Many-body localization in one dimensional optical lattice with speckle disorder
MMany-body localization in one dimensional optical lattice with speckle disorder
Artur Maksymov, Piotr Sierant,
1, 2 and Jakub Zakrzewski
1, 3 Institute of Theoretical Physics, Jagiellonian University in Krakow, (cid:32)Lojasiewicza 11, 30-348 Krak´ow, Poland ICFO- Institut de Sciences Fotoniques, The Barcelona Institute of Science and Technology,Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain Mark Kac Complex Systems Research Center, Jagiellonian University in Krakow, 30-348 Krak´ow, Poland. ∗ (Dated: October 1, 2020)The many-body localization transition for Heisenberg spin chain with a speckle disorder is studied.Such a model is equivalent to a system of spinless fermions in an optical lattice with an additionalspeckle field. Our numerical results show that the many-body localization transition in speckledisorder falls within the same universality class as the transition in an uncorrelated random disorder,in contrast to the quasiperiodic potential typically studied in experiments. This hints at possibilitiesof experimental studies of the role of rare Griffiths regions and of the interplay of ergodic andlocalized grains at the many-body localization transition. Moreover, the speckle potential allows oneto study the role of correlations in disorder on the transition. We study both spectral and dynamicalproperties of the system focusing on observables that are sensitive to the disorder type and itscorrelations. In particular, distributions of local imbalance at long times provide an experimentallyavailable tool that reveals the presence of small ergodic grains even deep in the many-body localizedphase in a correlated speckle disorder. I. INTRODUCTION
Isolated quantum many-body systems are genericallyexpected to reach thermal equilibrium according to eigen-state thermalization hypothesis [1–4]. The approach tothermal equilibrium may be precluded by strong disor-der resulting in phenomenon of Many-Body Localization(MBL) [5, 6] investigated in recent years both theoreti-cally and experimentally (for reviews see [7–10]). Furtherexamples of nonergodic many body-systems include mod-els with constrains [11–13], lattice gauge theories [14–19]often linked with periodic oscillatory behavior coined, inthe wake of well known quantum chaos notion [20, 21],as quantum scarring [22–26], as well as surprizingly basicsystems as interacting particles in tilted lattices (Stark-like localization) [27, 28] or even harmonic potentialsfeaturing coexistence of localized and delocalized phases[29–31].The theoretical studies of MBL typically consider uni-form uncorrelated random potential as a source of dis-order in the system. In contrast, the experimental se-tups used much easier to realize quasiperiodic potential[32, 33] correlated at arbitrary length scales. The behav-ior of one-dimensional (1D) models deep in the localizedphase is similar in both cases (leading e.g. to preserva-tion of the information about initial states in time dy-namics [32, 33]). The situation is more complicated inthe crossover between localized and extended phases. Itis claimed even that the observed behavior suggest dif-ferent universality classes of MBL transition dependingon the disorder type [34, 35]. For uncorrelated disorderone expects to observe the influence of rare events, the socalled Griffiths regions [36, 37] i.e. grains of the minority ∗ [email protected] phase on the either side of the transition (e.g. presenceof ergodic grains on the localized side). Those affect thetime dynamics and lead to e.g. subdiffusive transporton the delocalized side [38–43]. Even though the rareGriffiths regions are a priori absent in the deterministicquasiperiodic potential, the resulting dynamics are simi-lar as in the uncorrelated disorder featuring a power-lawdecay of time-correlators as well as a power-law growthof the entanglement entropy [42, 44] - for a recent reviewsee [45].Surpizingly much less is known about MBL in a ran-dom speckle potential, despite the fact that such a poten-tial has been successfully used in single particle physicse.g. for the experimental demonstration of Anderson lo-calization in cold atomic gases [46, 47]. For attractivelyinteracting bosons the bright soliton can be trapped ina speckle-disorder potential and get Anderson localized[48, 49]. A study of MBL in two-dimensional continuum[50] concludes that perturbation theory diverges for ar-bitrarily weak interactions in a speckle potential. More-over, it is not clear whether the insulating state of astrongly correlated atomic Hubbard gas in a speckle po-tential observed in center-of-mass velocity measurements[51] can be attributed to MBL since the phenomenon isbelieved to be not stable beyond one dimension [52]. Arecent theoretical study of MBL in a speckle potential[53] was limited to few particles only due to numericalrequirements of the continuum approach.The aim of this work is to provide an in depth studyof MBL in a speckle potential in a one dimensional chainwhich is the typical geometry in which MBL is stud-ied both experimentally and theoretically. While MBLhas been studied for spinless or spin-1/2 fermions [54]as well as for bosons in optical lattices potential [55–59]we choose to consider the simplest, paradigmatic modelused in MBL studies, namely, the disordered Heisenbergchain. There are at least two reasons underlying this a r X i v : . [ c ond - m a t . d i s - nn ] S e p x C(x) σ = 0σ = 1σ = 2 sinc (x/ σ )2 4 6 8 x -3 -2 -1 P ( x ) σ = 0 σ = 1σ = 2 FIG. 1. The speckle distribution (inset) and its correlationfunction for disorder strength W = 1 and different correlationlengths σ as indicated. choice. Firstly, for random uniform as well as quasiperi-odic disorder the Heisenberg spin chain has been quitedeeply analyzed already, thus a direct comparison withrandom uniform results allow us to better comprehendthe differences resulting from the nature of the specklepotential. Secondly, with the local on-site Hilbert spacedimension equal to 2, one may, using exact diagonaliza-tion approach reach system size of the order of L = 20straightforwardly. That allows us for an in-depth anal-ysis of properties of the system. That would be muchharder for bosons or spinful fermions – for the latter, inaddition, the inherent SU(2) symmetry affects deeply theMBL transition [60–64].The paper is organized as follows. Section II intro-duces the model and the speckle disorder, we review itsbasic properties there. With this knowledge we considerproperties of eigevalues and eigenstates of the model inSection III while the time dynamics is discussed in Sec-tion IV. Appendices provide additional discussion on spe-cialized topics. We summarize our findings in Section V. II. THE MODEL.
