Loschmidt echo singularities as dynamical signatures of strongly localized phases
Leonardo Benini, Piero Naldesi, Rudolf A. Römer, Tommaso Roscilde
LLoschmidt echo singularities as dynamical signatures of strongly localized phases
Leonardo Benini,
1, 2
Piero Naldesi, Rudolf A. R¨omer,
1, 4 and Tommaso Roscilde Department of Physics, University of Warwick, Coventry, CV4 7AL, UK Univ de Lyon, Ens de Lyon, Univ Claude Bernard,and CNRS, Laboratoire de Physique, F-69342 Lyon, France Universit´e Grenoble-Alpes, LPMMC and CNRS, F-38000 Grenoble, France CY Advanced Studies and LPTM (UMR8089 of CNRS),CY Cergy-Paris Universit´e, F-95302 Cergy-Pontoise, France
Quantum localization (single-body or many-body) comes with the emergence of local conservedquantities — whose conservation is precisely at the heart of the absence of transport through thesystem. In the case of fermionic systems and S = 1 / l -bits. While their existence is the definingfeature of localized phases, their direct experimental observation remains elusive. Here we show thatstrongly localized l -bits bear a dramatic universal signature, accessible to state-of-the-art quantumsimulators, in the form of periodic cusp singularities in the Loschmidt echo following a quantumquench from a N´eel/charge-density-wave state. Such singularities are perfectly captured by a simplemodel of Rabi oscillations of an ensemble of independent two-level systems, which also reproducesthe short-time behavior of the entanglement entropy and the imbalance dynamics. In the caseof interacting localized phases, the dynamics at longer times shows a crossover in the decay ofthe Loschmidt echo singularities, offering an experimentally accessible signature of the interactionsbetween l -bits. Introduction.
Localization phenomena in systems ofquantum particles offer striking evidence of the role ofinterference in quantum dynamics. Constructive inter-ference of paths bringing a particle back to its initiallocation in real space is at the heart of single-particle(or Anderson) localization (AL) [1, 2]; more recently asimilar phenomenon occurring in Hilbert space (dubbedmany-body localization - MBL) [3–6] has been shown toprevent many-body quantum systems from relaxing tothermal equilibrium, undermining the ergodic hypothesisin a large class of models of interacting quantum parti-cles. Localized phases are generally characterized in thenegative (absence of transport, of long-range order, ofspectral gaps, etc.), while positive characterizations aregenerally elusive. A crucial aspect of localization is thepersistence of initial conditions, which, in the case of ALof non-interacting particles, is related to the conserva-tion of populations in the localized single-particle eigen-states of the Hamiltonian. In the case of MBL, the analogphenomenon would be the appearance of local conservedquantities (called local integrals of motion or l -bits [7–10]) which are obtained by unitary transformations oflocal operators; and which, if extensive in number, con-strain the dynamics of the system to the point of prevent-ing relaxation. The existence of l -bits in disordered spinchains can be mathematically proven under the assump-tion of limited level attraction [9], and approximate l -bitsfor many-body systems can be constructed with a varietyof analytical as well as numerical methods [11–20]. Muchof the phenomenology of MBL dynamics (persistence oftraits of the initial state, logarithmic growth of entangle-ment entropies, etc.) – observed in numerical studies aswell as in experiments [4, 5] – can be directly explained interms of the existence of l -bits and interactions betweenthem. Yet observing l -bits directly is an arduous task, given that their expression is highly disorder-dependent(and generally unknown even in theory), and it wouldrequire high-precision measurements of local observablesin different local bases. An even more arduous task isthe one of probing directly the existence of interactionsamong l -bits, which is a defining feature distinguishingMBL from AL.The purpose of this Letter is to show that, in the case ofstrongly localized phases, the existence of l -bits can offerstriking signatures in the dynamics of the Loschmidt echo(LE), namely in the logarithm of the return probabilityto the initial state | ψ (cid:105) λ ( t ) = − L (cid:2) log |(cid:104) ψ | e − i H t | ψ (cid:105)| (cid:3) av . (1)Here H is the system’s Hamiltonian and L its size; [ ... ] av indicates the disorder average. When | ψ (cid:105) has a simplefactorized form and in the case of strong disorder, wefind that the LE displays periodic singularities, decayingvery slowly in amplitude – as illustrated using a model ofdisordered spinless fermions in 1d (corresponding to the S = 1 / l -bits. The sameminimal model captures quantitatively the dynamics ofthe entanglement entropy at short times as well as ofthe number entropy at longer times; and the dynamicsof the density imbalance characterizing the initial state.At longer times the deviation of the exact results for theMBL dynamics from the predictions of the 2LS ensembleoffers direct evidence of the interactions among the l -bitsin the form of a faster decay of the LE singularities and a r X i v : . [ c ond - m a t . d i s - nn ] A ug imbalance oscillations. As the LE is generally accessi-ble to quantum simulators measuring individual degreesof freedom [21, 22], our results show that strong signa-tures of l -bits dynamics are within the immediate reachof state-of-the-art experiments on disordered quantumsystems. Model.
