Anomalous dynamics in the ergodic side of the Many-Body Localization transition and the glassy phase of Directed Polymers in Random Media
AAnomalous dynamics in the ergodic side of the Many-Body Localization transitionand the glassy phase of Directed Polymers in Random Media
G. Biroli , and M. Tarzia Laboratoire de Physique de l’Ecole Normale Sup´erieure, ENS Universit´e PSL, CNRS, Sorbonne Universit´e, Universit´e de Paris, F-75005 Paris, France LPTMC, CNRS-UMR 7600, Sorbonne Universit´e, 4 Pl. Jussieu, F-75005 Paris, France Institut Universitaire de France, 1 rue Descartes, 75231 Paris Cedex 05, France
Using the non-interacting Anderson tight-binding model on the Bethe lattice as a toy model forthe many-body quantum dynamics, we propose a novel and transparent theoretical explanation ofthe anomalously slow dynamics that emerges in the bad metal phase preceding the Many-BodyLocalization transition. By mapping the time-decorrelation of many-body wave-functions onto Di-rected Polymers in Random Media, we show the existence of a glass transition within the extendedregime separating a metallic-like phase at small disorder, where delocalization occurs on an expo-nential number of paths, from a bad metal-like phase at intermediate disorder, where resonancesare formed on rare, specific, disorder dependent site orbitals on very distant generations. Thephysical interpretation of subdiffusion and non-exponential relaxation emerging from this picture iscomplementary to the Griffiths one, although both scenarios rely on the presence of heavy-taileddistribution of the escape times. We relate the dynamical evolution in the glassy phase to thedepinning transition of Directed Polymers, which results in macroscopic and abrupt jumps of thepreferred delocalizing paths when a parameter like the energy is varied, and produce a singularbehavior of the overlap correlation function between eigenstates at different energies. By comparingthe quantum dynamics on loop-less Cayley trees and Random Regular Graphs we discuss the effectof loops, showing that in the latter slow dynamics and apparent power-laws extend on a very largetime-window but are eventually cut-off on a time-scale that diverges at the MBL transition.
I. INTRODUCTION
The field of Many-Body Localization (MBL) startedabout 10 years ago with the work of Ref. [1] which studiedthe stability of the Anderson insulator with respect to theaddition of interactions via the so-called self-consistentBorn approximation for the one-particle Green’s func-tions, showing that isolated disordered many-body sys-tems can fail to thermalize even at finite energy density ifthe disorder is strong enough. MBL is a purely quantumphenomenon which occurs due to Anderson localizationin the Fock space as the result of the interplay of disorder,quantum fluctuations, and interactions, and gives rise toa completely new mechanism for ergodicity breaking:
Differently from (quantum or classical) integrable sys-tems, the MBL phase is stable to perturbations; Differ-ently from (classical or quantum) phase transitions, it isnot associated to any spontaneous symmetry breaking,and occurs without any signature in the static observ-ables (and in isolated systems only); The MBL state isalso different from—although it shares some similaritieswith—classical or quantum glasses; for instance, it es-tablishes also in 1 d models characterized by a not “too”complex energy landscape, and at infinite temperature.In fact the existence of the MBL transition was pre-dicted by Althsuler et al. in a seminal paper already 10years before the breakthrough of Refs. [1,8], by puttingforward a paradigmatic representation of MBL in termsof single-particle Anderson localization in Fock space. Inorder to explain this analogy, let us focus on the following disordered Ising spin chain as a reference model H MB = N (cid:88) i =1 (cid:0) J i ˆ σ zi ˆ σ zi +1 + h i ˆ σ zi (cid:1) + N (cid:88) i =1 Γ i ˆ σ xi , (1)with h i i.i.d in [ − h, h ], for which the existence of theMBL transition has been proven rigorously (under theminimal assumption of absence of level attraction). If onechooses as a basis the tensor product of the simultaneouseigenstates of the operators σ zi , the Fock space of themany-body Hamiltonian is a N -dimensional hyper-cubeof V = 2 N sites. The first part of H MB is by definitiondiagonal on this basis. Its diagonal elements correspondto correlated random energies associated to the sites ofthe hyper-cube: (cid:104){ σ zi }| N (cid:88) i =1 (cid:0) J i ˆ σ zi ˆ σ zi +1 + h i ˆ σ zi (cid:1) |{ σ zi }(cid:105) = (cid:15) ( { σ zi } ) , (2)while the interacting part of the Hamiltonian induces sin-gle spin flips on the configurations { σ zi } :Γ i ˆ σ xi | σ z , . . . , σ zi , . . . , σ zN (cid:105) = Γ i | σ z , . . . , − σ zi , . . . , σ zN (cid:105) , (3)and leads to hopping connecting neighboring sites of thehyper-cube. The many-body quantum dynamics canthen be thought as a tight binding model on a very high-dimensional disordered lattice. In large spatial dimen-sions the neighbors of a given site are organized in apeculiar way: their number grows very rapidly with thedistance and short loops among them are rare. Sincethese are distinctive features of tree-like structures, the a r X i v : . [ c ond - m a t . d i s - nn ] M a r authors of Ref. [7] argued that the (non-interacting) An-derson model on the Bethe lattice, originally introducedand studied in Ref. [10], can be used as a toy model forMBL (see also Refs. [8, 11, and 12] for a similar analy-sis and Ref. [13] for a quantitative investigation of suchmapping).Based on this idea, the existence of three distinctregimes was suggested. At strong disorder the many-body eigenfunctions are exponentially localized aroundsome specific site orbitals in the configuration space andare weak deformations of the non-interacting states: Thesystem is a perfect insulator (i.e., conductivity is strictlyzero) and is not ergodic, on-site energies are “good”quantum number (akin to the so-called local conservedquantities ), and the level statistics should be ofPoisson type. At weak disorder, instead, the wave-functions are extended over the whole accessible volume:the non-interacting states { σ zi } become effectively cou-pled to infinitely many other states (i.e., to a contin-uum of energy levels), the system provides its own bathand behaves as a normal metal (the statistics of energylevels should then be described by the GOE ensemble).Between the MBL and the metallic phase, the authorsalso predicted the possibility of the existence of an inter-mediate regime (nowdays called the “bad metal”) wherethe wave-functions might be delocalized but not ergodic:site orbitals in the Fock space may only hybridize withan infinitesimal fraction of the accessible volume. Thisregime is expected to be characterized by highly hetero-geneous transport and strong fluctuations (and, possibly,by anomalous level statistics).A huge amount of work has been done on this sub-ject in the latest years (see, e.g., Refs. [2–6] for recentreviews) and, as mentioned above, the existence of theMBL transition has been even established at a mathe-matical level for some specific models under some mini-mal assumptions. However, most of the studies have fo-cused either on the MBL phase itself or on the transitionpoint.The interest on the delocalized side of the transitionstarted recently, when it was observed that the delo-calized phase is actually very unusual:
In fact itwas found that in a broad range of parameter beforeMBL, transport is sub-diffusive and out-of-equilibriumrelaxation toward thermal equilibrium is anomalouslyslow and described by power-laws with exponents thatgradually approach zero at the transition. These fea-tures appear as remarkably robust: They were observedin Ref. [19], by solving numerically the equations ob-tained within the self-consistent Born approximation, in numerical simulations (using exact diagonalizations ortime-dependent matrix product states) of disordered spinchains of moderate sizes, as well as in recent ex-periments with cold atoms.
An appealing phenomenological interpretation of thesephenomena has been proposed in terms of Griffithseffects.
The idea is that a system close to MBL ishighly inhomogeneous (in real space) and is characterized by rare inclusions of the insulating phase with an anoma-lously large escape time (i.e., anomalously small localiza-tion length). In 1 d such insulating segments affect dra-matically the dynamics, since quantum excitations haveto go through broadly distributed effective barriers whichact as kinetic bottlenecks and give rise to sub-diffusionand slow relaxation, in a way which is very similar to thetrap model for glassy dynamics. However the Griffits picture is not completely satisfac-tory. In fact unusual transport and power-law relaxationshave been recently observed also in quasiperiodic 1 d anddisordered 2 d systems, both in experiments and nu-merical simulations, while on general grounds oneexpects that Griffits effects should only give a subdomi-nant contribution when the potential is correlated and/orthe dimension is larger than one. It is therefore natu-ral to seek for other mechanisms that might hold beyondthe specific case of 1 d disordered systems.In a recent paper, using the Anderson model on theBethe lattice as a pictorial representation for the many-body quantum dynamics (following Refs. [7] and [8, 11–13]), we proposed a possible complementary explanationof the slow and power-law-like relaxation observed inthe bad metal phase directly based on quantum dynam-ics in the Fock space. More precisely our toy model isthe tight-binding Hamiltonian for non-interacting spin-less fermions (introduced in Ref. [10]), H = V (cid:88) x =1 (cid:15) x | x (cid:105)(cid:104) x | + t (cid:88) (cid:104) x,y (cid:105) (cid:0) | x (cid:105)(cid:104) y | + | y (cid:105)(cid:104) x | (cid:1) , (4)where the on-site random energies are taken as i.i.d.random variable uniformly distributed in [ − W/ , W/ (cid:104) x, y (cid:105) denotes nearest-neighboring site on the Bethelattice. In Ref. [38] the Bethe lattice was taken as aRandom-Regular Graph (RRG), i.e., a random latticewhich has locally a tree-like structure but has loops whosetypical length scales as ln V ∝ N and no boundary. Inthe analogy with MBL discussed for the Hamiltonian (1),each site of the lattice should be interpreted as a many-body configurations, and on-site energies as (extensive)random energies of the N -body interacting system, seeEq. (2). Of course this analogy represents a drastic sim-plification of real systems, as one neglects the correlationbetween random energies as well as the specific struc-ture of the hyper-cube. Moreover, we considered Bethelattices of fixed connectivity (we set the total connectiv-ity k + 1 equal to 3 throughout, as in Ref. [38]), whilethe connectivity of the configuration space of the many-body system increases as N ∝ ln V . The counterpartof the MBL transition corresponds to Anderson localiza-tion which, for the non-interacting Hamiltonian (4) wefocus on, and for k + 1 = 3 and E = 0 (correspondingto the middle of the band, i.e., infinite temperature forthe many-body system), takes place at W L ≈ .