We consider a 1D Heisenberg spin chain, widely usedin MBL studies [38, 39, 65–68]. This model maps, viaJordan-Wigner transformation, to a system of interactingspinless fermions which allows us to make the connectionwith optical lattice experiments. Instead of quasiperi-odic disorder imposed by a secondary weak optical latticewith period incommensurate with the primary lattice (inwhich the tight binding approximation is inherently as-sumed) as in experiments [32, 44] we imagine that thedisorder is added by an additional optical potential dueto a speckle radiation. This additional light may operateon a different optical transition than the primary opti-cal lattice and, within the tight binding approximationfor the latter, leads to a desired speckle disorder. The resulting Hamiltonian of the system reads: H = J L − (cid:88) i =1 (cid:126)S i · (cid:126)S i +1 + L (cid:88) i =1 h i S zi , (1)where (cid:126)S i are spin-1/2 matrices, J = 1 will be consid-ered as the energy unit, open boundary conditions areassumed. The local magnetic fields, h i , are drawn fromthe speckle distribution P ( x ) = W exp ( − x/W ); x > x = W (an overbar denotes an average over dis-order realizations). Similarly, W is the standard devi-ation of the exponential distribution. Importantly, h i values may be correlated depending on their relative po-sition. The speckle is typically generated by transmissionof light through a ground glass plate [69]. The correla-tions in the speckle pattern result from interference oflight scattered by different parts of the plate and are,therefore, controlled by the aperture of the object. As-suming a rectangular plate, see [70] for more details, thecorrelation function takes a form (in discrete represen-tation) h i h j = W C ( | i − j | /σ ) with C ( y ) = [sin( y ) /y ] and | i − j | the distance (unit lattice constant assumed) -compare Fig. 1.Few remarks are in order. The disorder is asymmetricwith assumed positive x . We could also change the signof all h i to have an opposite case (in atomic implemen-tation a change of the sign corresponds to the change ofthe sign of the detuning on the atomic transition). Thissign is relevant and important for low lying states [48, 49](as the disorder corresponds to either peaks or valleys ofthe potential). However we shall consider highly excitedstates from the middle of the spectrum and this sign be-comes irrelevant. Second, in an optical implementation σ may be as low as 0 . µm [46] i.e. a fraction of the typicallattice spacing in experiment [32, 44]. In the tight bind-ing model we have then simply an uncorrelated disorder.Increasing σ we may study how finite correlations in thepotential affect MBL, the option apparently not availablefor other types of experimentally relevant disorder usedtill now.For reference, we shall use also the uniform random(UR) disorder, for which the fields h i are independentrandom variables drawn from uniform distribution on in-terval [ − W, W ]. III. PROPERTIES OF EIGENVALUES ANDEIGENSTATESA. Locating the transition
With the model defined we study first its spectral prop-erties to verify the presence of the ergodic-MBL transi-tion. Consider a mean gap ratio, r , calculated as anaverage of r i = min { s i +1 s i , s i s i +1 } , (2)where s i = E i +1 − E i (with E i being eigenvalues of thesystem) and the average is taken over a region of spec-trum and over disorder realizations. The mean gap ratiowas proposed as a simple probe of level statistics in [71]with r ≈ .
38 for Poisson statistics (PS) (correspondingto localized, integrable case) and r ≈ .
