Domain-wall melting as a probe of many-body localization
DDomain-wall melting as a probe of many-body localization
Johannes Hauschild, ∗ Fabian Heidrich-Meisner, and Frank Pollmann Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany Department of Physics and Arnold Sommerfeld Center for Theoretical Physics,Ludwig-Maximilians-Universit¨at M¨unchen, D-80333 M¨unchen, Germany
Motivated by a recent optical-lattice experiment by Choi et al. [Science , 1547 (2016)], wediscuss how domain-wall melting can be used to investigate many-body localization. First, byconsidering noninteracting fermion models, we demonstrate that experimentally accessible measuresare sensitive to localization and can thus be used to detect the delocalization-localization transition,including divergences of characteristic length scales. Second, using extensive time-dependent densitymatrix renormalization group simulations, we study fermions with repulsive interactions on a chainand a two-leg ladder. The extracted critical disorder strengths agree well with the ones found inexisting literature.
Introduction.
In pioneering works based on pertur-bation theory [1, 2], it was shown that Anderson lo-calization, i.e., perfectly insulating behavior even at fi-nite temperatures, can persist in the presence of inter-actions. Subsequent theoretical studies on mostly one-dimensional (1D) model systems have unveiled many fas-cinating properties of such a many-body localized (MBL)phase. The MBL phase is a dynamical phase of matterdefined in terms of the properties of highly excited many-body eigenstates. It is characterized by an area-law en-tanglement scaling in all eigenstates [3–5], a logarithmicincrease of entanglement in global quantum quenches [6–8], failure of the eigenstate thermalization hypothesis [9]and therefore, memory of initial conditions [10, 11]. Thephenomenology of MBL systems is connected to the ex-istence of a complete set of commuting (quasi) local in-tegrals of motion (so-called “l-bits”) that are believed toexist in systems in which all many-body eigenstates arelocalized [8, 12, 13]. These l-bits can be thought of asquasiparticles with an infinite lifetime, in close analogyto a zero-temperature Fermi liquid [1, 14]. Importantopen questions pertain to the nature of the MBL tran-sition and the existence of an MBL phase in higher di-mensions, for which there are only few results (see, e.g.,[15, 16]), mainly due to the fact that numerical simula-tions are extremely challenging in dimensions higher thanone for the MBL problem.The phenomenology of the MBL phase has mostly beenestablished for closed quantum systems. A sufficientlystrong coupling of a disordered, interacting system toa bath is expected to lead to thermalization (see, e.g.,[17, 18]). Thus, the most promising candidate systemsfor the experimental investigation of MBL physics arequantum simulators such as ultracold quantum gases inoptical lattices or ion traps. So far, the cleanest evidencefor MBL in an experiment has been reported for an in-teracting Fermi gas in an optical lattice with quasiperi-odicity, realizing the Aubry-Andr´e model [19, 20]. Otherquantum gas experiments used the same quasi-periodiclattices or laser speckles to investigate Anderson localiza-tion [21, 22] and the effect of interactions [23], however, at low energy densities. Experiments with ion traps pro-vide an alternative route, yet there, at most a dozen ofions can currently be studied [24].By using a novel experimental approach, a firstdemonstration and characterization of MBL in a two-dimensional (2D) optical-lattice system of interactingbosons with disorder has been presented by Choi etal. [25]. They start from a state that contains particles inonly one half of the system while the rest is empty. Oncetunneling is allowed, the particles from the initially oc-cupied region can spread out into the empty region (seeFig. 