2, asobtained from previous studies of the transmission prop-erties and dissipation propagation, and preciselydetermined in Ref. [43].Defining suitable proxies of the correlation functionsof local operators in real space of the original many-body problems (see Sec. III for a detailed explanation),we studied both the out-of-equilibrium and the (infinitetemperature) equilibrium dynamics, and showed that in abroad region of the phase diagram the counterpart of thespin imbalance and of the equilibrium correlation func-tion display slow relaxation and a power-law-like behav-ior strikingly similar (at least qualitatively) to the oneobserved in the bad metal phase of many-body systems,with apparent dynamical exponents that evolve continu-ously with the disorder and approach zero at the local-ization transition (see also Ref. [44] for a recent tightlyrelated investigation).Ref. [38] is far from being the only work addressingMBL-related questions using Anderson in terms of local-ization on the Bethe lattice. In fact, inspired by themapping of MBL onto single particle Anderson local-ization in a very high-dimensional space, in thelatest years an intense research activity has been de-voted to establish the existence of a non-ergodic delo-calized phase in the tight-binding model on the Bethelattice.
As a matter of fact, the slow dynamicsand power-law relaxation observed in Ref. [38] emergeprecisely in the region W T ≤ W ≤ W L where some ofthe previous studies have suggested that wave-functionsmight be extended but multifractal. Although re-cent results convincingly indicate that full ergodicity iseventually recovered on RRGs larger than a cross-overscale which diverges exponentially fast approaching thelocalization transition, these observations suggestthat the physical origin of the unusual slow and sub-diffusive dynamics observed in the bad metal phase istightly related to the apparent non-ergodic features ofthe spectral statistics that seem to emerge in the de-localized phase approaching the localization transition.Here we come back to this problem. The main outcomeof this paper is twofold. On the one hand, using a map-ping to directed polymer in random media, we obtaina clear, novel, and transparent physical interpretationof the unusual and slow dynamics observed in many-body isolated disordered system approaching the MBLtransition in terms of delocalization along rare, ramified,disorder-dependent paths in the Fock space. On the otherhand, we show that the apparent non-ergodic features ofthe delocalized phase that have been found for Andersonlocalization on random regular graphs are the vestiges ofthe truly non-ergodic delocalized phase present on Cay-ley trees.
We thus offer an explanation as well as aquantitative theory of the cross-over phenomena (alreadyextensively discussed in Ref. [52]) associated to the badmetal phase in RRGs.
Summary of results . In the next section, using thesingle-particle Anderson tight-binding model on the loop-less Cayley tree (4) as a toy model, we map the problemof ergodicity for quantum dynamics to the one of DirectedPolymers in Random Media (DPRM).
By analyz-ing the properties of the average free-energy of the DP, x (1) x (N−2) x (N) x (N−1) FIG. 1. Schematic representation of one of the k N paths onthe Cayley tree ( k = 2 here) going from the root 0 to a site ofthe boundary x ( N ) along which the resolvent matrix element G x ( N ) , is computed according to Eq. (6). we show the existence of a sharp transition to a glassyphase where delocalization can only occur on few specificdisorder-dependent paths. We discuss the manifestationof such freezing glass transition on the non-ergodicicityof the wave-functions, and in particular on the singularprobability distribution of the Local Density of States(LDoS) at the root of the Cayley tree. In Sec. III weprobe the dynamical evolution on the Cayley tree by mea-suring observables built as proxies for the imbalance andequilibrium correlation functions which display slowdynamics and power-laws in the glassy phase. We es-tablish a quantitative connection between the emergenceof the anomalously slow relaxation and the non-ergodicfeatures of the spectral statistics. In particular we high-light the relationship between the depinning transitionof the DP (i.e., of the preferred paths along which decor-relation can occur) in the glassy phase and the singu-lar behavior of the overlap correlation function betweeneigenstates at different energies (which is essentially theFourier transform to frequency space of the dynamicalcorrelation function). In Sec. IV we discuss the effectand the importance of loops by contrasting the dynam-ical evolution on the Cayley tree with the one observedon the RRG in Refs. [38] and [44], and offer an expla-nation as well as a quantitative theory of the cross-overphenomena associated to the non-ergodic-like behaviorin RRGs. Finally, in Sec. V we summarize the resultsfound and discuss their physical implications, providingsome perspectives for future work. Some technical as-pects are discussed in details in the appendices A-D. II. DELOCALIZATION IN FOCK SPACE ANDDIRECTED POLYMERS IN RANDOM MEDIA
Let us focus again on the many-body Hamiltonian (1)as a reference model, and imagine to start at time t = 0from a random (infinite temperature) magnetization pro-file with σ zi = ± / | ψ ( t = 0) (cid:105) = | x (cid:105) , whose random energy is close to zero)and let us ask ourselves the following question: What isthe probability that at large time the spin configurationhas completely decorrelated from the initial one? For asystem of N spins, this roughly corresponds to requir-ing that a finite fractions of them have flipped, i.e., thatat time t the system is found on a site | x (cid:105) of the con-figuration space which is order N steps (i.e., spin flips)away from the initial state. Here we address this ques-tion by using the pictorial representation of the quantummany-body dynamics in terms of a single-particle An-derson model on tree like structures . For sakeof simplicity, we start by focusing on (loop-less) Cayleytrees as the underlying lattice mimicking the Fock space,and address later the effect of loops by considering RRGs(see Sec. IV). A. From ergodicity of quantum dynamics todirected polymer in random media
Let us consider a loop-less Cayley tree of N genera-tions of fixed connectivity k + 1 = 3 and total number ofsites V ≈ N , as our toy model for the Fock space. Ameasure of the degree of ergodicity of quantum dynam-ics is the amount of spreading of a wave-function thatis initially localized at the root of the tree (labeled as0). More precisely, one wants to study the wave-functionamplitude (at large times) on the sites of the boundary ofthe tree | x ( N ) (cid:105) (i.e., at distance of order N ∝ ln V awayfrom the root) given the initial condition (cid:104) | ψ ( t ) (cid:105) = 1 at t = 0. The time evolution of the wave-function at time t , | ψ ( t ) (cid:105) , can be written in terms of the eigenvalues E α and eigenfunctions | α (cid:105) of the single-particle problem as: | ψ ( t ) (cid:105) = (cid:80) (cid:63)α | α (cid:105)(cid:104) α | (cid:105) e − iE α t/ (cid:126) (cid:113)(cid:80) (cid:63)α |(cid:104) α | (cid:105)| . In principle all eigenfunctions | α (cid:105) contribute to the sum.However in the many-body system, due to the scalingof the energies in the thermodynamic limit, the statesthat matters physically, even the virtual ones, have allthe same intensive energies (i.e., the states with an in-tensive energy different from zero will have a vanishingprojection on | (cid:105) ). Since our single-particle toy modellacks, of course, this concentration property, we have toimpose it as an extra-constraint. The star above thesum in the previous equation means that we are restrict-ing it to a subset of eigenstates of H which belongsto a small energy shell around the middle of the band( E α ∈ [ − ∆ E, ∆ E ]), and the denominator is a normal-ization factor that ensures that |(cid:104) ψ ( t ) | ψ ( t ) (cid:105)| = 1 (seeRef. [38] for a more extended discussion).The wave-function amplitude in the infinite time limiton the ( k + 1) k N − boundary site x ( N ) (when the timeevolution is constrained only on the states close to zeroenergy) W ( x ( N ) ) ≡ lim t →∞ (cid:12)(cid:12) (cid:104) x ( N ) | ψ ( t ) (cid:105) (cid:12)(cid:12) , has a verysimple spectral representation in terms of the elementsof the resolvent matrix of the non-interacting Hamilto-nian on the tree. The resolvent is defined as G ( z ) =( H − z I ) − , where H is given in Eq. (4), I is the iden-tity matrix, and z = E + iη , η being an infinitesimalimaginary regulator which smooths out the pole-like sin-gularities in the denominator. As detailed in App. A, one finds: W ( x ( N ) ) = (cid:80) (cid:63)α |(cid:104) x ( N ) | α (cid:105)| |(cid:104) α | (cid:105)| (cid:80) (cid:63)α |(cid:104) α | (cid:105)| ≈ lim η → + η |G x ( N ) , ( E = 0) | Im G , ( E = 0) . (5)(To simplify the notations we will set E = 0 through-out unless specified differently.) In the following, inthe analytical study we shall consider η = cδ where δ = 1 / ( V ρ ) is the mean level spacing, which is the nat-ural scale for the imaginary regulator, and take the si-multaneous limits V → ∞ and η → + . Thanks to hi-erarchical structure of the lattice, the matrix elementsof the resolvent on sites 0 and x ( N ) can be explicitlywritten in terms of the diagonal elements of the so-called“cavity” Green’s functions G x ( i ) → y ( i − (see fig. 1), whichis defined as the