53 for Gaussianorthogonal ensemble (GOE) [72, 73] of random matriceswell describing statistically an ergodic system.We determine the mean gap ratio as a function of thedisorder amplitude W for several system sizes. Since thetotal spin projection on the z axis is conserved, we con-sider the largest nontrivial sector of the Hamiltonian with (cid:80) S zi = 0 – that restricts system sizes considered to L even. Typically, we consider about N = 300 eigenener-gies from the middle of the spectrum, i.e. ε ≈ . ε = ( E − E min ) / ( E max − E min ) with E min ( E max ) be-ing the lowest (highest) eigenvalue for a given disorderrealization. The results are averaged over 1000 disorderrealizations or twice that number close to the estimatedtransition point. Data for L = 14 and L = 16 are ob-tained by full exact diagonalization, the ones for L = 18and L = 20 are obtained using POLFED algorithm [74].The results are shown in Fig. 2. Curves correspondingto different system sizes cross typically in the vicinity of r ≈ . W c for different disorders. Werefrain from using the procedure of a single parameterfinite size scaling [34, 39, 54] as it has became apparentrecently [75, 76] that the transition may be of Kosterlitz-Thouless type and such a finite size scaling approach maybe not valid.The critical values of disorder for ε ≈ . r _ L =14 L =16 L =18 L =20 w r _ w (a) UR (b) σ = 0 (d) σ = 2 (c) σ = 1 FIG. 2. Mean gap ratio as a function of the rescaled disorderamplitude w = W/W c – for W c values please see Table I.Panel (a) shows the reference behavior for a random uniformdisorder while the remaining panels correspond to differentscale of correlations of the speckle potential as indicated by σ values in the panels. The transition in a speckle potentialseems much broader with smaller differences between curvesfor different system sizes. Increasing correlations make thiseffect much stronger. TABLE I. The critical values of disorder for different disordertypes considered in this work obtained at the middle of thespectrum, ε = 0 .
5. UR σ = 0 σ = 1 σ = 2 W c disorder we get W c ≈ . W c ≈ . w = W/W c to facilitatea comparison between various models of disorder. Fromnow on we shall use, whenever possible, the rescaled dis-order, w .Comparison of Fig. 2(a) and Fig. 2(b) shows that thecrossover between ergodic and localized phases for URdisorder is sharper than for the uncorrelated speckle dis-order ( σ = 0). The disorder strength at which the av-erage gap ratio r departs from the GOE value reaches w ≈ . L = 20and w ≈ .
25 for speckle disorder with σ = 0. This canbe traced back to the unbounded on-site distribution ofthe speckle potential: the probability of having a fullyergodic system at e.g. w = 0 . h i is large on one of the sites. The effectof broadening of the crossover is further enhanced whenthe correlations ( σ >
0) are introduced in the specklepotential.This suggests that finite size effects are stronger forthe correlated disorder which is seemingly contradictedby the fact that curves for different sizes L are muchcloser to each other for the correlated speckle disorder.However, a weaker size dependence of ¯ r is a sign of strongfinite size effects – relatively small changes in system sizes L available to exact diagonalization are simply too smallto have a visible effect on r dependence for the correlatedspeckle disorder. B. Inter-sample randomness
A more detailed characteristic of ergodic to MBL tran-sition is obtained when one examines variation of systemproperties for individual disorder realizations [35]. Tothat end we average the gap ratios r i obtained from 300eigenvalues from the middle of the spectrum ( ε = 0 . r s . The distributions of ¯ r s (obtainedfrom calculating ¯ r s for many disorder realizations) vastlydiffer between UR disorder and quasiperiodic potential[35] pointing out the dominant role of inter-sample ran-domness in the MBL transition in the UR disorder. Thelarge inter-sample randomness, in turn, may be linked tothe existence of rare Griffiths regions. Hence, it was sug-gested [34] that MBL transitions for UR disorder andquasiperiodic potential belong to different universalityclasses.The distribution of the sample averaged gap ratio P (¯ r s ) obtained for the speckle disorder are comparedwith the UR disorder case in Fig. 3. Three values ofthe rescaled disorder are considered: w = 0 .
15 and w = 2 .
35 corresponding to delocalized and localizedphases, respectively, and intermediate one correspond-ing to w = 0 . w = 0 . r s distribution and in the thermodynamiclimit they tend to the respective critical values W c pro-vided that MBL persists in the thermodynamic limit(this issue is a topic of the current debate [74, 76, 78–80]).Observe that, as it could be expected, the distributionsfor the localized and delocalized cases are quite similarfor UR and for the speckle disorders with different corre-lation lengths. The situation is markedly different in thetransition regime. For the system with UR disorder the P (¯ r s ) distribution is unimodal, although broad. On theother hand, for the speckle disorder one can observe a bi-modal shape. With the increase of the correlation length σ the two peaks observed for speckle potential becomemore pronounced and overlap more the distributions fordelocalized and localized phases.Apparently, the sample averaged gap ratio distribu-tion catches an important difference between the speckleand UR disorder. The speckle distribution is exponen-tial favoring small values of disorder. Thus, it is quiteprobable to obtain nearby sites with very small differ-ence in the local potential – that facilitates transport.This observation may be quantified by finding the prob- P ( r _ s ) r _ s P ( r _ s ) r _ s (a) UR (b) σ = 0 (c) σ = 1 (d) σ = 2 FIG. 3. Distributions P ( r S ) of the sample-averaged gap ratioobtained for single realizations of disorder for random (a) andspeckle potential with σ = 0 (b), σ = 1 (c) and σ = 2 (d).The black circles and blue diamonds are for delocalized andlocalized phase with w = 0 .