1). The evolution of the particle density is trackedusing single-site resolution techniques [26, 27] and digitalmirror devices allow one to tune the disorder. The relax-ation dynamics provides evidence for the existence of anergodic and an MBL regime as disorder strength is var-ied, characterized via several observables such as densityprofiles, particle-number imbalances and measures of thelocalization length [25]. This experiment serves as themain motivation for our theoretical work.The term domain-wall melting is inherited from theequivalent problem in quantum magnetism (see, e.g.,[28–33]), corresponding to coupling two ferromagneticdomains with opposite spin orientation. Furthermore,the domain-wall melting describes the transient dynam-ics [34–36] of sudden-expansion experiments of interact-ing quantum gases in optical lattices (i.e., the release ofinitially trapped particles into an empty homogeneouslattice) [36–39]. Theoretically, the sudden expansion ofinteracting bosons in the presence of disorder was stud-
FIG. 1. Initial state (left) and density profile after a suffi-ciently long time (right) in the localized regime. The profiledevelops an exponential decay with distance n j ∝ exp( − jξ dw )in its tails away from the initial edge j = 0 of the domainwall. a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t ied in, e.g., [40, 41] for the expansion from the correlatedground state in the trap, while for MBL, higher energydensities are relevant.We use exact diagonalization (ED) and time-dependent density matrix renormalization group(tDMRG) methods [42–45] to clarify some key questionsof the domain-wall experiments. First, by consideringnoninteracting fermions in a 1D tight-binding modelwith diagonal disorder we demonstrate that it is possibleto extract the single-particle localization length ξ (1)loc asa function of disorder strength from such an experimentsince the density profiles develop exponential tailswith a length scale ξ dw (see Fig. 1). This domain-walldecay length ξ dw also captures the disorder drivenmetal-insulator transition in the Aubry-Andr´e modelwhen approached from the localized regime, exhibitinga divergence. Second, we study the case of spinlessfermions with nearest-neighbor repulsive interactionson chains and two-leg ladders, for which numericalestimates of the critical disorder strength W c of themetal-insulator transition are available [5, 9, 14, 46–49].For both models, essential features of the noninteractingcase carry over, namely, the steady-state profiles decayexponentially with distance in the localized regime W > W c (i.e., the expansion stops), while particlescontinue to spread in the ergodic regime W < W c .Moreover, we discuss experimentally accessible measuresto investigate the dynamics close to the transition for allmodels. Noninteracting cases.
We start by consideringfermions in a 1D tight-binding lattice with uncorrelateddiagonal disorder. The Hamiltonian reads: H = − J (cid:88) (cid:104) i,j (cid:105) (ˆ c † i ˆ c j + H.c. ) − (cid:88) j (cid:15) j ˆ n j , (1)where ˆ c † j denotes the creation operator on site j , ˆ n j =ˆ c † j ˆ c j is the number operator, n j = (cid:104) ˆ n j (cid:105) is density, and (cid:15) j ∈ [ − W, W ] is a random onsite potential ( L is thenumber of sites). We set the lattice spacing to unityand (cid:126) = 1. All single-particle eigenstates are localizedfor any nonzero W and thus the system is an Andersoninsulator at all energy densities [50, 51].Typical density profiles for the dynamics starting froma domain-wall initial state are shown for different timesin Fig. 2(a). Here, “typical” refers to the geometric mean¯ n j over disorder realizations (i.e., the arithmetic mean oflog n j ) (see [52]). The domain wall first melts slightly yetultimately stops expanding. The profiles clearly developan exponential tail ¯ n j ∝ exp( − j/ξ dw ) for j (cid:29)
0. Thecrucial question is now whether the length scale ξ dw isdirectly related to the single-particle localization lengthor not.We compare two ways of extracting ξ dw : First, a fitto the numerical data for ¯ n j in the tails j (cid:29) j − − − − − − ¯ n j (a) tJ = 10000 tJ = 100 − W/J ξ d w (b) tJ VA R n (c) tJ ∆ N (d) W/J = 0 . W/J = 2 . . . . W/J ξ d w FIG. 2. ED results ( L = 2000) for 1D noninteracting fermionswith uncorrelated diagonal disorder Eq. (1). (a) Represen-tative typical density profile [52] for W = 0 . J for times tJ = 100 , , , ,