diagonal element on site x ( i ) (belong-ing to the i -th generation of the tree) of the resolventof a modified (“cavity”) Hamiltonian H x ( i ) ↔ y ( i − wherethe edge between sites x ( i ) and y ( i − (belonging to the( i − G x ( i ) → y ( i − = ( H x ( i ) ↔ y ( i − − iη ) − x ( i ) ,x ( i ) . By progres-sively integrating out all the sites from the leaves to theroot in the following expression or using matrix identities,one finds (we set the hopping rate t equal to 1 through-out): G x ( N ) , = (cid:82) D ϕ ϕ ϕ x ( N ) e − (cid:80) x,y ϕ x ( H− iη ) x,y ϕ y (cid:82) D ϕ e − (cid:80) x,y ϕ x ( H− iη ) x,y ϕ y = G x ( N ) → x ( N − G x ( N − → x ( N − · · · G x (1) → G , , (6)Moreover, as shown in Ref. [10], on tree-like structuresthe diagonal elements of such cavity Green’s functionssatisfy the following exact recursion relation: G − x → y = (cid:15) x − iη − (cid:88) z ∈ ∂x/y G z → y , (7)where the sum runs over all the neighbors z of x exceptthe cavity site y . Hence, using Eqs. (5) and (6) W ( x ( N ) )can be finally expressed as: W ( x ( N ) ) ≈ η |G , | Im G , N (cid:89) i =1 | G x ( i ) → x ( i − | . A measure of the delocalization, or ergodicity, of thequantum dynamics can then be obtained as the wave-function amplitude at the boundary of the tree P ∞ B (seeApp. D) obtained by summing the previous expressionover all possible sites x ( N ) of the boundary of the tree (aproxy of the many-body configurations which are O ( N )spin flips away from the initial one): P ∞ B ≡ (cid:88) x ( N ) W ( x ( N ) ) ≈ η |G , | Im G , (cid:88) P N (cid:89) i =1 | G x ( i ) → x ( i − | , (8)where the sum in the numerator is over all directed paths P of length N connecting the leaves of the tree with theroot through the edges x ( i ) → x ( i − (one of those pathsis represented in fig. 1). A P ∞ B different from zero in thelarge- N limit is a signature that the system is delocalized,whereas on the contrary P ∞ B = 0 indicates localization.Now let us highlight a fact that is central to our work:The sum (8) is over an exponential number of paths, k N .In the large N limit there are hence two possible cases :(1) The sum is dominated by few paths only; (2) Thesum is dominated by an exponential number of paths k N with an effective branching ratio k less than k and dis-order dependent. In the following we show that withinthe delocalized phase, P ∞ B >
0, there exists a sharp phasetransition between these two regimes, and that such tran-sition is related to the glass transition of directed polymerin random media. Indeed, by introducing the edge-energies ω x ( i ) → x ( i − and the site-energy ω by setting e − ω x ( i ) → x ( i − = | G x ( i ) → x ( i − | and e − ω = |G , | , one can re-interpretthe numerator of Eq. (8) as the partition function of adirected polymer on the Cayley tree with N generationsin presence of quenched bond disorder (and quenchedon-site disorder on the root) at inverse “temperature” β = 1: (cid:88) x ( N ) W ( x ( N ) ) ≈ ηe − ω Im G , (cid:88) P N (cid:89) i =1 e − ω x ( i ) → x ( i − = ηZ DP ( β = 1 , N )Im G , (9)By similar arguments, see App. B, one can also relate theimaginary part of the resolvent Im G , to the partitionfunctions Z DP ( β = 1 , M ) of directed polymers of length M starting at the root of the Cayley tree and ending atthe M -th generation:Im G , = N (cid:88) M =0 ηZ DP ( β = 1 , M ) . (10)In conclusion, the thermodynamic properties of the as-sociated directed polymer problem are instrumental instudying the delocalization and ergodicity properties ofAnderson localization. In the following we will studythem in detail.
B. Glass transition of directed polymer in randommedia
In the original problem introduced by Derrida andSpohn the disordered consisted in i.i.d. onsite energiesonly. The DPRM can however be solved even in the caseof correlated onsite and link disorder (the | G x ( i ) → x ( i − | are correlated along a path) as we now recall.One has to compute the generalized average “free- energy” (also introduced in Refs. [50] and [61]): φ ( β ) = lim N →∞ βN (cid:42) ln (cid:32) |G , | β (cid:88) P N (cid:89) i =1 | G x ( i ) → x ( i − | β (cid:33)(cid:43) , (11)where the average is performed over the quenched ran-dom energies (cid:15) x of the non-interacting tight-binding toymodel (4), which, once the fixed point of Eqs. (7) isfound, yield the effective random energy landscape forthe DP. The average free-energy is a convex function of β and has a one-step RSB freezing glass transition, akin tothe one of the Random Energy Model (REM): bydecreasing the “temperature” 1 /β the generalized free-energy decreases until the critical point β (cid:63) , defined by ∂φ ( β ) /∂β | β (cid:63) = 0, is reached; for β > β (cid:63) the DP freezesand its free-energy remains constant: In this glass phasethe number of paths contributing to (11) is not exponen-tial in N , but instead O (1), implying that the DP canbe found only on few specific disorder-dependent pathswith probability of order 1, whereas for β < β (cid:63) there isan exponentially small probability of finding the polymeron an exponentially large number of conformations.The physical reason for that goes as follows: Denoting e − Nβf the contribution of a given path oflength N , one can rewrite the sum in Eq. (8) as an in-tegral over all paths giving a contribution characterizedby a value of f between f and f + d f times the numberof such paths. By denoting the latter exp( N Σ( f )), oneends up with the expression: Z DP ( β ) = (cid:90) d f e N [ − βf +Σ( f )] . The value of f that dominates the integral for N → ∞ depends on β . For small enough β , one finds that thesaddle point value of f (cid:63) ( β ) is such that Σ( f (cid:63) ) >
0. Inthis regime an exponential number of paths, k N (with k = e Σ( f (cid:63) ) ), contributes to the sum. By increasing β ,Σ( f (cid:63) ) decreases until the value β (cid:63) is reached. At thispoint the generalized entropy Σ( f (cid:63) ( β (cid:63) )) vanishes. Hencethe generalized average free-energy φ ( β ) is related to theLegendre transform of Σ( f ): φ ( β ) = − f (cid:63) + Σ( f (cid:63) ) β , and φ (cid:48) ( β ) = − Σ( f ( β )) /β , and allows one to find outwhether a finite (Σ( f (cid:63) ) = 0) or an exponential (Σ( f (cid:63) ) >
0) number of paths contributes to the partition function Z DP ( β ).Since the physical value of our proxy for decorrela-tion, Eq. (8), is obtained for β = 1, what matters here iswhether the freezing of the DP takes place at β (cid:63) aboveor below 1. In order to compute φ ( β ), on each edge ofthe lattice, and for a given value of β , we introduce thevariable y x ( i ) → x ( i − = (cid:88) P N − i N (cid:89) j = i | G x ( j ) → x ( j − | β , where P N − i are all the directed paths of length N − i connecting the site x ( i ) to the boundary of the tree,and x ( j ) → x ( j − are all the directed edges (including x ( i ) → x ( i − ) belonging to the path. It is straightfor-ward to derive the following exact recursion relation for y x ( i ) → x ( i − : y x ( i ) → x ( i − = | G x ( i ) → x ( i − | β (cid:88) x ( i +1) ∈ ∂x ( i ) y x ( i +1) → x ( i ) , (12)where | G x ( i ) → x ( i − | β can be computed using Eq. (7).Eqs. (7) and (12) naturally lead to an exact func-tional equation for the joint probability distributions W ( β ) i ( G, y ) at the i -th generation of the tree: W ( β ) i ( G, y ) = (cid:90) d p ( (cid:15) ) k (cid:89) (cid:96) =1 d W ( β ) i +1 ( G (cid:96) , y (cid:96) ) × δ (cid:34) G − + (cid:15) + iη + k (cid:88) (cid:96) =1 G (cid:96) (cid:35) δ (cid:34) y − | G (cid:96) | β k (cid:88) (cid:96) =1 y (cid:96) (cid:35) , where p ( (cid:15) ) = (1 /W ) θ ( − W/ ≤ (cid:15) ≤ W/ G , and Z DP at the root ofthe tree as: W ( β )0 ( G , Z ) = (cid:90) d p ( (cid:15) ) k +1 (cid:89) (cid:96) =1 d W ( β )1 ( G (cid:96) , y (cid:96) ) × δ (cid:34) G − + (cid:15) + iη + k +1 (cid:88) (cid:96) =1 G (cid:96) (cid:35) δ (cid:34) Z − | G (cid:96) | β k +1 (cid:88) (cid:96) =1 y (cid:96) (cid:35) . The equations above can be solved by iteration using apopulation dynamics algorithm with arbitrary numeri-cal precision. After N generations one can then com-pute φ ( β, N ) as the average value of the logarithm of Z over the distribution W ( β )0 ( G , Z ), divided by βN , as inEq. (11): φ ( β, N ) = (cid:104) ln Z (cid:105) W / ( βN ). C. Numerical results I: Glass transition of DPRMand delocalized non-ergodic phase
Performing the limit N → ∞ requires an extrapola-tion of the numerical results obtained at finite N . Inorder to avoid such extrapolation, the position of β (cid:63) canbe more conveniently found computing the logarithm ofthe average partition function instead of the average ofthe logarithm. This leads to the so-called annealed free-energy: φ ann ( β ) = lim N →∞ βN ln (cid:42) |G , | β (cid:88) P N (cid:89) i =1 | G x ( i ) → x ( i − | β (cid:43) = lim N →∞ ln (cid:104) Z (cid:105) W βN . (13) m φ ( m ) , φ a nn ( m ) W=6W=8W=10W=12W=14W=16W=18.3
FIG. 2. Quenched (continuous lines) and annealed (dashedlines) average free-energy, φ ( β ) and φ ann ( β ), Eqs. (11) and(13), as a function of the “inverse temperature” β of the DPon the Cayley tree associated to the delocalization of a par-ticle at the root through the effective random energy land-scape generated by the | G x i → x i − | ’s, for different values ofthe disorder strength across the delocalized phase up to thelocalization transition. The circles spot the position of thebreaking point β (cid:63) where ∂φ ( β ) /∂β | β (cid:63) = 0, which becomesgreater than one for W (cid:46) W T ≈ .
