Dynamical Scaling of Surface Roughness and Entanglement Entropy in Disordered Fermion Models
AAnomalous Dynamical Scaling of Roughness in Disordered Fermion Models
Kazuya Fujimoto,
1, 2
Ryusuke Hamazaki, and Yuki Kawaguchi Institute for Advanced Research, Nagoya University, Nagoya 464-8601, Japan Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan Nonequilibrium Quantum Statistical Mechanics RIKEN Hakubi Research Team,RIKEN Cluster for Pioneering Research (CPR), RIKEN iTHEMS, Wako, Saitama 351-0198, Japan (Dated: January 21, 2021)Localization is one of the most fundamental interference phenomena caused by randomness, andits universal aspects have been extensively explored from the perspective of one-parameter scalingmainly in equilibrium states. We theoretically study dynamics of fermions on disordered one-dimensional potentials exhibiting localization, and find that surface roughness and entanglemententropy show dynamical one-parameter-scaling in delocalized phases. The scaling of the roughnesscorresponds to the Family-Vicsek scaling in classical surface growth, and the associated universalscaling exponents depend on the type of disorder. Particularly, we find that partially localized statesin the delocalized phase of the random-dimer model lead to anomalous scaling exponents, which areabsent in classical systems and clean systems. Furthermore, we show that the surface roughness isapproximately proportional to the square root of the von Neumann entanglement entropy, and thendemonstrate that even the entanglement entropy obeys the Family-Vicsek-type scaling. This findingsuggests that the surface roughness becomes a reliable measure for the entanglement dynamics.
Introduction.
Since the pioneering work by P. W. An-derson [1], localization phenomena have been a long-standing topic drawing a lot of attention. The strik-ing role of disorder and its consequence of destructiveinterference have been uncovered in various fields, suchas solid-state physics, quantum optics, and classical me-chanics [2–5]. Indeed, the Anderson localization is ob-served in electronic systems [6, 7], ultracold quantumgases [8–15], light in photonic crystals [16–20], and ul-trasound in elastic materials [21, 22].Study of the Anderson localization has significantlybeen put forward in light of one-parameter scaling [3,23, 24], where physical quantities are scaled only by asingle parameter, such as a system size and a correlationlength. The example includes scaling for system-size de-pendence of conductance and for correlation functionsat localization transition points. Despite its importance,such universal one-parameter scaling has been focusedon equilibrium situations. According to previous liter-ature on Anderson and many-body localization [25–30],disorder is known to affect dynamics of quantum sys-tems profoundly, and entanglement dynamics [31–39] andtransport properties [40–46] are intensively investigatedin disordered models. To one’s surprise, however, dy-namical one-parameter scaling has little been discussedin quantum disordered systems. It is thus intriguing andfundamental to discuss whether disorder leads to uniqueuniversal one-parameter scaling even out of equilibriumfor isolated quantum systems.In this Letter, we theoretically demonstrate thatanomalous dynamical one-parameter scaling indeed ex-ists in fluctuation-growing dynamics of one-dimensional(1D) non-interacting fermions in a disordered poten-tial. We use the random model (RM), the random-dimermodel (RDM) [47], and the Aubry-Andr´e model (AAM) (a) (b) s u rf ace r oughn e ss time AAM RMRDM W delocalizedlocalized delocalizedlocalized localizedlocalized localized ∝ t β ∝ M α t sat ∝ M z Model FV scaling α β z
RDM ( W < 1)AAM ( W < 1) − − − 𝒪( M )𝒪( M )RMRDM ( W > 1)AAM ( W > 1) Decreasewith M S EE − t β t β (c) 𝒪( M )𝒪( M ) 𝒪(1) existexistnone
FIG. 1. (a) Phase diagram for the random model (RM), therandom-dimer model (RDM), and the Aubry-Andr´e model(AAM) as a function of the disorder strength W . Delocalizedphases appear for W < α , β , and z , whichrespectively capture system-size M dependence, power-lawgrowth, and a saturation time t sat of the surface roughness.This dynamical scaling is called the Family-Vicsek (FV) scal-ing (see Eq. (4)). The FV scaling does not emerge in thelocalized phase. (c) Summary of our results, including num-bers of delocalized eigenstates (DLESs) and localized eigen-states (LESs) and growth laws of von Neumann entanglemententropy S EE . [48], whose delocalized or localized phases are schemati-cally shown in Fig. 1(a) as a function of disorder strength W . Our numerical calculations elucidate that, in the de-localized phase, “quantum surface roughness” introducedin Refs. [49–51] obeys the Family-Vicsek (FV) scaling[52, 53], which is dynamical one-parameter scaling knownfor classical surface growth [54]. This scaling is charac- a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n terized by three universal exponents α , β , and z as shownin Fig. 1(b). We find the emergence of anomalous expo-nents ( α, β, γ ) (cid:39) (0 . , . , .
00) in the RDM, whichhave not been known for classical surface growth. Weargue that localized eigenstates (LESs) embedded in thedelocalized phase essentially alter the surface-roughnessdynamics and thus lead to such anomalous exponents be-cause the available delocalized eigenstates (DLESs) arelimited. Furthermore, we find that the surface roughnessis approximately proportional to the square root of thevon Neumann entanglement entropy (EE), and demon-strate the EE also exhibits FV-type scaling. Our findingsuggests that the surface roughness is a reliable measurefor EE. Table in Fig. 1(c) summarizes our results.
Theoretical models.
We consider non-interacting N -fermions on a 1D lattice with a disordered potential. Letus denote the annihilation and creation operators on asite j by ˆ f j and ˆ f † j ( j = 1 , · · · , M ), where M is thenumber of the lattice sites. Throughout this work, M isset to be even. Then, the Hamiltonian is given byˆ H = − J M (cid:88) j =1 (cid:16) ˆ f † j +1 ˆ f j + ˆ f † j ˆ f j +1 (cid:17) + M (cid:88) j =1 V j ˆ f † j ˆ f j (1)with a hopping constant J > V j .We use three potentials corresponding to the RM, theRDM, and the AAM. The RM consists of a random po-tential with no spatial correlation, where V j takes 0 or V ( >
0) following the probability function P RM ( V j ) = δ ( V j ) + δ ( V j − V ). The potential in the RDM [47]has a spatial correlation such that the probability func-tion is given by P RDM ( V j − , V j ) = δ ( V j ) δ ( V j − ) + δ ( V j − V ) δ ( V j − − V ) with j = 1 , , · · · , M/ V j = V cos(2 πθj ) with the irrational number θ = ( √ − / W = V / (2 J ). The mod-els have localized or delocalized phases depending on W [47, 56] as shown in Fig. 1(a). We assume the periodicboundary condition for the RM and the RDM, and theopen boundary condition for the AAM. In the RDM andthe RM, we take ensemble averages to calculate physicalquantities, and the sample number in all the calculationsis (cid:98) /M (cid:99) with the floor function (cid:98)· · · (cid:99) . Surface-height operator and the roughness.
To exploredynamical one-parameter scaling in the three disorderedmodels, we consider the surface roughness introduced inRefs. [49–51]. Extending the analogy between the clas-sical surface growth and the fluctuating hydrodynamics[58–61] to quantum systems, the quantum many-bodysurface-height operator is defined asˆ h j = j (cid:88) i =1 (cid:16) ˆ f † i ˆ f i − ν (cid:17) (2) with a filling factor ν = N/M . Then, the averagedsurface-height is given by h av ( t ) = M (cid:80) Mj =1 Tr[ˆ ρ ( t )ˆ h j ].We assume that the density matrix ˆ ρ ( t ) is averagedover many realizations of the random potentials for theRM and the RDM, respectively. The surface roughness w ( M, t ) is evaluated by the standard deviations of ˆ h j de-fined by w ( M, t ) = (cid:118)(cid:117)(cid:117)(cid:116) M M (cid:88) j =1 Tr[ˆ ρ ( t )(ˆ h j − h av ( t )) ] . (3)Our previous work [50] has found that the surfaceroughness in isolated quantum systems free from disorderexhibits the following FV scaling: w ( M, t ) = s − α w ( sM, s z t ) ∝ (cid:40) t β ( t (cid:28) t sat ); M α ( t sat (cid:28) t ) (4)with a parameter s and a saturation time t sat . Taking s = 1 /M , we obtain w ( M, t ) = M α f ( t/M z ) with a scal-ing function f ( x ) = w (1 , x ). This means that the sur-face roughness with different M is collapsed to a singlecurve after normalization of the ordinate and the ab-scissa by M α and M z . This dynamical one-parameterscaling is originally discussed in classical systems, andthe exponents α , β , and z classify universality of thesurface-roughness dynamics [54]. The famous classes arethe Edwards-Wilkinson class [62] and the Kardar-Parisi-Zhang class [63], for which the scaling exponents are( α, β, z ) = (1 / , / ,
2) and (1 / , / , / Surface-roughness dynamics.-
We numerically investi-gate the surface roughness to explore the universal FVscaling in the disordered models. Our numerical methodis based on Gaussian states [65] (see also Sec. I of Sup-plemental material (SM) [64]). The initial state is a stag-gered state | ψ (0) (cid:105) = (cid:81) Nj =1 ˆ f † j | (cid:105) with the total particlenumber N = M/
2. This initial state has small surfaceroughness, and thus is suitable to investigate the univer-sal aspect of the surface-roughness growth.Figures 2(a)-(c) show the time evolution of the sur-face roughness. In the delocalized phase ( W = 0 . α, β, z ) in the RDM and the AAM are(0 . , . , .
