Nonergodic dynamics in disorder-free potentials
NNonergodic dynamics in disorder-free potentials
Ruixiao Yao
Department of Physics and State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing,100084, China
Titas Chanda
Institute of Theoretical Physics, Jagiellonian University in Krak´ow, Łojasiewicza 11, 30-348 Krak´ow, Poland
Jakub Zakrzewski
Institute of Theoretical Physics, Jagiellonian University in Krak´ow, Łojasiewicza 11, 30-348 Krak´ow, PolandMark Kac Complex Systems Research Center, Jagiellonian University in Krakow, Krak´ow, Poland.
Abstract
We review the dynamics of interacting particles in disorder-free potentials concentrating on acombination of a harmonic binding with a constant tilt. We show that a simple picture of an ef-fective local tilt describes a variety of cases. Our examples include spinless fermions (as modeledby Heisenberg spin chain in a magnetic field), spinful fermions as well as bosons that enjoy alarger local on-site Hilbert space. We also discuss the domain-wall dynamics that reveals noner-godic features even for a relatively weak tilt as suggested by Doggen et. al. [arXiv:2012.13722].By adding a harmonic potential on top of the static field we confirm that the surprizing regulardynamics is not due to global conservation moments. It is rather strongly linked to inhibitedtransport within the domains of identically oriented spins. Once the spin-1 / ff ect of the harmonic potential is shown to lead mainly to an e ff ectivelocal tilt.
1. Introduction
Many-body localization (MBL) became a well established property of interacting many-bodysystems in su ffi ciently strongly disordered potentials. Seminal works [1, 2] started a chain of re-search resulting in hundreds of research papers partially summarized in recent reviews [3, 4, 5, 6].In a single sentence, MBL is proposed as a robust counterexample to thermalization as localiza-tion is supposed to lead to the sustainable memory of the initial state in the closed system andis in violent contradiction with earlier advocated eigenstate thermalization hypothesis [7, 8].Yet, a recent work [9] put the very existence of MBL in the thermodynamic limit in question Email address: [email protected] (Jakub Zakrzewski)
Preprint submitted to Annals of Physics January 28, 2021 a r X i v : . [ c ond - m a t . d i s - nn ] J a n timulating an intensive debate [10, 11, 12, 13, 14] mainly based on results coming from ex-act diagonalization studies. The second approach addresses time-dynamics with matrix productstates techniques [15, 16, 17, 18, 19, 20, 21, 22]. The claims on the transition to MBL proposedin these approaches mimics experimental approach [23, 24] addressing the long time dependenceof certain correlation functions. Still these claims are of limited value due to finite systems sizesand, importantly, rather short evolution times taken into consideration, times limited by the nu-merical methods based on matrix product states related techniques such as time-evolving blockdecimation (TEBD) [25, 26, 18] or a more popular time-dependent variational principle (TDVP)approach [27, 28].On the other hand, recent years brought several examples of strongly nonergodic behav-ior for specific initial states. For Rydberg atom systems such a behavior has been observedas pronounced long-time oscillations related to the so called scar states [29, 30]. Nonergodicdynamics also is being traced back to Hilbert-space fragmentation [31, 32, 33, 34], and lack-of-thermalization in gauge theories [35, 36, 37] where conservation laws a ff ect the dynamics evenin the absence of disorder. The spectacular example is a time crystal in driven systems. As shownin the seminal paper by Sacha [38] a strong resonant driving may lead to a creation of long livedcrystalline states where breaking of the discrete time translation symmetry leads to novel robuststates called the time crystals (for a review see [39]). Another possible source of disorder-freelocalization is the presence of frustration in the system [40].The seemingly simple and prominent example of disorder-free localization appears in tiltedlattices receiving the name of Stark many-body localization (SMBL) [41, 42]. The phenomenon,originally studied for Heisenberg spin chain (equivalent to a 1D lattice hosting spinless fermions)was also discussed for spinful fermions [43] and recently studied experimentally [44]. Similarlybosons were also shown to reveal SMBL [45, 46]. SMBL was discussed on a superconductingquantum processor [47] as well as for open systems [48, 49]. While the existence of the mobilityedge in the system seems apparent already from original works [41, 42] it was also addressed indetails [50]. Recently the fate of specially prepared domain wall states was studied [51] showingthat their dynamics reveal strongly nonergodic behavior also for a relatively small tilt, F , in theregime, where level statistics [42] indicates ergodic behavior. Their results were interpreted as asignature of “shattered Hilbert space”, the term introduced in [31].The latter work stresses the importance of global constants of the motion that may lead tothe so called fracton models [52, 53, 33]. To avoid global dipole moment conservation a smallquadratic potential is sometimes added to the tilt [41]. On the other hand a strong harmonicpotential may lead to a phase separation between localized and delocalized regimes [43, 46].The aim of this paper is to review the interplay between a harmonic and linear lattice poten-tial perturbation as described originally in [43, 46]. We shall also consider the e ff ect of harmonicconfinement on domain wall melting as discussed for SMBL in [51]. Adding a harmonic contri-bution to a tilt we point out that • The global conservation laws for the “charge”, “dipole moment” etc. are not very relevantfor the dynamics on a realistic (i.e. , experimentally feasible) time scales; • The mean field estimate of the local tilt is more relevant for describing the behavior of thesystem (although it does not carry all the necessary information); • the dynamics of the entanglement entropy growth is essential for an understanding the timedynamics of the system. 2n the subsequent section we provide results of the selected numerical simulations of theproblem that are further analyzed.
