradually decreases towards zeroas W grows, in consistency with Eq. (102).The function β ( ω ) for the localized phase is highlighted in the left panel of Fig. 12. The rightpanel shows the numerically determined exponent µ ( W ), as obtained from the data presented28n the left panel. These values were derived by the fit of numerical data by a pure power-lawdependence (100). At the critical point ( W =
18) the presence of an additional subleadinglogarithmic factor, together with finite-size e ff ects, lead to reduction of the obtained value of µ in comparison with its exact value, Eq. (101).
3. Many-body localization and its connections with localization on RRG
After the discussion of the main properties of the Anderson-localization problem on RRG,we now turn to the problem of many-body localization (MBL). In this section, we will reviewbasic features of the MBL transition in system with various interaction range: from quantumdots (unbounded interaction) to short–range interacting spin chains. As an intermediate case, wewill consider systems with power-law interactions, which interpolate between short and infiniteinteraction ranges. Our main focus will be on those features of the MBL transitions which canbe compared to the corresponding properties of the RRG model and, in some cases, can beinferred from an (approximate) mapping between the RRG and MBL problems. This includes,in particular, the localization criterion (i.e., the scaling of the localization transition) and thestatistical properties of many-body eigenstates and energy levels.
A model of disordered quantum dot can be described by the following Hamiltonian writtenin the basis of exact eigenstates of the non-interacting problem:ˆ H = (cid:88) i (cid:15) i ˆ c † i ˆ c i + (cid:88) i jkl V i jkl ˆ c † i ˆ c † j ˆ c k ˆ c l , (103)where single-particle orbital energies (cid:15) i and interaction matrix elements V i jkl are random quanti-ties, and we consider spinless fermions for simplicity. The single-particle spectrum (cid:15) i is charac-terized by mean level spacing ∆ , while the interaction matrix elements (which are zero in average)by their root–mean–square value V . The ratio of these two energy scales yields a dimensionlessinteraction strength: V ∆ = g − (cid:28) . (104)The parameter g (cid:29) In Ref. [24], a problem of a line width of a hot quasiparticle in a quantum dot characterizedby the Hamiltonian (103) was addressed. The authors pointed out inapplicability of the Fermigolden-rule calculation for decay rate at su ffi ciently low energies. They proposed an hierarchicalFock-space model for this problem and approximately reduced it to a tight-binding Andersonmodel on a Bethe lattice with a large coordination number. In this framework, they concludedthat there is an Anderson localization transition in Fock space as a function of the quasi-particleenergy. This basic physical picture was corroborated and further developed in subsequent works[69, 70, 71, 72, 73, 10]. Related physics was also considered in earlier numerical simulations ofa similar model in the nuclear-physics context [74, 75].29e use now this framework to study properties of typical many-body states at energy E (cid:29) ∆ above the ground state (filled Fermi see). Such a state contains typically ∼ n = √ E / ∆ (cid:29) (cid:15) i ∼ n ∆ . The problem is essentially equivalent toa model with n / n orbitals around the energy corresponding to the middleof the corresponding many-body spectrum. Eigenstates of the non-interacting problem ( V = ff erences with theRRG model (that will be discussed below), let us first discard them and estimate parameters ofthe e ff ective RRG model.The “volume” (number of sites) N of the random graph is determined by the Hilbert-spacevolume of the quantum dot model, N ∼ n . More accurately, there will be subleading prefactorsof power-law type with respect to n in this relation (in particular, since the number of electronsis fixed). They do not a ff ect, however, the leading behavior of ln N :ln N (cid:39) n ln 2 . (105)The coordination number m + n / n / (cid:39) n /
64. Discarding numerical factors, we thus have m ∼ n . (106)Since the energy of each particle in the quantum dot model is ∼ n ∆ , and since the interactionmoves only two particles, the spread of energies of states connected to a given basis state is ∼ n ∆ .This is associated with disorder of the RRG model: W ∼ n ∆ . (107)Finally, the hopping matrix element V of the RRG model is nothing but the interaction matrixelement of the quantum-dot model: V ∼ ∆ g . (108)With the e ff ective RRG model at hand, we can readily determine the corresponding localiza-tion transition point. Specifically, for RRG with hopping V and a large connectivity m +
1, thecritical disorder is given by W c ∼ Vm ln m . (109)Substituting here Eqs. (106), (107), and (108), we find the critical value of the parameter g forgiven n : g c ∼ n ln n . (110)Equivalently, this can be written as an equation for critical n at given g : n c ∼ g / ln − / g . (111)30o clarify the physics beyond Eqs. (109) and (110), it is worth recalling that, in the above identi-fication, W / m is the level spacing of states directly coupled by interaction to a given basis state.The condition V (cid:38) W / m is thus the condition of strong hybridization (i.e. “delocalization”) onthe first step of the hierarchy. The additional factor ln m comes from enhancement of delocaliza-tion in higher-order processes via virtual non-resonant states.This mapping on RRG is, however, only approximate. Indeed the e ff ective graph in the Fockspace of a quantum dot contains short-scale loops, at variance with RRG. For example, considera second-order process, in which we first move two electrons i , j (cid:55)→ k , l and then another pair ofelectrons i (cid:48) , j (cid:48) (cid:55)→ k (cid:48) , l (cid:48) . There is another second order process, in which the same is done in theopposite order. Clearly, we come to the same state, at variance with the tree-like RRG structureon short scales. Putting it di ff erently, there are correlations between amplitudes of higher-orderprocesses in the quantum-dot model that are discarded within the RRG approximation. Thepresence of such correlations is not surprising if one recalls that the number of independentrandom parameters grows exponentially with n in the RRG model but only as a power law of n in the quantum-dot model. This questions the direct applicability of Eq. (110) to the quantum-dot model. However, it turns out [10] that there is a mechanism in the quantum dot model thatgreatly reduces the e ff ect of the above correlations, thereby enhancing many-body delocalizationand making the quantum dot problem much more similar to the RRG problem than one mightthink. This mechanism is the spectral di ff usion: when an electron moves, energies of otherelectrons get modified, which leads to reshu ffl ing of energies of many-body states, thus makingthe problem more “RRG-like”. The corresponding detailed analysis was performed in Ref. [10],with the result that the equation (110) obtained from the mapping to RRG holds with respect tothe leading (power-law) factor but the exponent of the logarithmic factor is possibly modified: g c ∼ n ln µ n , (112)with − ≤ µ ≤
1. The upper bound on µ corresponds to the RRG result (110). The correspondingmodification of Eq. (111) is n c ∼ g / ln ˜ µ g , ˜ µ = − µ , (113)with − / ≤ ˜ µ ≤ /