65 and is equal to 1 / W L ≈ . φ (1 /
2) = 0). whereas the one in Eq. (11) is called quenched free-energy. As discussed in Ref. [61] (see also Refs. [62]and [63]) the two free-energies coincide for β ≤ β ∗ .Hence, the annealed and quenched free-energies can beequivalently used to identify the value of β (cid:63) , but the an-nealed one much less computationally demanding. Therefore one can use the annealed one to obtain φ ( β )for β ≤ β ∗ , and the value of β ∗ , and then impose that φ ( β ) = φ ( β ∗ ) for β > β ∗ .We have obtained φ ann ( β ), for several values of thedisorder across the whole delocalized side ( W ≤ W L ≈ . ) of the tight-binding Anderson model on the Cay-ley tree in the limit of large N , using the recursive equa-tions (7) and (12).We observe that φ (1) becomes positive below W L . Atthis point, delocalization takes place and the wavefunc-tion spreads far away from the root. As shown in App. C, φ ( β = 1) coincides asymptotically with the Lyapunov ex-ponent describing the growth Im G under the iteration re-lations (7), which is a decreasing function of W vanishingat W L . We find that β (cid:63) → . φ ( β (cid:63) ) → W → W L . This is expected, as it was rigorously provedin Refs. [61] and [64], and indirectly found in Ref. [10](see also Refs. [41], [40], and [50]). It is therefore agood check of our numerical method. When diminishing W below W L the value of β (cid:63) increases and eventuallyreaches 1 for W = W T ≈ .
65 (see Fig. 3), where theglass transition of the directed polymer takes place. Atweaker disorder,
W < W T , φ ( β ) is a smooth decreasingfunction of β in the whole range β ∈ [0 . , W β ∗ φ ( β * ) W k - W T W L FIG. 3. Top panel: Effective branching ratio k = e − φ (cid:48) ( β =1) (green) of the DP problem as a function of W . k vanishes at W = W T and remains zero in the glassy delocalized phase W T < W < W L . Bottom panel: Critical “inverse tem-perature” β (cid:63) (red) and average free-energy φ ( β (cid:63) ) (blue) asa function of W . At W T (turquoise dashed vertical line) β (cid:63) = 1, while at W L (orange dashed vertical line) β (cid:63) = 0 . φ ( β (cid:63) ) = 0. for 0 < W < W T the contribution to (cid:80) x ( N ) W ( x ( N ) )comes from an exponential number of paths, while for W T < W < W L delocalization occurs only on rare, ram-ified, specific paths which corresponds to the preferreddisorder-dependent configurations of the polymer in theglassy phase. In Fig. 2 we show plots of φ ( β ) for severalvalues of W across the delocalized phase, highlightingthe position of β (cid:63) . Since the number of paths of length N contributing to the sum in Eq. (8) scales as e N Σ( f (cid:63) ) ,the effective branching ratio k < k can be computed as k = e − φ (cid:48) ( β =1) . In the top panel of Fig. 3 we show that k associated to the exponential growth of the numberof paths vanishes at W = W T and remains zero in theglassy delocalized phase W T < W < W L . In the bot-tom panel of Fig. 3 we show the behavior of the critical“inverse temperature” β (cid:63) and of the average free-energy φ ( β (cid:63) ) of the associated DP problem as a function of W . D. Numerical results II: Glass transition ofDPRM, the singular statistics of the local density ofstates and multifractality
It was shown in Ref. [56] (see also Ref. [42]) that in thefreezing glass phase the DP partition function is a power-law tailed distributed random variable with an exponent1 + β (cid:63) (and possibly logarithmic corrections). In fig. 4 weconfirm this result in our case: we show the probabilitydistribution P ( ηZ DP ) (we will omit to specify that Z DP iscomputed at β = 1 henceforth to simplify the notation)for Cayley tree of different sizes ( N = 32 , . . . , W = 12 (top panel) and W = 4 (bottom panel), showingthat the statistics of ηZ DP is completely different below and above W T . For W < W T , P ( ηZ DP ) converges to a(size-independent) stable non-singular distribution whichdecreases fast to zero at large arguments. For W > W T ,instead, P ( ηZ DP ) has a singular behavior in the limit N → ∞ (and η → ηZ DP goesto zero in the large N limit, while its average stays fi-nite, and is dominated by the fat tails of the PDF whichare characterized by an exponent 1 + β (cid:63) (the power-lawbehavior is cut-off for ηZ DP ∼ /η ).This result has important implications on the distri-bution of the local density of states Im G , . In fact, asdiscussed in App. B, Im G , is directly connected to thepartition functions of the DPs via Eq. (10). Since in thedelocalized phase, W < W L , Z DP ( β = 1 , M ) grows with M one expect the sum in (10) to be dominated by the lastterm, i.e. that the main contribution to Im G , is givenby Z DP ( β = 1 , N ). This in turn implies that the singu-lar statistics found for Z DP ( β = 1 , N ) in the delocalizedglassy phase also holds for Im G , .In Fig. 4 we show that this is indeed the case: we plotthe probability distributions Q (Im G , ) of the imaginarypart of the Green’s functions at the root of the tree, whichas expected coincides essentially with P ( ηZ DP ).This singular statistics of the local density of states im-plies multi-fractal behavior for W T < W < W L . In fact,since β (cid:63) ≤
1, the tails of Q (Im G , ) give the leading con-tribution to the DoS and to all the moments (cid:104) (Im G , ) q (cid:105) with q > β (cid:63) , whereas the bulk part only yields a vanish-ing one. Conversely, all the moments (cid:104) (Im G , ) q (cid:105) with q < β (cid:63) are dominated by the behavior of the typicalvalue. (The probability distribution of the real part of G , instead converges for N → ∞ to a stationary non-singular distribution.) Since the moments of Im G , arerelated to the moments of the wave-functions’ amplitudesat the root of the Cayley tree, for W > W T one has abifractality scenario in the vicinity of the root which isexactly the same as the one recently found in Ref. [49] us-ing the supersymmetric non-linear σ -model approach forthe p -orbital Anderson model on the Cayley tree with p (cid:29)
1. Remarkably enough, the solution of the prob-lem in that case is found in terms of the Fisher-KPPequation, which was first introduced in Ref. [66], and isknown to emerge in a broad class of non-linear problemsdescribing propagation of a front between an unstableand a stable phase, including DPRM.
The transitiondiscussed here is also of the same kind as the one foundin Ref. [47] by mapping the iteration equation for theimaginary part of the Green’s function on the travelingwave problem, and using a RSB formalism for a slightlymodified distribution of the on-site random energies andin the large connectivity limit k → ∞ . η Z DP , ImG P ( η Z D P ) , Q (I m G , ) N=32N=48N=64N=80N=96N=112 η Z DP , ImG N=32N=48N=64N=80N=96N=112
W=4W=12
FIG. 4. Log-log plot of the probability distributions P ( ηZ DP ) (continuous lines) and Q (Im G , ) (dashed lines) forCayley trees of N generations (with N going from 32 to 112),and for W = 12 > W T (top panel) and W = 4 < W T (bottompanel). The dotted black straight line shows the exponent1 + β (cid:63) ≈ .
54 of the power-law tails of the distributions.
