Evidence for unbounded growth of the number entropy in many-body localized phases
Maximilian Kiefer-Emmanouilidis, Razmik Unanyan, Michael Fleischhauer, Jesko Sirker
EEvidence for unbounded growth of the number entropyin many-body localized phases
Maximilian Kiefer-Emmanouilidis,
1, 2
Razmik Unanyan, Michael Fleischhauer, and Jesko Sirker Department of Physics and Research Center OPTIMAS,University of Kaiserslautern, 67663 Kaiserslautern, Germany Department of Physics and Astronomy, University of Manitoba, Winnipeg R3T 2N2, Canada (Dated: March 25, 2020)We investigate the number entropy S N —which characterizes particle-number fluctuations betweensubsystems—following a quench in one-dimensional interacting many-body systems with potentialdisorder. We find evidence that in the regime which is expected to show many-body localization(MBL) and where the entanglement entropy grows as S ∼ ln t as function of time t , the numberentropy grows as S N ∼ ln ln t , indicating continuing particle transport at a very slow rate. Wedemonstrate that this growth is consistent with a relation between entanglement and number entropyrecently established for non-interacting systems. Introduction.—
The time dependence of the entangle-ment entropy S ( t ) after a quantum quench offers insightsinto the dynamics of quasi-particles and the influence ofconservation laws. Well studied are quenches startingfrom a product state in clean lattice models with short-range hoppings and interactions. In this case, the genericpicture is one of quasi-particles propagating through thesystem with a velocity bounded by the Lieb-Robinson ve-locity v LR [1–3]. The entanglement entropy is then pro-portional to the entangled region created by the quasi-particle excitations. For a subsystem with volume (cid:96) d in d dimensions, this leads to S ∼ (cid:96) d − t for times v LR t (cid:28) (cid:96) and a volume-law saturation, S ∼ (cid:96) d , at times v LR t (cid:29) (cid:96) .This picture has been confirmed in free scalar field theo-ries [4] and in one-dimensional systems which are confor-mally invariant [5]. An obvious exception from a linearincrease of the entanglement entropy after a quench andfrom a volume-law scaling at long times are disorderednon-interacting systems in an Anderson localized (AL)phase [6]. In this case, the spreading of excitations is lim-ited to the localization length ξ loc leading to an area law, S ∼ (cid:96) d − ξ loc , instead of a volume law at long times. Theincrease of the entanglement entropy after the quench istherefore bounded [7].In recent years, the question of localization in thepresence of interactions—termed many-body localization (MBL)—has attracted renewed interest [8–13]. For thespin- / Heisenberg chain with local magnetic fieldsdrawn from a box distribution, numerical data appearconsistent with a transition from an ergodic phase atsmall disorder to a non-ergodic MBL phase at strongdisorder [10, 14, 15]. One of the hallmarks of MBL ascompared to AL is the unbounded logarithmic growthof S after a quench [9, 16, 17]. Recently, evidence for S ∼ ln t has also been obtained in an experiment on coldatomic gases [18]. Here a quench in a one-dimensionalAubry-André model of interacting bosons was studiedwith single atom resolution. In such systems where thetotal particle number (or similarly the total magnetiza- tion) is conserved, the von Neumann entropy can be splitinto two parts, S = S N + S c [18–21]. Here S N = − (cid:88) n p ( n ) ln p ( n ) (1)is the number entropy with p ( n ) the probability of find-ing n atoms in the considered subsystem (also referredto as charge [20] or fluctuation entropy [21]). The config-urational entropy S c then contains the contributions toentanglement due to configurational correlations. Thissplitting of S is not only useful from an experimentalperspective because p ( n ) can be determined by single-site resolution atomic imaging [18] but also offers fur-ther insights into questions of localization and ergodicity.Very recently, we have shown that in any non-interactingfermionic system S (2) ∝ exp (cid:16) S (2) N (cid:17) where S (2) is the sec-ond Rényi entropy and S (2) N the corresponding numberentropy. I.e., a growth in the entanglement entropy isalways accompanied by a logarithmically slower growthin the number entropy [22].An exception to this picture of correlated dynamics ofentanglement and number entropies