4. We will discuss the spectral di ff usion as a mechanism of ergodization ofthe system in more detail in Sec. 3.2 in the context of systems with power-law interaction (forwhich the analysis turns out to be somewhat simpler).Recently, a closely related problem is considered in Refs. [76, 77] where a modified SYKmodel of 2 n Majorana fermions χ i was studied. The model is described by following Hamilto-nian: ˆ H = (cid:88) i j J i j ˆ χ i ˆ χ j + (cid:88) i jkl J i jkl ˆ χ i ˆ χ j ˆ χ k ˆ χ l , (114)where two-fermion and four-fermion couplings are random, with zero averages and variances (cid:68) J i j (cid:69) = δ n , (cid:68) J i jkl (cid:69) = J (2 n ) . (115)The system of 2 n Majorana fermions can be also represented as a system of conventional (com-plex) fermions, with n single-particle fermionic states. It is thus expected that the model (114)is essentially equivalent, in what concerns the MBL physics, to the quantum-dot problem (103)31iscussed above. The number of electronic orbitals was denoted by n in both models. Let us es-tablish a correspondence between the remaining parameters of the SYK model and those of theabove quantum-dot model ( ∆ and V , yielding the ratio g = ∆ / V ). The single-particle bandwidthof the SYK model is given by (cid:68) J i j (cid:69) / n / ∼ δ , so that the single-particle level spacing is ∆ ∼ δ n . (116)Further, the typical matrix element is of the order of V ∼ Jn / . (117)We thus find the following expression for the parameter g characterizing the interaction strengthin terms of the parameters of the model (114): g ∼ ∆ V ∼ δ n / J . (118)Therefore, the RRG transition-point condition (110), when expressed in terms of parameters ofthe modified SYK model, reads δ c ∼ Jn / ln n . (119)The authors of Ref. [77] choose at some point the normalization J ∼ n − / . Then Eq. (119)becomes δ c ∼ n ln n , which is exactly the result they find. In other words, the conclusion ofRef. [77] is that the RRG equation (110) [or, equivalently, (119) ] for the critical point holds pre-cisely, including the power of the logarithm. It should be mentioned, however, that the derivationin Ref. [77] employs an approximation of e ff ective-medium type that essentially approximatesthe structure of the graph by a locally tree-like structure. The status of this approximation is notfully clear to us at this stage. As was pointed out above, the analysis in Ref. [10] leaves a win-dow for the power of the logarithm, (112), which translates in the corresponding modification ofEq. (119): δ c ∼ Jn / ln µ n with − ≤ µ ≤
1. This question remains to be fully settled in futurework.Despite this uncertainty in the power of ln n in the formula for the critical disorder strength g c ( n ) [or, equivalently δ c ( n ) in the terminology of the perturbed SYK model], it is clear thatthe MBL transition in the quantum-dot problem is at least closely related to the localizationtransition in the RRG model. It is thus strongly expected that key properties of both phases(localized and delocalized) in the quantum-dot model are the same as on RRG. Specifically, weexpect the delocalized phase to be ergodic, implying, in particular, that asymptotically (for large n ) the IPR scales as − ln P (cid:39) n ln 2 and the level statistics takes the WD form (see also Ref. [78]for a recent related discussion). Further, the localized phase is characterized by P ∼ n limit at fixed value of the ratio g / g c ( n ). We briefly discuss now some of existing numerical results on the quantum-dot model. Theproblem of Fock–space localization on a quantum dot was studied numerically in a number ofpapers approximately two decades ago [69, 79, 80, 81, 82, 83, 84] and very recently in [77]. In32efs. [69, 79, 81] the scaling of the localization threshold n c with the quantum dot conductance g was found to be consistent with the power law n c ∼ g / , as in Eqs. (110) and (112). It shouldbe noted, however, that a logarithmic factor that was not included in the numerical analysis inRefs. [69, 79, 81] may play a quite substantial role for relatively small systems amenable for ED.In addition, we know that finite-size corrections are substantial in the RRG model, and they canbe expected to be equally important in the quantum-dot problem. It is thus not surprising thatan attempt of scaling analysis in Ref. [82] produced a rather broad window for a possible of theexponent characterizing the scaling of n c with g . The paper [80] came to the conclusion that n c ∼ g / but it used a Hamiltonian that di ff ers from Eq. (103) by omission of diagonal interac-tion terms, which suppresses the spectral di ff usion and thus favors localization. Summarizing,additional computational work appears to be needed to verify convincingly the analytically pre-dicted scaling of the MBL transition, Eq. (112). In particular, it would be very interesting to findout whether the exponent µ in the subleading logarithmic factor is consistent with its RRG value, µ = Spin models are very popular for investigation of MBL transitions in systems that are char-acterized by localization in real space in the absence of interaction, see Sec. 3.2 and 3.3 below.It is thus useful to consider also a spin analog of the quantum dot model. Such a “spin quantumdot” model was introduced and analyzed in Ref. [10]. It is defined by the Hamiltonian H = n (cid:88) i = (cid:15) i ˆ S zi + n (cid:88) i , j = (cid:88) α,β ∈{ x , y , z } v αβ i j ˆ S α i ˆ S β j . (120)Here ˆ S α i with i = , . . . , n and α = x , y , z are spin- operators, (cid:15) i are random fields with a boxdistribution on [ − W , W ], and the interactions v αβ i j and independent random variables with zeromean and with root-mean-square value (cid:104) (cid:16) v αβ i j (cid:17) (cid:105) / = V . (121)The model is thus characterized by two dimensionless parameters, n and W / V , and, in full anal-ogy with the fermion quantum dot, there is an MBL transition line in the plane spanned by thesetwo parameters. It is straightforward to generalize the above analysis performed for the fermionicmodel on the spin model. For the magnitude W of the random field and the characteristic value V of the interaction matrix element (that becomes the hopping in the random-graph interpretation),we already used the same notations as in the RRG formula (109). The coordination numberis m ∼ n , since the interaction operator a ff ects two spins. Thus, the analog of the RRG-likeformula (109) for the critical disorder W c reads W c ∼ Vn ln n . (122)As in the case of a fermionic quantum dot, the applicability of Eq. (122) to the spin quantum dotmodel is not obvious, in view of correlations between many-body states that are discarded by33he RRG model. Again, the spectral di ff usion strongly reduces the e ff ect of these correlations,restoring Eq. (122), possibly up to a di ff erent power of the logarithmic factor, ln n (cid:55)→ ln µ n , as inEq. (112) [10]. In this section, we discuss the MBL transition in a system with interaction that decays ac-cording to a power law with distance. A system of fermions with random potential, spatiallylocalized single-particle states, and a power-law interaction can be mapped onto a spin- modelin a random magnetic field with a power-law interaction [85, 86, 87]. We thus start form themodel in spin formulation (which also emerges as an e ff ective model of coupled two-level sys-tems in amorphous materials). The exposition in this section is largely based on Ref. [43] (seealso Refs. [88, 87]). We consider a spin- system described by the following lattice Hamiltonianin d -dimensional space (we assume lattice constant to be equal to unity):ˆ H = (cid:88) i (cid:15) i ˆ S zi + t (cid:88) i j u i j ˆ S zi ˆ S zj + v i j ( ˆ S + i S − j + ˆ S − i ˆ S + j ) r α i j . (123)Here ˆ S zi , ˆ S + i , and ˆ S − i are spin 1 / (cid:15) i are random with the box distribution on[ − W , W ], the interaction parameters u i j , v i j are independent random variables taking the values u i j , v i j = ±