III. QUANTUM DYNAMICS ON THE CAYLEYTREE AND THE DEPINNING TANSITION OFTHE DIRECTED POLYMERS
In this section we follow the ideas of Ref. [38] and studythe relaxation of proxies of average and correlation func-tions of local operators in real space on the delocalizedside of the phase diagram, by using the Anderson modelon the Cayley tree (4) as a toy model for the many-body dynamics. The observables we focus on are thecounterpart of the imbalance and of the (infinite tem-perature) equilibrium correlation function. The imbal-ance measures whether an initial random magnetizationprofile converges to its flat thermodynamic average orremains instead inhomogeneous even at very long time,keeping memory of the initial configuration. For a N -body disordered isolated quantum system, described forinstance by Eq. (1), this corresponds to check whether(1 /N ) (cid:80) i (cid:104) σ zi ( t ) (cid:105) tends to zero or to a positive resid-ual value at long times. Within our analogy betweenMBL and single-particle Anderson localizaiton in a high-dimensional space, the counterpart of a random initialstate corresponds to a wave-function at t = 0 local-ized on a particular site x of the lattice (that we willtake as the root of the tree) with energy close to zero: | ψ ( t = 0) (cid:105) = | (cid:105) , such that (cid:15) ≈
0. In order to studyaverages and correlations of spin operators in real space,we need to define Bethe lattice proxies of such local op-erators. The representation of ˆ σ zi in Fock space is simplyˆ σ zi = (cid:80) { σ zi } |{ σ zi }(cid:105)(cid:104){ σ zi }| f ( { σ zi } ), where f ( { σ zi } ) is equalto the value of σ zi in the configuration { σ zi } . The proper-ties of this function is that it varies in a rapid and scat-tered way along the hyper-cube and is equal to +1 or − O of the localoperator ˆ σ zi as: ˆ O ≡ (cid:88) x | x (cid:105)(cid:104) x | f ( x ) , (14)where f ( x ) is a i.i.d. random binary variable equal to ± /
2. By doing so, we only keep thestatistical properties of the coefficients f ( x ) but neglectits correlations and its specific structure. Using this def-inition we have that: (cid:104) O ( t ) (cid:105) = (cid:104) | e iHt/ (cid:126) ˆ O e − iHt/ (cid:126) | (cid:105) = V (cid:88) x =1 f ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:63) (cid:88) α (cid:104) x | α (cid:105)(cid:104) α | (cid:105) e − iE α t/ (cid:126) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where, as explained above, the sum over the eigenstatesof the single-particle Hamiltonian (4) is restricted only onthe eigenvectors with energies E α within a small band-width [ − ∆ E, ∆ E ] around zero energy, in order to mimicthe fact that in a many-body systems the only statesthat contribute to the time evolution have all the sameintensive energy. In a N -body system this restriction isautomatically enforced by the scaling of the energies inthe thermodynamic limit, but our toy model (4) lacks ofthis concentration properties and we need to enforce itby hand.Averaging over the random variables f ( x ) and on therandom on-site energies (cid:15) x we obtain the Bethe latticeproxy of the Imbalance: I ( t ) = (cid:80) V x =1 (cid:12)(cid:12)(cid:80) (cid:63)α (cid:104) x | α (cid:105)(cid:104) α | (cid:105) e − iE α t/ (cid:126) (cid:12)(cid:12) (cid:80) V x =1 (cid:12)(cid:12)(cid:80) (cid:63)α (cid:104) x | α (cid:105)(cid:104) α | (cid:105) (cid:12)(cid:12) . (15)Note that because of the constraint of the sum over theeigenstates the numerator is not equal to one for t = 0and we cure this pathology by normalizing I ( t ) by itsvalue at t = 0.Following the same kind of reasoning one can define theBethe lattice proxy of the (infinite temperature) equilib-rium correlation function: C ( t ) = (cid:104) ( O ( t ) O (0) + O (0) O ( t )) (cid:105) = (cid:80) (cid:63)α,β |(cid:104) | α (cid:105)| |(cid:104) | β (cid:105)| cos [( E α − E β ) t/ (cid:126) ] (cid:16)(cid:80) (cid:63)α |(cid:104) | α (cid:105)| (cid:17) , (16)where the average is performed over the random coeffi-cients f ( x ) of the local operator ˆ O and the random on-site energies (cid:15) x of the Anderson tight-binding toy model.Note that C ( t ) actually coincides with the so-called re-turn probability, which will be more extensively discussedin App. D and whose time dependence on the RRG hasbeen recently analyzed in Refs. [44] and [53].The time evolution of I ( t ) and C ( t ) has been stud-ied on the RRG in Ref. [38], where we showed thatat moderately large time they both display unusually t I( t ) N=10N=12N=14 1 10 100 t C ( t ) t C ( t ) t I( t ) W=12W=4
FIG. 5. I ( t ) (left panels) and C ( t ) (right panels) as a functionof t for Cayley tree of different sizes and for W = 12 (toppanels) and W = 4 (bottom panels). slow relaxations and power-law-like behaviors strikinglysimilar (at least visually and on moderately large time-scales) to the ones observed in recent experiments andsimulations in the bad metal phase of many-body disor-dered isolated quantum systems approaching the MBLtransition. This occurs in a broad range of disor-der where previous studies have suggested that the eigen-functions of the Anderson model on the RRG might bedelocalized but non-ergodic.
More recently, it wasshown that for larger system sizes and for larger times theapparent algebraic decay is in fact cut-off and replacedby an exponential one.
Here we repeat the analysis of Ref. [38] using the Cay-ley tree as the underlying lattice mimicking the Fockspace instead of the RRG, and we will come back onthe effect of the loops in Sec. IV. The dynamical evolu-tion of the imbalance and of the equilibrium dynamicalcorrelation function are plotted in fig. 5 for different sys-tem sizes and for two values of the disorder strength, W = 4, below the DP glass transition, and W = 12,above the DP glass transition, but still in the delocalizedphase (the data are averaged over 2 − N samples). Thefigures clearly show that for W > W T a regime of slowdynamics sets in where both observables exhibit a power-law behavior which extends to larger and larger times asthe system size is increased, whereas for W < W T therelaxation is fast and exponential (the plateau observedat large times is a finite size effect and goes to zero in thethermodynamic limit). A. Depinning transition of the Directed Polymersand relationship with the power-law relaxation
Such behavior of the quantum dynamics on the Cay-ley tree indicates that the emergence of the slow andunusual relaxation can not be simply related to the non- ergodicity of the wave-functions. In fact, all eigenstatesof the Anderson model on the Cayley tree are multifractalin the whole delocalized phase, even below W T , yet I ( t ) and C ( t ) exhibit a fast exponential decay in aregion 0 < W < W T (see the bottom panels of fig. 5).The slow dynamics only sets in for W > W T and thusmust be tightly related to the freezing glass transition ofthe paths along which the wave-function spreads. Moreformally, a direct link can be established between the de-pinning transition of the DP in the glass phase (e.g.,when a parameters like the energy is varied) and the thesingular behavior of the ( local ) overlap correlation func-tion, defined as follows: K (0)2 ( E ) = (cid:42) V (cid:63) (cid:88) α,β |(cid:104) | α (cid:105)| |(cid:104) | β (cid:105)| δ (cid:2) E − ( E α − E β ) (cid:3)(cid:43) , where (cid:104) | α (cid:105) is the amplitude of the eigenvector | α (cid:105) onthe root of the tree.The interest of introducing such spectral probe istwofold. On the one hand, this function displays dif-ferent scaling behaviors for the ergodic, localized, andmultifractal states, and can be thus used to probe thenon-ergodic delocalized phase: For eigenfunctionsof GOE matrices K (0)2 ( E ) = 1 identically, independentlyon E on the entire spectral band-width. In the stan-dard (ergodic) metallic phase K (0)2 ( E ) has a plateau atsmall energies ( E < E Th ), followed by a fast-decay whichis described by a power-law, K (0)2 ( E ) ∼ E − γ , with asystem-dependent exponent. The height of the plateauis larger than one, which implies an enhancement of cor-relations compared to the case of independently fluctuat-ing Gaussian wave-functions. The Thouless energy, E Th ,which separates the plateau from the power-law decay,stays finite in the thermodynamic limit and extends tolarger and larger energies as one goes deeply into themetallic phase, and corresponds to the energy range overwhich GOE-like correlations establish. The (expected)behavior of the overlap correlation function for multi-fractal eigenfunctions is instead drastically different: The plateau is present only in a narrow energy interval
E < E Th ∼ V D − which shrinks to zero in the thermo-dynamic limit, while its height grows V − D . This canbe interpreted recalling that multifractal wave-functionstypically occupy a fraction V D of the total sites, whichimplies the existence of an energy scale, E Th , which de-creases with V but stays much larger than the meanlevel spacing, beyond which eigenfunctions poorly over-lap with each other and the statistics is no longer GOE.On the other hand, K (0)2 ( E ) is essentially the Fouriertransform of our proxy for the equilibrium correlationfunction (i.e., the return probability), Eq. (16): C ( t ) ≈ C ∞ + (cid:82) Eδ d E K (0)2 ( E ) cos( Et/ (cid:126) ) (cid:16)(cid:80) (cid:63)α |(cid:104) | α (cid:105)| (cid:17) . (17)The behavior at large times of C ( t ) is thus tightly related0to the behavior at small E of K (0)2 ( E ), i.e., the power-laws observed in the time decay of C ( t ) are linked to thepower-laws in energy found for K (0)2 ( E ). (The statisticsof the infinite time limit of the return probability, C ∞ ,will be discussed separately in App. D.) For any given random instance of the Hamiltonian (4), the overlap cor-relation function at the root of the tree (6) can be easilyexpressed in terms of the Green’s functions on site 0 com-puted at energies ± E/ Since, as discussed aboveand in App. B, the imaginary part of the Green’s func-tion at the root of the tree coincides essentially with ( η times) the partition function of the DP, one finally finds: K (0)2 ( E ) = lim η → + (cid:42) V Im G , ( − E/