Our platform for the investigation of LE dy-namics is given by a paradigmatic model, namely the S = 1 / H = L − (cid:88) i =1 (cid:20) − J (cid:0) S + i S − i +1 + h . c . (cid:1) + J z S zi S zi +1 (cid:21) − L (cid:88) i =1 h i S zi = L − (cid:88) i =1 (cid:20) − J (cid:16) c † i c i +1 + h . c . (cid:17) + J z n i n i +1 (cid:21) − L (cid:88) i =1 h i n i , (2)where S αi ( α = x, y, z ) are spin operators and c i , c † i and n i = c † i c i are fermionic operators; the equality betweenthe two Hamiltonians is true up to an additive con-stant [24]. In the following the external field/potential h i is taken to be either quasi-periodic (QP), namely h i = ∆ cos(2 πκi + φ ) with κ = 0 .
721 (inspired by ex-periments on bichromatic optical lattices [25, 26]) and φ a random phase; or to be fully random (FR) and uni-formly distributed in the interval [ − ∆ , ∆]. We considerchains of length L (up to L = 22) with open boundaries,and we average our results over ∼ realizations of therandom phase (QP) or of the full random potential (FR).All the unitary evolutions considered in this study are ob-tained using exact diagonalization (ED), and they startfrom the charge-density wave state | ψ (cid:105) = | ... (cid:105) ,corresponding to a N´eel state for the spins.We shall focus on the case of interacting fermions J z = J (corresponding to an SU(2) invariant spin-spininteraction) and contrast it with the limit of free fermions J z = 0. In the latter case, the QP potential leads to atransition to fully localized single-particle eigenstates for∆ ≥ J , with an energy independent localization length ξ = 1 / log(∆ /J ); while the FR potential leads to AL ofthe whole spectrum at any infinitesimal value of disor-der. In the interacting case, instead, a QP potential ofstrength ∆ (cid:38) J [27] and a FR potential of strength∆ (cid:38) . J [28] are numerically found to lead to MBL, al-though the exact critical value ∆ c has been recently theobject of further investigation [28–35]. In the followingwe will conduct our discussion starting from the case ofthe QP potential, which has a simpler spatial structuredevoid of rare regions, leading to stronger localization ef-fects; and we shall later discuss how the picture shouldbe enriched in the case of the FR potential to accountfor the existence of rare regions. LE singularities.
Fig. 1 shows the dynamics of theLE, λ ( t ), along with that of the imbalance I ( t ) = (cid:80) i ( − i (2[ (cid:104) n i (cid:105) ] av − /L – the latter saturates to itsmaximum value of 1 in the initial state and probes the persistence of the initial density/spin pattern [36]. Weobserve that for both the QP and FR potentials, and fordisorder strengths compatible with the onset of the MBLregime, the LE displays a sequence of periodic cusp-likepeaks at times t n = (2 n + 1) π/J ( n = 0 , , , ... ). Thesetimes correspond to minima in the imbalance, as the sys-tem reaches instantaneous configurations which are thefarthest from the initial spin/density pattern. A closerinspection shows that, for sufficiently strong disorder, allthe peaks become sharp cusps, namely they representgenuine non-analyticities of the LE. They are rather re-markable given that they survive disorder averaging, andthey seemingly appear in a finite fraction of disorder re-alizations (see [37] for further details); and in particularthey decay very slowly in time, as we shall discuss indetail in the following. l -bits. All the essential details of theshort-time evolution of the LE can be captured with asurprisingly simple, yet rather insightful model. Thismodel is best understood (and justified) in the case ofthe QP potential, as illustrated in Fig. 2. In the caseof strong disorder, the fastest dynamics in the systemstarting from a Fock state will be offered by those par-ticles that sit on a site i which is nearly resonant withits unoccupied neighbor (say i + 1), because the hopping J/ δ i = h i +1 − h i (in the non-interacting case) or larger than the screenedoffset δ i − J z (in the presence of nearest-neighbor repul-sion). These 2-site clusters, representing nearly resonanttwo-level systems (2LS), have the property of being spa-tially isolated in the QP potential, because of the strong . . . . . . ∏ ( t ) . . . . . .
00 10 20 30 40 Jt . . . . . . I ( t ) Jt . . . . . .