1, and r i j is the distance between the sites i and j . For simplicity, we focus on thelimit of infinite temperature (which essentially means T (cid:29) W ) and measure energy in units of t ,taking t = α characterizing the decay of interaction with distance can in principle take anynon-negative value. The limiting case α = α = ∞ corresponds to the case of a short-range interaction, seeSec. 3.3 below. Here, we focus on an intermediate range, d < α < d . As explained below,the mechanism of many-body delocalization in this range is rather peculiar and permits a quiteintricate connection with the RRG model. We will briefly discuss the whole range of α in Sec. 4. We consider the regime of strong disorder. The starting point is a basis in the many-bodyHilbert space formed by eigenstates of the non-interacting part of the Hamiltonian [first term inEq. (123)]. Each basis state is characterized by definite z components of all spins, S zi = ± / S zi ˆ S zj part of the interaction is taken into account, the energy of each spin ¯ (cid:15) i getsrenormalized by interactions with other spins:¯ (cid:15) i = (cid:15) i + (cid:88) k r − α ik u ik S zk . (124)The ˆ S + i ˆ S − j part of the interaction generates matrix elements between basis states. If energies oftwo spins i and j are su ffi ciently close to each other, (cid:12)(cid:12)(cid:12) ¯ (cid:15) i − ¯ (cid:15) j (cid:12)(cid:12)(cid:12) (cid:46) r α i j , (125)they form a resonance pair. Two levels of this pair that have zero total z projection of spin getstrongly hybridized by interaction. This emergent two-level system is called “pseudospin”, with34he distance the sites i and j being the pseudospin size. The density of pseudospins of size ∼ R reads ρ PS ( R ) ∼ R d − α W . (126)If the interaction decays very slowly, α < d , the density ρ PS ( R ) increases with R . This means thatevery spin finds a divergent number of resonance partners, which implies delocalization. Thisis the “simple” mechanism of delocalization discussed in Ref. [1]. We will focus on the caseof interaction that decays faster, α > d . In this case, the pseudospin density ρ PS ( R ) decreaseswhen R increases. This means that a typical spin does not find any resonant partner at all (sincethe disorder is strong) and that the majority of pseudospins is of the size of the order of latticeconstant. Nevertheless, pseudospins at α > d may drive the many-body delocalization, as we aregoing to discuss.According to Eq. (126), the number of pseudospins of size ∼ L in a system of linear size ∼ L is N ( L ) ∼ L d ρ PS ( L ) ∼ W L d − α . (127)For α < d , the number N ( L ) increases with L . In a su ffi ciently small system, one has N ( L ) (cid:28)
1, i.e., there is essentially no pseudospins of size ∼ L . Pseudospins present in the system areof much smaller size and disconnected from each other, so that the system is in the localizedregime. On the other hand, for a su ffi ciently large system, we get N ( L ) >
1, i.e., there aremultiple pseudospins of size ∼ L . These pseudospins are coupled to each other, and processes oftheir flips lead to many-body delocalization very similar to delocalization of a particle on RRG,as we explain below.The system size at which large (size- L ) pseudospins appear is determined by the condition N ( L ) ∼
1, yielding the disorder-dependent scale [88, 87] L c ( W ) ∼ W d − α . (128)or, equivalently, expressing W through L , W c ( L ) ∼ L d − α . (129)Consider a system of size L a few times larger than L c ( W ). A typical basis state of such systemhas a few pseudospins of size ∼ L , i.e., it is resonantly coupled to a few other basis states byspin-flip interaction matrix elements. Flipping any of these pseudospins leads to another basisstate well coupled to the original one. This new state will in turn possess several pseudospins,so that the process can be repeated. Crucially, by virtue of spectral di ff usion, new resonancesare created in the process of exploring the Fock space via these flips. Indeed, after p steps ofspin flips, a typical distance from a given site i to the closest flipped spin can be estimated as ∼ Lp − / d . Therefore, the spin will experience a shift of the energy ¯ (cid:15) i of the order of [87] ∆ ( p ) ¯ (cid:15) i ∼ L − α p α/ d . (130)In the considered case α > d , the spectral di ff usion thus has in fact a “superballistic” character.This fast increase of ∆ ( p ) ¯ (cid:15) i with p guarantees that resonances are e ffi ciently reshu ffl ed (i.e., newresonances are created at every step), so that a tree-like network of many-body states coupled byresonances emerges [87]. This establishes a connection of the original many-body problem andthe RRG model, leading to the conclusion that systems of sizes L (cid:38) L c ( W ) should be ergodic. In35act, in analogy with the RRG model with large coordination number, the delocalization is furtherenhanced due to higher-order resonant processes that go via intermediate non-resonant states. Infull analogy with Eq. (109), this yields an additional logarithmic factor in W c in comparison withEq. (129), thus resulting in the following prediction for the position of the MBL transition [43]: W c ( L ) ∼ L d − α ln L d , (131)Equation (131) determines a line of the MBL transition in the W – L plane. It is seen that thecritical disorder W c has a power-law dependence on L (with a logarithmic correction), thus di-verging in the limit L → ∞ . Therefore, in order to study properties of the MBL transition, oneshould, with increasing L , simultaneously rescale disorder, i.e., to consider physical observablesas functions of W / W c ( L ). If one fixes W / W c ( L ) at considers the limit L → ∞ , the system will bein the delocalized phase for W / W c ( L ) <
1, at criticality for W / W c ( L ) =
1, and in the MBL phase W / W c ( L ) >
1. This procedure was termed a “non–standard thermodynamic limit” in Ref. [89].
We discuss now the scaling of the average inverse participation ratio P of many-body eigen-states (when considered in the basis of eigenstates of the non-interacting Hamiltonian character-ized by definite z components of all spins).Let us start with the localized phase. As was explained in Sec. 3.2.1, there are in general manysmall-size pseudospins (pairs of resonant spins) in the MBL phase, L < L c ( W ). Specifically,according to Eq. (126), the majority of pseudospins has a size of order unity, and their numberis N PS ∼ L d / W . While these resonances are not su ffi cient to globally delocalize the system, theya ff ect the IPR of many-body wave functions. Each resonance yields a factor of ∼ / − ln P ∼ N PS ∼ L d W . (132)The resulting IPR scaling has a form of fractality of eigenstates in the MBL phase. Indeed, thevolume N of the Hilbert space of the many-body problem is N = L d . (133)Therefore, we can rewrite Eq. (132) as P ∼ N − τ ( W ) , τ ( W ) ∼ W . (134)It is clear from this derivation that such fractal scaling of the IPR with N equally applies to theMBL phase of a system with short-range interaction, see also the corresponding discussion inSec. 3.3.We turn now to the delocalized phase, L > L c ( W ). In view of the connection to the RRGmodel, it is expected that the system becomes ergodic in the large- L limit in the delocalizedphase—i.e., in the limit L → ∞ taken at a fixed value of W / W c ( L ) <
1. This corresponds to theIPR proportional to the inverse volume of the Hilbert space 1 / N , i.e., − ln P (cid:39) L d ln 2 . (135)Finally, we consider the transition point, W = W c ( L ). In view of the relation to localizationtransition on RRG, it is expected that the critical point has localized character, i.e., its properties36 a) (b) Figure 13: Mean adjacent gap ratio r , Eq. (77), characterizing spectral correlations in a spin chain (123) with long-range interaction, α = /
2, as a function of disorder W for various system lengths L . (a) r ( W ) demonstrates ergodicity, r → r WD , at fixed W in the limit L → ∞ ; (b) r ( W ∗ ) plotted as a function of rescaled disorder W ∗ , Eq. (136). The driftingcrossing point is expected to converge, at L → ∞ , to a critical value W ∗ c . In the limit L → ∞ , the system is expectedto be ergodic for W ∗ < W ∗ c and localized for W ∗ > W ∗ c . An extrapolation (together with the data of Fig. 14) yields anestimate W ∗ c ≈ .