10 000 (bottom to top). (b) Domain-wall decay length ξ dw (extracted from VAR n ) for the sametimes as in (a), as a function of the disorder strength W . Thedashed line shows a fit to the expected scaling ξ dw ∝ W − [50]. (c) Variance VAR n of the distribution of expandedparticles for W/J = 0 . , . , . , . N ( t ). Inset in (b): ξ dw at tJ = 10 000 (circles) for the Aubry-Andr´e model [52], whichhas a delocalization-localization transition at W = J , com-pared with the analytical result ξ (1)loc = 1 / log (cid:0) WJ (cid:1) (red dottedline) [53]. ted into the originally empty region. For the latter, weview the density n j in the initially empty region j > (cid:104)·(cid:105) n ≡ (cid:16)(cid:80) j> n j · (cid:17) / ∆ N where∆ N = (cid:80) j> n j is the number of emitted particles. Thevariance VAR n = (cid:104) j (cid:105) n − (cid:104) j (cid:105) n of this particle distribu-tion is shown in Fig. 2(c) and approaches a stationaryregime on a timescale depending on W . For the timewindow plotted, only the curves with W ≥ J saturate,yet we checked that also the curves for W < J saturateat sufficiently long times. At short times, VAR n ∝ t signals a ballistic expansion of the particles as long asVAR n ( t ) (cid:28) ξ (1)loc .Assuming a strictly exponential distribution n j ∝ exp( − jξ dw ) for all j > n ≈ ξ for VAR n (cid:29)
1. We use that relation to extract ξ dw in the general caseas well and in addition, we introduce an explicit timedependence of ξ dw to illustrate the approach to the sta-tionary state. In general, this gives only a lower boundto W c since VAR n can be finite for diverging ξ dw if thedistribution is not exponential. Yet we find that bothmethods give similar results for the final profile and showonly ξ dw extracted from VAR n in Fig. 2(b).The known result for the localization length in the 1DAnderson model is ξ (1)loc = J − E ) W [50] for E = 0 (ourinitial state leads to that average energy for sufficientlylarge systems). Our data for ξ dw shown in Fig. 2(b)clearly exhibit the expected scaling ξ dw ∝ W − over awide range of W as suggested by a fit of ξ dw = a/W − to the data [dashed line in Fig. 2(b); the prefactor islarger by about a factor of 1.5 than the typical localiza-tion length ξ (1)loc ]. Deviations from the W − dependenceat small W , where ξ (1)loc ∼ O ( L ), are due to the finitesystem size. At large W , the discreteness of the latticemakes it impossible to resolve ξ dw that are much smallerthan the lattice spacing. We stress that fairly long timesneed to be reached to observe a good quantitative agree-ment with the W − dependence. For instance, for theparameters of Fig. 2(a), tJ ∼ ξ dw is a measure of the single-particle localization length, most importantly exhibitingthe same qualitative behavior.In Fig. 2(d), we introduce an alternative indicatorof localization, namely, the number of emitted particles∆ N ( t ) that have propagated across the edge j = 0 of theinitial domain wall at a time t . Due to particle conserva-tion, ∆ N is directly related to the imbalance I = N − NN analyzed in the experiment [25]. We observe that ∆ N shares qualitatively the same behavior with VAR n [notethe linear y scale in Fig. 2(d)], which will also apply tothe models discussed in the following.As a further test, we study the Aubry-Andr´e modelin the Appendix [52]. The comparison of ξ dw with theexactly known single-particle localization length [53] inthe inset of Fig. 2(b) demonstrates that the domain-wallmelting can resolve the delocalization-localization tran-sition at W = J . Interacting fermions on a chain.
Given the en-couraging results discussed above, we move on to study-ing the dynamics in a system with an MBL phase, namelyto the model of spinless fermions with repulsive nearest-neighbor interactions H int = H + V (cid:80) (cid:104) i,j (cid:105) ˆ n i ˆ n j , equiva-lent to the spin-1/2 XXZ chain. We focus on SU (2)symmetric exchange, i.e., V = J , for which numeri-cal studies predict a delocalization-localization transitionfrom an ergodic to the MBL phase at W c /J = (3 . ± j − − − − − ¯ n j (a) tJ = 30 tJ = 10 j (b) tJ = 200 tJ = 60 j − − − − − ¯ n j (c) tJ = 1000 tJ − VA R n (d) tJ . . . ∆ N (e) W/J = 0 . W/J = 6 . V = 0 W/J . . . C ( W ) FIG. 3. tDMRG results ( L = 60) for a chain of interactingspinless fermions with V = J . Top row: Typical densityprofiles for (a) W/J = 0 . tJ = 10 , ,