15 and w = 2 .
35, respectively; thered circles are for transient regime with w = 0 . w = 0 . L = 16, 4500 realizations and 300 levelsaround ε = 0 . ability distribution for the difference between consecutiverandom numbers which, for speckle disorder is also expo-nential. At the intermediate rescaled disorder, w = 0 . r s ≈ . h i resulting in a localized sample (¯ r s ≈ . P (¯ r s ) distribution for thespeckle disorder with σ = 0. This mechanism is furtherreinforced by the presence of correlations in the specklepotential ( σ = 1 , C. Participation entropies of eigenstates andmultifractality
While gap ratio statistics provides statistical informa-tion about eigenvalues of the model, additional informa-tion may be gained from eigenvector properties. Thosemay be probed via e.g. participation entropies. Follow-ing [82] we consider participation entropies S q definedvia the q -th moments of wavefunction | Ψ (cid:105) following theexprespages = 036206,sion: S q (Ψ) = 11 − q ln (cid:32) N (cid:88) i =1 | c i | q (cid:33) , (3)where c i are the coefficients of wavefunction | Ψ (cid:105) in thebasis state | n (cid:105) , i.e. | Ψ (cid:105) = (cid:80) Ni =1 c i | n (cid:105) , N is the dimensionof Hilbert space. While providing supplementary infor-mation to that hidden in eigenvalues one should remem-ber that the participation entropies are basis dependent(becoming trivial in e.g. the eigenbasis of the Hamilto-nian). We shall consider the eigenbasis of S zi operators,equivalent to the basis of Fock states in the language ofspinless fermions. On the delocalized side this basis isto a large extent unbiased. On the localized size, sincethe so called local integrals of motion of the Heisenbergchain [7, 83–88] can be thought of as dressed S zi opera-tors this basis is rather close to the eigenbasis (for anyreasonable measure of the basis distance [89]). Whilethis may be considered as a drawback, this choice as-sures that participation entropies are sensitive to the lo-calization transition. We considered only the lowest mo-ments q = 1 and q = 2: S (Ψ) = − (cid:80) Ni =1 | c i | ln | c i | and S (Ψ) = − ln (cid:16)(cid:80) Ni =1 | c i | (cid:17) that are equal to to theShannon entropy of | c i | distribution and logarithm ofthe inverse participation ratio (IPR), respectively.To probe the distributions of participation entropieswe again consider L = 16 where we have accumulateddata for 4500 disorder realizations for all cases studied.We take 300 eigenfunctions corresponding to the middleof the spectrum around ε = 0 .
5. We show here the resultsfor S - the logarithm of IPR - the celebrated measure oflocalization studies in single particle physics [90, 91] asreviewed in [92]. S -3 -2 -1 P ( S ) σ = 0σ = 1σ = 2 UR (a) (a) S -4 -3 -2 -1 P ( S ) σ = 0σ = 1σ = 2 UR (b) (b)FIG. 4. The distribution of the participation entropy S fordeeply localized (empty markers) and delocalized (filled mark-ers) phases (a) and for the transition regime (b). The data fordelocalized phase are obtained for w = 0 .
15, whereas the onesfor localized state are for w = 2 .
35. The transition regime cor-responds to maximal inter-sample randomness i.e. w = 0 . w = 0 . The S distributions are shown in Fig. 4. Top panelcompares UR disorder with the speckle for localized anddelocalized regimes highlighting differences in the tworegimes. In the delocalized case, for UR disorder we ob-serve a narrow, almost symmetric gaussian-like distribu-tion. This is not the case for speckle potential despitethe fact that w = 0 .
