How periodic driving stabilises and destabilises Anderson localisation on random trees
HHow periodic driving stabilises and destabilises Anderson localisation on random trees
Sthitadhi Roy,
1, 2
Roderich Moessner, and Achilleas Lazarides Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory,Oxford University, Parks Road, Oxford OX1 3PU, United Kingdom Physical and Theoretical Chemistry, Oxford University,South Parks Road, Oxford OX1 3QZ, United Kingdom Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Straße 38, 01187 Dresden, Germany Interdisciplinary Centre for Mathematical Modelling and Department of Mathematical Sciences,Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom
Motivated by the link between Anderson localisation on high-dimensional graphs and many-bodylocalisation, we study the effect of periodic driving on Anderson localisation on random trees. Thetime dependence is eliminated in favour of an extra dimension, resulting in an extended graphwherein the disorder is correlated along the new dimension. The extra dimension increases thenumber of paths between any two sites and allows for interference between their amplitudes. Westudy the localisation problem within the forward scattering approximation (FSA) which we adaptto this extended graph. At low frequency, this favours delocalisation as the availability of a largenumber of extra paths dominates. By contrast, at high frequency, it stabilises localisation com-pared to the static system. These lead to a regime of re-entrant localisation in the phase diagram.Analysing the statistics of path amplitudes within the FSA, we provide a detailed theoretical pictureof the physical mechanisms governing the phase diagram.
Localisation in quantum systems [1–6] renders invalidthe framework of statistical mechanics and paves the wayto novel quantum phases and order in excited eigen-states [7]. Perhaps the most remarkable examples ofthese are the discrete time crystals in a periodically-driven (Floquet) setting [8–11]. This spatiotemporal or-dering is an inherently non-equilibrium phenomenon [12]apparently impossible in systems governed by staticHamiltonians [13]. While ergodic Floquet systems gener-ically heat up to a featureless “infinite-temperature”state [14, 15], breaking ergodicity robustly via disorder-induced localisation prevents this heat death [16–19].Understanding the mechanisms that might stabilise ordestabilise localisation in the presence of periodic driv-ing is therefore a question of immanent interest.While the problem of many-body localisation (drivenor otherwise) can also be equivalently viewed as one ofsingle-particle localisation on the high-dimensional Fock-space graph, strong correlations in the latter render itqualitatively different from conventional Anderson local-isation on such graphs [20, 21]. Nevertheless, a naturalstep towards a theoretical understanding of the mecha-nism governing localisation in Floquet systems is to an-alytically study a simpler problem in a more controlledsetting – the fate of Anderson transitions and localisa-tion on high-dimensional graphs. We, therefore, studyFloquet tight-binding models defined on random high-dimensional graphs which are generally tree-like locally.In particular, the model we consider is described by atime-periodic (with period T ≡ π/ω ) Hamiltonian, H ( t ) = H + H cos( ωt ) , (1)with H = H hop + H dis and H = H hop . H hop describes n = − n = − n = 0 n = 1 n = 2 ω Γ c Γ c ≈ ae − b/ω Γ c ≈ Γ s c e c/ω Γ s c LocalisedDelocalisedΓ c ≈ ae − b/ω α α · · · α d n (a) (b)(c) FIG. 1. (a) The Shirley picture for a driven tight-bindingmodel on a tree. The different copies of the tree are indexedby n . The blue line shows the static path between the root siteand another, say α and α d , whereas the different colouredlines show some of the driving-induced paths between the rootsite and the same site’s copy, | α d , n (cid:105) , on other Shirley layers.