is expected to occurin many-body localized (MBL) phases. Here S N is be-lieved to saturate after a quantum quench, indicating lo-calization, while S continues to grow in time. It has beenargued that MBL systems are described at long times byeffective Hamiltonians [23, 24] H = (cid:88) i ε i η i + (cid:88) i,j J ij η i η j + · · · (2)with exponentially many local conserved charges [ H, η i ] = 0 , random energies ε i , and amplitudes J ij whichdecay exponentially with distance between these charges.As a consequence of the coupling terms ∼ J ij , a regionof length (cid:96) will become entangled over time t ∼ e (cid:96) . Sincethe entanglement entropy is extensive, one then expects S ∼ (cid:96) ∼ ln t [25] consistent with the numerical and ex-perimental observations. If Eq. (2) is a valid effective a r X i v : . [ c ond - m a t . d i s - nn ] M a r discription of MBL phases of matter, then the increase inentanglement at long times is entirely due to the continu-ing buildup of configurational entanglement S c . Since theconserved charges η j are local, the number entropy S N has to be bounded, reflecting the expected localized andnon-ergodic character of this phase. On the other hand,the experimental data for the number entropy in Ref. 18appear to show a slow increase, although a detailed analy-sis of the number entropy as a function of system size anddisorder strength has not yet been performed. Further-more, it has recently been suggested that paradigmaticmodels expected to show MBL phases might ultimatelybe ergodic at very long times [26, 27].These recent results motivate us to investigate thenumber entropy in systems believed to show MBL. Inthis letter we provide evidence that the picture of MBLphases based on effective Hamiltonians (2) might be in-complete. For all system sizes and times we can accessnumerically, we find that the number entropy grows as S N ∼ ln ln t even at strong disorder and does not showany signs of saturating. We, furthermore, present evi-dence that the relation S (2) ∝ exp (cid:16) S (2) N (cid:17) , proven for freefermionic systems in [22], also appears to hold in the in-teracting case, both in the ergodic and in the MBL phase,with proportionality factors renormalized by interactionsand disorder. Number entropies.—
If we split a one-dimensional sys-tem S into two parts, A and B , then the Rényi entangle-ment entropies are given by S ( α ) = (1 − α ) − ln tr ρ αA (3)where ρ A is the reduced density matrix of the consid-ered subsystem. The von-Neumann entanglement en-tropy is given by S ≡ S (1) = lim α → S ( α ) . If the totalparticle number is conserved, then we can write S ( α ) =(1 − α ) − ln( (cid:80) n p α ( n ) tr ρ αA ( n )) where ρ A ( n ) is the blockof the reduced density matrix with particle number n nor-malized such that tr ρ A ( n ) = 1 . If there is only a singleconfiguration for each n then tr ρ αA ( n ) = tr ρ A ( n ) = 1 .We thus call S ( α ) N = (1 − α ) − ln (cid:80) n p α ( n ) the Rényinumber entropy, generalizing Eq. (1). Any additionalentanglement is due to different configurations in eachparticle sector having finite probability and is thus partof what we call the Rényi configurational entropy. System.—
To be concrete, let us consider a half-filledfermionic model H = − J (cid:88) j ( c † j c j +1 + h.c. ) + (cid:88) j D j n j + V (cid:88) j n j n j +1 , (4)with nearest-neighbor hopping amplitude J , interaction V , and onsite disorder D j ∈ [ − D/ , D/ . Here n j = c † j c j is the particle number at site j . Using a Jordan-Wigner transformation, this model can be mapped ontoa spin- / XXZ chain with magnetic field disorder. For V = 2 J , in particular, one obtains the isotropic Heisen-berg model which is the most studied system to inves-tigate MBL physics. We set J = 1 and (cid:126) = 1 in thefollowing. Thermalization.—
If such a system after a quantumquench thermalizes to a high temperature state, then aregion of size (cid:96) will contain (cid:96) particles on average andevery arrangement of particles will approximately haveequal probability. If we now cut the thermalized region inhalf, then the probability to find n particles in one half is p ( n ) = (cid:0) (cid:96)n (cid:1)(cid:0) (cid:96)(cid:96) − n (cid:1) / (cid:0) (cid:96)(cid:96) (cid:1) . For large n, (cid:96) this distribution canbe approximated by a continuous distribution and onefinds for all Rényi number entropies (including α → ) inthe ergodic case S ( α ) N = const + ln (cid:96) with S ( α ) N > S ( α +1) N [22]. If the excitations in the system spread as t ν afterthe quench then the thermalized regions have size (cid:96) ∼ t ν and we obtain S ( α ) N ( t ) = const + ν t . (5) Localization.—