01) and (0 . , . , . M and does not exhibitclear power-law growth as shown in Figs. 2(c) for all the FIG. 2. Surface-roughness dynamics and FV scaling for delocalized phases of (a) RDM ( W = 0 .
5) and (b) AAM ( W = 0 . W = 1 . τ = (cid:126) /J . In (a) and (b), the main panelsshow w ( M, t ) with M = 200 , , , M/ α and ( M/ z ,and the insets show the corresponding raw data. The estimated power exponents in (a) and (b) are ( α, β, z ) = (0 . , . , . . , . , . α , β , and z on W , respectively, for the AAM and the RDM in the delocalized phases( W < models, indicating the absence of the FV scaling.We systematically investigate disorder dependence ofthe exponents ( α, β, z ) by changing W in the delocalizedphases. As shown in Figs. 2(d)-(f), we find that the expo-nents in the RDM and the AAM are almost independentof W . Thus, we conclude that the RDM and the AAMin the delocalized phase show the FV scaling with theuniversal exponents ( α, β, z ) (cid:39) (0 . , . , .
00) and(0 . , . , . W .The exponents in the AAM are close to ( α, β, z ) (cid:39) (0 . , . , .
00) for the non-interacting fermion modelwithout disorder [50]. This coincidence can be under-stood by considering the numbers of the DLESs and theLESs for the single-particle eigenstate of ˆ H . Accordingto Sec. III of SM [64], the numbers of the DLESs andthe LESs in the AAM with W < M and O ( M ), respectively. Thus, we conjecture that theeffect of the remaining LESs is too weak and that theexponents are almost the same as the ones for fermionsystems without disorder.The situation drastically changes in the RDM with theanomalous exponent α (cid:39) . W < √ M and M , respectively. In stark contrast to theAAM, the RDM supports many LESs even in the delo-calized phase, and they can strongly affect the surface-roughness dynamics. Indeed, just from the information about the eigenstates and the initial state, we can numer-ically reproduce the exponent α (cid:39) .
33 and 0 . w ave ( M ) using the approximated diagonal ensemble[66–69] (see Sec. IV of SM [64]). Since we use the sameinitial state for the RDM and the AAM, the result inFig. 3 implies that the difference in α originates fromthe statistical property of the eigenstates. Furthermore,we can analytically derive the non-anomalous exponent α = 0 . M w a v e ( M ) ∝ M . ∝ M . AAM ( W = 0 . W = 0 . M . M . FIG. 3. Saturated surface roughness w ave ( M ) obtained bythe approximated diagonal ensemble for the AAM and theRDM with W = 0 . FIG. 4. FV scaling for von Neumann EE. The upper andlower main panels show S EE ( M, t ) for the RDM and the AAM,respectively, with W = 0 . M = 200, 300, 500, and 800,where the time is normalized by τ = (cid:126) /J , and the ordinateand the abscissa are normalized by ( M/ α and ( M/ z with the exponents α and z obtained in Fig. 2. The insetsshow time evolutions of (cid:112) S EE ( M, t ) / w ( M, t ) for M =800, which confirm the success of Eq. (6) in the early stagesof the dynamics. ber of the DLESs and a large number of LESs. Entanglement entropy and surface roughness.
We findthat the surface roughness is related to von NeumannEE through a nontrivial relation. The EE quantifiesquantum entanglement in a pure state by dividing thesystem into two subsystems. Here, we divide the M -site system into subsystems A = { j | ≤ j ≤ M/ } and B = { j | M/ < j ≤ M } , and define the reduceddensity matrix ˆ ρ re ( t ) = tr B [ˆ ρ pure ( t )], where ˆ ρ pure ( t ) is adensity matrix for a single realization of the disorderedmodels. Then, the EE is calculated by S EE ( M, t ) = − Tr A [ˆ ρ re ( t ) log ˆ ρ re ( t )], where the overline denotes the en-semble average in the RDM.To derive the relation between S EE ( M, t ) and w ( M, t ), we assume (i) h av ( t ) (cid:39)
0, (ii) w ( M, t ) (cid:39) Tr (cid:104) ˆ ρ ( t )(ˆ h M/ − h av ( t )) (cid:105) , and (iii) (cid:80) M/ j =1 Tr [ˆ ρ ( t )ˆ n j ] (cid:39) νM/
2. The validity of these assumptions is numericallyconfirmed in Sec. IV of SM [64]. The assumptions (i)and (ii) lead to w ( M, t ) (cid:39) Tr ˆ ρ ( t ) M/ (cid:88) j =1 ˆ f † j ˆ f j − M ν . (5) Equation (5) means that w ( M, t ) can be approximatedby the particle-number fluctuation in the half of the sys-tem from the averaged number νM/
2, and one can seethat both w ( M, t ) and S EE ( M, t ) are defined in the samebipartite system. We then find the following relation: S EE ( M, t ) (cid:39) w ( M, t ) , (6)where we use (iii) and the additional assump-tion that eigenvalues of the correlation matrixTr[ˆ ρ pure ( t ) ˆ f † i ˆ f j ] ( i, j ∈ A ) are uniformly distributedbetween zero and unity. The detailed derivation ofEq. (6) is shown in Sec. V of SM [64].Substituting Eq. (6) into Eq. (4), we obtain the FV-type scaling in the delocalized phases: S EE ( M, t ) = s − α S EE ( sM, s z t ) ∝ (cid:40) t β ( t (cid:28) t sat ); M α ( t sat (cid:28) t ) . (7)Figure 4 shows time evolutions of S EE ( M, t ) in the RDMand the AAM with W = 0 .
5. Our numerical resultsclearly reveal that the EE well obeys the FV-type scaling(7). The insets of Fig. 4 compare the both sides of Eq. (6),showing that the relation works quite well especially inthe early stages of the dynamics. Although they deviatefrom each other in the late stages, the FV-type scalingstill hold with the expected exponent (2 α, β, z ).This finding suggests that the surface roughness maybecome a possible measure for entanglement and its uni-versal scaling. Furthermore, we rigorously prove in Sec.V of SM [64] that if the bipartite number fluctuationTr[ˆ ρ ( t )ˆ h M/ ] [70–72] with the assumption (iii) exhibitspower-law growth t β , S EE ( M, t ) also grows as t β in thethermodynamic limit (and vice versa). Finally, we com-ment the entanglement dynamics studied in view of thesurface roughness [73, 74]. Nahum et al. show that theEE itself behaves as effective surface height. Their find-ing is different from our result of Eq. (6). Moreover, westress that the clear FV scalings have not been observedin Refs. [73, 74]. Discussion.
We first discuss that the RDM exhibitsanomalous behavior in single-particle transport proper-ties as well as the FV scaling with the anomalous expo-nents. As already mentioned, in the RDM with
W < √ M . Ow-ing to this √ M dependence, the standard deviation ∆ x of the position of a particle initially localized at a cer-tain site grows as t / [47], differently from t growthin a disorder-free system. Similarly, the saturated valueof ∆ x at late time is found to be proportional to M / rather than M (see Sec. VI of SM [64] for the detailedcalculation). It is an interesting future problem to in-vestigate whether these single-particle anomalous powerexponents are related to α = 0 .
352 and β = 0 .
337 in theFV scaling with many particles.Next, we comment on experimental possibility. TheAnderson localization in 1D systems has been observedusing a quasi-periodic potential [8] and a random specklepotentials [9–14] in cold atoms. The former case corre-sponds to the AAM, and our prediction can be accessibleby using a microscope. On the other hand, the RDM canin principle be realized using digital micrometer devices,by which various kinds of potentials including a randomone have already been made in a highly controllable man-ner [75–77].
Conclusion.