30 (bottom to top),(b)
W/J = 3 and additional data for tJ = 60 , W/J = 6, additional data for tJ = 1000 (on top of the datafor shorter times). (d) Variance VAR n of the distribution ofexpanded particles for W/J = 0 . , , , , , , V = 0. Error bars are smaller thansymbol sizes and omitted. (e) Number of emitted particles∆ N ( t ). Inset: C ( W ) from fit of ∆ N ( t ) to Eq. (2) for tJ > bated in the recent literature (see, e.g., [54, 55]).Typical time evolutions of density profiles in the er-godic and MBL phase are shown in Figs. 3(a)-3(c), ob-tained from tDMRG simulations [42–44]. We use a timestep of dt = 0 . /J and a bond dimension of up to χ = 1000 and keep the discarded weight in each timestep under 10 − . The disorder average is performed overabout 500 realizations. These profiles show a crucial dif-ference between the dynamics in the localized and the de-localized regime. Deep in the localized regime, Fig. 3(c),similar to the noninteracting models discussed before, thedensity profiles quickly become stationary with an expo-nential decay even close to j = 0. In the ergodic phase,however, the density profiles never become stationary onthe simulated time scales and for the values of interac-tions considered here. For W = 0 . J shown in Fig. 3(a),the particles spread over the whole considered system.Remarkably, we find a regime of slow dynamics [47, 56–59] at intermediate disorder W < W c in Fig. 3(b), wherethere seems to persist an exponential decay of n j at finitetimes, but with a continuously growing ξ dw ( t ). We notethat ξ dw ( t ) at the shortest time scales is on the orderof the single-particle localization length. An explana- tJ − VA R n (a) W/J = 4
W/J = 10 tJ . . . . . . . ∆ N (b) W/J . . C ( W ) FIG. 4. tDMRG results for a two-leg ladder ( L = 60) ofinteracting spinless fermions with V = J . (a) Variance VAR n at W/J = 4 , , ,
10 (top to bottom). (b) Number of emittedparticles ∆ N ( t ). Inset: C ( W ) from fit to Eq. (2) for tJ > tion can thus be obtained in this picture: On short timescales, single particles can quickly expand into the right,empty side within the single-particle localization length,thus leading to the exponential form of n j . The interac-tion comes into play by scattering events at larger times,ultimately allowing the expansion over the whole systemfor infinite times.The slow regime is also reflected in the quantitiesVAR n and ∆ N in Figs. 3(d) and 3(e), which behave qual-itatively in the same way. While both quantities saturatefor W > W c and the results hardly differ from the nonin-teracting case shown by the dotted lines, the slow growthbecomes evident for W (cid:46) W c at the intermediate timescales accessible to us. The slow growth of both VAR n and ∆ N is, for W (cid:46) W c , the best described by (yet hardto distinguish from a power-law)∆ N ( t ) , VAR n ( t ) = C ( W ) log( tJ ) + const . (2)This growth is qualitatively different from the non-interacting case, where a saturation sets in after a fasterinitial increase. The inset of Fig. 3(e) shows the pref-actor C ( W ) extracted from a fit to the data of ∆ N ( t )for tJ >
10. This allows us to extract W c since C ( W >W c ) = 0 for the stationary profiles in the localized phase.Our result for W c is compatible with the literature value W c /J = 3 . ± Interacting fermions on a ladder.
As a first steptowards 2D systems, we present results for the dynamicsof interacting spinless fermions on a two-leg ladder inthe presence of diagonal disorder. The simulations aredone with a variant of tDMRG suitable for long-rangeinteractions [45], with a time step dt = 0 . /J . Figures4(a) and 4(b) show the variance VAR n and ∆ N for V = J , respectively. As for the chain, we observe that boththe variance and ∆ N have a tendency to saturate forlarge disorder strength, while they keep growing for smalldisorder. The data are best described by Eq. (2) andwe extract C ( W ) from fits of the data for tJ >
10 to Eq. (2). The results of these fits shown in the inset ofFig. 4 suggest a critical disorder strength 8 (cid:46) W c /J (cid:46)
10, in good agreement with the value of W c /J = 8 . ± . Summary and outlook.