The presence of disorder (i.e. D (cid:54) = 0 )can prevent thermalization and lead to localized states.The simple scaling argument why free particles ( V = 0 )on a lattice with short-range hoppings become localizedfor strong potential disorder works as follows [28–30]: Areal hopping process requires a resonance, i.e., an energymatching between the two sites involved in the hoppingprocess. The smallest mismatch in energy on a subsys-tem of volume (cid:96) d decreases as (cid:96) − d in d dimensions onaverage. The transport between quasi-degenerate statesneeds on the order of n ∼ (cid:96) hopping processes and theamplitude for such a virtual n -site hopping process fallsof exponentially with distance. Therefore distant reso-nances have a vanishingly small probability to proliferateand to delocalize the system. A non-interacting systemat sufficiently strong disorder will therefore be in an ALphase and both S and S N will saturate. In one dimen-sion, even arbitrarily weak disorder is sufficient to localizeall states. The crucial question then is what influence in-teractions have on the probability of distant resonances.If the model (4) is in an AL phase for V = 0 , a localizedbasis {| ψ l (cid:105)} exists such that the non-interacting Hamil-tonian becomes diagonal, H = (cid:80) l ε l η l = (cid:80) l ε l d † l d l .We can transform (4) to this localized basis using c † j = (cid:80) l (cid:104) ψ l | φ j (cid:105) d † l , where | φ j (cid:105) is the original Wannier basis.Here l can be understood as the index of the site aroundwhich the localized single-particle wavefunction is cen-tered, i.e., |(cid:104) ψ l | φ j (cid:105)| ∼ exp( −| l − j | /ξ loc ) where ξ loc isthe localization length. If we transform the interactionpart to the new basis, we find contributions describingdensity-density interactions between localized orbitals aswell as hopping processes between these orbitals. Thedensity-density part is given by H (1) int = (cid:80) l,l (cid:48) J ll (cid:48) η l η l (cid:48) withan amplitude which decays exponentially with distancebetween the orbitals, J ll (cid:48) ∼ V exp( | l − l (cid:48) | /ξ loc ) . If thiswould be the only relevant correction due to interactions,then particles would remain localized with H (1) int causinga logarithmic buildup of configurational entanglement.However, the interaction also leads to a correlated hop-ping between the single-particle orbitals | ψ l (cid:105) H (2) int = (cid:88) l,l (cid:48) ,k,k (cid:48) K ll (cid:48) kk (cid:48) d † l d l (cid:48) d † k d k (cid:48) (6)with unequal lattice sites and exponentially decaying am-plitude K ll (cid:48) kk (cid:48) . Similar to the AL case, one then has toconsider the possibility of resonances destroying localiza-tion. In contrast, hopping processes are now long-rangedso that both direct and virtual transitions to distant sitesare possible. The smallest expected average mismatch inenergy, ∆ ε = ε l − ε l (cid:48) + ε k − ε k (cid:48) , on a subsystem of length (cid:96) now decreases as (cid:96) − . Without taking the renormaliza-tion of the bare energies ε l into account, one would thusstill conclude that distant resonances do not proliferate.On the other hand, numerical and experimental data [31]indicate that for small disorder interactions do destroythe localized phase. I.e., in this case energy renormaliza-tions do seem to lead to a proliferation of resonant hop-ping processes. For strong disorder, on the other hand,it has been argued that the processes (6) are irrelevantand the particles are localized [32, 33]. However, theseresults are based on approximations. The proof of MBLfor weak interactions in Ref. 33, in particular, is basedon an assumption about limited level attraction in thestatistics of energy eigenvalues. Numerical results.—
Since the question about the rel-evance of resonances ultimately cannot be decided an-alytically, we investigate the number entropy for themodel (4) by exact diagonalization (ED). We concentrateon