We have theoretically studied out-of-equilibrium phenomena in the three disordered fermionmodels, namely the RM, the RDM, and the AAM fromthe perspective of dynamical one-parameter scaling. Inthe delocalized phases, we have found the universal FVscaling, which is one of the well-known dynamical one-parameter scalings, in the surface roughness of the RDMand the AAM. We have pointed out that the LESs give asignificant impact on the universal aspects of the FV scal-ing and lead to the anomalous exponents for the RDM.On the other hand, in the localized phases, our numericalsimulations have found no FV scaling in all the models.Furthermore, exploring the relation between the EE andthe surface roughness, we have discovered the nontriv-ial relation (6), and have elucidated that the FV-typescaling emerges even in the entanglement dynamics.Our study opens an unexplored avenue for pursuingunexpected relation between Anderson localization andsurface growth physics through the FV scaling and theEE. From this viewpoint, it is interesting to investigateuniversality class of the FV scaling in many-body local-ization [27–30, 35–37, 45] and the Anderson localizationwith long-range interactions. Another interesting direc-tion is to explore effects of Anderson localization on otherknown results in classical surface growth, such as a di-rected percolation transition [54, 78–80].This work was supported by JST-CREST (GrantNo. JPMJCR16F2), JSPS KAKENHI (GrantNos. JP18K03538, JP19H01824, JP19K14628, and20H01843), and the Program for Fostering Researchersfor the Next Generation (IAR, Nagoya University) andBuilding of Consortia for the Development of HumanResources in Science and Technology (MEXT). [1] P. W. Anderson, Absence of diffusion in certain randomlattices, Phys. Rev. , 1492 (1958).[2] B. Kramer and A. MacKinnon, Localization: theory andexperiment, Reports on Progress in Physics , 1469(1993).[3] F. Evers and A. D. Mirlin, Anderson transitions, Rev.Mod. Phys. , 1355 (2008).[4] F. Izrailev, A. Krokhin, and N. Makarov, Anomalous lo-calization in low-dimensional systems with correlated dis-order, Physics Reports , 125 (2012).[5] D. Thouless, Electrons in disordered systems and the the- ory of localization, Physics Reports , 93 (1974).[6] M. Cutler and N. F. Mott, Observation of anderson local-ization in an electron gas, Phys. Rev. , 1336 (1969).[7] M. Evaldsson, I. V. Zozoulenko, H. Xu, and T. Heinzel,Edge-disorder-induced anderson localization and conduc-tion gap in graphene nanoribbons, Phys. Rev. B ,161407 (2008).[8] G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort,M. Zaccanti, G. Modugno, M. Modugno, and M. In-guscio, Anderson localization of a non-interacting bose–einstein condensate, Nature , 895 (2008).[9] J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht,P. Lugan, D. Cl´ement, L. Sanchez-Palencia, P. Bouyer,and A. Aspect, Direct observation of anderson localiza-tion of matter waves in a controlled disorder, Nature ,891 (2008).[10] S. S. Kondov, W. R. McGehee, J. J. Zirbel, and B. De-Marco, Three-dimensional anderson localization of ultra-cold matter, Science , 66 (2011).[11] F. Jendrzejewski, A. Bernard, K. M¨uller, P. Cheinet,V. Josse, M. Piraud, L. Pezz´e, L. Sanchez-Palencia,A. Aspect, and P. Bouyer, Three-dimensional localiza-tion of ultracold atoms in an optical disordered potential,Nature Physics , 398 (2012).[12] F. Jendrzejewski, K. M¨uller, J. Richard, A. Date, T. Plis-son, P. Bouyer, A. Aspect, and V. Josse, Coherentbackscattering of ultracold atoms, Phys. Rev. Lett. ,195302 (2012).[13] G. Semeghini, M. Landini, P. Castilho, S. Roy, G. Spag-nolli, A. Trenkwalder, M. Fattori, M. Inguscio, andG. Modugno, Measurement of the mobility edge for 3danderson localization, Nature Physics , 554 (2015).[14] D. H. White, T. A. Haase, D. J. Brown, M. D. Hooger-land, M. S. Najafabadi, J. L. Helm, C. Gies, D. Schu-mayer, and D. A. W. Hutchinson, Observation of two-dimensional anderson localisation of ultracold atoms, Na-ture Communications , 4942 (2020).[15] G. Modugno, Anderson localization in bose–einstein con-densates, Reports on Progress in Physics , 102401(2010).[16] D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righ-ini, Localization of light in a disordered medium, Nature , 671 (1997).[17] M. St¨orzer, P. Gross, C. M. Aegerter, and G. Maret, Ob-servation of the critical regime near anderson localizationof light, Phys. Rev. Lett. , 063904 (2006).[18] T. Schwartz, G. Bartal, S. Fishman, and M. Segev,Transport and anderson localization in disordered two-dimensional photonic lattices, Nature , 52 (2007).[19] Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti,D. N. Christodoulides, and Y. Silberberg, Anderson lo-calization and nonlinearity in one-dimensional disorderedphotonic lattices, Phys. Rev. Lett. , 013906 (2008).[20] M. Segev, Y. Silberberg, and D. N. Christodoulides, An-derson localization of light, Nature Photonics , 197(2013).[21] R. Weaver, Anderson localization of ultrasound, WaveMotion , 129 (1990).[22] H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov,and B. A. van Tiggelen, Localization of ultrasound in athree-dimensional elastic network, Nature Physics , 945(2008).[23] E. Abrahams, P. W. Anderson, D. C. Licciardello, andT. V. Ramakrishnan, Scaling theory of localization: Ab- sence of quantum diffusion in two dimensions, Phys. Rev.Lett. , 673 (1979).[24] P. A. Lee and T. V. Ramakrishnan, Disordered electronicsystems, Rev. Mod. Phys. , 287 (1985).[25] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalat-tore, Colloquium: Nonequilibrium dynamics of closed in-teracting quantum systems, Rev. Mod. Phys. , 863(2011).[26] J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many-body systems out of equilibrium, Nature Physics , 124EP (2015).[27] M. ˇZnidariˇc, T. c. v. Prosen, and P. Prelovˇsek, Many-body localization in the heisenberg xxz magnet in a ran-dom field, Phys. Rev. B , 064426 (2008).[28] A. Pal and D. A. Huse, Many-body localization phasetransition, Phys. Rev. B , 174411 (2010).[29] J. A. Kj¨all, J. H. Bardarson, and F. Pollmann, Many-body localization in a disordered quantum ising chain,Phys. Rev. Lett. , 107204 (2014).[30] R. Nandkishore and D. A. Huse, Many-body localizationand thermalization in quantum statistical mechanics, An-nual Review of Condensed Matter Physics , 15 (2015).[31] Y. Zhao, F. Andraschko, and J. Sirker, Entanglemententropy of disordered quantum chains following a globalquench, Phys. Rev. B , 205146 (2016).[32] Y. Zhao and J. Sirker, Logarithmic entanglement growthin two-dimensional disordered fermionic systems, Phys.Rev. B , 014203 (2019).[33] Y. Zhao, D. Feng, Y. Hu, S. Guo, and J. Sirker, Entangle-ment dynamics in the three-dimensional anderson model,Phys. Rev. B , 195132 (2020).[34] M. Kiefer-Emmanouilidis, R. Unanyan, J. Sirker, andM. Fleischhauer, Bounds on the entanglement entropyby the number entropy in non-interacting fermionic sys-tems, SciPost Phys. , 83 (2020).[35] J. H. Bardarson, F. Pollmann, and J. E. Moore, Un-bounded growth of entanglement in models of many-bodylocalization, Phys. Rev. Lett. , 017202 (2012).[36] S. Iyer, V. Oganesyan, G. Refael, and D. A. Huse, Many-body localization in a quasiperiodic system, Phys. Rev.B , 134202 (2013).[37] M. Serbyn, Z. Papi´c, and D. A. Abanin, Universal slowgrowth of entanglement in interacting strongly disorderedsystems, Phys. Rev. Lett. , 260601 (2013).[38] A. Nahum, J. Ruhman, and D. A. Huse, Dynamics ofentanglement and transport in one-dimensional systemswith quenched randomness, Phys. Rev. B , 035118(2018).[39] M. J. Gullans and D. A. Huse, Localization as an en-tanglement phase transition in boundary-driven ander-son models, Phys. Rev. Lett. , 110601 (2019).[40] G. S. Ng and T. Kottos, Wavepacket dynamics of thenonlinear harper model, Phys. Rev. B , 205120 (2007).