02 4 6 8 ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ( a ) ( b ) ( d ) ( c ) Figure 1. Loschmidt echo and imbalance dynamics for an L = 22 chain with QP potential (a-c) and FR potential (b-d),for various disorder strengths ∆ = 2 , . . . , (1) (2) ( a )( b ) (1) J z = 0 J /2 J z δ − J z J /2 δ (2) J z ≠ 0 δ Figure 2. (a) Example of a L = 22 chain in a QP poten-tial (lines) in the initial CDW state | ... (cid:105) . Particles aredenoted as coloured balls. (b) Zoom on two quasi-resonantregions (shaded areas): in the case of non-interacting parti-cles ( J z = 0) the region (1) presents a pair of quasi-resonantsites for the particle in orange; in the case of interacting par-ticles, region (2) shows two quasi-resonant sites for the orangeparticle, thanks to the partial screening of disorder offered bythe interaction with the red particle. anticorrelation among two consecutive energy offsets ( δ i and δ i ± – see [37]). As a consequence, a nearly resonant2-site system will be generally surrounded by highly non-resonant pairs of sites, which can be considered as nearlyfrozen to the initial state. This invites us to write for theevolved state the following 2LS Ansatz | ψ ( t ) (cid:105) ≈ (cid:16) ⊗ n | ψ ( n )2 LS ( t ) (cid:105) (cid:17) ⊗ ( ⊗ (cid:48) i | ψ ,i (cid:105) ) , (3)where the first tensor product ⊗ n runs over the nearlyresonant 2LS, while the second tensor product ⊗ (cid:48) i runsover the leftover sites (we have taken the freedom ofreordering the sites arbitrarily in the tensor product). | ψ ( n )2 LS ( t ) (cid:105) is the evolved state of the n -th (isolated) 2LSsystem, corresponding to two states split by an energydifference δ (cid:48) n = δ n − J z and connected by a Rabi cou-pling J [37]; while | ψ ,i (cid:105) is the (persistent) initial stateof the site i belonging to the remainder of the system.The LE for such a system is readily calculated as λ ( t ) = − L (cid:88) n log [1 − p ( δ (cid:48) n , J, t )] , (4)with p ( δ, Ω , t ) = (Ω / Ω (cid:48) ) sin (Ω (cid:48) t/
2) (and Ω (cid:48) = √ Ω + δ ) the well-known probability of finding the 2LSin the state orthogonal to the initial one while perform-ing Rabi oscillations [38]. When averaging Eq. (4) overdisorder, it is immediate to obtain the following simpleexpression λ ( t ) = − (cid:90) P ( δ (cid:48) + J z ) log [1 − p ( δ (cid:48) , J, t )] . (5)Here P ( x ) is the probability that the energy offset be-tween two neighboring sites takes the value x [39]. Eq. (5) is an analytical integral formula which depends uniquelyon the (known) statistics of the disorder potential viathe P distribution. In the case of the QP potential P ( x ) = [1 − ( x/ ˜∆) ] − / /π ˜∆ with ˜∆ = ∆ sin( πκ ) [40];while for the FR potential P ( x ) is the normalized tri-angular distribution defined on the [ − , any adjustableparameter. In particular the cusp singularities of the EDresults are easily explained as descending from the di-vergent singularity of the LE for a fully resonant 2LSwith Ω = Ω (cid:48) = J , reaching a state orthogonal to theinitial one after odd multiples of half a Rabi oscillation t n = (2 n + 1) π/ Ω. These divergences are smoothenedinto cusp singularities due to the fact that such resonant2LSs are a set of zero measure in the disorder statis-tics. This result has important consequences. Indeed thenearly resonant 2LSs captured by the model are clearlyan ensemble of approximate l -bits with Hamiltonian H ≈ (cid:88) n K n τ n , (6)where τ n = ( δ (cid:48) n /K n ) σ zn − J/ ( K n ) σ xn is a Pauli matrixexpressed as a rotation of the Pauli operators σ zn = S zi +1 − S zi (when projected onto the subspace with S zi + S zi +1 =0) and σ x = S + i S − i +1 + h . c . , built from the original spinoperators for the pair n = ( i, i + 1); and K n = (cid:112) δ n + J is the l -bit splitting. Hence the LE singularities are astriking manifestation of the existence of such (nearlyfree) l -bits, to be found in the short-time dynamics ofthe system.It is worthwhile to mention at this point that the ex-istence of singularities in the quench dynamics of the LEis currently the subject of several theoretical and exper-imental investigations, as they represent the main signa-ture of so-called dynamical quantum phase transitions,studied both in non-random systems [22, 41–44] as well Jt . . . . ∏ ( t ) ¢= 4 . Jt . . . . . ¢= 6 . Jt . . . . ¢= 8 . ED2LS3LS Jt . . . ∏ ( t ) ¢= 4 . Jt . . . . ¢= 6 . Jt . . . ¢= 8 . ED2LS3LS ( f )( e )( d ) FR ( a ) ( b ) ( c ) QP Figure 3. Comparison between the LE λ ( t ) for and L = 22chain and the predictions of the 2LS and 3LS models: (a-c)QP potential; (d-f) FR potential. as in disordered quantum systems [45–47]. Nonethelessour observation of LE singularities is fully explained by amodel of individual 2LS, without the need of any many-body effect; therefore we shall refrain from associatingthem to any form of time-dependent transition. From 2LS to 3LS.