(b) Schematic of the FSA amplitude on sites | α d , n (cid:105) start-ing from | α , (cid:105) . The amplitudes on the former can be bothpositive and negative so they can interfere to give the totalamplitude ψ α d ∝ (cid:80) n ψ α d ,n . (c) Summary of the localisa-tion phase diagram in the (Γ , ω ) plane for the model in (a).At high frequencies, driving favours localisation and enhancesthe critical hopping compared to the static case, Γ s c , whereasat low frequencies, it favours delocalisation and Γ c decreaseswith decreasing ω . the hopping while H dis the onsite disorder, H hop = Γ (cid:88) (cid:104) α,β (cid:105) | α (cid:105) (cid:104) β | + H . c . ; H dis = (cid:88) α (cid:15) α | α (cid:105) (cid:104) α | , (2)respectively where (cid:104) α, β (cid:105) denotes links between pairs ofsites on the graph, and the random potentials are drawnfrom a distribution with width W . Undriven cousins of a r X i v : . [ c ond - m a t . d i s - nn ] D ec the model in Eq. 1, i.e. disordered tight-binding modelson trees and random regular graphs have long served asarchetypes for localisation transitions and related phe-nomena on high-dimensional graphs, allowing for a re-markable amount of theoretical progress [22–37].Following the approach pioneered by Shirley [38, 39],the periodic time-dependence of a Hamiltonian of theform (1) can be eliminated in favour of an extra dimen-sion. Generalising the Forward Scattering Approxima-tion (FSA) [40] to this Shirley picture, we focus on theadditional paths (compared to the static case) the ex-tra dimension allows for between any two physical siteson the graph. The interplay between these extra pathsbecoming available and their interference results in anamplification or attenuation of the amplitude relative tothat of the static one. We provide analytical estimatesfor the locations of Anderson transitions in the hoppingstrength–frequency plane, Fig. 1. The phase diagram ex-hibits a characteristic re-entrant localised portion.The remainder of this paper is organised as follows.We first recapitulate the Shirley formalism, showing howthe time dependence is eliminated in favour of an ex-tra dimension, then generalise the FSA to this situation.We then provide numerical evidence for the localisationphase diagram (Fig. 1(c)), followed by analytical argu-ments elucidating the mechanisms underpinning it.The problem in the Shirley picture maps to a new un-driven system on an infinite ladder where each rung cor-responds to a copy of the static part of the Hamiltonian H with the n th rung shifted in energy by nω , and thedriving Hamiltonian, H , mediating hopping between therungs. Thus the new system lives on an extended graphwith one additional dimension. This time-independentHamiltonian, for the model described by Eqs. 1 and 2,takes the form H F =Γ (cid:88) (cid:104) α,β (cid:105) ,n (cid:88) s =0 , ± [ | α, n (cid:105) (cid:104) β, n + s | + H . c]+ (cid:88) α,n ( (cid:15) α + nω ) | α, n (cid:105) (cid:104) α, n | , (3)where | α, n (cid:105) = | α (cid:105)⊗| n (cid:105) denotes a state localised on phys-ical site α and Shirley rung n . Solutions of the time-dependent Schr¨odinger equation ( i∂ t − H ( t )) | φ ( t ) (cid:105) = 0can be written as | φ a ( t ) (cid:105) = e i Ω a t (cid:80) n | ϕ ( n ) a (cid:105) e inωt , where | ϕ a (cid:105) = (cid:80) ∞ n = −∞ | ϕ ( n ) a (cid:105) ⊗ | n (cid:105) is an eigenstate of the time-independent Hamiltonian (3) with eigenvalue Ω a and | ϕ ( n ) a (cid:105) is a state living in a single rung.Physically, the hopping between different rungs corre-sponds to the system gaining or losing energy in integermultiples of ω . Thus additional processes, or in the time-independent picture, additional paths are made availableto hop and delocalise away from a site α by the driving,see Fig. 1 for a visual representation. The question of lo-calisation for the driven system (1) is now mapped ontoone for the time-independent system (3) with one extra dimension, which we will study within the FSA.We begin by briefly describing the FSA for a staticsystem. For a state initially (in the absence of hopping)localised at α , the FSA estimates the amplitude on someother site, α d at distance d , upon perturbatively includ-ing the effects of hopping by summing over all the short-est paths between the two sites. It is thus a stabilityanalysis of the localised phase at Γ = 0 for Γ (cid:54) = 0 [40].The amplitude