[41] A. S. Pikovsky and D. L. Shepelyansky, Destruction ofanderson localization by a weak nonlinearity, Phys. Rev.Lett. , 094101 (2008).[42] S. E. Skipetrov, A. Minguzzi, B. A. van Tiggelen, andB. Shapiro, Anderson localization of a bose-einstein con-densate in a 3d random potential, Phys. Rev. Lett. ,165301 (2008).[43] T. Devakul and D. A. Huse, Anderson localization tran-sitions with and without random potentials, Phys. Rev.B , 214201 (2017).[44] M. J. Gullans and D. A. Huse, Entanglement structure of current-driven diffusive fermion systems, Phys. Rev. X , 021007 (2019).[45] M. Schreiber, S. S. Hodgman, P. Bordia, H. P. L¨uschen,M. H. Fischer, R. Vosk, E. Altman, U. Schneider, andI. Bloch, Observation of many-body localization of inter-acting fermions in a quasirandom optical lattice, Science , 842 (2015).[46] M. ˇZnidariˇc, A. Scardicchio, and V. K. Varma, Diffusiveand subdiffusive spin transport in the ergodic phase ofa many-body localizable system, Phys. Rev. Lett. ,040601 (2016).[47] D. H. Dunlap, H.-L. Wu, and P. W. Phillips, Absence oflocalization in a random-dimer model, Phys. Rev. Lett. , 88 (1990).[48] S. Aubry and G. Andr´e, Analyticity breaking and ander-son localization in incommensurate lattices, Ann. IsraelPhys. Soc , 18 (1980).[49] M. den Nijs and K. Rommelse, Preroughening transitionsin crystal surfaces and valence-bond phases in quantumspin chains, Phys. Rev. B , 4709 (1989).[50] K. Fujimoto, R. Hamazaki, and Y. Kawaguchi, Family-vicsek scaling of roughness growth in a strongly interact-ing bose gas, Phys. Rev. Lett. , 210604 (2020).[51] T. Jin, A. Krajenbrink, and D. Bernard, From stochas-tic spin chains to quantum kardar-parisi-zhang dynamics,Phys. Rev. Lett. , 040603 (2020).[52] T. Vicsek and F. Family, Dynamic scaling for aggregationof clusters, Phys. Rev. Lett. , 1669 (1984).[53] F. Family and T. Vicsek, Scaling of the active zone in theeden process on percolation networks and the ballisticdeposition model, Journal of Physics A: Mathematicaland General , L75 (1985).[54] A.-L. Barab´asi and H. E. Stanley, Fractal concepts insurface growth (Cambridge university press, 1995).[55] J. C. Flores and M. Hilke, Absence of localization in dis-ordered systems with local correlation, Journal of PhysicsA: Mathematical and General , L1255 (1993).[56] J. Biddle, B. Wang, D. J. Priour, and S. Das Sarma,Localization in one-dimensional incommensurate latticesbeyond the aubry-andr´e model, Phys. Rev. A , 021603(2009).[57] S. Ganeshan, K. Sun, and S. Das Sarma, Topologi-cal zero-energy modes in gapless commensurate aubry-andr´e-harper models, Phys. Rev. Lett. , 180403(2013).[58] H. Spohn, Nonlinear fluctuating hydrodynamics for an-harmonic chains, Journal of Statistical Physics , 1191(2014).[59] H. Spohn, Fluctuating hydrodynamics approach to equi-librium time correlations for anharmonic chains, in Ther-mal Transport in Low Dimensions: From StatisticalPhysics to Nanoscale Heat Transfer , edited by S. Lepri(Springer International Publishing, Cham, 2016) pp.107–158.[60] M. Kulkarni, D. A. Huse, and H. Spohn, Fluctuatinghydrodynamics for a discrete gross-pitaevskii equation:Mapping onto the kardar-parisi-zhang universality class,Phys. Rev. A , 043612 (2015).[61] C. B. Mendl and H. Spohn, Low temperature dynam-ics of the one-dimensional discrete nonlinear schr¨odingerequation, Journal of Statistical Mechanics: Theory andExperiment , P08028 (2015).[62] S. F. Edwards and D. R. Wilkinson, The surface statisticsof a granular aggregate, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences , 17 (1982).[63] M. Kardar, G. Parisi, and Y.-C. Zhang, Dynamic scalingof growing interfaces, Phys. Rev. Lett. , 889 (1986).[64] See Supplemental Material for (I) Numerical method, (II)Numerical data for the surface-roughness dynamics, (III)Numbers of delocalized and localized eigenstates in thedisordered fermion models, (IV) Family-Vicsek-scalingexponent α in the approximated diagonal ensemble, (V)Relation between the von Neumann entanglement en-tropy and the surface roughness, and (VI) Anomalousbehavior of a single particle transport in the RDM. Thisincludes Ref. [81].[65] X. Cao, A. Tilloy, and A. D. Luca, Entanglement ina fermion chain under continuous monitoring, SciPostPhys. , 24 (2019).[66] M. Rigol, V. Dunjko, and M. Olshanii, Thermalizationand its mechanism for generic isolated quantum systems,Nature , 854 (2008).[67] M. Kollar and M. Eckstein, Relaxation of a one-dimensional mott insulator after an interaction quench,Phys. Rev. A , 013626 (2008).[68] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol,From quantum chaos and eigenstate thermalization tostatistical mechanics and thermodynamics, Advances inPhysics , 239 (2016).[69] T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda,Thermalization and prethermalization in isolated quan-tum systems: a theoretical overview, Journal of PhysicsB: Atomic, Molecular and Optical Physics , 112001(2018).[70] H. F. Song, S. Rachel, C. Flindt, I. Klich, N. Laflorencie,and K. Le Hur, Bipartite fluctuations as a probe of many-body entanglement, Phys. Rev. B , 035409 (2012).[71] D. J. Luitz, N. Laflorencie, and F. Alet, Many-body local-ization edge in the random-field heisenberg chain, Phys. Rev. B , 081103 (2015).[72] R. Singh, J. H. Bardarson, and F. Pollmann, Signaturesof the many-body localization transition in the dynamicsof entanglement and bipartite fluctuations, New Journalof Physics , 023046 (2016).[73] A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantumentanglement growth under random unitary dynamics,Phys. Rev. X , 031016 (2017).[74] T. Zhou and A. Nahum, Emergent statistical mechanicsof entanglement in random unitary circuits, Phys. Rev.B , 174205 (2019).[75] J.-y. Choi, S. Hild, J. Zeiher, P. Schauß, A. Rubio-Abadal, T. Yefsah, V. Khemani, D. A. Huse, I. Bloch,and C. Gross, Exploring the many-body localizationtransition in two dimensions, Science , 1547 (2016).[76] M. Aidelsburger, J. L. Ville, R. Saint-Jalm,S. Nascimb`ene, J. Dalibard, and J. Beugnon, Re-laxation dynamics in the merging of n independentcondensates, Phys. Rev. Lett. , 190403 (2017).[77] R. Saint-Jalm, P. C. M. Castilho, E. Le Cerf, B. Bakkali-Hassani, J.-L. Ville, S. Nascimbene, J. Beugnon, andJ. Dalibard, Dynamical symmetry and breathers in a two-dimensional bose gas, Phys. Rev. X , 021035 (2019).[78] H. H. L. S. Henkel, Malte, Non-Equilibrium Phase Tran-sitions (Springer Netherlands, 2008).[79] S. V. Buldyrev, A.-L. Barab´asi, F. Caserta, S. Havlin,H. E. Stanley, and T. Vicsek, Anomalous interface rough-ening in porous media: Experiment and model, Phys.Rev. A , R8313 (1992).[80] V. Ahlers and A. Pikovsky, Critical properties of the syn-chronization transition in space-time chaos, Phys. Rev.Lett. , 254101 (2002).[81] A. Pi˜neiro Orioli, K. Boguslavski, and J. Berges, Univer-sal self-similar dynamics of relativistic and nonrelativisticfield theories near nonthermal fixed points, Phys. Rev. D , 025041 (2015). SUPPLEMENTAL MATERIAL FOR “ANOMALOUS DYNAMICAL SCALING OF ROUGHNESS INDISORDERED FERMION MODELS”
This supplemental material describes the following topics:(I) Numerical method,(II) Numerical data for the surface-roughness dynamics,(III) Numbers of delocalized and localized eigenstates in the disordered fermion models,(IV) Family-Vicsek-scaling exponent α in the approximated diagonal ensemble,(V) Relation between the von Neumann entanglement entropy and the surface roughness,(VI) Anomalous behavior of a single particle transport in the RDM. NUMERICAL METHOD