Fig. 3(d-f) shows that, in the case ofthe FR potential, the 2LS model of Eq. (5) still predictsthe correct frequency of the LE singularities, but not thecorrect height and it also misses a global offset. Thisis not surprising, as in the case of the FR potential theassumption of anticorrelation between the energy offsetof contiguous pairs is no longer valid, namely the poten-tial can host “rare” regions in which contiguous pairs ofsites – ( i − , i ) and ( i, i + 1) – are nearly resonant atthe same time. To take those regions into account (atleast partially) one can easily promote the 2LS model toa model of 3-site systems (amounting to effective three-level systems – 3LS), and approximate the evolved stateas that of a collection of independent 3LS. Unlike thecase of the 2LS model, the 3LS model requires a fullynumerical treatment (detailed in [37]), which amounts tonumerically scanning the ensemble of 3-site systems in asingle realization of the disorder potential in an arbitrar-ily large chain. As shown by Fig. 3(d-f), the improvementoffered by the 3LS model for the FR potential is substan-tial; these results can further be improved by moving to4-site clusters etc. albeit at an exponential cost. Slow dephasing and l -bit interactions. A significantfeature of the LE singularities is their slow decay in time– which is remarkable given that they result from theRabi oscillations of a collection of 2LS with a distribu-tion of frequencies that can be a priori expected to leadto fast dephasing. The reason behind the slow decay isalso captured by the 2LS model, Eq. (5), namely by thefact that the integral expressing LE takes contributionsfrom a small window of detunings δ (cid:48) around zero, thesmaller the longer the time. When looking at the sin-gularity times t = t n , a direct inspection of the functionlog(1 − p ( δ (cid:48) , Ω , t n )) seen as a function of δ (cid:48) shows that ithas a large peak centered on δ (cid:48) = 0 with a width depend-ing on time as t − / n [37]. The singularity in the averageLE comes from the integral of this peak, while the rest ofthe integral contributes essentially to the regular part ofthe LE; hence it is immediate to predict that the heightof the cusp singularity should decay as the peak width,(namely as t − / n ). Fig. 4(a) shows the time evolution ofsingularity peaks in the LE for free as well as interactingfermions in the QP potential, compared to the predictionof the 2LS model (for the interacting case): we observethat the t − / decay is indeed confirmed by the ED datafor free fermions, as well as by the ED data for interact-ing fermions at sufficiently short times ( tJ (cid:46) t ∗ ≈
100 for∆ = 8 J ). On the other hand, at longer times the inter-acting data are found to display a strong deviation fromthe 2LS model prediction, exhibiting a much faster de-cay. This crossover to an interaction-induced dephasing(IID) regime clearly shows the limits of the 2LS modelas a model of free l -bits expressed by Eq. (6), and it Jt ° ° ° ∏ ( t n ) ° ∏ QP J z = 0 J z = 1 ª / p t Jt ° ° ° FR J z = 0 J z = 1 ª / p t IIDIID
Figure 4. Decay of the peak heights of the LE, λ ( t n ) − ¯ λ (¯ λ stands for the time-averaged LE). Left: QP potential; Right:FR potential. The data are obtained for ∆ = 8 J ; the 2LS and3LS predictions are for J z = J . The grey-shaded area marksthe interaction-induced dephasing (IID) regime exhibited bythe exact data for J z = J . marks a fundamental difference between AL and MBLin the QP system. Indeed the faster decay of the LEmust be related to the effect of l -bit interactions, whichare a defining feature of MBL, and which add terms ofthe kind (cid:80) nm U nm τ n τ m + (cid:80) nml V nml τ n τ m τ l + ... to theeffective l -bit Hamiltonian. Such terms are responsiblefor the persistent growth of entanglement entropy in thesystem [7] as the logarithm of time, and indeed the onsetof the log t growth of entanglement occurs at a time com-patible with t ∗ (see [37]). A similar crossover from a slowpower-law decay of the LE peak height to a faster decay,dictated by the presence of interactions, is also exhibitedby the comparison between the ED data for interactingfermions in the FR potential with the same data for non-interacting fermions and for the 3LS model – as shownin Fig. 4(b). Further predictions.
The 2LS and 3LS models allowus not only to predict the dynamics of the LE, but alsothat of the imbalance and of the entanglement entropy– which in the case of the 2LS model acquire an explicitintegral formula similar to Eq. (5) for the LE dynamics.The results (detailed in [37]) for the imbalance dynam-ics show that the oscillations exhibited by the imbalance(see Fig. 1(a,c)) decay in amplitude as t − / similarly towhat is seen for the LE singular peaks, crossing over toa faster decay due to the l -bit interactions in the case ofinteracting fermions. The entanglement entropy is alsocorrectly captured by the 2LS (3LS) models (as the av-erage entropy across a bipartition of the 2-site (3-site)cluster) at all times for free fermions, and at short timesfor interacting ones. In the case of interacting fermionsat longer times, the models in question only capture thenumber entropy component of the entanglement entropy[48, 49], something which is readily understood in thatthe residual entropy component (the so-called configura-tional entropy) precisely originates from the l -bit inter-actions. Conclusions.
In this work we have shown that sharpcusp-like singularities in the Loschmidt echo (LE) are ageneric feature of the localized dynamics of an extendedquantum system initialized in a factorized state. Thesefeatures can be fully explained by the dynamics of asimple model, describing an ensemble of effective inde-pendent two-level (or even three-level) systems, offeringan explicit approximation to the conserved l -bits in theMBL regime. Such a model predicts very accurately theLE singularities for strongly disordered systems as wellas their decay; the faster decay in the dynamics com-pared to that predicted by the model is a direct manifes-tation of the interactions between the l -bits, namely thedefining feature of many-body localization (MBL) withrespect to Anderson localization (AL). Based on our re-sults, we can conclude that experimental evidence of l -bitdynamics and of their interactions is readily accessibleto state-of-the-art quantum simulators which have direct access to the Loschmidt echo, such as e.g. trapped ions[21, 22], quantum-gas microscopes [48] or superconduct-ing circuits [50]. Acknowledgements.