15 lays deeply in the delocalizedregime. Distributions of S show a pronounced asymme-try with a broad tail extending towards smaller values of S . The tail significantly grows with the speckle correla-tion length. The presence of this tail indicates relativelyrare situations where a partial localization occurs within ln N S σ = 0 σ = 1 σ = 2 ln N S σ = 0 σ = 1 σ = 2 (a) (b) FIG. 5. Finite size scaling of participation entropies S – panel (a) and S – panel (b) for deeply localized phase( w = 2 . ε = 0 . D q (visible as coefficients in front of x ≡ ln N ) indicatemultifractal character of eigenstates even in this deeply local-ized regime for all considered types of disorder. the sample. A reversed trend is observed on the localizedside. Here the participation entropy S for an UR disor-der shows a characteristic shape with a tail decaying as S − α , for this value of w we find α ∼
2. The uncorrelatedspeckle disorder leads to a similar distribution but with,again, the tail which decays more slowly. The tail getsheavier when the correlations in speckle disorder are in-troduced. In the vicinity of the transition (see bottompanel of Fig. 4) the distributions become very broad forboth types of disorder, corroborating our claims aboutthe same universality class for MBL transition in URand speckle disorder, even in presence of a finite rangecorrelations in the latter case. We also note that for thespeckle potential an apparent excess of large S corre-sponding to delocalized samples occurs.The distribution shapes differ for UR and speckle dis-order. A quantitative analysis is obtained by finding the S q scaling with the system size - here we follow closelythe similar analysis performed for RU disorder [82]. Itwas shown that in that case to a good approximation S q = D q ln N + b q , where N is the Hilbert space dimen-sion of the system studied. The eigenstates are multifrac-tal if the fractal dimensions D q differ among themselves.This is to be contrasted with D q = 0 / w = 2 . σ = 0. The increase of speckle correlationlength shows a further increase of multifractal dimen-sions revealing that localization is somehow “weaker” forthe speckle potential at same, linearly rescaled disorder.Interestingly, observe, however, that the subleading b q coefficient remains positive for all cases considered (sug-gesting a localization as noted in [82]). IV. TIME DYNAMICSA. Time evolution of the imbalance
While eigenvalues and eigenvector properties provideus with an understanding of the difference between ran-dom and speckle potential they are not directly ac-cessible in experiments. Standard MBL experiments[32, 44, 93, 94] consider instead the dynamics inferringthe information from time evolution of appropriately cho-sen initial states. The first and simple conceptually ap-proach [32] considers the evolution of the density-wavelike state with every second site occupied and every sec-ond empty (for spinful fermions). Analogously, for theHeisenberg spin chain one may consider a N´eel state | ψ (0) (cid:105) with spins up/down on even/odd sites, respec-tively (or vice versa). In time evolution starting fromthe state | ψ (0) (cid:105) , the spin correlation function defined as I ( t ) = D L (cid:88) i =1 (cid:104) ψ ( t ) | S zi | ψ ( t ) (cid:105)(cid:104) ψ (0) | S zi | ψ (0) (cid:105) , (4)where D is a normalization constant (so that I (0) = 1) isfollowed. The spin correlation function I ( t ) for the N´eelstate is mapped, via Jordan-Wigner transformation, tothe difference of populations of spinless fermions at evenand odd sites at time t , hence we refer to it as an im-balance. While in the delocalized regime the imbalancerapidly decays to zero in accordance with eigenstate ther-malization hypothesis [3, 4], in fully localized case, afteran initial transient, it saturates to a certain value depen-dent on the disorder amplitude. Theoretical and exper-imental studies [42, 44] addressed the time dependenceof imbalance also in the transition regime observing typ-ically its power-law decay. This effect has been used toestimate the critical disorder for MBL transition for largesystem sizes [56, 77, 95, 96].The exemplary behavior of the imbalance (4) for thestudied disorder types is depicted in Fig. 6. Results areobtained for L = 20 system using Chebyshev propagationtechnique [97, 98]. The curves indeed show the algebraicdecay with I ( t ) ∝ t − α that persists to long times anddisorder strengths. The algebraic decay describes wellthe time dependence of I ( t ) for both UR and speckledisorder.Fig. 7 shows the fitted exponents α as a function of thescaled disorder amplitude for system sizes L = 16 , , α observedfor system sizes 20 (cid:54) L (cid:54)
200 for UR [77, 95, 96] appearsfor the speckle disorder as well. Another interesting ob-servation is that the power α changes differently withdisorder amplitude for various cases depicted in Fig. 7.The functional form of α ( w ) for UR disorder is well ap-proximated by an exponential decay (a straight line in I(t)
10 100 t I(t)
10 100 1000 t (a) UR (b) σ = 0 (d) σ = 2 (c) σ = 1 FIG. 6. The dynamics of imbalance for the system withrandom (a) and speckle potential (b)-(d) at disorder values(from bottom to top in each subpanel): w = 0 . w = 0 . w = 0 . w = 1, w = 2. The data are obtained for N´eel stateby averaging over 600 disorder realizations for L = 20 sitessystem. The α is extracted by power law fitting of imbalancecurves within the range 100 ≤ t ≤