ψ α d is ψ α d = (cid:88) p : α (cid:1) α d (cid:89) β ∈ p Γ (cid:15) β − (cid:15) α , (4)where the sum is over all the shortest paths from α to α d . Breakdown of localisation is indicated by the proba-bility (over disorder realisations) of the state spreading toat least one site at arbitrarily large distance d approach-ing unity. Defining | ψ d | = max α d | ψ α d | , the maximumamplitude over all sites at a given distance, the delocali-sation criterion islim d →∞ P (cid:18) ln | ψ d | d > − ξ − (cid:19) → , (5)where ξ is the localisation length. Defining Λ α d =ln | ψ α d | / d − ln(Γ /W ) and Λ d = max α d Λ α d , the crit-ical point (Γ /W ) c can be expressed, using Eq. 5, aslim d →∞ P (Λ d > − ln(Γ /W ) c ) → ⇒ lim d →∞ C (Λ d = − ln(Γ /W ) c ) → , (6)where C (Λ d ) is the cumulative distribution and we haveused the fact that the localisation length ξ diverges atthe transition. In other words, if P (Λ d ) is peaked atsome value (cid:104) Λ d (cid:105) then the critical point is at ln(Γ /W ) c = − (cid:104) Λ d (cid:105) . Note that, as in the static case, taking the maxi-mum amplitude over all sites at distance d overestimatesthe critical disorder, resulting in an upper bound.For the driven system at the Γ = 0 limit, the | ϕ a (cid:105) areall localised on a single physical site and a single rung.Switching on Γ allows for hopping to other sites andrungs. There is however one difference from the staticcase: for stroboscopic time evolution, t = kT with in-teger k , | φ a ( kT ) (cid:105) = e i Ω a kT (cid:80) n | ϕ ( n ) a (cid:105) . Thus, the ampli-tude at a physical site α will depend on the amplitudeson all rungs, (cid:80) n | ϕ ( n ) a (cid:105) , implying that the probabilityamplitude of the state spreading to physical site α d is ψ α d = (cid:80) n ψ α d ,n where ψ α d ,n is the amplitude on site α d and Shirley rung n .To each path in the static system between two sites, say α (cid:1) α (cid:1) · · · α d , there correspond 3 d paths in the drivensystem because at each hop in physical space the rung canchange by 0 or ± | α, (cid:105) andterminate at | α d , n d (cid:105) with n d ∈ [ − d, d ] [41]. − . − . . . . . . . d . . . . . . C Λ d d − P Λ d . . . . . ω . . . . Γ c K = 1 . K = 1 . K = 1 . K = 1 . K = 1 . ω = 0 . ω = 2 . ω = 4 . ω = 10 . FIG. 2. (a) The cumulative distribution of the rescaled FSA amplitudes, C (Λ d ) (see Eqs. 5 and 6) for values of d = 8 , , · · · , ω labelled by the four colour-maps. The crossing points of the curves indicate the ln( W/ Γ) c forthe corresponding ω . The inset shows the distributions P (Λ d ) for ω = 0 . ω = 2 which get sharper with increasing d .Results are for K = 1 .
05 and the disordered potential was drawn from a standard Normal distribution (cid:15) α ∼ N (0 , W = 1). (b)The critical Γ c /W as a function of ω/W for different values of K obtained from the crossing of C (Λ d ) exemplified in (a) andobtained using Eq. 8. Note the non-monotonicity in Γ c /W as a function of ω/W . Statistics for all data were obtained over5 × realisations. The total amplitude, ψ α d due to all these paths canthen be written as ψ α d = (cid:88) p : α (cid:1) α d d (cid:88) q =1 (cid:89) α i ∈ p Γ∆ i + n ( q ) α i ω , (7)where ∆ i = (cid:15) α i − (cid:15) α , the first sum (over p ) is over thephysical paths between α and α d , the second sum (over q ) is over the Shirley paths corresponding to the physicalpath p , α i is the physical site on the i th step of the path,and n ( q ) i is the rung index of the path q at step i [42].The delocalisation criterion for the driven system is thenidentical to that in Eq. 5 but with the FSA amplitudeobtained in the Shirley picture from Eq. 7. In the nota-tion used in Eq. 7, the path present in the static case isthe one for which n ( q ) α i = 0 for all i .It is important to distinguish between the various inter-ference effects at play here. For a given n , ψ α d ,n receivescontributions from multiple paths, each of which can bewritten as | α , n (cid:105) → . . . → | α j , n j (cid:105) → . . . → | α d , n d (cid:105) with n = 0 and n d = n . In particular, there are (i) dif-ferent physical paths that follow the same Shirley rungs,so the