3. From Ref.[43]. are obtained continuously from the localized phase, W → W c ( L ) + P ∼ N − τ ( W c ) .Implications of the relation to RRG for the level statistics are straightforward. The levelstatistics is expected to be of WD form in the delocalized phase, reflecting its ergodicity, and ofPoisson form in the localized phase and at criticality.All the above analytical predictions for the model with long-range interaction are supportedby numerical simulations, as we are going to discuss. The MBL transition in the model (123) was studied by numerical simulations (ED) in Refs. [88,43]. We focus on results of Ref. [43] where a detailed analysis of the level and eigenfunctionstatistics around the MBL transition was performed.A 1D spin chain with the interaction exponent α = / r of two consecutive spacings, Eq. (77)—areshown in Fig. 13a. With increasing system size L , the curve r ( W ) rapidly moves to the right, sothat for a fixed W the gap ratio r tends to its ergodic (WD) value r WD = .
530 at L → ∞ . Thisconfirm the analytical expectation that for a fixed disorder W the system is in the ergodic phasein the limit of large L . In order to check the predicted scaling (131) of the critical disorder W c ( L ),results for r are replotted in Fig. 13b as a function of rescaled disorder, W ∗ = WL d − α ln L . (136)The data exhibit now a behavior similar to the one found for the RRG model, see left panel ofFig. 8. The curves become steeper with increasing L and show a crossing point (between curveswith consecutive values of L ). As is seen in the inset, this crossing point drifts towards largervalues of W ∗ , with the drift slowing down when L increases. (The ln L factor in the denominatorof Eq. (136) is important; if it is discarded, the drift accelerates, signalling a divergence at L →∞ .) Thus, the level-statistics data corroborate the analytical prediction (131) for the scaling ofthe MBL transition point: there exists a critical value W ∗ c separating the delocalized, ergodic37 a) (b) Figure 14: Average IPR P of many-body eigenstates of the 1D spin chain (123) with long-range-interaction, α = / L . (a) − ln P as a function of disorder W without rescaling. If W is not too large, the systemreaches ergodicity for these values of L . For large W , the system is still in the localized phase for these system sizes.Inset: − ln P as a function of W for various L . The fractal scaling (132) is clearly seen at large W . (b) − ln P / L α − d as afunction of the rescaled disorder W ∗ , Eq. (136). A crossing point slowly drifting towards larger values of W ∗ is observed.It is expected that this drift converges to the critical value W ∗ c in the limit L → ∞ ; an extrapolation yields an estimate W ∗ c ≈ .
3. From Ref. [43]. phase at W ∗ < W ∗ c from the localized phase at W ∗ > W ∗ c . The finite drift of the crossing point(after a rescaling of disorder to W ∗ ) is analogous to its drift in the RRG model and is related tothe localized nature of the critical point, see Sec. 2.4.1. An extrapolation towards L → ∞ ofthe data for the level statistics, together with those for the eigenfunction statistics (see text belowand Fig. 14b), yields [43] an estimate W ∗ c ≈ . α = /
2. Numerical results for the dependence of IPR P on disorder W are shown in Fig. 14a.For not too strong W , the ergodic behavior (135) is reached already for relatively small systemsizes L accessible to ED. For strong disorder W , the system is still on the localized side of thetransition for these values of L . As shown in the inset of Fig. 14a, the numerical data fullyconfirm the behavior predicted analytically for the localized phase, Eq. (132), i.e., the fractalscaling (134). To determine the position of the transition, in Fig. 14b the rescaled logarithm of theIPR, − ln P / L α − d , is plotted as a function of the rescaled disorder W ∗ , Eq. (136). The rescalingof the vertical axis is such that the plotted quantity increases with L on the delocalized side ofthe transition ( W ∗ < W ∗ c ) and decreases in the localized phase and at criticality ( W ∗ ≥ W ∗ c )according to the analytical results in Sec. 3.2.2. Thus, according to the analytical theory and inanalogy with Fig. 13b for the levels statistics, there should be a crossing point drifting towardslarger W ∗ and converging to W ∗ c at L → ∞ . The data in Fig. 14b fully confirm this expectation.As pointed out above, they were used, together with those from Fig. 13b, to estimate the criticalvalue of the rescaled disorder, W ∗ c ≈ . P ∼ .3. Systems with short-range interaction In this section, we discuss the MBL transition in systems with spatial localization (in theabsence of interaction) and short-range interaction. While originally the corresponding modelswere formulated in terms of fermions [3, 4], the same physics can be addressed in terms of spinmodels. In particular, the S = Heisenberg chain in a random magnetic field has become aparadigmatic models for the investigation of the MBL physics [64, 90]. In its simplest version,the model is equivalent to the fermionic model with random potential and nearest-neighbor inter-action. The model is governed by the Hamiltonian ( S L + ≡ S for periodic boundary conditions) H = L (cid:88) i = S i · S i + − (cid:15) i S zi , (137)with random fields (cid:15) i drawn from a uniform distribution [ − W , W ]. A brief review of approaches to the MBL transition in a model of the type (137) was givenin Sec. 1; we focus here on relations to localization on RRG. Connections of the many-bodyperturbation series for this class of models with the Anderson localization on tree-like graphswas pointed out, or is implicit, in a number of papers, see Refs. [3, 4, 9, 91, 10, 39]. We onlybriefly sketch the key point. Let | j (cid:105) be a typical basis many-body state, i.e., an eigenstate of alloperators S zi . The spin-flipping part of interaction, (cid:80) i ( S + i S − i + + S − i S + i + ), directly connects thisbasis state to ∼ L basis states with energies within the window ∼ W around the energy of thestate | j (cid:105) . Each of these states is obtained from | j (cid:105) by flipping a pair of adjacent spins. Proceedingto the second order, one finds, at first sight, L states. However, almost all of them are formed bytwo remote pairs of flipped spins. Combining amplitudes for flipping these remote pairs of spinsin di ff erent orders, one finds that they largely cancel, so that such processes do not contributeto many-body delocalization, see Ref. [10] and references therein. The spectral di ff usion isnot operative in this case, since remote pairs of spins “do not talk to each other”. As a result,such second-oder processes in fact decouple into two independent pieces (first-order processes).This is fully analogous to short-scale resonances (“pseudospins”) in the MBL phase of modelswith power-law interaction, see discussion in Sec. 3.2. Therefore, to potentially yield a second-order resonance, two spin pairs should form a connected cluster. Consequently, the number ofprocesses that can lead to second-order resonances is ∼ mL , where m does not depend on L . Forthe spin chain (137) one has m ∼
1; a model with parametrically large (but still L -independent) m can be constructed by considering a chain of coupled “spin quantum dots” (120), see Ref. [10].Extending this argument to higher orders, one finds that the number of relevant processes in theorder k is N k ; L ∼ Lm k . (138)In Ref. [91], the parameter m in this formula was denoted by e s , where s was termed “configu-ration entropy per flipped spin of the possibly resonant clusters”. This is the same behavior ason RRG with coordination number m +
1, up to the overall factor L . Thus, for m ∼ W c ∼ , (139)independent of L , as on RRG with m ∼