To solve the Schr¨odinger equation in non-interacting fermion models, we use the numerical method described inRef. [65]. In this section, we review the details of how to implement it.The system considered here is N -fermions on a one-dimensional lattice. We denote by ˆ f j and ˆ f † j ( j = 1 , · · · , M ) thefermionic annihilation and creation operators at a site j with the number M of the lattice sites, respectively. Then,all the Hamiltonians used in this work are given by the following quadratic form:ˆ H = M (cid:88) i,j =1 ˆ f † i H ij ˆ f j (S-1)with an M × M hermitian matrix H ij . Numerically solving the Schr¨odinger equation with Eq. (S-1), we utilize theGaussian state defined by N (cid:89) α =1 M (cid:88) j =1 ˆ f † j G jα | (cid:105) , (S-2)where G jα is a M × N matrix and | (cid:105) is a vacuum state satisfying ˆ f j | (cid:105) = 0 for ∀ j . Here, the matrix is assumed tosatisfy (cid:80) j G ∗ jα G jβ = δ αβ , which ensures that the operator (cid:80) Mj =1 ˆ f † j G jα obeys the fermionic anticommutator relation.The unitary time evolution with Eq. (S-1) can keep this Gaussian-state structure in time if an initial state is theGaussian state. Thus, we can track the time-evolved state only by calculating the matrix G jα .We here show that the Gaussian-state structure is kept in time under Eq. (S-1). The initial state | ψ (0) (cid:105) is assumedto be the Gaussian state: | ψ (0) (cid:105) = N (cid:89) α =1 M (cid:88) j =1 ˆ f † j U jα (0) | (cid:105) , (S-3)where U jα (0) is a M × N matrix. In the main text, we use U jα (0) = δ j, α corresponding to a staggered state (cid:81) M/ j =1 ˆ f † j | (cid:105) . Applying the unitary operator ˆ U ( t ) = exp ( − i ˆ Ht/ (cid:126) ) to Eq. (S-3), we obtain the time-evolved state | ψ ( t ) (cid:105) at a time t : | ψ ( t ) (cid:105) = ˆ U ( t ) N (cid:89) α =1 M (cid:88) j =1 ˆ f † j U jα (0) | (cid:105) (S-4)= N (cid:89) α =1 M (cid:88) j =1 ˆ U ( t ) ˆ f † j ˆ U † ( t ) U jα (0) | (cid:105) . (S-5)Here, we use ˆ U ( t ) ˆ U † ( t ) = 1 and ˆ H | (cid:105) = 0 to derive the last line. Calculating ˆ U ( t ) ˆ f † j ˆ U † ( t ) by means of the Baker-Campbell-Hausdorff formula, we find ˆ U ( t ) ˆ f † j ˆ U † ( t ) = M (cid:88) k =1 ˆ f † k A kj ( t ) (S-6)with coefficients A kj ( t ) ( j, k = 1 , · · · , M ) depending on H ij . As a result, the quantum state (S-5) is expressed by | ψ ( t ) (cid:105) = N (cid:89) α =1 (cid:32) M (cid:88) k =1 ˆ f † k U kα ( t ) (cid:33) | (cid:105) (S-7)with U kα ( t ) = (cid:80) Mj =1 A kj ( t ) U jα (0). Obviously, Eq. (S-7) has the Gaussian-state structure (S-2). Thus, we caninvestigate the quantum dynamics only by calculating the time evolution of U kα ( t ) starting from the initial coefficient U kα (0). In all our numerical simulations, we calculate U kα ( t ) using the Crank-Nicolson method, which conserves thenorm (cid:104) ψ ( t ) | ψ ( t ) (cid:105) = 1 at least up to the order of 10 − .Finally, we comment on a correlation matrix. Using Eq. (S-7), we express the correlation matrix D ij ( t ) with U iα ( t ): D ij ( t ) := Tr (cid:104) ˆ ρ ( t ) ˆ f † i ˆ f j (cid:105) (S-8)= N (cid:88) α =1 U ∗ iα ( t ) U jα ( t ) . (S-9)By definition, the occupation number at a site- j is given by D jj = (cid:104) ˆ f † j f j (cid:105) . Similarly, we can deriveTr (cid:104) ˆ ρ ( t ) ˆ f † i ˆ f i ˆ f † j ˆ f j (cid:105) = (cid:40) D ii D jj − D ij D ji ( i (cid:54) = j ); D ii ( i = j ) . (S-10)We numerically calculate Eqs. (S-9) and (S-10), and then obtain the surface roughness investigated in the main text. NUMERICAL DATA FOR THE SURFACE-ROUGHNESS DYNAMICS
We show all the numerical results for time evolution of the surface roughness, which are used to extract the universalpower exponents featuring the Family-Vicsek (FV) scaling in Figs. 2 and 3 of the main text. Figures S-1 and S-2show time evolution of all the surface roughness in the AAM and the RDM with
W <
1, respectively. Our methodfor calculating the power exponents is based on Ref. [50, 81], and the tables S-1 and S-2 summarize the time regionused for the calculation.0 W system size M time region for α and z time region for β , , , , τ, τ ] [15 τ, τ ]0.4 200 , , , , τ, τ ] [15 τ, τ ]0.5 200 , , , , τ, τ ] [15 τ, τ ]0.6 300 , , , τ, τ ] [40 τ, τ ]0.7 500 , , τ, τ ] [40 τ, τ ] TABLE S-1. Fitting information for the AAM in the delocalized phase (
W < W , and second and third ones show the system sizes M and the fitting time regions used for the evaluation of α and z . Toextract the exponent β , we use the data for M = 1200 and the time region given in the fourth column.FIG. S-1. Surface-roughness dynamics in the AAM with (a) W = 0 .
3, (b) W = 0 .
4, (c) W = 0 .
5, (d) W = 0 .
6, and (e) W = 0 .
7. The upper and lower panels show the raw data and the data with the nor-malized ordinate and abscissa by ( M/ α and ( M/ z , respectively. The extracted exponents are ( α, β, z ) =(0 . , . , . , (0 . , . , . , (0 . , . , . , (0 . , . , . . , . , .
07) for (a), (b), (c), (d), and(e), respectively. W system size M time region for α and z time region for β , , τ, τ ] [7 τ, τ ]0.4 300 , , , τ, τ ] [7 τ, τ ]0.5 300 , , , τ, τ ] [8 τ, τ ]0.6 200 , , , , τ, τ ] [10 τ, τ ]0.7 200 , , , , τ, τ ] [12 τ, τ ] TABLE S-2. Fitting information for the RDM in the delocalized phase (
W < W , and second and third ones show the system sizes M and the fitting time regions used for the evaluation of α and z . Toextract the exponent β , we use the data for M = 1200 and the time region given in the fourth column.FIG. S-2. Surface-roughness dynamics in the RDM with (a) W = 0 .
3, (b) W = 0 .
4, (c) W = 0 .
5, (d) W = 0 .
6, and (e) W = 0 .
7. The upper and lower panels show the raw data and the data with the nor-malized ordinate and abscissa by ( M/ α and ( M/ z , respectively. The extracted exponents are ( α, β, z ) =(0 . , . , . , (0 . , . , . , (0 . , . , . , (0 . , . , . . , . , .
00) for (a), (b), (c), (d), and(e), respectively. NUMBERS OF DELOCALIZED AND LOCALIZED EIGENSTATES IN THE DISORDERED FERMIONMODELS
This section addresses numbers of delocalized eigenstates (DLESs) and localized eigenstates (LESs) in the threedisordered fermion models, namely the random model (RM), the random-dimer model (RDM), and the Aubry-Andr´e model (AAM). First, we numerically solve the stationary Schr¨odinger equation in the single-particle Fock basis { ˆ f † i | (cid:105) | i = 1 , , · · · , M } : M (cid:88) j =1 H ij v jα = (cid:15) α v iα , (S-11)where v iα and (cid:15) α are an eigenvector and an eigenvalue labeled by a quantum number α = 1 , , · · · , M . The matrixelement H ij is given by H = V − J A − J V − J − J V − J . . . . . . . . .. . . . . . . . . A − J V M . (S-12)Here, A is a constant depending on the boundary conditions, and becomes 0 and − J in the open and periodic boundarycondition, respectively. Next, we define the inverse participation ratio R as R α = M (cid:88) j =1 | v jα | M (cid:88) j =1 | v jα | . (S-13)If an eigenvector is delocalized, the ratio R α is proportional to 1 /M . On the other hand, R α becomes O (1) if aneigenvector is localized. In this work, we identify DLESs by the condition that R α is smaller than 10 /M , and countthe number N DLES ( M ) of the DLESs. Then, the number of the LESs is defined by N LES ( M ) := M − N DLES ( M ).Figure S-3 shows N DLES ( M ) and N LES ( M ) in the RM, the RDM, and the AAM. The left and right panels are theresults for W = 0 . W = 1 .
1. The RDM with W = 0 . √ M and M , respectively. This result is consistent with Ref. [47]. On the otherhand, the AAM shows N LES ( M ) (cid:28)
1, which means N DLES ( M ) (cid:39) M . Thus, most of the eigenstates are extended tothe entire system, and the LESs do not affect the surface-growth dynamics. In the localized phase, N DLES ( M ) and N LES ( M ) in all the models approaches to O (1) and M , respectively, when M is much larger than unity. FAMILY-VICSEK-SCALING EXPONENT α IN THE APPROXIMATED DIAGONAL ENSEMBLE
Employing an approximated diagonal ensemble [66–69], we numerically obtain α = 0 .
33 and 0 .
50 in the RDMand the AAM without directly solving the time-dependent Schr¨odinger equation. This section explains the detailedcalculations.