α j are different for each path for j (cid:54) = 0 , d but the n j the same for all–this is present for the static case too;(ii) the same physical path on different rungs, so the α j are the same for all but the n j different. In addition,(iii) since ψ α d = (cid:80) n ψ α d ,n , the FSA amplitudes ψ α d ,n on different rungs interfere with each other. Crucially,neither of (ii) and (iii) is present for the static case [43].Eq. 7 manifestly takes all of these kinds of interferenceinto account.In order to single out the effect of periodic-driving andinterference between the Shirley paths, we take our phys-ical graph to be a rooted tree, which has no loops. There then exists a unique shortest physical path between anytwo sites of the graph, removing the interference effectlabelled (i) above and leaving only the driving-inducedeffects (ii) and (iii).For a tree with average branching number K [44], thereare K d sites at distance d from the root and one needsthe probability distribution of the maximum amplitudeover these K d sites, see above Eq. 5. As the amplitudeson each of these sites are independent and identically dis-tributed with P , denoting this distribution by P max (Λ d ),one finds P max (Λ d ) = K d [ C (Λ d )] K d − P (Λ d ) , (8)with P, C the distribution and cumulative distributionfor a single site. Equivalently, C max (Λ d ) = [ C (Λ d )] K d .It is thus possible to compute P max (Λ d ) or C max (Λ d ) forarbitrary K by considering only single physical paths oflength d . We, therefore, use the notation Λ d for boththe amplitude of a single physical path and the maxi-mum amplitude. The FSA for the former in the Shirleypicture can be efficiently implemented using a transfermatrix [45].In Fig. 2 we show the numerical results from the FSAin the Shirley picture. The inset in panel (a) shows thatthe distributions P (Λ d ) sharpen with increasing d . Thisis manifested in the cumulative distributions, C (Λ d ) forvarious d crossing at a particular value of Λ d as evidencedin panel (a). The crossing point from Eq. 6 can be in-ferred to be ln( W/ Γ) c , yielding the critical point. Thecritical Γ c /W so obtained is shown as a function of thefrequency ω in panel (b). There are two crucial featuresof note: (i) For ω/W (cid:29)
1, Γ c /W is larger than that ofthe undriven system, Γ s c /W . Thus, in this regime, theperiodic driving enhances the localised phase. (ii) In theregime of ω/W (cid:47)
1, Γ c /W decreases with decreasing ω and is, in fact, much smaller than Γ s c /W . Hence, inthis regime, the driving suppresses the localised phase,favouring delocalisation, and parametrically so with de-creasing ω .In the following we present analytical arguments whichgive insight into the aforementioned behaviour of Γ c with ω . As the distributions for a system with arbitrary K aresimply related to that for a single physical path via Eq. 8,which leads to the qualitative behaviour of Γ c with ω be-ing the same for all K , it suffices to analyse the FSAamplitudes for different Shirley paths corresponding toa single physical path α (cid:1) α (cid:1) · · · α d . It will be use-ful to write Λ d = ln[( (cid:80) d q =1 w q ) ] / d , where (Γ /W ) d w q isthe amplitude of Shirley path q . Further, since the dis-tribution of Λ d gets sharper with increasing d , one maytake lim d →∞ (cid:104) Λ d (cid:105) = − ln(Γ /W ) c . It will also be usefulto decompose Λ d into ‘direct’ and ‘interference’ terms,Λ d = 12 d ln (cid:88) q w q (cid:124) (cid:123)(cid:122) (cid:125) Λ dir d + 12 d ln (cid:32) (cid:80) q (cid:54) = q (cid:48) w q w q (cid:48) (cid:80) q w q (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) Λ int d . (9)We first turn to the low-frequency regime, ω/W (cid:28) d ≈ Λ dir d in this regime. This is justified be-cause [45], firstly, for ω/W (cid:28) | Λ int d | (cid:28) | Λ dir d | and sec-ondly Λ int d < W/ Γ) c ,consistent with the general bounds placed by the FSA.For a given path q , | n ( q ) α i − n ( q ) α i +1 | = 0 or 1, so a pathselected at random from all possible paths is a simplerandom walk with α playing the role of time. On theother hand, the (cid:15) α are uncorrelated for different α . Wedefine a