300 (shown by dashedlines). the lin-log plot). A similar dependence is apparent foran uncorrelated speckle potential, with one difference:the exponent α decreases significantly more slowly with w . The presence of correlations in the speckle disorderfurther enhances the slow decay of imbalance at largedisorder strengths as we observe a non-zero exponent α even for w >
2. This suggest that a linear rescaling ofthe disorder (by the critical disorder value) does not fullycompensate for correlations in the disorder, the transition α L =16 L =18 L =20 w α w (a) UR (b) σ = 0 (d) σ = 2 (c) σ = 1 FIG. 7. Exponent α of the spin imbalance decay as a functionof disorder strength w , as extracted from fits for differentsystem sizes for random potential (a), speckle potential with σ = 0 (b), σ = 1 and σ = 2. The data are obtained for500 realizations. Standard bootstrapping approach is used toestimate the errors shown in the figure. for larger correlation length is “broader” with larger tran-sition region. This parallels a similar observation madefor participation entropies as well as the dependence ofthe fractal dimensions on the speckle correlation length σ and can be linked to a formation of small ergodic grainsdeep in the MBL phase as we show in the next section. B. Local imbalances
Is it possible to reveal in a more pronounced way thedifferent properties of MBL for UR and speckle disor-ders in time dynamics? After all those differences werequite strikingly visible in the eigenvector spectral proper-ties was well as in participation entropies. The answer ispositive. What we need is a measure of local localizationproperties, as the broad entropic distributions discussedabove convincingly revealed the presence, even in thesame sample, of regions seemingly localized to a differentdegree. Such a measure may be constructed as a localimbalance I k , k ∈ [1 , L/
2] involving spins on 2 k − , k sites: I k ( t ) = 2[ (cid:104) ψ ( t ) | S z k − | ψ ( t ) (cid:105)(cid:104) ψ (0) | S z k − | ψ (0) (cid:105) + (cid:104) ψ ( t ) | S z k | ψ ( t ) (cid:105)(cid:104) ψ (0) | S z k | ψ (0) (cid:105) ] . (5)The global imbalance is just a sum of I k . The local im-balances provide information about local scrambling ofspin degrees of freedom.We consider a set of local imbalances { I k ( t i ) } k = L/ − k =2 at times t i = 1000 − i (in the J − units) for i = 0 , , ..., { I k ( t i ) } for a largenumber of disorder realizations we plot the resulting dis-tributions of local imbalance, P ( I ), in Fig. 8. Consider P (I ) w = 0.15w = 0.6w = 1.1w = 2.35 w = 0.15w = 0.6w = 1.34w = 2.35 -1 -0.5 0 0.5 1 I P (I ) w = 0.15w = 0.6w = 1.5w = 2.35 -0.5 0 0.5 1 I w = 0.15w = 0.6w = 1.5w = 2.35 (a) UR (b) σ = 0 (c) σ = 1 (d) σ = 2 FIG. 8. The distribution of local imbalance I for variousconsidered types of disorder. Observe the pronounced effectthe speckle correlation has on that distribution. The systemsize is L = 16, the distributions are obtained for time t = 1000and from more than 2000 disorder realizations. first the distribution for UR disorder shown in panel (a):for large disorder the distribution is sharply peaked nearits maximal unit value indicating an almost complete lo-calization and a good memory of the initial state. Onthe contrary, for small disorder we observe a smoothgaussian-like profile centered at I = 0 - a signatureof a lost memory of the initial state. With increasingdisorder this distribution sharpens, becomes asymmet-ric (as a total imbalance becomes positive) developing atail at positive values of I . Around a critical disordervalue the distribution of I is broad reaching the edge atunity. For speckle uncorrelated disorder – Fig. 8(b) – thecurves look similar although a careful inspection revealsthat for the localized case the tail extending to small I is higher than for UR disorder. The broadest distribu-tion, extending almost uniformly between zero and unityis obtained at slightly different w value, in comparisonto the UR case. The picture changes for the correlatedspeckle disorder. We observe a spectacular narrowing ofdistributions in the delocalized case. This may be eas-ily understood, once some disorder value is chosen for agiven site, the next correlated site has, with a large prob-ability, a similar disorder value facilitating delocalization.Interestingly the broadest distributions (at values of dis-order shifting towards slightly larger values) develop lo-cal maxima at I = 0 , σ = 2 wheneven for large w corresponding globally to the deep MBLcase, a noticeable maximum at I = 0 still exists point-ing towards the existence of small regions that locallythermalize.Note the real resemblance between bimodal distribu-tions observed for local imbalance with the similarlyshaped characteristics of the sample averaged gap ratioshown in Fig. 3. The local imbalance allows us to geta similar understanding of the system behavior as eigen-value statistics not only on the level of the global prop-erties reflected by the imbalance but on a deeper, locallevel. Let us stress that the measurement of local imbal-ances requires a single site resolution - however in coldatomic systems as well as in spin models such resolutionis already achieved experimentally [99]. V. CONCLUSIONS