Diagonalization of the Hamiltonian
In this subsection, we diagonalize the quadratic Hamiltonian (S-12) and give a relation between the bare-fermionoperators { ˆ f i , ˆ f † i } i =1 , ··· ,M and the quasi-fermion ones { ˆ F α , ˆ F † α } α =1 , ··· ,M (see Eqs. (S-16) and (S-17)). Solving aneigenvalue problem with the Hermitian matrix H of Eq. (S-12), we obtain the eigenvalue (cid:15) α and the corresponding3 system size M N D L E S ( M ) (a) Number of DLESs ( W = 0 . ∝ M . ∝ M AAM ( W = 0 . W = 0 . W = 0 . ∝ M ∝ M . system size M N L E S ( M ) (b) Number of LESs ( W = 0 . ∝ M AAM ( W = 0 . W = 0 . W = 0 . N LES ( M ) = M system size M N D L E S ( M ) (c) Number of DLESs ( W = 1 . AAM ( W = 1 . W = 1 . W = 1 . system size M N L E S ( M ) (d) Number of LESs ( W = 1 . ∝ M AAM ( W = 1 . W = 1 . W = 1 . ∝ M FIG. S-3. (a,b) N DLES ( M ) and N LES ( M ) in the RDM, the AAM, and the the RM with W = 0 .
5. The numbers of the DLESsin the RDM and the AAM obey the √ M and M power-law growth, while one in the RM does not increase. On the one hand,the numbers of the LESs in the RDM and the RM grows with increasing M , while one in the AAM is almost to zero. (c,d) N DLES ( M ) and N LES ( M ) in the RDM, the AAM, and the RM with W = 1 .
1. The numbers of the DLESs approach to zero inlarge M , while those of the LESs grows with the power law N LES ( M ) ∝ M in large M . eigenvector v α with the label α = 1 , , · · · , M . Then, the Hamiltonian is diagonalized as V † HV = diag( (cid:15) , (cid:15) , · · · , (cid:15) M ) . (S-14)Here, we define a unitary matrix V = ( v , v , · · · , v M ). Using all these results, we finally obtainˆ H = M (cid:88) α =1 (cid:15) α F † α ˆ F α , (S-15)where ˆ F α and ˆ F † α are annihilation and creation operators for the quasi-fermions defined byˆ F α = M (cid:88) j =1 v ∗ jα ˆ f j , (S-16)ˆ F † α = M (cid:88) j =1 v jα ˆ f † j . (S-17)4The inverse transformations for Eqs. (S-16) and (S-17) becomeˆ f j = M (cid:88) α =1 v jα ˆ F α , (S-18)ˆ f † j = M (cid:88) α =1 v ∗ jα ˆ F † α . (S-19)In what follows, employing the transformations, we use the diagonal ensemble to investigate the surface roughness inthe stationary state. Approximated expression of the surface roughness
Before applying the diagonal ensemble to the surface roughness, we first approximate the surface roughness byimposing the following assumption:Assumption (i) : h av ( t ) (cid:39) , Assumption (ii) : w ( M, t ) (cid:39) Tr (cid:104) ˆ ρ ( t )(ˆ h M/ − h av ( t )) (cid:105) , Assumption (iii) : M/ (cid:88) j =1 Tr [ˆ ρ ( t )ˆ n j ] (cid:39) νM . While the first and second assumptions are difficult to prove rigorously, our numerical simulations confirm theirvalidity as shown in Figs. S-4 and S-5 as discussed later. The third assumption means that the particle numberin the half of the system is equal to half of the total particle number, and we expect it to be valid in our systembecause the initial state is the staggered state and all the delocalized modes can spread over the whole system. Indeed,we numerically confirm the validity of assumption (iii) as shown in Fig. S-4. We also note that assumption (iii) isanalytically justified for the RDM after the average over disorder.Under the assumptions (i) and (ii) and the definition of the surface-height operator, we obtain w ( M, t ) (cid:39) Tr ˆ ρ ( t ) M/ (cid:88) j =1 ˆ n j − M ν , (S-20)= M/ (cid:88) k,l =1 Tr (cid:2) ˆ ρ ( t )(ˆ n k ˆ n l − ν ˆ n k − ν ˆ n l + ν ) (cid:3) , (S-21)It is worthy of mentioning here that Eq. (S-20) is equivalent to a square of the bipartite fluctuation [70–72], whichquantifies the particle-number fluctuation in the half of the system.Finally, applying the Wick decomposition (S-10) to Eq. (S-21) and use the assumption (iii), we find w ( M, t ) (cid:39) M/ (cid:88) j =1 D jj ( t ) − M/ (cid:88) i,j =1 D ij ( t ) D ji ( t ) + M/ (cid:88) j =1 D jj ( t ) − ν M (cid:39) M/ (cid:88) j =1 D jj ( t ) − M/ (cid:88) i,j =1 D ij ( t ) D ji ( t ) . (S-23)Here, to derive the last line, we utilize (cid:16)(cid:80) M/ j =1 D jj ( t ) (cid:17) = (cid:16)(cid:80) M/ j =1 Tr [ˆ ρ ( t )ˆ n j ] (cid:17) = ν M / A ( M, t ) = M/ (cid:88) j =1 D jj ( t ) , (S-24)5 t/τ − . − . − . . . . . h a v ( M , t ) / w ( M , t ) RDM ( W = 0 . t/τ . . . . . . P M / j = T r [ ˆ ρ ( t ) ˆ n j ] (b) 10 t/τ − . − . − . . h a v ( M , t ) / w ( M , t ) AAM ( W = 0 . t/τ . . . . . . P M / j = T r [ ˆ ρ ( t ) ˆ n j ] (d) FIG. S-4. Numerical test of the assumptions (i) and (iii) for (a,b) the RDM and (c,d) the AAM with W = 0 . M = 800.(a,c) Time evolution of h av ( t ) /w ( M, t ). We find that, in both of the models, the ratios become smaller than unity as time goesby, and then the assumption (i) works well. (b,d) Time evolution of (cid:80) M/ j =1 Tr [ˆ ρ ( t )ˆ n j ]. One can see that the values are close to N/ M/ B ( M, t ) = M/ (cid:88) i,j =1 D ij ( t ) D ji ( t ) . (S-25)Then, we finally obtain the approximated surface roughness w app ( M, t ), which is defined as w app ( M, t ) := A ( M, t ) − B ( M, t ) . (S-26)We numerically investigate the validity of the three assumptions given in the beginning of this section, and checkwhether or not Eq. (S-23) works well. Figure (S-4) shows the time evolution for h av ( t ) /w ( M, t ) and (cid:80) M/ j =1 Tr [ˆ ρ ( t )ˆ n j ],from which we find that the assumptions (i) and (iii) are valid. Note that, in the AAM, the averaged surface-heightis not much smaller than unity but becomes smaller as time goes by.In the upper panels of Fig. S-5, in order to consider the assumption (ii), we plot the site-dependent surface roughnessdefined by w j ( M, t ) = (cid:104) ˆ ρ ( t )(ˆ h j − h av ( t )) (cid:105) . (S-27)For both of the AAM and the RDM, w j ( M, t ) in the early stage of the dynamics is almost independent of the site j except for the edges. The non-uniformity, however, appears even at the center as time goes by, and thus theassumption (ii) becomes worse in the late stage. Actually, as shown in the lower panels of Fig. S-5, w app ( M, t ) beginsto deviate from w ( M, t ) in t > τ and t > τ for the RDM and the AAM, respectively. Note that, as described inFig. 