We analyzed the domain-wall melting of fermions in the presence of diagonal dis-order, motivated by a recent experiment [25] that wasfirst in using this setup for interacting bosons in 2D. Ourmain result is that several quantities accessible to exper-imentalists (such as the number of propagating particlesand the variance of their particle density) are sensitive tolocalization and can be used to locate the disorder-drivenmetal-insulator transition, based on our analysis of sev-eral models of noninteracting and interacting fermionsfor which the phase diagrams are known. Notably, thisencompasses a two-leg ladder as a first step towards nu-merically simulating the dynamics of interacting systemswith disorder in the 1D-2D crossover. Our work furtherindicates that care must be taken in extracting quantita-tive results from finite systems or finite times since theapproach to the stationary regime can be slow. Interest-ingly, we observe a slow dynamics in the ergodic phase ofinteracting models as the transition to the MBL phase isapproached, which deserves further investigation.The domain-wall melting thus is a viable approach fortheoretically and experimentally studying disordered in-teracting systems, and we hope that our work will in-fluence future experiments on quasi-1D systems wherea direct comparison with theory is feasible. Concern-ing 2D systems, where numerical simulations of real-timedynamics face severe limitations, our results for two-legladders provide confidence that the domain-wall melt-ing is still a reliable detector of localization as well, asevidenced in the experiment of [25]. Even for clean sys-tems, experimental studies of domain-wall melting in thepresence of interactions could provide valuable insightsinto the nonequilibrium transport properties of interact-ing quantum gases [28, 29, 34, 35, 37, 38]. For instance,even for the isotropic spin-1/2 chain ( V = 1 in our case),the qualitative nature of transport is still an open issue[60–67]. Moreover, the measurement of diffusion con-stants would be desirable [68]. Acknowledgments.
We thank I. Bloch, J. Choi,G. De Tomasi, and C. Gross for useful and stimulat-ing discussions. F.P. and F.H.-M. were supported bythe DFG (Deutsche Forschungsgemeinschaft) ResearchUnit FOR 1807 through Grants No. PO 1370/2-1 andNo. HE 5242/3-2. This research was supported in partby Perimeter Institute for Theoretical Physics. Researchat Perimeter Institute is supported by the Government ofCanada through Industry Canada and by the Province ofOntario through the Ministry of Economic Development& Innovation. ∗ E-mail: [email protected][1] D. Basko, I. Aleiner, and B. Altshuler, Ann. Phys. (NY) , 1126 (2006).[2] I. V. Gornyi, A. D. Mirlin, and D. G. Polyakov, Phys.Rev. Lett. , 206603 (2005).[3] B. Bauer and C. Nayak, J. Stat. Mech. , P09005(2013).[4] J. A. Kj¨all, J. H. Bardarson, and F. Pollmann, Phys.Rev. Lett. , 107204 (2014).[5] D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B , 081103 (2015).[6] J. H. Bardarson, F. Pollmann, and J. E. Moore, Phys.Rev. Lett. , 017202 (2012).[7] M. ˇZnidariˇc, T. Prosen, and P. Prelovˇsek, Phys. Rev. B , 064426 (2008).[8] M. Serbyn, Z. Papi´c, and D. A. Abanin, Phys. Rev. Lett. , 260601 (2013).[9] A. Pal and D. A. Huse, Phys. Rev. B , 174411 (2010).[10] E. Altman and R. Vosk, Annu. Rev. Condens. MatterPhys. , 383 (2015).[11] R. Nandkishore and D. Huse, Annu. Rev. Condens. Mat-ter Phys. , 15 (2015).[12] D. A. Huse, R. Nandkishore, and V. Oganesyan, Phys.Rev. B , 174202 (2014).[13] A. Chandran, I. H. Kim, G. Vidal, and D. A. Abanin,Phys. Rev. B , 085425 (2015).[14] S. Bera, H. Schomerus, F. Heidrich-Meisner, and J. H.Bardarson, Phys. Rev. Lett. , 046603 (2015).[15] S. Inglis and L. Pollet, Phys. Rev. Lett. , 120402(2016).