physical site α i to be resonant on Shirley rung n ∗ α i if (cid:12)(cid:12) ∆ i − n ∗ α i ω (cid:12)(cid:12) < ω such that all the resonances liewithin a strip of width ∝ W/ω in the Shirley direction(see Fig. 3). Hence, for a typical path, the probability ofa resonance vanishes after (cid:29) ( W/ω ) steps since by thenthe random walk is well outside the strip of resonances.Although in the limit of d → ∞ , a fraction tending to1 of these paths eventually do escape this strip eliminat-ing the possibility of a resonance, driving can still causedelocalisation as Λ d consists of a sum of of the contribu-tions of the paths and not an average. One thus needs toconsider an effective number of paths N eff q which is thesum, over all the paths, of the fraction of each path spentinside the strip.For simplicity, consider a single path lying entirelywithin the strip. As it meanders along the strip (Fig. 3)the probability it picks up a resonance per step is con-stant, ρ ( ω ), the same for all steps (due to the uncorre-lated nature of the (cid:15) α ). A path of length d will there-fore pick up µ = ρ ( ω ) d resonances on average. Overthe ensemble of all paths lying entirely in the strip, the probability that a given path has r resonances is Pois-son, P r = exp( − µ ) µ r /r !. Finally, for a given path q , w q = (cid:81) di =1 (∆ i + n ( q ) α i ) − , we replace the factor for α i with1 /W if α i is off-resonant, 1 /ω if resonant; thus a pathwith r resonances will have w q ≈ ( W/ω ) r . From all theabove together, we have Λ d ≈ d ln (cid:16) N eff q (cid:80) dr =0 P r W r ω r (cid:17) .In the limit of d → ∞ ,lim d →∞ Λ d ≈ c − ρ ( ω ) (cid:18) − W ω (cid:19) , (10)where c ≡ lim d →∞ (ln N eff q ) / d is an O (1) constant [45].As a site α i on some path q is resonant when | ∆ i + n ( q ) α i | is within ω , a rough estimate for the probability of a res-onance ρ ( ω ) is ω/W . With this estimate, the localisationcritical point is(Γ /W ) c ≈ a exp( − bW/ω ) , (11)where a and b are constants. Thus, in this regime, drivingdestabilises the localised phase (suppressing Γ c ) as onemight expect [16, 17, 19]. Note that this is a result ofthe competition between the increase in the number ofresonances (since ρ ( ω ) = ω/W ) and the reduction of theirstrength (1 /ω ) as ω increases.Next we turn to the high-frequency regime, ω/W (cid:29) c compared to Γ s c . We first note that for ω = ∞ , only the static path contributes, the amplitudefor which we denote as w s . We will therefore use it asa reference and consider the effect of deviations from it.For large ω/W (cid:29)
1, Shirley paths that deviate from thestatic path at x sites have an amplitude relative to thestatic one suppressed parametrically as O (( W/ω ) x ). Theleading order contribution (at O (( W/ω ) ) ) can be shownto be [45]Λ d ( ω ) ≈ d ln (cid:34) w (cid:32) − ω d (cid:88) i =1 ∆ i + ∆ d − ∆ d ω (cid:33)(cid:35) . (12)Equation 12 already shows that Λ d in the presence ofdriving is smaller than that of the static case, Λ s d =ln w / d , for any disorder realisation indicating an en-hancement of the localised phase. Using the fact thatthe ∆ i ’s are independent of each other and ω/ ∆ i (cid:29) (cid:104) Λ d ( ω ) (cid:105) d (cid:29) ≈ (cid:104) Λ s d (cid:105) − W /ω . (13)Since the distribution of Λ d sharpens with increasing d (see Fig. 2(a)), it seems reasonable to assume thatlim d →∞ (cid:104) Λ d (cid:105) = − ln(Γ /W ) c , as such Eq. 13 yields,Γ c ( ω ) ≈ Γ s c e W /ω . (14) dn Wω dn Wω FIG. 3. Schematic picture for the Shirley resonances in thelow-frequency regime, ω (cid:28) W . In the plots above ω > ω .The resonances denoted by the red dots, where the effectivesite-energies | ∆ i + n ∗ α i ω | are within ω typically stay within astrip (shaded in orange) in the Shirley direction of width W/ω .The larger ω/W , the greater the likelihood of a resonanceoccurring per unit length. A path q is resonant if the n ( p ) α passes within a distance 1 of these resonances, as indicatedby the blue-shaded “swept area” of each path. The result in Eq. 14 explicitly shows the increase in Γ c from Γ s c and the enhancement of the localised phase.The mechanism behind the driving-induced suppres-sion of the total FSA amplitude can be understood asfollows. For every site α i on the physical path, thereexists a pair of Shirley paths with n ( q ) α j = ± δ ij . If we con-sider, without loss generality, ∆ i >