4 of the main text, the FV-type scaling of the EE still holds.6 j w j ( M , t ) RDM ( W = 0 . t/τ = 300 t/τ = 80 t/τ = 45 t/τ = 20 t/τ = 5 t/τ w ( M , t ) , w a pp ( M , t ) (b) w app ( M, t ) w ( M, t ) j w j ( M , t ) AAM ( W = 0 . t/τ = 400 t/τ = 200 t/τ = 100 t/τ = 60 t/τ = 20 t/τ w ( M , t ) , w a pp ( M , t ) (d) w app ( M, t ) w ( M, t ) t/τ − . . e rr o r t/τ − . . e rr o r FIG. S-5. Numerical test of the assumption (ii) and Eq. (S-26) in (a,b) the RDM and (c,d) the AAM with W = 0 . M = 800. (a,c) Time evolution of w j ( M, t ). In the early stage of the dynamics, the distribution is uniform far from theboundaries, but the uniformity breaks down in the late dynamics. (b,d) Time evolution of w ( M, t ) and w app ( M, t ). The insetsplot the error ( w − w app ) /w . The maximum deviation is about − . w app ( M, t ) on the diagonal ensemble We apply the diagonal ensemble to Eq. (S-26). For this purpose, we first consider time dependence of the correlationmatrix, which is determined by the eigenvalues (cid:15) α ( α = 1 , · · · , M ). Using the operators for the quasi-particles, wecan obtain D ij ( t ) = Tr (cid:104) ˆ ρ (0) ˆ f † i ( t ) ˆ f j ( t ) (cid:105) (S-28)= M (cid:88) α,β =1 v ∗ iα v jβ Tr (cid:104) ˆ ρ (0) ˆ F † α ( t ) ˆ F β ( t ) (cid:105) (S-29)= M (cid:88) α,β =1 v ∗ iα v jβ Tr (cid:104) ˆ ρ (0) ˆ F † α ˆ F β (cid:105) e i ( (cid:15) α − (cid:15) β ) t/ (cid:126) . (S-30)7Then, substituting the exact result (S-30) into A ( t ) and taking the long time average by assuming no degeneracy, weobtain A ave := lim T →∞ T (cid:90) T dtA ( t ) (S-31)= lim T →∞ T (cid:90) T dt M/ (cid:88) j =1 D jj ( t ) (S-32)= M/ (cid:88) j =1 M (cid:88) α,β =1 v ∗ jα v jβ Tr (cid:104) ˆ ρ (0) ˆ F † α ˆ F β (cid:105) δ αβ (S-33)= M/ (cid:88) j =1 M (cid:88) α =1 v ∗ jα v jα Tr (cid:104) ˆ ρ (0) ˆ F † α ˆ F α (cid:105) . (S-34)Defining I αβ = (cid:80) M/ j =1 v ∗ jα v jβ , we obtain A ave = M (cid:88) α =1 I αα Tr (cid:104) ˆ ρ (0) ˆ F † α ˆ F α (cid:105) . (S-35)Following the same way, we calculate the time-averaged value of B ( t ) as B ave := lim T →∞ T (cid:90) T dtB ( t ) (S-36)= M (cid:88) α,β,µ,ν =1 I αν I µβ Tr (cid:104) ˆ ρ (0) ˆ F † α ˆ F β (cid:105) Tr (cid:104) ˆ ρ (0) ˆ F † µ ˆ F ν (cid:105) δ α + µ,β + ν . (S-37)Thus, the substitution of Eqs. (S-35) and (S-37) into Eq. (S-26) leads to the surface roughness in the stationary state: w ave ( M ) := lim T →∞ T (cid:90) T dtw app ( M, t ) (S-38)= A ave − B ave . (S-39)Thus, instead of directly solving the time-dependent Schr¨odinger equation, we can calculate the surface-roughness inthe stationary state. Derivation of the FV scaling exponent α = 0 . in a fermion system free from disorder. Before discussing disordered systems, we show that normal exponent α = 1 / V j = 0 by using Eq. (S-39). In this model,the eigenfunctions v jα are expressed by the plane waves: v j = 1 √ M (S-40) v jM = 1 √ M ( − j (S-41) v even jα = (cid:114) M cos (2 παj/M ) ( α = 1 , , · · · , M/ − , (S-42) v odd jα = (cid:114) M sin (2 παj/M ) ( α = 1 , , · · · , M/ − , (S-43)where the corresponding eigenvalue is given by (cid:15) α = − J cos(2 πα/M ) with the integer label α . Using this expression,we can evaluate I αβ = (cid:80) M/ j =1 v ∗ jα v jβ , but do not show the concrete expression since they are too complicated. As8
100 200 300 400 500 ↵ . . . . . . . ↵ . . . . . . . I αβ I αβ FIG. S-6. Numerical result of I αβ in the non-interacting fermion with V j = 0 and M = 500. The left panel shows all the data,and the right one is the enlarged figure. shown in Fig. S-6, in large M , I αβ is well approximated to be I αβ (cid:39) δ αβ . (S-44)We comment on negative and positive values of the off-diagonal component I αβ in Fig. S-6. Their contribution inEqs. (S-35) and (S-37) will be small because the summation leads to the cancelation.We substitute Eq. (S-44) into Eqs. (S-35) and (S-37), and then obtain A ave (cid:39) M (cid:88) α =1 Tr (cid:104) ˆ ρ (0) ˆ F † α ˆ F α (cid:105) . (S-45) B ave (cid:39) M (cid:88) α,β =1 (cid:12)(cid:12)(cid:12) Tr (cid:104) ˆ ρ (0) ˆ F † α ˆ F β (cid:105)(cid:12)(cid:12)(cid:12) . (S-46)Next, using Eqs. (S-16), (S-17), and (S-43), we evaluate Tr (cid:104) ˆ ρ (0) ˆ F † α ˆ F β (cid:105) asTr (cid:104) ˆ ρ (0) ˆ F † α ˆ F β (cid:105) = M (cid:88) i =1 M (cid:88) j =1 v iα v ∗ jβ Tr (cid:104) ˆ ρ (0) ˆ f † i ˆ f j (cid:105) (S-47)= M/ (cid:88) j =1 v (2 j ) α v ∗ (2 j ) β (S-48)Substituting Eq. (S-48) into Eqs. (S-45) and (S-46), we derive A ave (cid:39) M , (S-49) B ave (cid:39) M w ave ( M ) (cid:39) M. (S-51)9 system size M w a v e ( M ) , A a v e ( M ) , B a v e ( M ) w ave ( M ) A ave ( M ) B ave ( M ) M/ M/ FIG. S-7. Numerical test of Eqs. (S-49), (S-50), and (S-51). Solving the eigenvalue problem (S-11) with V j = 0 under theperiodic boundary condition, we calculate Eqs. (S-35), (S-37), and (S-39). One can see that Eqs. (S-49) shows the excellentagreement with the numerical results, while Eqs. (S-50) and (S-51) deviates from the numerical ones but correctly reproducethe M -power-law dependence. This deviation is attributed to the approximation of Eq. (S-44). Note that the degeneracy of the eigenvalue (cid:15) α exists, which seems to be inconsistent with the assumption for thediagonal ensemble. However, Eqs. (S-35) and (S-37) still hold because of Eq. (S-44). Actually, only by using Eq. (S-44),we can directly derive Eqs. (S-35) and (S-37) without utilizing the non-degeneracy assumption.We check the validity of Eqs. (S-49), (S-50), and (S-51) by numerically calculating Eqs. (S-35), (S-37), and (S-39).Figure (S-7) shows that Eqs. (S-49) works well, while Eqs. (S-50) and (S-51) does not. This deviation comes fromthe approximation of Eq. (S-44). However, all the analytical results correctly reproduce the M -power law, and thusthe free-fermion system free from disorder potentials have a normal exponent α = 1 /
2. Moreover, as discussed inRef. [50], systems satisfying the eigenstate thermalization hypothesis also have α = 1 /
2. Therefore, we can concludethat generic delocalized systems have α = 1 / Numerical results for the AAM and the RDM
Next, we consider disordered systems. Figure 3 of the main text shows the dependence of w ave on the system size M , which is numerically obtained by diagonalizing the matrix H of Eq. (S-12) and using Eqs. (S-39). We find thatthe stationary surface-roughness in the RDM and the AAM scales as M . and M . , which are consistent with theexponents α = 0 .
33 and 0 . α = 1 / α = 0 . RELATION BETWEEN THE VON NEUMANN ENTANGLEMENT ENTROPY AND THE SURFACEROUGHNESS
In the main text, we investigate the entanglement dynamics and its dynamical one-parameter scaling by consideringthe relation between the von Neumann entanglement entropy (EE) and the surface roughness. Here, we describe howto derive the relation and a related useful inequality.