L.B. gratefully acknowledges hos-pitality and financial support from the Laboratoire dePhysique of the ENS Lyon. R.A.R. and L.B. ac-knowledge funding from the CY Initiative of Excel-lence (grant”Investissements d’Avenir” ANR-16-IDEX-0008) where this work developed during R.A.R.’s stayat the CY Advanced Studies. Part of the exact diag-onalization simulations were performed using routinescontained in the
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Loschmidt echo singularities as dynamicalsignatures of strongly localized phases
I. TWO-SITE SYSTEM AS TWO-LEVELSYSTEM AND ITS RABI OSCILLATIONS
Let us isolate a two-site system ( i, i + 1) hosting oneparticle in the fermionic chain, with Hamiltonian H − site = − J (cid:16) c † i c i +1 + c † i +1 c i (cid:17) + h i n i +( h i +1 + J z ) n i +1 , (S1)where we assume that the site i + 2 is occupied by a(pinned) particle, while size i − σ z = n i − n i +1 ,σ x = c † i c i +1 + c † i +1 , c i , (S2)the Hamiltonian becomes simply H − site = − J σ x + δ σ z + const ., (S3)namely, a two-level system (2LS) with splitting δ andRabi frequency J . If the system starts from the | (cid:105) state, the return probability is given by the well-knownformula for the probability of persistence in the initialstate during Rabi oscillations [38], namely 1 − p ( δ ; t ),where p ( δ ; t ) = 11 + ( δ/J ) sin (cid:32) (cid:112) δ/J ) tJ (cid:33) . (S4) II. SPATIAL CORRELATIONS IN THEQUASI-PERIODIC VS. FULLY RANDOMPOTENTIAL
A fundamental assumption of the 2LS model describedin the main text is that quasi-resonant two-site sys-tems are spatially isolated in a (quasi)-disordered chain– namely, if a pair of sites ( i, i + 1) is quasi-resonant forthe motion of a particle, the two adjacent pairs of sites( i − , i ) and ( i + 1 , i + 2) are not resonant. Defining δ = h i +1 − h i as the energy difference of the two sites inquestion, and δ = h i +2 − h i +1 as that of the followingpairs of sites, in the case of non-interacting fermions, theabove condition requires that the two energy differencesdo not vanish simultaneously.Such a form of correlation is indeed observed in thecase of the quasiperiodic (QP) potential: Fig. S1(a)shows the joint probability P ( δ , δ ) for two adjacent en-ergy differences, displaying a dip for δ = δ = 0 – anaspect which prevents two successive pairs of sites frombeing resonant simultaneously. In the case of interacting fermions, on the other hand, the above condition requiresthat if, e.g. J z ± δ ≈
0, then J z ∓ δ is non-zero, or viceversa – this prevents a state of the type | (cid:105) on the sites( i, i +1 , i +2) from being simultaneously (quasi-)resonantwith | (cid:105) and | (cid:105) , or, similarly, the state | (cid:105) frombeing quasi-resonant with | (cid:105) and | (cid:105) . This is indeedguaranteed by the fact that P ( δ, − δ ) is nearly vanishingfor any finite δ , except for δ ≈ .
25∆ – but the lattersituation does not lead to consecutive resonances when∆ > . J , which is always the case in our study.On the other hand the uniform potential has no corre-lations between two consecutive energy differences, andthe P ( δ , δ ) distribution is the product of two triangu-lar distributions for δ and δ – shown in Fig. S1(b).This implies that a fundamental assumption behind the2LS model description is not guaranteed to be satisfied– while it is more likely to have two adjacent pairs ofsites with different energy offsets than with similar ones,one cannot exclude the existence of “rare” regions withconsecutive nearly resonant pairs. This requires to im-prove the 2LS model to a three-site (three-level) one, asdetailed in Sec. IV. ° ° ± / ¢ ° ° ± / ¢ . . . . ° ° ± / ¢ ° ° ± / ¢ . . . . FRQP
Figure S1. Numerically sampled probability distribution P ( δ , δ ) for two energy differences δ and δ on contiguouspairs of sites. Left panel: QP potential; Right panel: FRpotential. III. DECAY OF THE LOSCHMIDT-ECHOSINGULARITIES AND IMBALANCEOSCILLATIONS FROM THE 2LS PREDICTION
As seen in the main text, the 2LS predicts the LE inthe form of the integral λ ( t ) = (cid:90) dδ P ( δ ) f ( δ ; t ) (S5)with f ( δ ; t ) = − log [1 − p ( δ, t )] (S6)and p ( δ ; t ) as given in Eq. (S4).The cusp singularities in λ ( t ) at times t n J = (2 n +1) π , n = 0 , , , . . . , descend from the fact that the function f ( δ ; t n ), seen as a function of δ , develops a logarithmicsingularity at δ = 0, as shown in Fig. S2, while it is fully f ( ; t ) tJ= π tJ=5 π tJ=25 π -6 -4 -2 0 2 4 6 tJ= π tJ=5 π tJ=25 π tJ=41 π p tJ /J /J Figure S2. Function f ( δ ; t ) vs. δ at different singularity times t n J = (2 n + 1) π ; as shown in the right panel, for large t n thewidth of the central peak becomes time-independent when f is plotted as a function of √ t δ . regular at any other time. The singular peak centered at δ = 0 has a support shrinking with t n as t − / n – as seenin Fig. S2 when plotting the function f ( δ ; t n ) vs δ √ t n ,which leads to a collapse of the peak widths at differenttimes (when n (cid:29) f function out-side the peak contributes to the regular part of the LE,while the integral of the peak dictates fundamentally theheight of the cusps above the regular background (esti-mated as the long-time average ¯ λ ), namely the quantity λ P ( t n ) = λ ( t n ) − ¯ λ . Therefore one can expect that for n (cid:29)