0, then ∆ i ± ω ≷ | ∆ i + ω | > | ∆ i − ω | . Hence, the path whose am-plitude has the opposite (same) sign to the static pathhas a higher (lower) magnitude. The total amplitude isthus suppressed compared to the static case. However,the correction turns out to be O (1 /ω ) so that one hasto take into account all paths which deviate at up to twosites from the static path.The physical picture and the derivation of Eq. 12 sug-gest that this generalises to pairs of paths that deviateat multiple sites from the static path, but with oppositesigns of the Shirley rung indices. While the quantita-tive result in Eq. 14 can change upon including higher-order (in 1 /ω ) contributions, localisation enhancement isexpected to stay robust; however, a concrete analyticaldemonstration remains for future work.In summary, we have addressed the problem of An-derson localisation on driven, disordered trees using theFSA on an extended Shirley graph by eliminating thetime-dependence of the Hamiltonian in favour of an ex-tra dimension. The interplay between the availability ofadditional paths due to the driving and their interferenceyields a non-trivial localisation phase diagram; in thehigh-frequency regime, the localised phase is stabilisedrelative to the undriven case, so that there is localisationfor weaker disorder than in the static case, whereas inthe low-frequency regime driving destabilises localisationand increases the strength of the disorder required for localisation.A natural next step towards addressing the stabilitytowards driving of many-body localisation due to eithercorrelations in Fock-space disorder [20] or constraints inthe Fock-space graph [46] would be to incorporate the ef-fect of multiple static paths between any two sites, ratherthan focus on purely tree-like structures as we have donehere. 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Supplementary material: How periodic driving stabilises and destabilises Anderson localisation onrandom trees
Transfer matrix for FSA in Shirley picture
We briefly describe the transfer matrix implementationof the FSA in the Shirley picture for a single physicalpath α (cid:1) α (cid:1) · · · α d . The transfer matrix acts on aspace of ( d + 1) × (2 d + 1) sites on the Shirley graph asthere are d + 1 physical sites and 2 d + 1 Shirley rungs n ∈ [ − d, d ]. We first introduce a matrix A proportionalto the adjacency matrix of the directed Shirley graph.The directed-ness of the graph stems from the fact thatamplitude from site ( α i , n ) can only be transported to( α i +1 , n + s ) with s = ± , A = Γ d − (cid:88) i =0 d (cid:88) n = − d (cid:88) s = ± , | α i +1 , n + s (cid:105) (cid:104) α i , n | . (S1)Additionally, we introduce a diagonal matrix defined onthe vertices of the graph as D = d (cid:88) i =0 d (cid:88) n = − d ( (cid:15) α i + nω − (cid:15) α ) − | α i , n (cid:105) (cid:104) α i , n | . (S2)The transfer matrix then given by T = DA , (S3)such that for the initial state | ψ (cid:105) = | α , (cid:105) , we have T x | ψ (cid:105) = x (cid:88) n = − x ψ α x ,n | α x , n (cid:105) (S4)where ψ α x ,n ’s are the FSA amplitudes of interest. FSA amplitudes at high-frequency
In this section, we present the details of the derivationof Eq. 12 at leading order. Since, the Shirley Hamilto-nian (3) is invariant under ω → − ω , the leading ordercorrection is expected to be O (1 /ω ); hence we considerall the paths that deviate from the static path at at mosttwo sites. As a matter of notation, the amplitude of apath that deviates from the static path at site α i , suchthat n ( q ) α i = ± w i ± . The paths thatdeviate at exactly two sites from the static path are alsoforced to have n ( q ) α i = ± n ( q ) α d − = ± n ( q ) α d = ±