2) Im G , ( E/ (cid:80) V x =1 Im G x,x ( − E/ (cid:80) V x =1 Im G x,x ( E/ (cid:43) ≈ lim η → + ( πρ ) − (cid:104) ηZ DP ( − E/ ηZ DP ( E/ (cid:105) . (18)Thus, the equilibrium dynamical correlation function C ( t ), defined in Eq. (16) and studied in the previous section,is essentially the Fourier transform of the correlation function of Z DP computed at two different values of the energyfor the same disorder realization (and then averaged over the disorder). As discussed previously, it is the behavior atsmall energy of K (0)2 ( E ) that plays a key role in determining the correlation function at long-times. A GOE-like trendfor which K (0)2 (0) is finite leads to a fast decorrelation in time, whereas a power law with negative exponent and,hence, a divergent K (0)2 (0), is associated to effective power laws for C ( t ). By using our result that the distributionof ηZ DP (0) is well behaved and has a finite second moment for W < W T , whereas it is a power law with a divergingsecond moment for W T < W < W L , we can therefore directly link the anomalously slow dynamics in the bad metalphase to the glassy regime found for W T < W < W L .A more physical insight can be gained analyzing bystudying the behavior of the imaginary part of theGreen’s function. We start by plotting in the top panelof fig. 6 the variation of Im G , at the root of a Cayley(of N = 16 generations) when the energy is continuouslyvaried on the scale of the mean level spacing δ close tothe band center for four independent realization of thedisorder. We notice that Im G , is roughly constant (andsmall) over broad energy intervals (typically much largerthan δ ) and has abrupt spikes around some specific valuesof E . Such spikes correspond to the existence of preferredconformations of the DP giving a large contribution tothe partition function Z DP (and hence to Im G , ) at thatparticular value of the energy. As the energy is varied, thepolymer is pulled away from such preferred conformationuntil a new one is found. (A similar behavior has beenrecently found in 2 d Anderson localization in the strongdisorder regime. ) This is shown in the bottom panelof fig. 6, where we plot the imaginary part of the cavityGreen’s function for two specific values of the energy forwhich a maximum of Im G , is found (for a given disorderrealization), focusing on the sites x (5) = 1 , . . . , V of the5-th generation of the tree (where V i = 3 · i − is the totalnumber of sites belonging to the i -th generation of thetree). Im G x (5) → x (4) is pinned (and small) on most of thesites, and differs only on few points between the two con-figurations. This is the manifestation of the role playedby rare events in determining Im G , , and the depinningtransition of the DP when the energy is varied, resultingin a macroscopic jump (i.e., an “avalanche”) between twopreferred directed paths: The preferred path which con-tribute the most to the first spike of Im G , (i.e., of thepartition function Z DP ) passes mainly through x (5) = 15 and x (5) = 16, while the preferred path contributing tothe second spike passes mainly through x (5) = 39 and x (5) = 40. For W < W T , instead, Im G , is a smoothfunction of E .We suspect that is the existence of such depinning tran-sition of the DP , which results in the macroscopic re-arrangements of the conformation of the directed paths,which contribute the most to the sum of Eq. (8), pro-duces a singular behavior (in the thermodynamic limit)of (cid:104) ηZ DP ( − E/ ηZ DP ( E/ (cid:105) and of the overlap correla-tion function K (0)2 ( E ), Eq. (18), at small energy differ-ence, E ∼ δ . This is also a direct manifestation of thefact that wave-functions closeby in energy display anoma-lously large correlations: In the glassy phase of the DPproblem, W > W T , eigenfunctions whose energy distanceis of the order of few level spacing occupy typically thesame paths on the tree, while eigenstates whose energyseparation is larger than the typical distance between twospikes of fig. 6 poorly overlap.The change of behavior of K (0)2 ( E ) in the glassy andnormal regimes of the delocalized phase is clearly visiblein fig. 7: For W = 4 < W T the overlap correlation func-tion approaches a size-independent value of order 1 whenthe energy is of order δ . For W = 12 > W T , instead,the value reached by K (0)2 ( E ) at small energy separa-tion grows with the system size. As shown in Fig. 8, thecurves obtained for different N collapse when the energyis rescaled by the mean level spacing δ and K (0)2 ( E ) by V − D , with a spectral fractal dimension D ≈ . Moreover, the overlap correlation function decays as apower-law, K (0)2 ( E ) ∼ E − γ , with γ ≈ E (cid:46) δ ) and γ ≈ − κ ≈ . -5-4-3-2-1 -30 -20 -10 0 10 20 30 I m G , E/ δ -6-5-4-3-2-10 5 10 15 20 25 30 35 40 45 I m G x ( ) - > x ( ) sites x (5) of the 5 th generation FIG. 6. Top panel: (log of the) Imaginary part of theGreen’s function Im G , at the root of a Cayley tree of N = 16generations as a function of the energy E (measured in unitsof the mean level spacing δ ) at disorder W = 12 > W T forfive independent realizations of the disorder. Im G , is pinnedmost of the time but jumps abruptly to very different valuesat some specific values of E which depend on the specificdisorder realization. Bottom panel: (log of the) Imagi-nary part of the (cavity) Green’s function Im G x (5) → x (4) onthe sites x (5) = 1 , . . . ,
48 of the 5-th generation of the tree forthree specific values of the energy spotted by the red, orange,and yellow circles in the top panel (and for a the disorderrealization which corresponds to the magenta curve of thetop panel).The difference between the two configurations ofIm G x (5) → x (4) is very small on most of the sites and of O (1)on few sites only. This corresponds to a macroscopic jumpbetween two preferred paths of the DP. describes the decay in time of C ( t ) (see previous section).Interestingly, it was shown that systems that areasymptotically in a thermal state, yet exhibit anoma-lous relaxation and subdiffusion, must satisfy a modi-fied version of the Eigenstate Thermalization Hypothesisansatz for the off-diagonal matrix elements of local op-erators (cid:104) α | O | β (cid:105) . Taking our definition (14) for Bethelattice proxies of local observables in real space, such off-diagonal elements, written in the eigenbasis of the Hamil-tonian, read: (cid:104) α | O | β (cid:105) = V (cid:88) x =1 |(cid:104) α | x (cid:105)| |(cid:104) x | β (cid:105)| f ( x ) . In Ref. [24] a general connection between the scaling ofthe variance of this object (which is tightly related to theoverlap correlation function introduced above) and thenon-exponential decay of dynamical correlation functionswas derived. In particular, it was found that for subd-iffusively systems the variance exhibits an anomalouslyslow scaling with system size than expected for diffusivesystems, which corresponds to the singular behavior of K (0)2 ( E ) discussed above. Within our interpretation interms of the freezing transition of DPRM, such unusual E K ( ) ( E ) N=16N=14N=12N=10 E K ( ) ( E ) N=16N=14N=12N=10
W=12W=4
FIG. 7. Overlap correlation function K (0)2 ( E ) as a functionof the energy for Cayley trees of several system sizes ( N = 10,12, 14, and 16 generations) and for W = 12 > W T (top panel)and W = 4 < W T (bottom panel). E/ δ K ( ) ( E ) / V - D N=10N=12N=14N=16
FIG. 8. Collapse of the data at W = 12 obtained whenrescaling the energy by the mean level spacing δ and K (0)2 ( E )by V − D , with D ≈ .
18. The dashed black line is a fit of thepower-law behavior observed at moderate energy separationas K (0)2 ( E ) ∼ /E − κ , with κ ≈ .
2, see fig. 5. scaling is tracked back to the ramified structure of thewave-functions in the Fock space within the delocalizedglassy phase. IV. THE EFFECT OF LOOPS: CAYLEY TREEVS RANDOM REGULAR GRAPH
All the results presented so far have been obtained inabsence of loops, i.e., by considering the loop-less Cayleytree as the underlying lattice mimicking the Fock space.This has the advantage that the mapping to DPRM canbe carried out without resorting to any approximation,the DP average free-energy, Eqs. (11) and (13), is well2defined and can be computed with very high numericalprecision, and the transition between the two delocalizedphases taking place at W T can be established accurately.However, the configuration space of N -body systems ismore appropriately represented by RRGs, which do notposses boundaries (differently from the Cayley tree, allsite of the RRG are statistically equivalent after averag-ing over the disorder) and have loops at all scale (whosetypical length is of order ln V ∝ N ). A crucial questionthat naturally arises is therefore to what extent the sce-nario discussed above is modified when the effect of theseloops is taken into account.A first piece of the answer can be obtained by compar-ing the quantum dynamics on the Cayley tree and on theRRG, recently analyzed in Refs. [38] and [44], at the samedisorder strength. This comparison is shown in Fig. 9.The plots indicate that for moderately large times andmoderately large system sizes, the time dependence ofthe imbalance and of the correlation function on the Cay-ley tree is very similar to the one previously found on theRRG (shown in gray in fig. 9). In particular for W = 12we find very similar apparent exponents (within our nu-merical accuracy) describing the power-law decay of theimbalance and of the correlation function as I ( t ) ∼ t − ζ and C ( t ) ∼ t − κ , ζ ≈ . κ ≈ .
2, as the ones reportedin Ref. [38] for the RRG. However, as mentioned above,on longer time scales the apparent power-laws observedon the RRG in Ref. [38] are actually cut-off and replacedby an exponential decay.
This occurs on a timescalewhich grows very fast as the localization transition isapproached, as τ ergo ∼ e B/ √ W L − W , and is very largealready far from W L . One then needs to simulate verylarge samples to observe the crossover from the algebraicdecay to the (stretched) exponential one. We argue thatthis is due to the fact that the single-particle Andersonmodel on the RRG becomes eventually fully ergodic ona characteristic volume which diverges exponentially asone approaches W L as V ergo ∼ e A/ √ W L − W . Instead the Anderson model on the Cayley tree displaysa genuine non-ergodic behavior, with multifractal wave-functions in the whole delocalized phase.