2. Heisenberg spin chain in a harmonic confinement
Harmonic confinement is typical in cold atom studies although sometimes the e ff ect is min-imized [24]. Quadratic dependence on the coordinate is quite common in quasi-1D situationsrealized in optical lattices [54, 55], where a tight confinement in directions perpendicular to achosen one is due to strong laser beams with Gaussian transverse profiles. Those profiles may bereasonably approximated as a harmonic trap along the considered direction [54]. In recent works[43, 46] we take a more brutal approach and assume a relatively strong harmonic binding poten-tial. We show that, surprizingly, such a harmonic binding may lead to a true separation of thespace for one-dimensional interacting particles with a central region being seemingly delocalizedwith the edges, on the other hand, being strongly localized.Consider at first the spinless interacting fermions equivalent in 1D, via Jordan-Wigner trans-formation to interacting XXZ spin chain. Taking, as quite common in MBL studies, the hoppingand interaction strength constants to be equal we arrive at Hamiltonian: H = J L / − (cid:88) l = − L / (cid:126) S l · (cid:126) S l + + L / (cid:88) l = − L / FlS zl + A L / (cid:88) l = − L / l S zi , (1)where (cid:126) S l are 1 / ff erent sites, A is the curvature of the harmonic potential and F is themagnitude of the possible additional tilt of the lattice. From now on we shall assume J = F = A obeys a quadrupole conservationlaw, while for A = F the dipole conservation appears [31]. In these limits themodel belongs to a class of fracton systems where generically slow subdi ff usive approach tothermalization is predicted [33, 34]. We shall restrict ourselves to moderate times of the order ofthousands of hopping times and in advance we state that we did not observe any traces of suchslow processes in the model studied.For a discussion of spectral properties of the system (e.g. , the gap-ratio statistics) we referthe reader to [43]. Here, we visualize the time dynamics for moderate system size, L = hybrid variation of the TDVP schemementioned in [28, 58, 36, 21], where we first use two-site version of TDVP to dynamically growthe bond dimension up to a prescribed value, say χ max . After saturation occurs, we shift to theone-site version (avoiding errors due to a truncation is singular values in the two-site version[28, 58]). The final results are produced with χ max = z -axis).Consider first the motion with no static field to observe the e ff ects coming from a strongharmonic confinement. The time dynamics of the system is presented in Fig. 1 for two di ff er-ent values of harmonic potential curvature A . Observe that while at the center of the trap spindelocalize, the pattern of consecutive up and down spins is to a large extend preserved at bothsides of the trap. This indicates that the system remembers in these regions the initial state andis localized.Bearing in mind localization in the tilted lattice, SMBL the explanation of the observedphenomenon readily comes. For a site located at the distance l from the center of the trap3 . . Time15101520253035404550 S i t e s A = 0 . ° . ° . ° . ° . ° . . . . . . . . . Time15101520253035404550 S i t e s A = 0 . ° . ° . ° . ° . ° . . . . . . . . . Time15101520253035404549 B o nd s A = 0 . . . Time15101520253035404549 B o nd s A = 0 . Time0123456 S A = 0 . Bond 25Bond 30Bond 32Bond 38
Time0123456 S A = 0 . Bond 25Bond 28Bond 32 . . Time15101520253035404550 S i t e s A = 0 . ° . ° . ° . ° . ° . . . . . . . . . Time15101520253035404550 S i t e s A = 0 . ° . ° . ° . ° . ° . . . . . . . . . Time15101520253035404549 B o nd s A = 0 . . . Time15101520253035404549 B o nd s A = 0 . Time0123456 S A = 0 . Bond 25Bond 30Bond 32Bond 38
Time0123456 S A = 0 . Bond 25Bond 28Bond 32
Figure 1: The Heisenberg chain in the harmonic trap. Site-resolved spin dynamics, as measured by the local expectationvalues (cid:104) S zl (cid:105) for L =
50 Heisenberg chain with no disorder – left column for two values of the curvature A . One clearlyobserves the coexistence of delocalized (in the center) and localized regions (at the edges). The black dashed linesgive the border of localization as given by Stark localization prediction with F c ≈ χ max = the chemical potential is µ ( l ) = Al leading to the appearance of the e ff ective local static field F = ∂∂ l (cid:16) A l (cid:17) = l A . The localization length in the physical space for MBL and SMBL is quitesmall (otherwise the initial staggered pattern of spins would be washed out) so such a local tiltwell describes a situation around l . When this local field exceeds the Stark localization border[42, 41] – the system behaves in this region as localized. The modulus of the local e ff ective fieldincreases towards the sides of the system so both the outer regions are localized while the regionclose to the center remains extended. Note that one can always find localized regions for anyfinite values of A for large enough systems under harmonic trapping potential - the situation notappearing for a pure Stark case. The dashed lines in Fig. 1 give the Stark localization border, aspredicted in [41] to be F ≈