1. This argument does not include, in fact, a potentiale ff ect of exponentially rare regions with atypically strong or atypically weak disorder [11, 12].39owever, the role of such rare regions turns out to be less important in 1D geometry, and theconclusion about W c ∼ ff ected. (In higher-dimensional systems, the avalancheinstability due to rare regions lead to a slow (slower than any power law) increase of W c withsystem size.) In 1D, the rare regions and the resulting avalanche instability are essential ingredi-ents of phenomenological renormalization-group approaches to the scaling at the MBL transition[13, 14]. The fact that the asymptotic critical behavior at the MBL transition in systems with spa-tial localization and short-range interaction is di ff erent from that on RRG is clear already fromthe fact that the RRG value of the exponent ν del = / ν ≥ / d . According to the analytical arguments discussed in Sec. 3.3.1, the model (137) undergoesan MBL transition at a critical disorder W c that stays finite in the thermodynamic limit L → ∞ .ED studies in systems up to L =
24 yield estimates W c = . . W with increasing system size.This behavior is similar to that seen for the RRG model. Indeed, the experience with RRGteaches us that the actual (thermodynamic-limit) critical disorder is considerably larger, due tofinite-size e ff ects, than the value suggested by the ED. Specifically, for RRG the ED wouldsuggest W c ≈
15 [e.g., as the position of the crossing point in spectral statistics, Fig. 8 (leftpanel), or based on maximum correlations of adjacent eigenstates, see Fig. 7 (right panel)], whilethe actual value is W c = .
17, see analysis in Sec. 2.2. In genuine interacting MBL models(such as the random-field spin chain), finite-size e ff ects are expected to be further enhanceddue to rare-region physics. This expectation has been supported by an analysis based on thetime-dependent variational principle with matrix product states that was carried out in Ref. [20],where the dynamics (imbalance relaxation) was explored for systems of much larger size, up to L = W c with system size L was found, whichhas, however, essentially saturated at L = W c ≈ . L → ∞ ). Interestingly, anadvanced ED procedure supplemented by an extrapolation to L → ∞ in recent Ref. [94] resultedin a very similar estimate W c ≈ . ν are in the range0 . −
1, in conflict with the Harris criterion that requires ν ≥
2. This shows that the system sizesthat can be treated via ED are too small to access the asymptotic critical behavior. Estimatesin Refs. [17, 18, 19] indicate that spin chains of length L (cid:38)
50 – 100 are needed to access theultimate critical scaling. Interestingly, the values of the index ν obtained on the basis of ED dataare quite close to the value ν del = / ff ects in small systems make the apparent ν del larger, see Sec. 2.2. Thus, systems of intermediatesizes (relevant to experiments) may exhibit the critical behavior similar to that in the RRG model. We begin the discussion of properties of exact many-body eigenstates ψ k by the analysis oftheir most conventional characteristics: the inverse participation ratio (IPR) P (defined as beforein the basis of eigenstates j of all S zi operators), P = (cid:68)(cid:80) j | ψ k ( j ) | (cid:69) .40he volume of the many-body space is N ∼ L . More accurately, if we take into account thatthe total spin is conserved and consider the sector with total S z =
0, the volume of the many-bodyHilbert space is found to be N = L ! / [( L / (cid:39) L √ /π L . The subleading power-law factordoes not, play, however, any essential role.At strong disorder W (cid:29) ∼ N L / W ∼ L / W . The number of second-order resonances is ∼ N L / W ∼ L / W , and higher-order resonances are suppressed by still higher powers of W , so that they all can be neglected.Exactly like in the case of a long-range interaction, Sec. 3.2.2, each of first-order resonancesbrings a factor ∼ / P . This results in a fractal scaling of the IPR [10, 43] that has thesame form as Eq. (134): P ∼ N − τ ( W ) , τ ( W ) ∼ W . (140)This fractal scaling of IPR was indeed found numerically [96, 16] in the MBL phase of the spin-chain model (137). Furthermore, the numerical analysis in Ref. [16] led to the conclusion thatthis behavior holds up to the critical point, with τ ( W c ) = lim W → W c + τ ( W ). This means that thecritical point of the MBL problem has properties of the localized phase, in analogy with the RRGmodel. Also, it was found in Ref. [16] that the scaling τ ( W ) ∼ / W that is derived analyticallyfor W (cid:29) W c , with τ ( W c ) ≈ . P ∝ N . (141)Indeed, the ergodic scaling (141) of the IPR was found in numerical simulations (ED) in Ref. [16]. Let us now consider correlations of di ff erent many-body eigenstates. For the RRG model,the corresponding dynamical correlation function β ( ω ) was analyzed in Sec. 2.3.4 and 2.5. Werecall that β ( ω ) is the average overlap of two eigenstates separated in energy by ω , β ( ω ) = (cid:42)(cid:88) j | ψ k ( j ) ψ l ( j ) | (cid:43) , ω = E k − E l . (142)It was found in Ref. [39] that properties of the correlation function β ( ω ) for the MBL problemare remarkably similar (although not identical) to those for RRG. We briefly review below thecorresponding analytical and numerical results.We begin with analytical consideration. On the delocalized side of the transition, the ergodicbehavior is expected, which should have the same form as the first line of Eq. (66) derived forthe RRG model: N β ( ω ) ∼ N ξ , ω < ω ξ . (143)Here N ξ , which has a meaning of Hilbert-space correlation volume, depends on disorder W only(i.e., does not depend on N and on ω ). For higher frequencies, ω > ω ξ , a critical behavior isexpected, in analogy with the second line of Eq. (66). As we discuss below, it is characterizedby a power-law dependence of β ( ω ) on ω , as on RRG.41 − − − − ω β ( ω ) N Figure 15: Dynamical eigenstate correlation function β ( ω ) for spin chain of size L =
16 and disorder W = . , . , , , , ,
10 (from cyan to magenta). The three smallest values of W are well in the ergodic phase, the nexttwo are also on the delocalized side but correspond to the critical regime for system sizes accessible to ED, and the twolargest values are in the MBL phase. The figure is a counterpart of Fig. 11 for the RRG model. From Ref. [39]. We turn now to the analysis of eigenstates correlations in the MBL phase. It largely parallelsthe corresponding calculation for the RRG model, Sec. 2.5, with the correlation function β ( ω )being governed by Mott–type resonances. The result reads [39] N β ( ω ) ∼ ω − µ ( W ) (log N ) / N τ ( W ) . (144)Equation (144) has largely the same form as Eq. (100) for RRG; the key factor is the power-law dependence on frequency, ω − µ ( W ) , with a disorder-dependent exponent µ ( W ). Specifically, µ ( W ) decreases logarthmically at large W as in the RRG model, Eq. (102). The additional factor N − τ ( W ) in Eq. (144) originates from the suppression of resonant overlap, which is of order unityin the RRG model, see Eq. (94) and becomes (cid:88) j | ψ k ( j ) ψ l ( j ) | ∼ N − τ ( W ) (145)for the genuine many-body problem (spin chain), for the same reason as the IPR, Eq. (140).The exponent τ ( W ) is parametrically small ( ∼ / W ) in the MBL phase and remains quite smallnumerically at the critical point [ τ ( W c ) ≈ . ff erence between the results for the RRG and spin-chain models is not so significant.We discuss now ED results for the dynamical eigenstate correlations in the model (137),which corroborate analytical expectations. In Fig. 15, the correlation function β ( ω ) is shown fordisorder strength from W = . W =
10, i.e., across the MBL transition. The behavior of β ( ω )is very similar to that for the RRG model, Fig. 11. For relatively weak disorder, W = .