Putting all these observations together, one then comesto the following physical interpretation: On finite timescales the dynamical evolution can only explore finite re-gions of the Bethe lattice. Since RRGs look locally asloop-less trees, it is natural to expect that the dynam-ics on RRGs at moderately large times is well describedby the dynamics on Cayley trees. Moreover, RRGs ofmoderately large sizes, i.e., smaller than the correlationvolume V ergo , do not possess loops that are large enoughto restore the ergodicity and behave as if they were in adelocalized non-ergodic phase for all practical purposes. Hence the dynamical behavior on RRGs smaller than V ergo ( W ) is essentially the same as the one on the Cayleytree. The difference between the two lattices can only beseen at very large sizes and very large times ( V > V ergo and t > τ ergo ): On the Cayley tree the power-law regimeswill persist up to arbitrary large times, while on the RRG t I( t ) RRG (V=2 )CT (N=14) t C ( t ) FIG. 9. I ( t ) (left panel) and C ( t ) (right panel) as a func-tion of t for W = 12. The magenta squares show the datafor Cayley trees of N = 14 generations while the gray curvecorresponds to the numerical results previously reported inRef. [38] when using RRGs of V = 2 sites as the underlyinglattice. they are cut-off on a huge crossover scale. In other words,while a sharp transition takes place at W T in the limitof infinite Cayley trees, this transition becomes only acrossover on RRGs, and is smeared out if V (cid:29) V ergo .In order to understand the effect of loops on the quan-tum dynamics, one can also contrast the properties of theoverlap correlation function K (0)2 ( E ) on the Cayley treesdiscussed in the previous section with the ones found onRRGs of (about) the same sizes and at the same disorderstrength. In fig. 10 we show the overlap correlationfunction K ( E ) computed on the RRG with the ap-proximate technique of Ref. [52] for W = 12 and for sev-eral system sizes V = 2 N , with N = 10 , . . . ,
26. For sys-tem sizes smaller than the ergodic crossover scale V ergo (e.g., V (cid:46) ) the overlap correlation function on theRRG behaves very similarly to the Cayley tree. How-ever, for larger system sizes the dependence of K ( E )on the volume saturates and the curves converge to asize-independent limiting non-singular function charac-terized by a plateau at small energy followed by a fastdecrease at larger energy. As discussed above, this is thetypical metallic behavior found on the (fully ergodic) de-localized side of the Anderson transition. The energyscale E Th over which the plateau extends stays finitein the thermodynamic limit and represents the width ofthe energy band within which GOE-like correlations areestablished. These findings confirm the scenario discussed inthe previous section: RRGs smaller than the ergodiccrossover scale behaves for all practical purposes as (thebulk of) Cayley trees, while full ergodicity and standardmetallic behavior is recovered for larger samples on an en-ergy scale E Th which vanishes exponentially fast close tothe Anderson localization. From Eq. (17) one then3 E K ( E ) N=10N=12N=14N=16N=18N=20N=22N=24N=26N=10 (CT)N=12 (CT)
FIG. 10. Overlap correlation function K ( E ) as a func-tion of E for RRGs of several system sizes ( V = 2 N , with N = 10 , . . . ,
26) obtained with the approximate techniquedescribed in Ref. [52] for W = 12. The overlap correlationfunction on RRGs of size smaller than the correlation volume V ergo is very similar to the one found on the root of Cayleytrees of (about) the same size (dashed curves). expects a crossover to a standard exponential decay ofdynamical correlations on RRGs of size larger than V ergo and on time scales larger than τ ergo = (cid:126) /E Th . Con-versely on Cayley trees K (0)2 ( E ) is characterized by agenuinely singular limit in the whole delocalized glassyphase, and the power-laws in the dynamics persist to in-finitely long times.A more detailed discussion of the crossover phenomenaassociated to the non-ergodic-like regime in RRGs can befound in Ref. [52]. V. CONCLUSIONS AND PERSPECTIVES
In conclusion, using the non-interacting Andersonmodel on the Cayley tree as a toy model for the quan-tum many-body dynamics, we have proposed atransparent theoretical explanation of the subdiffusivebehavior and anomalously slow relaxation observed onthe thermal side of isolated disordered many-body sys-tems, in terms of delocalization along rare and rami-fied paths in the Fock space. In particular, we haveshown the existence of a glass transition (in the thermo-dynamic limit) separating two different extended phase:A metallic-like phase at weak disorder (0 ≤ W ≤ W T )where delocalization occurs on an exponential number ofpaths, and a bad metal-like phase at intermediate disor-der ( W T ≤ W ≤ W L ) where delocalization takes placeonly on few, specific, ramified paths. Translating our re-sults to the many-body problem, this means that at weakdisorder the number of site orbitals in the Fock space towhich the initial state is effectively coupled grows expo-nentially with the distance, while in the intermediateglassy phase resonances are formed only on rare site or- bitals on very distant generations, implying that energyand spin transport is highly heterogeneous, precisely aspredicted in the pioneering work of Ref. [7].The physical interpretation of the unusual relaxationobserved in the bad metal phase emerging from this pic-ture is complementary to the Griffiths one. In both sce-narios subdiffusion and non-exponential relaxation arethe result of the presence of heavy-tailed distribution(which are often associated to the failure of the cen-tral limit theorem). However, according to the Griffithspicture, the unusual slow dynamics follows from expo-nentially rare inclusions in real space which acts as ki-netic bottlenecks and yield effective barriers with expo-nentially large relaxation times. According to theperspective proposed here, instead, the slow dynamics istracked back to very heterogeneous delocalization of localexcitations along rare, disorder-dependent paths in Fockspace with a singular and heavy-tailed probability distri-bution of the decorrelation probability. In many practi-cal situations (especially for systems of limited size) botheffects could be at play simultaneously. Studying the be-havior of typical versus average correlation functions, asrecently done in Ref. [81] for a Floquet model, should bea good probe to distinguish and disentangle them.Notice that within our approach the unusual slow dy-namics and power-law behavior is not just related tothe non-ergodicity of wave-functions. In fact, the eigen-functions of the non-interacting Anderson model on theCayley tree are multifractal in the whole delocalizedphase, whereas the dynamical observables dis-play a fast exponential decay for
W < W T , as shown infig. 5. The emergence of algebraic decays is instead as-sociated to the freezing glass transition of the delocaliz-ing paths and to the singular statistical properties of thedecorrelation probability along these paths. This can beunderstood in terms of the depinning transition of theDP in the glassy phase, which yield abrupt rearrange-ments of the preferred conformations of the delocalizingpaths when a parameter like the energy is varied. Thisresults in a singular behavior of the overlap correlationfunction between eigenstates at different energies, whichis essentially the Fourier transform to frequency domainof the dynamical correlation function.Of course, our toy model is the result of a series ofextreme (over)simplifications. More work is needed togo beyond the approximations considered here and es-tablish a precise connection between the spreading of thewave-packet on a complicated graph and the many-bodydynamics of a realistic system. In this respect it wouldbe important to establish a direct quantitative relation-ship between the exponents describing the power-law de-cay of the dynamical observables such as the Bethe lat-tice proxies of the imbalance and of the equilibrium cor-relation function and the statistical properties of thedecorrelation probability along the paths, which becomessingular above W T . It would be also interesting to un-derstand how the properties of the intermediate glassyphase depend on the connectivity of the lattice (which,4to mimic the structure of the configuration space of a N -body interacting system should increase as N ), and onthe scaling of the random on-site energies (which shouldbe thought as the counterpart of extensive energies of the N -body system). It is in fact well established that thelocalization transition scales as W L ∼ k ln k , andit would be interesting to check whether W T follows thesame asymptotic behavior or a different one. (A naiveestimation of W T obtained using the forward-scatteringapproximation on the Bethe lattice seem to suggest that W T ∝ √ k .) A step forward in these directions would beto adapt the present analysis to the quantum version ofthe REM, which is possibly the simplest system exhibit-ing a MBL transition, as it allows to retain the localconnectivity of the configuration space and the scalingwith N of the random energies, and yet to neglect thecorrelations between random energies on different site or-bitals. Finally, the glass transition of the DP analyzedhere takes place as a front propagating from the rootof the tree towards the boundary, as already noticed inRef. [49]. It would be interesting to study the extendedphase diagram of the problem as a function of the di-mensionless distance from the root which, in terms ofdynamical evolution, is akin to time.A crucial point, partially discussed here, concerns towhat extent the details of the specific structure of theunderlying lattice mimicking the configuration space areimportant. In particular in this paper we have focusedon the effects of loops, which are completely disregardedwhen one considers loop-less Cayley trees as the un-derlying lattice for our toy model. Doing this has theadvantages that the mapping to DPRM can be carriedout without resorting to any approximation and that theglass transition found at W T survives in the thermody-namic limit. However, the configuration space of N -bodysystems is more appropriately represented by RRGs,which do not posses boundaries (differently from the Cay-ley tree, all site of the RRG are statistically equivalentafter averaging over the disorder) and have loops at allscale (whose typical length is of order ln V ∝ N ). How-ever, as discussed above, full ergodicity is (most likely)eventually recovered in the whole delocalized phase ofthe Anderson model on the RRG for system larger thana correlation volume which diverges exponentially (as V ergo ∼ e A/ √ W L − W , see Refs. [69], [43], and [70]) ap-proaching the Anderson transition. One then ex-pects that the apparent power-law decay of dynamicalcorrelation functions observed on the RRG should beeventually cut-off for sizes larger than V ergo , and shouldbe replaced by a standard exponential decay. Yet, thetime scale at which the decay of the correlation functionscan be distinguished by an algebraic one also diverges ex-ponentially (as τ ergo = (cid:126) /E Th ∼ e B/ √ W L − W ) approach-ing the Anderson transition, and is already very large farfrom it. Since on large but finite times the dynamics canonly explore a large but finite volume, close enough tothe localization transition, the dynamics of the system isslow and unusual for many decades (and well described by the Cayley tree), although it becomes eventually er-godic at large times.The Fock space of a realistic many-body Hamilto-nian, such as the disordered spin chain of Eq. (1), is a N -dimensional hyper-cube, with the extra complicationthat on-site random energies are strongly correlated. Aninteresting possibility would be then to diagonalize nu-merically the many-body Hamiltonian and analyze thestatistical properties of the delocalizing pahts in the con-figuration space (similarly to the recent analysis of [82])together with the scaling behavior of the spectral probe K ( E ) as a function of the system size, and benchmarkthe results onto the quantum (equilibrium and out-of-equilibrium) many-body dynamics. This opens a newtheoretical perspective to investigate the MBL transitionand to characterize the properties of the bad metal phase. ACKNOWLEDGMENTS
We acknowledge support from the Simons Foundation(
Appendix A: Spectral representation of W ( x ( N ) ) In order to obtain the spectral representation of W ( x ( N ) ) in terms of the elements of the resolvent matrixone needs to define the following correlation function: γ ( x, E ) = (cid:42)(cid:88) α |(cid:104) x | α (cid:105)| |(cid:104) α | (cid:105)| δ ( E − E α ) (cid:43) , and the local DoS on a site x : ρ x ( E ) = (cid:88) α |(cid:104) α | x (cid:105)| δ ( E − E α ) , whose spectral representations are simply given by: γ ( x, E ) = lim η → + η |G ,x ( E ) | πρ ( E ) ,ρ x ( E ) = lim η → + Im G x,x ( E ) π , where ρ ( E ) = (1 / V ) (cid:80) x ρ x ( E ) is the total DoS. The in-finite time limit of the wave-function amplitude on theboundary site x ( N ) starting from the root of the tree canbe then written as: W ( x ( N ) ) ≡ lim t →∞ (cid:12)(cid:12)(cid:12) (cid:104) x ( N ) | ψ ( t ) (cid:105) (cid:12)(cid:12)(cid:12) = (cid:80) (cid:63)α |(cid:104) x ( N ) | α (cid:105)| |(cid:104) α | (cid:105)| (cid:80) (cid:63)α |(cid:104) α | (cid:105)| = (cid:82) +∆ E − ∆ E ρ ( E ) γ ( x, E )d E (cid:82) +∆ E − ∆ E ρ ( E )d E .