V /J = 2 corresponding to the isotropic Heisenbergmodel. In our notation, the critical coupling for the tran-sition from the ergodic into the MBL phase is D c /J ≈ [10, 14]. We study quenches starting from half-filled ran-dom product states. If not stated otherwise the datashown for L ≤ are obtained by standard full diagonal-izations of the Hamiltonian, averaged over 10000 disorderrealizations for L ≤ and 3000 realizations for L > ,while a second order Trotter-Suzuki decomposition of thetime evolution operator is used for L = 24 , see App. Afor details.Let us first consider the regime D < D c where there isconsensus that the system is ergodic. ED [34, 35], large-scale density-matrix renormalization group (DMRG) cal-culations [36, 37], and phenomenological numerical renor-malization groups [38, 39] furthermore find subdiffusivetransport either all the way down to zero disorder or up toa second critical disorder below which transport becomesdiffusive. In contrast to the linear-in-time spreading ofexcitations in the clean case, it now takes time t ∼ (cid:96) /ν for excitations to spread across a region of length (cid:96) with ν = 1 / corresponding to diffusion. We therefore expect S ∼ (cid:96) ∼ t ν with S N given by Eq. (5). This scaling of S ( t ) -1 t S N ( a ) D=5D=7.5D=10D=12.5fit L S N (t →∞ ) ( b ) D=5D=10 D ν , b ( c ) ν b FIG. 1. S N for D < D c ≈ : (a) S N ( t ) for L = 24 , with 500disorder realizations and a logarithmic fit, S N = ν ln t + b ,for D = 10 with ν = 0 . , b = 0 . . (b) S N ( t → ∞ ) fordifferent system sizes L . (c) Prefactors ν and constants b ofthe logarithmic fits as a function of D for L = 24 . -1 t S N ( a ) D=15D=17.5D=20fit 4 6 8 10 14 18 L S N (t →∞ ) ( b ) D=15 D=20
16 18 20 D ν , b ( c ) ν b FIG. 2. S N for D > D c : (a) S N ( t ) for L = 24 and doublelogarithmic fit, S N = ν ln ln t + b , for D = 20 with ν =0 . , b = 0 . . (b) S N ( t → ∞ ) for different system sizes L .(c) Prefactor ν and constant b of the double logarithmic fitsas a function of D for L = 24 , see also App. B. in the ergodic regime is consistent with DMRG calcula-tions for infinite chains with binary disorder [37] and withED [35] for box disorder. In Fig. 1, results for the num-ber entropy of model (4) at various disorder strengths D < D c are shown. Here we consider systems of length L with open boundary conditions which are split into twoequal halfs, (cid:96) = L/ . We find that S N ( t ) grows logarith-mically consistent with Eq. (5) and thus ergodic behavior.This is also supported by the close to linear scaling of thesaturation value S N ( t → ∞ ) with system size. Finally,we note that the prefactor ν decreases continuously as afunction of disorder D and appears to approach zero for D → D c . The results for the number entropy are qual-itatively consistent with previous results for the scalingof the current [36] and of the bipartite particle numberfluctuations ∆ n [16, 40].Turning to the case D > D c , it is expected that it thentakes time t ∼ e (cid:96) to entangle regions over a distance (cid:96) .The resulting scaling of the von-Neumann entropy S ∼ ln t has been demonstrated already by various methodsand for a number of different models and our results are FIG. 3. S (2) N and bound (7) for (a) D < D c , and (b) D >D c . The renormalization parameter γ appears to decreasemonotonically with increasing D . consistent with such a scaling as well. Our main newresult are the data for the number entropy presented inFig. 2.We find that the number entropy continues to increaseas S N ∼ ln ln t and that the saturation value continuesto grow as a function of length as in the ergodic case D < D c , however, now only approximately logarithmi-cally. For the numerically accessible times and lengths wefind no indications for a saturation of the number entropyas would be expected if the system is localized. Note that S N ∼ ln ln t is exactly the scaling which is anticipated ifthe system is ultimately ergodic and t ∼ e (cid:96) is not onlythe relevant scaling for the buildup of configurational en-tanglement but also for the spreading of particles (seederivation of Eq. (5)). As a function of disorder strength D we find that the prefactor ν of the double logarithmicgrowth is decreasing continuously. There are no indi-cations for a sharp transition. Let us also comment onthe bipartite particle fluctuations ∆ n investigated pre-viously [16, 40]. Our results (not shown) are consistentwith ∆ n ( t ) growing without bounds and ∆ n ( t → ∞ , L ) increasing with increasing system size L .To provide