1, the height of the cusps in the LE decay with timeas λ P ( t n ) ∼ t − / n . This prediction is confirmed in Fig. 4of the main text.The 2LS model prediction for the imbalance is verysimilar to that of the LE, as the imbalance is simplyrelated to the persistence probability of the initial state( | (cid:105) or | (cid:105) ) on the 2-site cluster – given that the orthog-onal state contributes zero to the imbalance. Thereforethe 2LS expression for the imbalance simply reads I ( t ) = (cid:90) dδ P ( δ ) [1 − p ( δ, t )] . (S7)The times t n giving cusp singularities in the LE corre-spond to dips in the imbalance, and these dips come fromlocal dips in the g ( δ ; t ) = 1 − p ( δ, t ) function centeredaround δ = 0 and touching zero for t = t n . The width ofthese dips is also shrinking in time as t − / n . Thereforeone expects the depth of the minima in the fluctuationsof the imbalance to decay to the long-time average as t − / n as well – this prediction will be verified in Sec. V. IV. THREE-SITE MODEL
The Hamiltonian of a three-site system ( i, i + 1 , i +2) containing two interacting fermions in an initial | (cid:105) state is explicitly given by H = − J (cid:16) c † i c i +1 + c † i +1 c i +2 + h . c . (cid:17) + δ i n i +1 + ( δ i +1 + δ i ) n + J z ( n n + n n ) + const . (S8)Here all single-site energies are referred to the energyof site i , and δ i = h i +1 − h i . The above Hamiltonianassumes that the sites i − i +3 remain empty duringthe time evolution. The Hilbert space of the 3-site systemis restricted to the three states | (cid:105) , | (cid:105) and | (cid:105) ,making of it a three-level system (3LS), with a generictime-dependent wavefunction | ψ ( t ) (cid:105) = α ( t ) | (cid:105) + β ( t ) | (cid:105) + γ ( t ) | (cid:105) . (S9)Its explicit form can be easily calculated numerically forany specific choice of the energy differences δ i .A similar calculation can be done for a three-site sys-tem hosting a single particle, and starting from the | (cid:105) configuration, with Hamiltonian H = − J (cid:16) c † i c i +1 + c † i +1 c i +2 + h . c . (cid:17) + J z n i + δ i n i +1 + ( δ i +1 + δ i + J z ) n i +2 , (S10)which assumes that the sites i − i +3 host two pinnedparticles. The Hilbert space | (cid:105) , | (cid:105) , | (cid:105) defines a3LS, whose instantaneous state takes the generic form | ψ ( t ) (cid:105) = ˜ α ( t ) | (cid:105) + ˜ β ( t ) | (cid:105) + ˜ γ ( t ) | (cid:105) . (S11)For both types of clusters, the main quantity of interest tous is the Loschmidt echo λ ( t ; δ i , δ i +1 ) = − log | β ( t ) | and λ ( t ; δ i , δ i +1 ) = − log | ˜ β ( t ) | .We can then model a chain in a QP or FR potentialas an ensemble of independent 3LSs by generating se-quences of energy offsets δ i , δ i +1 between adjacent sitepairs according to the distribution P ( δ i , δ i +1 ). The 3LSprediction for the Loschmidt echo of the ensemble is λ ( t ) (S12)= 12 (cid:90) dδ dδ P ( δ , δ ) [ λ ( t ; δ , δ ) + λ ( t ; δ , δ )] . In practice, the above integral can be sampled numeri-cally by simply averaging over a large number of differ-ent realizations of the potential on 3-site systems, suchas those offered by a very long chain, namely λ ( t ) ≈ L L (cid:88) i =1 λ α i ( t ; δ i , δ i +1 ) , (S13)where α i = 101 if i is odd and 010 if i is even, and L (cid:29) I ( t ) ≈ L (cid:88) i (cid:0) −| α i | + 3 | β i | − | γ i | (cid:1) (S14)with α i , β i , γ i = α ( t ) , β ( t ) , γ ( t ) or ˜ α ( t ) , ˜ β ( t ) , ˜ γ ( t ) depend-ing on whether i is odd or even.The above expression for the LE (Eq. (S13)) has theapparent drawback of triple-counting each site. Nonethe-less, similarly to what is done in the main text for the2LS case, it is fair to assume (and it can be numericallytested) that, out of the three clusters containing each site,only one at most will contribute significantly to the LE.As a consequence the triple counting has a mild effecton the final result. One could avoid triple counting bythoughtfully decomposing a chain into non-overlappingclusters of up to 3 sites, in such a way as to maximizethe LE; yet this procedure introduces significant com-plications which are not justified a posteriori, given thequality of the results offered already by the naive ensem-ble average (see Fig. 3 of the main text and further resultsin Sec. VI). V. IMBALANCE DYNAMICS
Fig. S3 shows the comparison between the ED resultsfor the imbalance dynamics of interacting fermions im-mersed in a QP and fully random potentials of variablestrength, compared with the predictions of the 2LS andthe 3LS models. For sufficiently strong disorder (∆ (cid:38) J )the 2LS predictions are already rather accurate in thecase of the QP potential, and the 3LS model offers fur-ther improvement. On the other hand in the case of theFR potential the 3LS model offers a more substantial im-provement, fixing an overall offset (for sufficiently strongdisorder) which is seen in the 2LS predictions. Fig. S4shows the evolution of the depth of the minima of theimbalance at times t = t n , taken with respect to the long-time average, namely the quantity I M ( t n ) = ¯ I − I ( t n ).We observe that the predictions of the 2LS system forthe fermionic chain immersed in the QP potential showsa clear power-law decay at long times, compatible with t − / , which is indeed reproduced in the case of non-interacting fermions. In the case of interacting fermions, Jt . . . . I ( t ) ¢= 4 . Jt . . . ¢= 6 . Jt . . . ¢= 8 . ED2LS3LS J z t . . . I ( t ) ¢= 4 . J z t . . . ¢= 6 . J z t . . . ¢= 8 . ED2LS3LS ( f )( e )( d ) FR ( a ) ( b ) ( c ) QP Figure S3. Comparison between the imbalance I ( t ) for a L =22 chain and the predictions of the 2LS and 3LS models givenby (S7), (S14): (a-c) QP potential; (d-f) FR potential. Jt ° ° ° I ° I ( t n ) QP J z = 0 J z = 1 ª / p t Jt ° ° ° FR J z = 0 J z = 1 ª / p t IIDIID