[16] Y. B. Lev and D. R. Reichman, Europhys. Lett. ,46001 (2016).[17] R. Nandkishore, S. Gopalakrishnan, and D. A. Huse,Phys. Rev. B , 064203 (2014).[18] S. Johri, R. Nandkishore, and R. N. Bhatt, Phys. Rev.Lett. , 117401 (2015).[19] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨uschen,M. H. Fischer, R. Vosk, E. Altman, U. Schneider, andI. Bloch, Science , 842 (2015).[20] P. Bordia, H. P. L¨uschen, S. S. Hodgman, M. Schreiber,I. Bloch, and U. Schneider, Phys. Rev. Lett. , 140401(2016).[21] J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht,P. Lugan, D. Clement, L. Sanchez-Palencia, P. Bouyer,and A. Aspect, Nature (London) , 891 (2008).[22] G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort,M. Zaccanti, G. Modugno, M. Modugno, and M. Ingus-cio, Nature (London) , 895 (2008).[23] C. D’Errico, E. Lucioni, L. Tanzi, L. Gori, G. Roux, I. P.McCulloch, T. Giamarchi, M. Inguscio, and G. Mod-ugno, Phys. Rev. Lett. , 095301 (2014).[24] J. Smith, A. Lee, P. Richerme, B. Neyenhuis, P. W. Hess,P. Hauke, M. Heyl, D. A. Huse, and C. Monroe, Nat.Phys. , 907 (2016).[25] J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch,and C. Gross, Science , 1547 (2016).[26] W. S. Bakr, J. I. Gillen, A. Peng, S. Foelling, andM. Greiner, Nature (London) , 74 (2009).[27] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau,I. Bloch, and S. Kuhr, Nature (London) , 68 (2010). [28] T. Antal, Z. R´acz, A. R´akos, and G. M. Sch¨utz, Phys.Rev. E , 4912 (1999).[29] D. Gobert, C. Kollath, U. Schollw¨ock, and G. Sch¨utz,Phys. Rev. E , 036102 (2005).[30] R. Steinigeweg, J. Gemmer, and M. Michel, Europhys.Lett. , 406 (2006).[31] J. Lancaster and A. Mitra, Phys. Rev. E , 061134(2010).[32] V. Eisler and Z. R´acz, Phys. Rev. Lett. , 060602(2013).[33] L. F. Santos, Phys. Rev. E , 031125 (2008).[34] L. Vidmar, S. Langer, I. P. McCulloch, U. Schneider,U. Schollw¨ock, and F. Heidrich-Meisner, Phys. Rev. B , 235117 (2013).[35] J. Hauschild, F. Pollmann, and F. Heidrich-Meisner,Phys. Rev. A , 053629 (2015).[36] L. Vidmar, J. P. Ronzheimer, M. Schreiber, S. Braun,S. S. Hodgman, S. Langer, F. Heidrich-Meisner, I. Bloch,and U. Schneider, Phys. Rev. Lett. , 175301 (2015).[37] U. Schneider, L. Hackerm¨uller, J. P. Ronzheimer, S. Will,S. Braun, T. Best, I. Bloch, E. Demler, S. Mandt,D. Rasch, and A. Rosch, Nat. Phys. , 213 (2012).[38] J. P. Ronzheimer, M. Schreiber, S. Braun, S. S. Hodg-man, S. Langer, I. P. McCulloch, F. Heidrich-Meisner,I. Bloch, and U. Schneider, Phys. Rev. Lett. , 205301(2013).[39] L. Xia, L. A. Zundel, J. Carrasquilla, A. Reinhard, J. M.Wilson, M. Rigol, and D. S. Weiss, Nat. Phys. , 316(2015).[40] G. Roux, T. Barthel, I. P. McCulloch, C. Kollath,U. Schollw¨ock, and T. Giamarchi, Phys. Rev. A ,023628 (2008).[41] P. Ribeiro, M. Haque, and A. Lazarides, Phys. Rev. A , 043635 (2013).[42] S. R. White and A. E. Feiguin, Phys. Rev. Lett. ,076401 (2004).[43] G. Vidal, Phys. Rev. Lett. , 040502 (2004).[44] A. Daley, C. Kollath, U. Schollw¨ock, and G. Vidal, J.Stat. Mech. , P04005 (2004).[45] M. P. Zaletel, R. S. K. Mong, C. Karrasch, J. E. Moore,and F. Pollmann, Phys. Rev. B , 165112 (2015).[46] V. Oganesyan and D. A. Huse, Phys. Rev. B , 155111(2007).[47] Y. Bar Lev, G. Cohen, and D. R. Reichman, Phys. Rev.Lett. , 100601 (2015).[48] T. Devakul and R. R. P. Singh, Phys. Rev. Lett. ,187201 (2015).[49] E. Baygan, S. P. Lim, and D. N. Sheng, Phys. Rev. B , 195153 (2015).[50] B. Kramer and A. MacKinnon, Reports on Progress inPhysics , 1469 (1993).[51] F. Evers and A. D. Mirlin, Rev. Mod. Phys. , 1355(2008).[52] See Supplemental Material for a discussion of disorderstatistics and the dynamics of domain-wall melting inthe Aubry-Andr´e model.[53] S. Aubry and G. Andr´e, Ann. Israel Phys. Soc. , 18(1980).[54] A. Chandran, A. Pal, C. R. Laumann, and A. Scardic-chio, Phys. Rev. B , 144203 (2016).