5, 1.7,2 (i.e., deeply in the delocalized phase), β ( ω ) exhibits a power-law critical behavior at higherfrequencies and saturates at lower frequencies, in agreement with the expectation (143). The sat-uration value is fully consistent with the N -independence of N ξ in Eq. (143), thus confirming theergodicity of the delocalized phase, as discussed below. For W = β ( ω ) shows a clear bending towards saturation at small ω but the saturation is not reached. Thisimplies that these values of W are on the delocalized side of the transition (in the limit L → ∞ )but the accessible system sizes are insu ffi cient to observe ergodic behavior: for these values of42 − − − − ω β ( ω ) N W . . . . µ ( W ) Figure 16: Dynamical eigenstate correlations for spin chain of size L = Left:
Correlation function β ( ω ) for disorderstrengths W = , , ,
10 (from cyan to magenta).
Right:
Disorder dependence of the exponent µ ( W ) governing thepower-law scaling β ( ω ) ∝ ω − µ ( W ) , see Eq. (144). The figure is a counterpart of Fig. 12 for the RRG model. From Ref.[39]. L the system is still in the critical regime. At strong disorder, W = β ( ω )shows a power-law scaling in the full range of ω , as predicted for the MBL phase, Eq. (144).Figure 16 displays the data for stronger disorder, from W = W c ) to W =
10 (well in the MBL phase). It is very similar to its RRG counterpart, Fig. 12.Data in the left panel clearly demonstrate the power-law frequency scaling β ( ω ) ∝ ω − µ ( W ) , inagreement with Eq. (144). The disorder dependence of the exponent µ ( W ) is shown in the rightpanel. As for the RRG model [see Sec. 2.3.6], it is useful to consider the correlation function ofadjacent-in-energy eigenstates, Eq. (73). The corresponding data are presented in Fig. 17, whichis very similar to its RRG analog, Fig. 7. In the left panel of Fig. 17, the function β nn ( W ) isshown in a broad interval of disorder W across the MBL transition, for several values of L . For W < W c , the data approach, with increasing L , a limiting curve [ N ξ ( W ) in Eq. (143)], whichis a manifestation of the ergodic character of the delocalized phase. For values of L accessibleto ED, this large- L ergodic behavior is reached for disorder strengths W (cid:46)
2. As for the RRGmodel, the β nn ( W ) curves have a maximum near W ≈
3. It serves as a finite-size estimate for thetransition point, drifting slowly towards the true ( L → ∞ ) value of W c .The right panel of Fig. 17 presents the flowing exponent µ nn defined by Eq. (76) (with thereplacement N → N ). This figure is also similar to its RRG counterpart, right panel of Fig. 7.There is, however, a di ff erence: while for the RRG model the maximum value of µ nn for thelargest L is unity with a good accuracy (as expected analytically), for the spin-chain model themaximum value is ≈ .
75. This is partly due to finite-size e ff ects (which are stronger for the spinchain), but there is also a deeper reason. In full analogy with the IPR scaling at the transitionpoint, P ∼ N − τ ( W c ) [see Eq. (140)], we expect the overlap of two adjacent states at criticality toexhibit the same scaling, N β nn ∼ N − τ ( W c ) [cf. Eq. (145)], so that µ nn ( W c ) = − τ ( W c ) . (146)Thus, in the limit N → ∞ , we have µ nn ( W c ) ≈ .
8, which is the maximum of µ nn ( W ). This resultcan be also obtained from Eq. (144) by extending it from the MBL phase to the transition point43 W β nn ( W ) W c N W . . . . . . µ nn ( W ) W c µ nn,max = 1 − τ ( W c ) Figure 17: Correlation of adjacent eigenstates for spin chain with system sizes L = , , ,
18 (from cyan to magenta).
Left:
Correlation function β nn ( W ). Dashed vertical line marks an estimated value of the MBL transition in the thermody-namic limit, W c (cid:39)
5, as obtained by an approach based on matrix-product states for large chains [20].