We investigate an optical speckle field placed on topof a quasi one-dimensional optical lattice which allowsus to go beyond the continuum approaches consideredso far and to model the system within a tight bindingdescription in which the speckle field gives rise to a on-site perturbation. The speckle disorder obtained in thatmanner has unique features enabling a control over itscorrelation length. On one hand, the speckle disorderallows us to study MBL transition in an uncorrelated“trully random” disorder, as opposed to the routinelyrealized experimentally case of quasiperiodic potentialwhich leads to MBL transition of different universalityclass [34]. On the other hand, the speckle field opensup the possibility of studying the influence of correla-tions in disorder on many-body localization transition.A specific exponential distribution function of the uncor-related speckle already leads to certain differences in thesystem behavior as compared to usually studied randomuniform disorder, the effects become amplified when thespeckle correlation length is increased. We observe morepronounced finite size effects that are, surprizingly, hid-den in the reduced sensitivity of the system response tochanges of its size. This suggests that system sizes of fewhundreds sites, available in current experiments in opti-cal lattices may be needed to reach the thermodynamiclimit. MBL in the speckle potential has increased sam-ple to sample variation as compared to random uniformdisorder, as visualized in the gap ratio analysis as well asin the study of eigenvector properties. With increasingcorrelations the critical regime of transition broadens.Additional insights may be obtained from time dynam-ics. Global imbalance decay confirms the broadeningof the transition while a local imbalance, a tool intro-duced in this work, allows us to visualize the origin ofthe resistance against localization observed in (partic-ularly correlated) speckle potential. Apparently it fa-vors creation, even for large disorder amplitude, of smallgrains that locally thermalize. That resembles the be-havior expected of rare Griffiths regions for uncorrelateddisorder but, surprisingly, the correlations of finite rangeactually enhance their importance. Excitingly, such localimbalances seem to be readily accessible experimentallyoffering possible experimental verification of the resultspresented.
ACKNOWLEDGMENTS
We thank Dominique Delande for providing the codefor the speckle potential and discussions. The lat-ter were also enjoyed with Titas Chanda. We ac-knowledge support by PL-Grid Infrastructure. This re-search has been supported by National Science Cen-tre (Poland) under projects 2015/19/B/ST2/01028 (P.S.and A.M.), 2018/28/T/ST2/00401 (doctoral scholarship– P.S.) and 2019/35/B/ST2/00034 (J.Z.). Partial sup-port by the Foundation for Polish Science under Polish-French Maria Sk(cid:32)lodowska and Pierre Curie Polish-FrenchScience Award is also acknowledged. P.S. acknowledgessupport by the Foundation for Polish Science (FNP)through scholarship START.
Appendix A: Appendix
We present here additional numerical results for themodel. Those results, while not essential for the conclu-
FIG. 9. The mean gap ratio, r , plotted as function of disorderstrength W and the rescaled energy ε . Observe that whileuncorrelated, σ = 0 plot resembles, qualitatively, the situationfor random uniform disorder [39]. For correlated disorder thelobe becomes asymmetric with lower lying states being lesslocalized. Data for L = 16 Heisenberg chain. sions reached in the main text, supplement them withadditional numerical evidence.
1. Energy dependence of the transition and densityof states
The mean gap ratio r as a function of the disorderamplitude and the relative energy is presented in Fig. 9.The latter is defined as ε = ( E − E min ) / ( E max − E min )with E min ( E max ) being the lowest (highest) eigenvaluefor a given disorder realization. We take slices in ε withsize 0.05 obtaining 20 bins for energy. For σ = 0 i.e.an uncorrelated disorder we observe a rather symmetricin energy lobe resembling the one shown in [39]. Thecorrelation in disorder makes low lying states being moreresistant to localization as clearly visible for σ = 2 plotin Fig. 9.This, at a first glance, surprizing behavior may be par-tially explained by the energy dependence of the densityof states - compare Fig. 10. The raw (unscaled) density isperfectly symmetric with respect to origin. Scaling, per-formed for each diagonalization separately, shifts slightlythe maximum of the density of states to (cid:15) below 0 .
5. Fora finite disorder value the maximum of the density con-centrates around (cid:15) = 0 . r which, for a correlateddisorder has a tip at around (cid:15) = 0 .
25. Still, however, for σ = 2 the maximum of DOS does not corresponds to thelobe in the energy gap ratio observed at ε = 0 .
25. Thecentral part of DOS distribution for scaled eigenenergies[Fig. 10 (b)] is Gaussian for the system with the uniformrandom and the uncorrelated speckle potential (at leastwithin the range 0 . (cid:47) ε (cid:47) . σ . The -75 -50 -25 0 25 50 75 Energy -5 -4 -3 -2 -1 DO S W=3.4; RandW=2.25; σ=0
W=4; σ=1
W=6.5; σ=2 (a) ε -2 -1 DO S W=3.4; RandW=2.25; σ=0
W=4; σ=1
W=6.5; σ=2 (b)
FIG. 10. The density of states [unscaled in (a) and scaled in(b)] for the system with the speckle potential. The dashedlines give Gaussian fits of the data. The data are obtained for1000 realization for the system of size L = 16 and disordervalues indicated in the figure - corresponding to our estimateof the critical values. non-Gaussian character of DOS distribution is more pro-nounced for unscaled eigenenergies where for correlatedspeckle potential the exponential tails are observed.