[55] W. De Roeck, F. Huveneers, M. M¨uller, and M. Schiulaz,Phys. Rev. B , 014203 (2016).[56] R. Vosk, D. A. Huse, and E. Altman, Phys. Rev. X ,031032 (2015). − − − − − − − − − log ( n ( tJ = 400)) b i n c oun t s arithmetic meangeometric meanmediandata FIG. S1. Distribution of n j on a logarithmic scale, exemplaryfor the free fermions at site j = 200, tJ = 400 and W = 0 . J .[57] A. C. Potter, R. Vasseur, and S. A. Parameswaran, Phys.Rev. X , 031033 (2015).[58] D. J. Luitz, Phys. Rev. B , 134201 (2016).[59] D. J. Luitz, N. Laflorencie, and F. Alet, Phys. Rev. B , 060201 (2016).[60] J. Herbrych, P. Prelovˇsek, and X. Zotos, Phys. Rev. B , 155125 (2011).[61] S. Grossjohann and W. Brenig, Phys. Rev. B , 012404(2010).[62] M. ˇZnidariˇc, Phys. Rev. Lett. , 220601 (2011).[63] C. Karrasch, J. Hauschild, S. Langer, and F. Heidrich-Meisner, Phys. Rev. B , 245128 (2013).[64] J. Sirker, R. G. Pereira, and I. Affleck, Phys. Rev. B ,035115 (2011).[65] B. Bertini, M. Collura, J. D. Nardis, and M. Fagotti,(2016), arXiv:1605.09790.[66] T. Prosen and E. Ilievski, Phys. Rev. Lett. , 057203(2013).[67] R. Steinigeweg, J. Gemmer, and W. Brenig, Phys. Rev.Lett. , 120601 (2014).[68] C. Karrasch, J. E. Moore, and F. Heidrich-Meisner,Phys. Rev. B , 075139 (2014). SUPPLEMENTAL MATERIAL
Disorder Statistics
An exemplary distribution of n j for the free fermioncase is shown in Fig. S1. In a rough approximation, theprobability for a particle to hop the j sites out of thedomain wall can be seen as a product of the hoppingprobabilities to neighboring sites, which depend on thespecific disorder realization. The geometric mean ¯ n j isthus a natural choice for the average over different dis-order realizations. As evident from Fig. S1, it coincideswith the median and represents the typical value. In contrast, the arithmetic mean is an order of magnitudelarger as it puts a large weight in the upper tail of thedistribution.Although the geometric mean is a good choice for n j , itis reasonable to use the arithmetic mean for other quan-tities such as VAR n and ∆ N : they represent quantitiesintegrated over j for a given disorder realization. Wechecked that the arithmetic mean is close to typical val-ues for these quantities. Aubry-Andr´e model
We now focus on the dynamics in the Aubry-Andr´emodel, where a quasiperiodic modulation is introducedin Eq. (1) via (cid:15) j = W cos(2 πrj + φ ) (employed in theMBL experiments of [19, 20]). We set the irrational ratio r to r = ( √ − / . . . . and perform the equiv-alent to disorder averages by sampling over the valueof the phase φ ∈ [0 , π ). This noninteracting modelhas a delocalization-localization transition at W c /J = 1,where the single-particle localization length diverges as ξ (1)loc = log (cid:0) WJ (cid:1) [53]. Similar to the previously consid-ered Anderson model, the density profiles become sta-tionary with an exponential tail in the localized phasefor W > W c . As W is varied, a clear transition is visiblein the time dependence of both VAR n and ∆ N shownin Figs. S2(a) and S2(b), respectively, which become sta-tionary for W > W c , while growing with a power law for W < W c . The corresponding domain-wall decay length ξ dw (see the inset of Fig. 2(b)) diverges as W c is ap-proached from above, in excellent agreement with thesingle-particle localization length of that model [53]. Themaximum value of ξ dw in the extended phase reached atlong times diverges with L . tJ VA R n (a) W/J = 0 . W/J = 1 . W/J = 1 . tJ ∆ N (b) FIG. S2. ED results ( L = 2000) for the Aubry-Andr´emodel with a localization transition at W c = J (indi-cated by the thick lines). (a) Variance VAR n for W/J =0 . , . , . , . . . , .
25 (top to bottom). (b) Number of emit-ted particles ∆ N ( tt