Assuming that ∆ E is small enough such that the de-pendence of γ ( x, E ) and ρ ( E ) on the energy is weak,one can approximate the integrals by the values at themiddle of the band, which finally yields the approximateexpression given in Eq. (5).5 Appendix B: Relationship between Im G , and thepartition functions of directed polymers on theCayley tree The recursive equation (7) for the imaginary part ofthe cavity Green’s functions can be telescoped as:Im G x → y = | G x → y | η + (cid:88) z ∈ ∂x/y Im G z → x . (B1)From this equation, the Green’s function at the root ofthe tree can be re-expressed as:Im G , = η |G , | N − (cid:88) M =0 (cid:88) P M M (cid:89) i =1 | G x ( i ) → x ( i − | + |G , | (cid:88) P N − (cid:32) N − (cid:89) i =1 | G x ( i ) → x ( i − | (cid:33) Im G x ( N ) → x ( N − (B2)where the sums are over all directed paths P M of length M connecting the root of the tree with the sites of the M -th generation, and x ( i ) → x ( i − are all the edges belong-ing to P M connecting the site x ( i ) of the i -th generation tothe site x ( i − of the ( i − η times) the partition functions of DPs of length M originating from the root of the Cayley tree in presenceof the quenched random energy landscape generated bythe | G x ( i ) → x ( i − | . Since the cavity Green’s function onsite x ( N ) of the boundary of the tree in absence of its onlyneighbor x ( N − is simply G x ( N ) → x ( N − = ( (cid:15) x ( N ) − iη ) − ,one has that Im G x ( N ) → x ( N − = η | G x ( N ) → x ( N − | . Thusthe second line of the r.h.s. of Eq. (B2) exactly coincideswith the r.h.s. of Eq. (8). One then obtains Eq. (10) ofthe main text.The term M = N of (B2) gives the leading contribu-tion to Im G , , as also confirmed by fig. 4, which showsthat asymptotically P ( ηZ DP ( N )) ∼ Q (Im G , ).This is the result that one would obtain neglecting theimaginary regulator inside the barkets of Eq. (B1) fromthe beginning. In fact, the natural scale of the imagi-nary regulator is the mean level spacing, η = cδ , with δ = 1 / ( V ρ ). Hence η behaves as W/ V , while the typi-cal value of Im G x ( i ) → x ( i − grows under iteration in thedelocalized phase. This result has an obvious physicalinterpretation: the spreading of the level width of localexciatations (described essentially by Im G , ) is tightlyrelated to the probability that such excitations travel faraway on the tree. Appendix C: Relationship between φ ( β = 1) and theLyapunov exponent Let us focus on the iteration relations describing thepropagation of the imaginary part of the cavity Green’sfunction from the leaves to the root of a Cayley tree of N generations. As mentioned above, on the bound-ary of the tree Im G x ( N ) → x ( N − = η/ ( (cid:15) x ( N ) + η ). Thetypical value of Im G is thus of order η , Im G typ N = e (cid:104) ln Im G x ( N ) → x ( N − (cid:105) ≈ η . In the whole delocalized phaseIm G typ grows (by definition) under iteration. Suchgrowth can be characterized by a Lyapunov exponentΛ( W ) as Im G typ i − = e Λ Im G typ i . Λ( W ) is positive inthe delocalized phase, decreases as W is increased,and vanishes at the Anderson localization transition at W L . In the large N limit one then has that:Im G typ0 , ≈ e N Λ Im G typ N ≈ ηe N Λ . (C1)Moreover, since Im G , (cid:39) ηZ DP (see App. B and fig. 4),we have: φ ( β ) (cid:39) βN (cid:68) ln (Im G , /η ) β (cid:69) , which implies that:Im G typ0 , (cid:39) ηe Nφ ( β =1) . (C2)In conclusion, from Eqs. (C1) and (C2) one obtainsthat in the thermodynamic limit φ ( β = 1) coincidesasymptotically with the Lyapunov exponent Λ( W ). For W > W T we find that β (cid:63) <
1. Since φ ( β ) remains con-stant for β > β (cid:63) , one has that in the intermediate phase φ ( β = 1) = φ ( β (cid:63) ). We indeed find that at the localizationtransition φ ( β (cid:63) ) = 0 and β (cid:63) = 1 / Appendix D: Return probability
The correlation function defined in Eq. (16) is actuallyequivalent to the so-called return probability, recentlystudied in Ref. [44] on the RRG. The return probabil-ity (constructed using eigenstate within the bandwidth[ − ∆ E, ∆ E ]) is defined as: P R ( t ) = |(cid:104) | ψ ( t ) (cid:105)| = (cid:12)(cid:12)(cid:12) (cid:104) | e − i H t/ (cid:126) | (cid:105) (cid:12)(cid:12)(cid:12) = (cid:80) (cid:63)α,β |(cid:104) α | (cid:105)| |(cid:104) β | (cid:105)| e − i ( E α − E β ) t/ (cid:126) (cid:16)(cid:80) (cid:63)α |(cid:104) | α (cid:105)| (cid:17) . (The normalization factor in the denominator ensuresthat P R (0) = 1.)In the long time limit P ∞ R ≡ P R ( t → ∞ ) measure theprobability that the system stays localized around theroot of the tree | (cid:105) and keeps memory for infinite time ofthe initial configuration, P ∞ R = (cid:80) (cid:63)α |(cid:104) α | (cid:105)| (cid:16)(cid:80) (cid:63)α |(cid:104) | α (cid:105)| (cid:17) , and decays to zero with V in the whole delocalized phase.Using the expressions of the correlation function γ ( x, E )6and of the LDoS ρ x ( E ) introduced in App. A, P ∞ R canbe rewritten as: P ∞ R = (cid:82) +∆ E − ∆ E ρ ( E ) γ (0 , E )d E (cid:16)(cid:82) +∆ E − ∆ E ρ ( E )d E (cid:17) . Assuming once again that ∆ E is small enough such thatthe dependence of γ (0 , E ) and ρ ( E ) on the energy isweak (and assuming that ρ (0) (cid:39) ρ (0)), and using thespectral representation of γ (0 , E ) one finally obtains: P ∞ R ≈ lim η → + η |G , | Eρ . The probability distributions of the long time limit ofthe return probability, R ( P ∞ R ), are plotted in fig. 11 forseveral system sizes ( N = 32 , . . . , W = 4 and W = 12), showing thatthey behave in a completely different way on the two sidesof the transition: For W = 4 < W T the probability distri-butions of the long time limit of the return probability fol-lows a trivial scaling behavior, R N ( P ∞ R ) = V R ∞ ( V P ∞ R ),where R ∞ ( x ) is a narrow function which decays fast tozero. Hence both the typical value and the average valueof P ∞ R go to zero with the same exponent as V − . Con-versely, for W = 12 > W T the probability distributions R N ( P ∞ R ) are multifractal (i.e., do not follow a simplescaling behavior) and are characterized by heavy-tailswhich correspond to anomalously large rare values of P ∞ R . The average and the typical value go to zero as (cid:104) P ∞ R (cid:105) ∼ V − ˜ D and e ln (cid:104) P ∞ R (cid:105) ∼ V − ˜ D typ2 with the exponents˜ D ≈ .
514 and ˜ D typ2 ≈ .
517 both smaller than one.(The fractal exponent D describing the scaling of thetypical value of the imaginary part of the Green’s func-tion at the root of the Cayley tree as e (cid:104) ln Im G , (cid:105) ∼ V − D is D ≈ .
483 for W = 12.) This is a manifestation of thenon-ergodicity of the wave-functions. A similar resulthas also been recently found for the Anderson model onthe RRG. However, repeating the same analysis on theRRG we find that in this case the non-ergodic behavior isestablished only for systems smaller than the correlationvolume, while full ergodicity is restored in the limit ofvery large samples (i.e., ˜ D = ˜ D typ2 = 1 in the whole de-localized phase provided that V (cid:29) V ergo ). See Refs. [52]and [53] for more details.In a similar way, one can define the probability thata particle that sits at the root of the tree at time 0is found on the boundary of the tree at time t (whichis the situation analyzed in the main text). Using thefact that the wave-functions’ amplitudes can be writ-ten in terms of the matrix elements of the resolventas |(cid:104) x | ψ ( t → ∞ ) (cid:105)| ≈ lim η → + η |G x, | / Im G , , usingEq. (6) to express the correlation functions G x, in terms of the cavity Green’s functions on the edges connecting P Roo R ( P R oo ) N=32N=48N=64N=80N=96N=112 P Roo
N=32N=48N=64N=80N=96N=112
W=4W=12
FIG. 11. Log-log plot of the probability distributions of thelong-time limit of the return probability R ( P ∞ R ) for Cayleytrees of N generations (with N going from 32 to 112), and for W = 12 > W T (top panel) and W = 4 < W T (bottom panel).The black dotted straight line in the top panel correspondsto a power-law fit of the data as R ( P ∞ R ) ∼ ( P ∞ R ) − ˜ D with˜ D ≈ . sites 0 and x , and recalling Eq. (10), one finally obtains: P B ( t ) = (cid:88) x ( N ) (cid:12)(cid:12)(cid:12) (cid:104) x ( N ) | ψ ( t ) (cid:105) (cid:12)(cid:12)(cid:12) = (cid:88) x ( N ) (cid:12)(cid:12)(cid:12) (cid:104) x ( N ) | e − i H t/ (cid:126) | (cid:105) (cid:12)(cid:12)(cid:12) = (cid:88) x ( N ) (cid:63) (cid:88) α,β (cid:104) α | (cid:105)(cid:104) x ( N ) | α (cid:105)(cid:104) β | (cid:105)(cid:104) x ( N ) | β (cid:105) e − i ( E α − E β ) t/ (cid:126) . The long time limit of this object, defined in Eq. 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