Figure S4. Decay of the depth of the imbalance minima I ( t n )with respect to the average value I from ED simulations ( J z =0 ,
1) and from the 2LS (QP) and 3LS (FR) models. Thedotted black line is a guide for the eye with slope 1 / √ t . Thedata are obtained for ∆ = 8 J ; the 2LS and 3LS predictionsare for J z = J . The grey-shaded area marks the interaction-induced dephasing (IID) regime exhibited by the exact datafor J z = J . on the other hand, a crossover is observed at long times( t (cid:38) t ∗ ≈ l -bits.A similar picture is offered by the case of the FR po-tential. There the ED results are compared with thepredictions from the 3LS model; the latter model pre-dicts correctly the decay of the minima depth in the non-interacting case at all times, while the exact results forthe interacting system show a clear crossover towards afaster decay for times t (cid:38) t ∗ ≈
50; this crossover to anIID regime is again fully compatible with that observedin the decay of the cusp maxima of the LE (see maintext), as well as with the onset of the logarithmic growthin the entanglement entropy (see Sec. VI).
VI. ENTANGLEMENT DYNAMICS: FULLENTROPY VS. NUMBER ENTROPYA. Entanglement entropy from the 2LS and 3LSmodel
The 2LS and 3LS models allow for a simple calcula-tion of the entanglement entropy of a A / B bipartition ofthe system into two adjacent chains, defined as the vonNeumann entropy of the reduced density matrix S A ( t ) = − Tr [ ρ A ( t ) log ρ A ( t )] , (S15)where ρ A ( t ) = Tr B | ψ ( t ) (cid:105)(cid:104) ψ ( t ) | is the partial trace (overthe degrees of freedom in B ) of the instantaneous pure-state density matrix associated with the evolved state | ψ ( t ) (cid:105) .0 ° Jt S L / ( t ) ¢= 8 . ° Jt ¢ = 15 . ED2LS3LS10 ° Jt S L / ( t ) ¢= 8 . ° Jt ¢ = 15 . ED2LS3LS
IIDIID QP ( a ) ( b ) FR ( c ) ( d ) Figure S5. Half-chain entanglement entropy of interactingfermions in a QP potential (a-b) and FR potential (c-d) fora chain of size L = 16, compared with the prediction for the2LS/3LS models for two different disorder strengths (∆ /J = 8and 15). Within those models, the entanglement associatedwith such a bipartition simply comes from the entangle-ment inside the 2-site or 3-site cluster which contains thecut defining the bipartition. In the case of the 2LS modelthe disorder-averaged entanglement entropy of a bipar-tition is simply predicted as the entropy of the reducedstate of one site in the 2-site cluster, namely S A ( t ) = (cid:90) dδ P ( δ ) h [ p ( δ ; t )] , (S16)and h [ x ] = − x log x − (1 − x ) log(1 − x ). Notice that,unlike for the formulas of the LE and of the imbalance,no-double counting is implied in the above formula, sincethe entanglement is referred to a cut of the chain, andthere is one unique cut per 2-site cluster.On the other hand a 3-site cluster can be cut in twodifferent ways, that we will indicate as ◦ | ◦ ◦ and ◦ ◦ | ◦ in the following (where ◦ stands for a site and | standsfor the cut). The reduced density matrices for the twocuts are readily obtained from the cluster wavefunctionsdescribed in Sec. IV; e.g. for a 101 cluster the reduceddensity matrix associated with the ◦ | ◦ ◦ cut reads ρ ◦|◦◦ ( t ) = (cid:18) | α ( t ) | | β ( t ) | + | γ ( t ) | (cid:19) (S17)with associated entanglement entropy S ◦|◦◦ = −| α ( t ) | log | α ( t ) | − (cid:0) | β ( t ) | + | γ ( t ) | (cid:1) log (cid:0) | β ( t ) | + | γ ( t ) | (cid:1) ;(S18)the one associated with the ◦ ◦ | ◦ cut reads ρ ◦◦|◦ ( t ) = (cid:18) | γ ( t ) | | α ( t ) | + | β ( t ) | (cid:19) . (S19) The density matrices ρ ◦|◦◦ and ρ ◦◦|◦ and related entropiesassociated with a 010 cluster can be calculated similarly.The disorder-averaged entanglement entropy of thewhole system within the 3LS model is then given by S ( t ) = 12 L (cid:88) i S i ( t ) , (S20)where S i = (cid:40) S ◦|◦◦ ( t ) + S ◦◦|◦ ( t ) if i odd ,S ◦|◦◦ ( t ) + S ◦◦|◦ ( t ) if i even . (S21)The factor 1 / B. Entanglement entropy vs. number entropy