Right:
Exponent µ nn that characterizes the N scaling of adjacent-state correlation function, see Eq. (76) (with the replacement N → N ).Dashed line shows expected N → ∞ behavior, see Eq. (74) for the ergodic phase and Eq. (75) for the MBL phase (thissection of the line is schematic). The figure is a counterpart of Fig. 7 for the RRG model. From Ref. [39]. and setting µ ( W c ) = ω ∼ / N in Eq. (144),one gets a relation between the exponents in the MBL phase, µ nn ( W ) = µ ( W ) − τ ( W ) . (147)At strong disorder, the exponent τ ( W ) is small (it decays with increasing disorder as ∼ / W ,while µ ( W ) decays only logarithmically), so that τ ( W ) (cid:28) µ ( W ) and thus µ nn ( W ) ≈ µ ( W ) . (148)Numerically, this remains valid with reasonable accuracy up to the transition point, since τ ( W c )is rather small. It should be also mentioned that logarithmic corrections to scaling, such as thelogarithmic factor in Eq. (144), as well as further finite-size e ff ects, significantly a ff ect numericalvalues of MBL-phase exponents as obtained by ED.The expected behavior of µ nn ( W ) in the limit L → ∞ is shown by a dashed line in the rightpanel of Fig. 17. In analogy with the β nn peak, the position of the maximum of µ nn yields a finite-size estimate for the critical point and drifts, with increasing L , towards the true (thermodynamic-limit) transition point W c . The drift is approximately linear with system size L ; the accessiblesystem sizes are way too small to allow for a reliable estimate of the L → ∞ critical disorder W c .In similarity with the RRG model, a substantial part of the delocalized phase belongs to a broadcritical regime, 2 . (cid:46) W (cid:46)
5, for sizes L accessible to ED. Finally, we briefly discuss connections between the eigenstate correlation function β ( ω ) andother dynamical observables. In particular, Ref. [97] (see also a recent paper [98]) studied matrixelements of local (in real space) operator S iz , F ( ω ) = (cid:28)(cid:12)(cid:12)(cid:12)(cid:12)(cid:16) S zi (cid:17) kl (cid:12)(cid:12)(cid:12)(cid:12) (cid:29) ≡ (cid:68) |(cid:104) ψ k | S zi | ψ l (cid:105)| (cid:69) , (149)as a function of the frequency ω = E k − E l . The ω → t Fourier transform of Eq. (149) has ameaning of the return probability in real space. At the same time, the Fourier transform of β ( ω )44 d d α ∞ quantum dot short-range interaction L ln α − d L L ln √ c e ∼ c W α L µ ln α − d L Figure 18: Evolution of L -scaling of the critical disorder W c of the MBL transition in the spin model (123) with thepower-law-interaction exponent α . The range d ≤ α < d is considered in Sec. 3.2, and the corresponding criticaldisorder is given by Eq. (131). Extreme cases are the limits of infinite-range interaction ( α =
0, quantum dot, Sec. 3.1)and of short-range interaction ( α = ∞ , Sec. 3.3). In the range 0 ≤ α < d the mechanism of ergodization is analogous tothat in the quantum dot ( α =
0) model discussed in Sec. 3.1. In the range 2 d ≤ α < ∞ , the delocalization is expected totake place due to rare ergodic spots [12, 99, 100]. Adapted from Ref. [43]. yields the return probability in the many-body space p ( t ) [cf. Sec. 2.3.5] , which is clearly verydi ff erent in general. Indeed, in the ergodic phase the behavior of β ( ω ) and F ( ω ) is very di ff erent:while β ( ω ) is a constant at not too large frequency, F ( ω ) has a power-law behavior reflectingdi ff usive or subdi ff usive transport. Let us note that, contrary to β ( ω ), the function F ( ω ) doesnot have a direct counterpart in the RRG model, since the latter mimics the many-body spacebut not the real space. Nevertheless, there is a remarkable similarity in the behavior of β ( ω ) and F ( ω ) in the MBL phase (and at criticality): they both exhibit a power-law dependence on ω ,with a continuously varying exponent. In Ref. [91], a related power-law frequency scaling of theconductivity (which is another real-space-related observable) was found, σ ( ω ) ∼ ω α , with theexponent α varying in the range 1 < α < β ( ω )) and in the real space (like F ( ω ) or σ ( ω )) areclosely related in the MBL phase, since both these classes of correlation functions are governedby Mott-type resonances. More work is needed to better understand these relations.
4. Summary
In this article, we have reviewed properties of Anderson localization in the RRG model and itsconnection to a broad class of MBL problems, i.e. to interacting many-body models that exhibita localization transition in Fock space. For models that are in the spatially localized phase inthe absence of interaction, this MBL transition manifests itself also in real-space localizationproperties.The RRG model has a great advantage that it can be treated analytically in a controllableway within a field-theoretical approach using supersymmetry. Many of the physical observablescharacterizing eigenstate and level statistics can be then expressed in terms of a solution of a self-consistency equation. Key properties of the solution of this equation are understood analytically.Furthermore, this equation can be very e ffi ciently solved numerically by means of PD. Remark-ably, this allows one to proceed e ff ectively to system sizes N (Hilbert-space volume) as huge as N ∼ , which is many orders of magnitude larger than systems that can be studied via ED( N (cid:46) ). This permits to determine the position of the critical point with an outstanding accu-racy; in particular, for the most frequently studied model with coordination number m + = W c = . ± . • The delocalized phase, W < W c , is ergodic. The ergodicity implies, in particular, the 1 / N asymptotic scaling of the IPR P as well as the universal, WD form of the level statistics(for not too large frequencies ω ). In the vicinity of the transition, the ergodicity is reachedfor system sizes N (cid:29) N ξ , where ξ is the correlation length and N ξ ∼ m ξ is the correlationvolume. The correlation length diverges according to a power law, ξ ∼ ( W c − W ) − ν del , withthe critical index ν del = , so that N ξ diverges exponentially fast. In view of this, there isa sizeable range of disorders on the ergodic side of the transition for which the condition N (cid:29) N ξ cannot be reached with su ffi cient margin in ED, so that the ED data can not fullyreveal the ergodic behavior. • The critical point of the Anderson transition, W = W c has a localized character. Theleading behavior of various observables at criticality can be obtained by an extrapolationfrom the localized phase to the critical point, W → W c +
0. For example, the IPR inthe critical point is P ∼ d dimensions where the corresponding behavior is in a sense intermediatebetween those in localized and delocalized phases. • The localized character of the critical point leads to strong finite-size e ff ects. A systemin the ergodic phase ( W < W c ) but close to the transition first evolves, with increasingsize N , towards criticality, i.e., towards localized behavior of energy-level and eigenstatestatistics. Only when N exceeds N ξ , the flow changes direction and the system startsevolving towards ergodicity. This non-monotonic N -dependence of various observablesmanifests itself in a substantial drift of the apparent transition point (as obtained on thebasis of ED data) towards stronger disorder with increasing system size.In Sec. 3, we have briefly reviewed some of main properties of the MBL transitions. Wehave considered three classes of models. In Sec. 3.1, “quantum-dot” models were discussed,with single-particle states spread over the whole system. In these models, the MBL transitionhappens in the Fock space only. Sections 3.2 and 3.3 deal with models that are characterized bysingle-particle states that are localized in real space, with long-range and short-range interaction,respectively. For each of these three classes, there are closely related models formulated in termsof fermions and in terms of spins. Since our main focus is on relations between the RRG andMBL problems, we discussed those observables in MBL models that have direct counterparts inthe RRG model. These include many of key properties of the MBL systems, such as the scalingof MBL transition points, level statistics, as well as fluctuations and correlations of many-bodyeigenstates. For all of the considered MBL models, there are strong connections with the RRGproblem, and in some cases approximate mapping to RRG is used to infer physical properties.Main results can be summarized as follows: • For the quantum-dot model (or, equivalently, the SYK model perturbed by a kinetic-energyterm), Sec. 3.1, the Fock-space MBL transition takes place in the plane spanned by twodimensionless parameters: the number n (cid:29) g − = V / ∆ (cid:28) V to the single-particle level spacing ∆ .46he model has approximately a structure of an RRG model with the Hilbert-space volume N ∼ n , disorder strength W ∼ n ∆ , hopping V ∼ ∆ / g and coordination number m + ∼ n .The RRG position of the critical point, W / V ∼ m ln m , translates, in this approximation,into the following line of the MBL transition in the g – n plane: g ∼ n ln n , Eq. (110). Infact, the accuracy of this approximation is not fully clear at this stage. The reason is that amodel with two-body interaction involves correlations between many-body states coupledby interaction that are not present in the RRG model. It turns out, however, that spectraldi ff usion e ffi ciently reduces the e ff ect of these correlations in the quantum-dot problem.Specifically, the analysis in Ref. [10] comes to the same result as the RRG model suggestsbut with possibly smaller power of the logarithm: g ∼ n ln µ n with µ ≤