further support for an unbounded growthof the number entropy, we now show that the numericalresults are consistent with a relation recently proven inthe non-interacting case [22]. There we found that S (2) N ≥ γ (cid:26) S (2) − ln (cid:20) I (cid:18) S (2) (cid:19)(cid:21)(cid:27) + b (7)provides a tight bound with γ = 1 , b = 0 , and I beingthe modified Bessel function. I.e., a growth of the secondRényi entropy S (2) is always accompanied by a growth,albeit logarithmically slower, of the corresponding num-ber entropy S (2) N . In Fig. 3 we show that this bound witha renormalized γ (and curves shifted by b > for easeof presentation) appears to remain valid in the interact-ing case both for D < D c and D > D c . Note that in theinteracting case, i.e. V (cid:54) = 0 , γ appears to decrease contin-uously with increasing D but does not show indicationsof a sharp transition. -1 t S ( ) N γ = 1 , b = 0 γ = 0 . ,b = 0 . (2)N Eq . (7) FIG. 4. S (2) N for model (4) with binary disorder D = 20 andsegments with equal D limited to four sites (1000 disorderrealizations). For t < t Th an unrenormalized bound ( γ = 1 )holds while γ is renormalized for t > t Th , see text. Finally, we want to consider a system with very strongdisorder to check whether the increase of the number en-tropy is transient. To this end, we consider the model (4)with binary disorder D j ∈ {− D/ , D/ } . For D → ∞ this will result in finite segments which are coupled bythe interaction term but not by hopping processes. We,furthermore, limit the size of segments with equal poten-tial ± D/ to four lattice sites. In this case, the disorderis no longer uncorrelated but this should only help inreducing the time scale where S ( α ) N potentially saturates.Numerically, we find that the bound (7) also holds in thiscase, see Fig. 4. For t (cid:46) t Th , where t Th ∼ exp( D/ Ω) L [26] is the Thouless time with Ω being a constant, theoriginal bound in the non-interacting case ( γ = 1 ) holds,while a renormalized bound holds for longer times (seealso App. C). In the thermodynamic limit, S (2) N thus ap-pears to grow without bounds in this model as well. Conclusions.—
The slow increase of the number en-tropy S N ∼ ln ln t and the increase of the saturationvalue as a function of system size, found in our numericalsimulations, are not expected in an MBL phase. Thereare at least two different possible interpretations of thesedata. First, it cannot be excluded that the observed be-havior after all is transient and that S N in the thermody-namic limit does saturate at very long times. While thisinterpretation would not challenge the established phe-nomenology of MBL phases, it is then an open questionto understand the origin of such a long-time transientbehavior as well as the time and length scales where par-ticle fluctuations ultimately cease to grow. While finite-size effects have been suggested to strongly affect nu-merical studies of MBL [41], we note that in contrast toRef. 26 our data—which also challenge the establishedMBL phenomenology—are obtained at strong disorder.Second, it is possible that hopping processes introducedby the interaction term (6) are relevant and resonancesdo exist. A possible scenario would be that for D > D c the dynamic scaling t ∼ e (cid:96) does hold, leading to a loga-rithmic growth of the entanglement entropy but that thesame dynamical scaling also holds for the spreading ofparticles resulting in an unbounded growth S N ∼ ln ln t .While this implies that the system is ultimately ergodicat very long time scales, it will not drastically alter thebehavior on experimentally accessible time scales: MBLsystems would still be good quantum memories and theHamiltonian (2) an effective description.J.S. acknowledges support by the Natural Sciences andEngineering Research Council (NSERC, Canada) andby the Deutsche Forschungsgemeinschaft (DFG) via Re-search Unit FOR 2316. We are grateful for the comput-ing resources and support provided by Compute Canadaand Westgrid. M.K., R.U. and M.F. acknowledge finan-cial support from the Deutsche Forschungsgemeinschaft(DFG) via SFB TR 185, project number 277625399. Thesimulations were (partly) executed on the high perfor-mance cluster "Elwetritsch" at the University of Kaiser-slautern which is part of the "Alliance of High Perfor-mance Computing Rheinland-Pfalz" (AHRP). We kindlyacknowledge the support of RHRK. M. K. would liketo thank J. Léonard and M. Greiner for hospitality andfruitful discussions and J. Otterbach for advice for theoptimization of algorithms to GPU’s. Appendix A: Numerical methods