The 2LS and 3LS models picture the entanglement be-tween two adjacent subsystems as arising uniquely fromthe coherent motion of particles within the restricted sizeof the clusters they describe. When starting from a fac-torized state, this picture is certainly valid at short times.At long times it remains valid only if particles remainlocalized within the size of the clusters (namely if thelocalization length is smaller than the cluster size), andif this is a sufficient condition for entanglement not tospread any further. The latter aspect is true in the caseof non-interacting fermions, for which the only mecha-nism behind entanglement of different spatial partitionsis particle motion between them. On the other hand,in the case of interacting fermions in the MBL regime,entanglement keeps growing due to the interactions be-tween l -bits, and distant degrees of freedom can becomeentangled even without any net particle exchange.In this context it is useful to decompose the entangle-ment entropy of a subsystem A into a number entropycontribution, and a remainder part (called the configura-tional entropy), S A = S A,N + S A,c [48, 49]. The numberentropy is given by S A,N = − (cid:88) N A p N A log p N A , (S22)where p N A is the probability of having N A particles insubsystem A , and it accounts for the particle number un-certainty appearing in subsystem A because of the coher-ent exchange of particles with its complement B . On theother hand the configurational entropy accounts for cor-relations establishing between the particle arrangementsin A and B once the partitioning of the particles between A and B has been fixed. The 2LS and 3LS models,lacking completely any form of correlations among theclusters, can only capture the number entropy contribu-tion in systems with a localization length smaller thanthe cluster size; as we shall see in the next section, this1 − Jt . . . . . S ( t ) QP , J z = 0 − Jt QP , J z = 1 − Jt S ( t ) FR , J z = 0 − Jt FR , J z = 1 S L/ ( t ) P N ( t ) S LS ( t ) S LS ( t ) S L/ ( t ) P N ( t ) S LS ( t ) S LS ( t ) Figure S6. Half-chain entanglement entropy and number entropy of free and interacting fermions on a L = 16 chain, comparedwith the 2LS and 3LS predictions: (a) non-interacting fermions in a QP potential; (b) interacting fermions ( J z = 1) in a QPpotential; (c) non-interacting fermions in a FR potential; (d) interacting fermions ( J z = 1) in a FR potential. For all the panelsthe disorder strength is ∆ = 8 J . limited picture still offers a faithful description of entan-glement in the MBL regime for short times (the longerthe stronger disorder is), while it can describe entangle-ment at all times for strongly localized non-interactingparticles. C. 2LS/3LS entropy vs. exact entanglement andnumber entropy
Fig. S5 shows a comparison between the entanglemententropy of interacting fermions in a QP potential and the2LS prediction. We observe that at moderate disorder inthe MBL phase (∆ = 8 J ) the 2LS and 3LS models onlycapture the initial rise of the entanglement entropy and(partly) the first maximum; in particular the very exis-tence of a maximum is explained by the models as theresult of nearly resonant small clusters returning close tothe initially factorized state – albeit at different timesdue to the inhomogeneously broadened local frequencies,which explains why the entanglement entropy does notcome back to (nearly) zero. The 2LS and 3LS models onthe other hand completely miss the long-time logarithmicgrowth of the entanglement entropy – something whichis fully expected, given that such a growth is the con-sequence of interactions between l -bits, not included inthe 2LS and 3LS models by construction. On the otherhand, at stronger disorder (∆ = 20 J ) the interactions be-tween l -bits are parametrically suppressed, and the 2LSand 3LS description of entanglement becomes accurateup to very long times.As suggested in the previous section, a more appropri-ate comparison with the entanglement entropies of the 2LS and 3LS models would involve the number entropyfrom the ED data – shown in Fig. S6. For non-interactingfermions in a QP potential of strength ∆ = 8 J the num-ber entropy is found to nearly coincide with the full en-tanglement entropy, and to be very well described by the2LS prediction – see Fig. S6(a). When adding the inter-actions, the agreement between the number entropy andthe 2LS entropy deteriorates mostly at long times, seem-ingly due to the ∼ log log t growth of the number entropyobserved in the MBL phase [49]. Similar considerationscan be made in the case of the FR potential – the 3LSmodels describe well the entanglement and number en-tropy in the non-interacting case, and they miss the slowlong-time growth of the number entropy in the interact-ing case. VII. LOSCHMIDT-ECHO DYNAMICS FORDIFFERENT DISORDER REALIZATIONS
Figs. S7 and S8 show the disorder average of the LEfor a chain of L = 22 sites, along with all the disorderrealizations ( > ) contributing the average, for var-ious strengths (∆ /J = 1 , , ...,
10) of the QP and FRpotential, respectively. We observe that sharp cusp sin-gularities are exhibited by a signification portion of thedisorder realizations, and that for sufficiently strong dis-order these realizations are a finite fraction of the disorderstatistics (in the asymptotic limit), so that cusp singu-larities persist in the disorder-averaged results as well.These plots also suggest the fact that cusp singularitiescan be observed with a limited disorder statistics, underrealistic experimental conditions.2
Figure S7. Averaged λ ( t ) (black line) at different disorder strengths ∆ = 1 , . . . ,
10 for a chain of L = 22 sites in a QP, plottedalong with all the realizations used for the averaging procedure (grey lines). The dotted line represents a typical individualrealization exhibiting singular behavior. Figure S8. Averaged λ ( t ) (black line) at different disorder strengths ∆ = 1 , . . . ,