2. Correlations between participation entropy andthe sample averaged gap ratio
While in the main text we have shown the participa-tion entropy S distributions in different regimes, herewe show, compare Fig. 11 that similar picture we obtainfor the Shannon entropy S . The relatively broad distri-butions obtained, in particular in the transition regimecalls for comparing S q with gap ratio distributions.Having in mind the sample-to-sample randomness asexhibited by P (¯ r s ) one may consider the similar proper-ties on the entropy level. Fig. 12 shows the distributionof S q obtained for disorder realizations with given sam-ple averaged gap ratio ¯ r s ± δ , we take δ = 0 . S q and the value of the sample averagedgap ratio ¯ r s . To quantify those correlations we considerthe eigenvectors for 300 eigenvalues around ε = 0 .
5, findtheir participation entropies and average them for eachdisorder realization separately. Let as denote such an av-erage of for s - sample of S q values as ¯ S sq . The resultingcorrelation between ¯ r s and sample averaged Shannon en-tropy ¯ S sq is shown in Fig. 13. The correlations are biggerfor the speckle than for the random uniform disorder andfurther increase with the correlation length of the specklepotential. The increase of correlations between the sam-ple averaged quanities ¯ S sq and ¯ r s with correlation length σ can be understood as an effect of diminishing number ofuncorrelated random fields h i in a given disorder sample:for larger values of σ , the potential fluctuations acrossa given sample are smaller and hence properties of thesample, reflected either by ¯ S sq or ¯ r s , vary less yielding thelarger ¯ S sq , ¯ r s correlation. For uniform disorder the max-imum of the ¯ S sq , ¯ r s correlation occurs at the scaled dis-order w = W/W c ≈ . w = 0 . σ = 2 considered by us. Withan increase of the system size the correlation maximum S -3 -2 -1 P ( S ) σ = 0σ = 1σ = 2 UR (a) S -4 -3 -2 -1 P ( S ) σ = 0σ = 1σ = 2 UR (b) FIG. 11. The Shannon entropy distribution for deeply local-ized (empty markers) and delocalized (filled markers) phases(a) and for the transition regime (b). The data for delocal-ized phase are obtained for w = 0 .
15, whereas the ones forlocalized state are for w = 2 .
35. The transition regime corre-sponds to the maximal ¯ r s − ¯ S correlations –compare Fig. 13i.e. w = 0 . w = 0 . shifts towards higher disorder values. One may specu-late that, provided MBL persists in the thermodynamiclimit, eventually the maximum shifts towards w = 1 i.e.the critical point. The dependence of the position of thecorrelation maximum on system size was checked for sys-tems with L = 10 to L = 20 (not shown). It was observedthat the shift of the maximum is slower with increasing σ supporting the observation made for the gap ratio thatfinite size effects increase with correlation length of thespeckle potential.
3. Bipartite entanglement entropy of eigenstates
To complete the analysis of entropic properties let usconsider a different measure, the bipartite entanglemententropy defined as S e = − Tr ρ A ln ρ A when the system isdevided into two parts A and B . The entanglement en-tropy is evaluated from Schmidt decomposition as follow S e = − n (cid:88) i =1 λ i ln λ i , (A1)0 FIG. 12. The partial distributions of the Shanon entropy S asa function of ¯ r s for uncorrelated speckle potential ( σ = 0).Thedata are obtained for the system size L = 16, 300 levelsaround ε = 0 . w = 1 . / .
25 = 2 /
3. Sharp distribu-tions for “delocalized” samples shift and broaden for lower ¯ r s values. w < r _ S S _ S > - < r _ S >< S _ SS > σ = 0σ = 1σ = 2 UR FIG. 13. The correlations between the sample mean gap ratio¯ r s and the mean Shannon entropy ¯ S s for speckle and uniformrandom potentials (similar correlations are observed for ¯ S s ).The data are obtained for the system size L = 16, 300 levelsaround ε = 0 . where λ i are singular values obtained from decomposi-tion. The size of the subsystem A over which the partialtrace ρ A is taken has beens chosen as a half-size of theentire chain, i.e. L A = L/ S E re-gion, i.e. there are non-negligible number of values of S E S E -3 -2 -1 P ( S E ) σ = 0σ = 1σ = 2 UR (a) S E -3 -2 -1 P ( S E ) σ = 0σ = 1σ = 2 UR (b) FIG. 14. The distribution of bipartite entanglement entropyfor localized (open markers) and delocalized phase (filledmarkers) (a) and for transient regime (b) for speckle (corre-lated and uncorrelated) and uniform random potential. 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