2. We denote the amplitude ofpaths that deviate at sites α i and α j with n ( q ) α i = ± n ( q ) α j = ± w i ± j ± . Additionally, we denote the ampli-tudes of the two paths with n ( q ) α d − = ± n ( p ) α d = ± w (2)+ and w (2) − respectively. With this notation, to O (1 /ω ), G = ( (cid:80) q w q ) can be written as G = w + ( d (cid:88) i =1 w i + ) + ( d (cid:88) i =1 w i − ) (S5a)+ 2 w s d (cid:88) i =1 ( w i + + w i − ) + 2 (cid:88) i,j w i + w j − (S5b)+ 2 w s (cid:88) i,j (cid:88) η,η (cid:48) = ± w i η j η (cid:48) + w (2)+ + w (2) − . (S5c)Using ∆ i ≡ (cid:15) i − (cid:15) , the amplitudes w i ± and w i ± j ± canbe related to w s as w i ± = w s ∆ i ∆ i ± ω , w i ± j ± = w s ∆ i ∆ j (∆ i ± ω )(∆ j ± ω ) , (S6)and w (2) ± = w s ∆ d − ∆ d (∆ d − ± ω )(∆ d ± ω ) . (S7)Using Eqs. S6 and S7, and ω (cid:29) ∆ i in Eq. S5, we have G = w + 2 w d (cid:88) i =1 ∆ i ω + 2 w (cid:88) i (cid:54) = j ∆ i ∆ j ω (S8a) − w d (cid:88) i =1 ∆ i ω − w (cid:88) i,j ∆ i ∆ j ω (S8b)+ w ∆ d − ∆ d ω . (S8c)The terms in Eqs. S8a, S8b, and S8c correspond ex-actly to their counterparts in Eqs. S5a,S5b, and S5c re-spectively (The summation in Eq. S5c vanishes.). Equa-tion S8 can be reorganised to yield G = w (cid:32) − ω d (cid:88) i =1 ∆ i + ∆ d − ∆ d ω (cid:33) , (S9)which directly leads to Eq. 12. More on the statistics of Shirley path amplitudes
In this section, we present some more results on thestatistics of Shirley path amplitudes. We first show evi-dence that with decreasing ω , the number of paths thatcontribute to the FSA amplitudes increases qualitatively.To this end, we define an effective inverse participationratio I = (cid:88) q (cid:32) | w q | (cid:80) q (cid:48) | w q (cid:48) | (cid:33) . (S10)2 N = 3 d − − − − − I (a) ω = 10 . ∼ N − . ω = 0 . ∼ N − . ω . . . . h Λ d i (b) h Λ d ih Λ dir d ih Λ int d i FIG. S1. (a) The IPR defined in Eq. S10. The scaling with N = 3 d shows that with decreasing ω , qualitatively morepaths contribute to the total amplitude. (b) The behaviourof (cid:104) Λ d (cid:105) (blue), (cid:104) Λ dir d (cid:105) (red), and (cid:104) Λ int d (cid:105) (green) as a functionof ω . The different markers and colour-intensities show d = 8to d = 16, all of which are well converged with d . In the limit of ω →
0, when there is only one path (thestatic path) with a finite amplitude, we expect
I ∼ N with N = 3 d the total number of paths. On the otherhand, in the limit of ω →
0, where all the Shirley pathsare equivalent, we expect
I ∼ N − . The results shownin Fig. S1(a) suggest that I ∼ N − α with α growing withdecreasing ω . This implies that the total FSA amplitudeis more and more delocalised over all the Shirley pathsas ω is decreased.In the main text, we had decomposed Λ d = Λ dir d +Λ int d ,where the first term is the direct term and the secondterm encodes the interferences between the paths. InFig. S1(b), we show the behaviour of (cid:104) Λ d (cid:105) , (cid:104) Λ dir d (cid:105) , and (cid:104) Λ int d (cid:105) with ω . With increasing ω , since fewer paths con-tribute and the strength of each resonance also weak-ens, (cid:104) Λ dir d (cid:105) decreases monotonically. On the other hand, (cid:104) Λ int d (cid:105) is a non-monotonic function of ω , which leads tothe eventual non-monotonicity in Γ c as a function of ω as explain in the main text. Note that for ω/W (cid:28) | (cid:104) Λ int d (cid:105) | (cid:28) | (cid:104) Λ dir d (cid:105) | and also (cid:104) Λ int d (cid:105) < d ≈ Λ dir d in the low-frequency regime. Effective number of paths in low-frequency regime
We show, in this section, that lim d →∞ ln( N eff q ) / d = c where c is a constant and N eff q is the total length of allpaths spent inside the resonant strip, | n | < W/ω . Wecompute N eff q as follows. At distance x , let us the denotethe probability that a path passes through rung n x as P n x , which is given by the trinomial distribution P n x ( n ) = x (cid:88) a,b =0 δ a − b,n x x ! a ! b ! ( x − a − b )! x (cid:29) ≈ √ πσ e − n / σ ; σ = 2 x/ . (S11) The total probability that a path is inside the strip ofresonances at distance x is given by m x = (cid:90) W/ω − W/ω dn P n x ( n ) ≈ Erf (cid:34) Wω (cid:114) x (cid:35) . (S12)Since, the total number of paths is 3 d , the effective num-ber of paths N eff q is N eff q = 3 d d (cid:90) d dx m x , (S13)as such in the limit of d → ∞ , we havelim d →∞ d ln N eff q = ln √ ≡ c .c .