Relation between S EE ( M, t ) and w ( M, t ) The EE in our models can be expressed by the eigenvalues of the correlation matrix D ij . This expression is usefulto derive the relation between S EE ( M, t ) and w ( M, t ). In the main text, the EE is defined by S EE ( M, t ) = − Tr (cid:48) [ˆ ρ re ( t ) log ˆ ρ re ( t )] , (S-52)where a reduced density matrix is ˆ ρ re ( t ) = tr B ˆ ρ pure ( t ) with the set B = { M/ , · · · , M } and Tr (cid:48) denotes a partialtrace excluding B . Here, ˆ ρ pure ( t ) is a density matrix for a single realization, and we take the ensemble average for S EE ( M, t ) in the RDM. According to the previous work [65], S EE ( M, t ) becomes S EE ( M, t ) = − M/ (cid:88) n =1 λ n ( t ) log λ n ( t ) − M/ (cid:88) n =1 (1 − λ n ( t )) log (1 − λ n ( t )) , (S-53)where λ n ( t ) ∈ [0 ,
1] ( n = 1 , · · · , M/
2) are eigenvalues of the correlation matrix D ij ( t ) ( i, j = 1 , · · · , M/ w ( M, t ) (cid:39) M/ (cid:88) n =1 λ n ( t ) (1 − λ n ( t )) (S-54)= w app ( M, t ) ( ∵ (S-26)) . (S-55)To connect S EE ( M, t ) with w ( M, t ), we introduce the probability function P ( λ, t ) for λ n ( t ). Then, we can rewrite S EE ( M, t ) and w ( M, t ) as S EE ( M, t ) = − (cid:90) dλP ( λ, t ) { λ log λ + (1 − λ ) log (1 − λ ) } , (S-56) w ( M, t ) (cid:39) (cid:90) dλP ( λ, t ) λ (1 − λ ) . (S-57)Here, we assume P ( λ, t ) (cid:39) p ( t ) δ ( λ ) + p ( t ) θ (1 − λ ) θ ( λ ) + p ( t ) δ ( λ − , (S-58)where p j ( t ) ( j = 1 , ,
3) is a time-dependent weight independent of λ and θ ( · ) is the Heaviside step function. Thecrucial assumption given here is that the probability distribution is uniform for 0 < λ <
1. As shown in Fig. S-8, theassumption is well satisfied in both the RDM and the AAM. Then, using this assumption, we get S EE ( M, t ) (cid:39) − p ( t ) (cid:90) dλ { λ log λ + (1 − λ ) log (1 − λ ) } , (S-59) w ( M, t ) (cid:39) p ( t ) (cid:90) dλλ (1 − λ ) . (S-60)Finally, we note the following integral formula: − (cid:90) ( x log x + (1 − x ) log(1 − x )) dx = (cid:90) x (1 − x ) dx. (S-61)We use the formula in Eqs. (S-59) and (S-60), which leads to S EE ( M, t ) (cid:39) w ( M, t ) . (S-62)1 . . . . . . λ P ( λ , t ) t/τ = 20 t/τ = 50 t/τ = 150 t/τ = 300 . . . . . . λ P ( λ , t ) t/τ = 10 t/τ = 20 t/τ = 60 t/τ = 150 . . . . λ . . . P ( λ , t ) . . . . λ . . . P ( λ , t ) FIG. S-8. Probability density P ( λ, t ) for the eigenvalues λ n ( t ) of the correlation matrix D ij ( t ) ( i, j = 1 , · · · , M/ W = 0 . M = 800, respectively. The insetsare the enlarged figures for λ ∈ [0 . , . The insets of Fig. 4 in the main text show that Eq. (S-62) works well especially in the early stage of the dynamics. Inthe late stage, the relation becomes a little worse because Eq. (S-55) becomes worse, but the roughness still capturesthe qualitative behavior of the von Neumann EE.
Power-law growth of S EE ( M, t ) and w app ( M, t ) Our numerical results demonstrate that the EE S EE ( M, t ) and the approximated surface-roughness w app ( M, t ) obeythe power-law growth. We here derive the relation between the two power exponents in the thermodynamic limit.2 t/τ w app ( M, t ) log(2) S EE ( M, t )log( M ) w app ( M, t ) + 1 t/τ w app ( M, t ) log(2) S EE ( M, t )log( M ) w app ( M, t ) + 1 FIG. S-9. Numerical verification of the inequality (S-65). The upper and lower panels show the results for the RDM and theAAM with W = 0 . M = 800, respectively. Inequlity of S EE ( M, t ) and w app ( M, t ) We first prove a useful inequality for S EE ( t, M ). The approximated surface-roughness w app ( M, t ) is expressed by w app ( M, t ) = M/ (cid:88) n =1 λ n ( t ) (1 − λ n ( t )) . (S-63)We note that the following inequality is derived for x ∈ [0 ,
1] and
M > x (1 − x ) log(2) < − x log( x ) − (1 − x ) log(1 − x ) < x (1 − x ) log( M ) + 2 M . (S-64)Finally, by using Eqs. (S-53), (S-63), and (S-64), we obtain4 w app ( M, t ) log(2) < S EE ( M, t ) < w app ( M, t ) log( M ) + 1 . (S-65)3Figure S-9 numerically checks Eq. (S-65) in the RDM and the AAM. Power exponents in the thermodynamic limit
According to our numerical results, the surface roughness shows the FV scaling characterized by the universalexponents ( α, β, z ) > for sufficiently large systems. Then, it is reasonable to assume that, for any positive and areal number (cid:15) , there exists an integer M such that the surface roughness satisfies (cid:12)(cid:12)(cid:12)(cid:12) log w app ( M, t )log t − β (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15) (S-66)for any M > M and aM u < t < bM u with given positive constants a , b , and u < u < z . Under this assumption,we can prove the following proposition. Proposition 1
For any positive and a real number δ , there exists an integer M such that the EE satisfies (cid:12)(cid:12)(cid:12)(cid:12) log S EE ( M, t )log t − β (cid:12)(cid:12)(cid:12)(cid:12) < δ (S-67) for any M > M and aM u < t < bM u .Proof. The assumption (S-66) leads to t β − (cid:15) < w app ( M, t ) < t β + (cid:15) . (S-68)Substituting t = aM u into the lower bound of Eq. (S-68), we obtain a β − (cid:15) M ( β − (cid:15) ) u < w app ( M, t ) , (S-69)from which we can always take an integer M ( > M ) ensuring 1 < w app ( M, t ) and 0 < log t for M > M because theleft-hand side of Eq. (S-69) and the lower bound aM u of the time regime increase with M .Using the inequality of Eq. (S-65), we derive2 log w app ( M, t ) + log(4 log 2)log t < log S EE ( M, t )log t < log( w app ( M, t ) (log M ) + 1)log t . (S-70)The fact 1 < w app ( M, t ) leads tolog( w app ( M, t ) (log M ) + 1)log t < w app ( M, t ) + log(log( M ) + 1)log t . (S-71)Then, for M > M we obtain2 log w app ( M, t ) + log(4 log 2)log t < log S EE ( M, t )log t < w app ( M, t ) + log(log( M ) + 1)log t . (S-72)Using Eqs. (S-68) and (S-72), we obtain − (cid:15) + log(4 log 2)log t < log S EE ( M, t )log t − β < (cid:15) + log(log( M ) + 1)log t . (S-73)For any positive and a real number δ , considering the condition aM u < t < bM u and setting (cid:15) = δ/
4, we can alwaystake an integer M such that max (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:15) + log(4 log 2)log t (cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) + log(log( M ) + 1)log t (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) < δ. (S-74)for M > M . Thus, we finally obtain (cid:12)(cid:12)(cid:12)(cid:12) log S EE ( M, t )log t − β (cid:12)(cid:12)(cid:12)(cid:12) < δ (S-75)4for M > max( M , M ) := M . This completes the proof. (cid:4) This proposition means that the EE grows with t β in the thermodynamic limit if the approximated surface-roughness (which is also approximated by the bipartite particle fluctuation) grows with t β . ANOMALOUS BEHAVIOR OF A SINGLE PARTICLE TRANSPORT IN THE RDM
The number of the DLESs in the RDM with
W < √ M . This fact essentially alters the surface-roughness dynamics as we have discussed throughout the main text and this supplemental material so far. We hereillustrate another example for anomalous single-particle diffusion in the RDM. The similar observation was reportedin Ref. [47].We consider a single-particle dynamics starting from the following state: | ψ single (cid:105) = ˆ f † M/ | (cid:105) . (S-76)Using the Schr¨odinger equation with this initial state, we calculate the deviation (cid:52) x ( t ) from the center defined by (cid:52) x ( t ) = M (cid:88) j =1 (cid:18) j − M (cid:19) (cid:104) ψ single | ˆ f † j ( t ) ˆ f j ( t ) | ψ single (cid:105) . (S-77)In the quasi-particle representation with Eqs. (S-16) and (S-17), this is expressed by (cid:52) x ( t ) = M (cid:88) j =1 M (cid:88) α =1 M (cid:88) β =1 (cid:18) j − M (cid:19) u ∗ jα u jβ × (cid:104) ψ single | ˆ F † α ˆ F β | ψ single (cid:105) e i (cid:126) ( (cid:15) α − (cid:15) β ) t . (S-78)Just as the calculation of the diagonal ensemble, we apply the long-time average, and then obtain the stationarydeviation: (cid:52) x := lim T →∞ T (cid:90) T (cid:52) x ( t ) dt (S-79)= M (cid:88) j =1 M (cid:88) α =1 (cid:18) j − M (cid:19) | u jα | (cid:104) ψ single | ˆ F † α ˆ F α | ψ single (cid:105) . (S-80)Here, using the initial state, we derive (cid:104) ψ single | ˆ F † α ˆ F α | ψ single (cid:105) = | u M/ ,α | . (S-81)Thus, this leads to (cid:52) x = M (cid:88) j =1 M (cid:88) α =1 (cid:18) j − M (cid:19) | u jα | | u M/ ,α | . (S-82)We estimate Eq. (S-82) by noting the fact that the RDM with W < M and √ M , respectively. Let us denote a set of labels α for the localized(delocalized) states by L ( D ). This notation gives (cid:52) x = M (cid:88) j =1 (cid:88) α ∈L (cid:18) j − M (cid:19) | u jα | | u M/ ,α | + M (cid:88) j =1 (cid:88) α ∈D (cid:18) j − M (cid:19) | u jα | | u M/ ,α | . (S-83)In the first term on the right hand side of Eq. (S-83), the product of the eigenfunctions has large values around j = M/ α = α M/ whose eigenstates are spatially localized around j = M/