1. Furthermore,Ref. [77] proposes that the RRG result even holds exactly, including the power of thelogarithm. The argument is based, however, on an approximation of e ff ective-mediumtype that, in our view, has to be studied more accurately. Independently of these details,the MBL transition in the quantum-dot (perturbed-SYK) model is closely related to thelocalization transition on RRG. This concerns not only the scaling of the MBL transitionbut also properties of both phases and of the critical point. • In Sec. 3.2, we have considered the MBL transition in a model with random long-range in-teraction decaying with distance as 1 / r α . While we discussed (following Ref. [43]) a spinmodel with random interaction and random on-site magnetic field, interacting fermionicmodels can be (approximately) mapped on a spin model of this type and have similarproperties. Parameters characterizing the model are the linear size L , the spatial dimen-sionality d , the power-law exponent α , and the strength W of the random field (normalizedto the nearest-neighbor interaction strength). Depending on the value of the exponent α ,there are three distinct situations, see Fig. 18. For α < d , the number of direct resonancesthat each spin finds increases without bound with increasing L . This can be viewed as a“quantum-dot-like” situation. The analysis can be performed analogously to the quantum-dot model, with the critical disorder of the MBL transition behaving like W c ∼ L d − α ln µ L .For d < α < d , only rare spins form resonances (“pseudospins”). At the same time,the total number of such pseudospins increases with L , so that they eventually proliferateand start to interact. It turns out that the system of interacting pseudospins can be approxi-mately mapped on the RRG model, yielding the result for the critical point W c ∼ L d − α ln L ,Eq. (131). Finally, for α > d , such mechanism of delocalization is not operative. Thisregime is similar to the case of short-range interaction. Delocalization at large L (and fixedlarge W ) is expected to happen due to rare events only (as in short-range-interaction mod-els in d > W c increases with L in a very slow fashion (more slowly than anypower law).We have focussed in Sec. 3.2 on the case d < α < d . The mapping of the MBL model withlong-range interaction with such α onto RRG model strongly suggests that most of keyproperties of the RRG model hold also for the MBL problem. This includes, in particularthe ergodic character of the delocalized phase and the localized nature of the critical point W c (i.e., properties of the critical point are obtained continuously by an extrapolation W → W c + ff erence between the MBL models that involvelocalization in real space and their RRG counterparts. The scaling of the IPR in the local-ized phase of the MBL problems is of multifractal form, P ∼ N − τ ( W ) , with the disorderdependent exponent τ ∼ / W , at variance with P ∼ / W . While such resonances are not su ffi cient to establish delo-calization for W > W c , each of them yields a factor ∼ / P , thus leading to a fractalscaling. This equally applies to models with power-law interaction (with α > d ) and withshort-range interaction.The analytical predictions resulting from the mapping of a model with d < α < d onRRG have been verified by numerical simulations (for d = α = / W c ( L ).Further, numerical results for the IPR scaling confirm the ergodicity of the delocalizedphase and the fractal scaling of IPR in the localized phase. • In Sec. 3.3 we discussed the MBL problem with a short-range interaction. Specifically, wehave focussed on 1D S = spin chain in random field that represents the most popularMBL model. For a short-range interaction model, the connection to RRG is somewhatmore delicate than in the case of, e.g., quantum-dot model. Indeed, a number of states towhich each basis many-body state (eigenstate of the non-interacting problem) is connectedby interaction grows ∼ L , where L is the chain length. At first sight, one could think that themodel is similar to RRG with a coordination number ∼ L . This is not true, however in viewof a combination of two reasons. First, there are strong correlations between states coupledby the interaction. Second, distant resonances “do not talk to each other” in a model withshort-range interaction, so that the spectral di ff usion does not play a role that it plays inthe quantum-dot model. The number of resonances growing as ∼ L leads only to fractalbehavior of IPR, P ∼ N − τ ( W ) , as in the case of power-law interaction. At the same time,the number of potential resonances that can be responsible for many-body delocalizationgrows with the order n of the perturbation theory as N n ; L ∼ Lm n , with m ∼
1, Eq. (138).The delocalization is thus similar to that on RRG with a coordination number m ∼
1, sothat the critical disorder is L -independent, W c ∼
1, in the limit L → ∞ . This argumentdoes not include a possible e ff ect of rare ergodic regions. However, in 1D geometry, rareregions do not a ff ect the conclusion about the L -independence of W c at large L (althoughthey do a ff ect the exact value of W c and probably the critical behavior).Numerical results support the analytical conclusion that W c saturates at an L -independentvalue in the large- L limit. Furthermore, they indicate that qualitative properties of the MBLtransition are in many key respects analogous to those of the RRG model. Specifically, thedelocalized phase is found to be ergodic, the critical point has the localized characacter,and the apparent critical disorder W c exhibits a sizeable drift towards larger values withincreasing L . Of course, the ED is limited by systems of moderate length (typically L ≤ W c belongs to a critical region.As an important characteristic of the many-body state, we have analyzed in Sec. 3.3 dy-namical correlations of eigenstates β ( ω ) across the MBL transition. The correlations reveala remarkable similarity between the MBL and RRG problems. In particular, the Figs. 15,16, and 17 for the spin-chain problem are very similar to their RRG counterparts, Figs. 11,12, and 7. On the delocalized side of the MBL transitions, these correlations support (alongwith various other observables) ergodicity of the system. On the MBL side, W > W c , theyyield a power-law dynamical scaling, β ( ω ) ∝ ω − µ ( W ) , with a disorder-dependent exponent µ ( W ), in analogy with the RRG problem.Clearly, the RRG model is not equivalent to any of the MBL models discussed above. There48re many observables in the MBL problems (such as properties of single-particle excitationsor real-space observables) that do not have direct counterparts in the RRG model. Indeed, theHilbert space of the RRG problem, although mimicking important features of the Fock space ofthe many-body problem, is not the Fock space. For the same reason, notions of rare ergodic orlocalized spots that play an important role in phenomenological theories of the MBL transitiondo not have their counterparts in the RRG model either. The RRG critical behavior cannot bethe asymptotic behavior for MBL problems since it would violate the Harris criterion. All thesedi ff erences are manifestations of the fact that the RRG model is only a toy model representinga simplistic version of the genuine MBL problem. Nevertheless, as was discussed in the presentreview and emphasized in this section, the RRG model captures many key physical propertiesof the MBL problem. In models with long-range interaction, the relation between the MBL andRRG problems can be even made quantitative.
5. Acknowledgments
We acknowledge collaboration with M. Skvortsov on Ref. [29], which was the paper thatstarted our investigations of the RRG model and its relations to the MBL problem. In courseof these studies, we enjoyed useful discussions with many colleagues, including A. Altland, A.Burin, I. P. Castillo, F. Evers, M. V. Feigelman, A. Knowles, V. E. Kravtsov, N. Laflorencie, G.Lemari´e, F. L. Metz, A. Scardicchio, M. Serbyn, and M. Tarzia.
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