We first discuss the numerical methods used in moredetail. If not stated otherwise, our results are obtainedby standard ED methods for open boundary conditionsfor system sizes L ≤ using particle number conserva-tion. For system sizes L > , we employ a second orderTrotter-Suzuki decomposition (TSD)[42–44] of the timeevolution operator to evolve the full many-body state e − i( ˆ A + ˆ B ) δt = e − i ˆ Aδt e − i ˆ Bδt e − i ˆ Aδt + O ( δt ) , (8)where ˆ A, ˆ B are non-commuting operators given by theeven and odd parts of the Hamiltonian. Since we applythe decomposed time-evolution operator to the full state,no truncations are needed as, for example, in TEBD al-gorithms [45]. The cumulative error in the time evolutionof a system using a second order TSD is thus quadraticin δt . Therefore, the simulation time is restricted by thestep size of the TSD. We use δt = 0 . for computa-tions with D < D c and δt = 0 . for computations with D > D c , because the latter is the regime we are mostinterested in. In order to reach times beyond hop-ping amplitudes, see Fig. 5, higher-order Trotter-Suzukidecompositions would be more efficient. The second or-der TSD is, however, sufficient for our purposes. TheTSD method can be easily parallelized on a large arrayof graphical computing units (GPU’s), making systemsizes of up to L = 24 accessible. FIG. 5. S N for D > D c ≈ with V = 0 . and L = 16 :(a) S N ( t ) computed by ED (dots) compared to second orderTSD (shaded areas), averaged over 1000 disorder realizations.The lines show fits ∼ ln ln t . (b) The same as in (a) but forstronger disorder and different numbers of realizations, seelegend. The shaded areas represent a rolling average whichcontain 99% of the TSD data with errorbars. Appendix B: Scaling deep in the MBL phase
For weak interactions it has been shown that the criti-cal disorder strength D c , needed to localize a many-bodysystem, is small. In the following, we consider V = 0 . where the critical disorder strength has been estimatedto be D c ≈ , see Ref. [35]. We note, however, that ithas been argued that for very large system sizes L > the critical disorder strength might actually be about afactor of larger [46]. We therefore investigate the be-havior of the prefactor ν when fitting the number en-tropy according to S N ∼ ν ln ln t for disorder strengths D ∈ [5 , . The larger MBL phase expected at weakinteractions gives us better access to the scaling behav-ior of S N . As can be seen from Fig. 6, ν is decreasingwith increasing disorder but not faster than a power law,even for very strong disorder about 15 times the criticaldisorder strength. Most importantly, we do not find anyindications for a sudden drop or a phase transition. FIG. 6. Parameter ν extracted from fits S N ≈ ν ln ln t + b as function of disorder strength D for V = 0 . and L = 12 .Error bars represent uncertainties related to varying ν , b , andthe fit interval. Appendix C: Renormalization of bounds andThouless time -1 t S ( ) N γ =1 ,b =0 D=7.5 γ =0 . ,b =0 . (2)N Eq . (7) -1 t S ( ) N γ =1 ,b =0 D=12.5 γ =0 . ,b =0 . (2)N Eq . (7) -1 t S ( ) N γ =1 ,b =0 D=16.25 γ =0 . ,b =0 . (2)N Eq . (7) -1 t S ( ) N γ =1 ,b =0 D=20 γ =0 . ,b =0 . (2)N Eq . (7) FIG. 7. Lower bound, Eq.(7), and number entropy S (2) N for V = 2 . The boxes indicate where the bound with γ = 1 and b = 0 crosses S (2) N for the first time. In the main text, we have seen that the a lowerbound for the number entropy recently obtained for freefermions [22] appears to hold also in the interacting casebut with parameters renormalized by interactions anddisorder. This bound is given by Eq. (7). For binarydisorder, see Fig. 4 in the main text, we found that atsmall times the free-fermion bound applies directly, whilebeyond a certain time t x a renormalized value of γ had tobe used. We now give further evidence for this behaviorin the case of box disorder and V = 2 , see Fig. 7, bothfor D < D c ≈ and D > D c . The figure shows that wecan find perfect fits in this case as well. We define thecrossover time t x as the time where Eq. (7) with unrenor-malized parameters crosses S (2) N . For t > t x , we have torenormalize γ in order for the bound to hold.
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