2, which nicely fits numerical data.At each bond linking sites l and l + S l = − (cid:80) λ i ln λ i where λ i are eigenvalues of the density matrix of one of the parts (A or B) obtained by tracing outthe remaining part. The bond dependent time dynamics of the entanglement entropy is presentedin middle and right columns of Fig. 1. The rapid growth of the entropy at the center of the trapin delocalized regime allows us to expect that the results are not fully converged for χ max = χ max =
384 (not shown) leadsto fully converged results. Interestingly, the growth in the “delocalized” regime, close to thelocalization border seems logarithmic in time (compare curves for bonds 30 and 32 for A = . ff er much as the local Hilbert space dimension is equal to 2.Thus logarithmic slow growth in time in localized region is “exported” to sites being the closeneighborhood of the border between localized and delocalized sites.One could wonder whether the observation for L =
50 are su ffi cient to draw conclusionsfor larger system sizes, in the thermodynamic limit. Observe that adding sites (on both sidessymmetrically) to the system leads to an increase of regions with large value of the local electricfield - the central region is not a ff ected by this procedure. Thus contrary to the case of MBLin random disorder where the thermodynamic limit is vividly discussed as mentioned in theintroduction, in our case the thermodynamic limit is almost trivial. . . Time15101520253035404550 S i t e s A = 0 . ° . ° . ° . ° . ° . . . . . . . . . Time15101520253035404550 S i t e s A = 0 . ° . ° . ° . ° . ° . . . . . . . . . Time15101520253035404549 B o nd s A = 0 . . . Time15101520253035404549 B o nd s A = 0 . Time0123456 S A = 0 . Bond 15Bond 20Bond 22Bond 28
Time0123456 S A = 0 . Bond 20Bond 23Bond 28 . . Time15101520253035404550 S i t e s A = 0 . ° . ° . ° . ° . ° . . . . . . . . . Time15101520253035404550 S i t e s A = 0 . ° . ° . ° . ° . ° . . . . . . . . . Time15101520253035404549 B o nd s A = 0 . . . Time15101520253035404549 B o nd s A = 0 . Time0123456 S A = 0 . Bond 15Bond 20Bond 22Bond 28
Time0123456 S A = 0 . Bond 20Bond 23Bond 28
Figure 2: The Heisenberg chain with combined static field and the harmonic potential. Here we set static field F to be 2,which shifts the minima of the e ff ective potential from the center of the system. However, the dynamics remains exactlyequivalent to the scenario of Fig. 1 but with shifted delocalized region. Other descriptions are same as in Fig. 1. Let us now add an additional static electric field. We expect that the delocalized region willshift to center at the position where the sum of the external field and the e ff ective local tilt comingfrom the harmonic trap cancel. And indeed it is so as shown in Fig. 2.
3. Spinful fermions in a harmonic trap
Let us now discuss briefly the analogous physics for spinful fermions. Contrary to the spin-less case, spinful fermions can interact on a given site making strong interactions feasible. Forthat reason experimental results for disorder driven MBL was presented for spinful fermions[23, 24]. Similarly one may consider SMBL in a tilted lattice as described in [43] and recentlystudied experimentally [44]. Referring the reader to these works, let us just review here theharmonic confinement related e ff ects.The Hamiltonian discussed reads: H = − J (cid:88) l ,σ (cid:16) ˆ c † l σ ˆ c l + σ + h . c . (cid:17) + U (cid:88) l ˆ n l ↑ ˆ n l ↓ + (cid:88) l ,σ h l ˆ n l σ + A L / (cid:88) l = − L / l (ˆ n l ↑ + ˆ n l ↓ ) , (2)5 . . Time148121620242832 S i t e s A = 0 . . . . . . . . . . . . . . Time148121620242832 S i t e s A = 1 . . . . . . . . . . . . . . Time148121620242831 B o nd s A = 0 . . . Time148121620242831 B o nd s A = 1 . Time0123456 S A = 0 . Bond 16Bond 20Bond 24
Time0123456 S A = 1 . Bond 16Bond 18Bond 22Bond 24 . . Time148121620242832 S i t e s A = 0 . . . . . . . . . . . . . . Time148121620242832 S i t e s A = 1 . . . . . . . . . . . . . . Time148121620242831 B o nd s A = 0 . . . Time148121620242831 B o nd s A = 1 . Time0123456 S A = 0 . Bond 16Bond 20Bond 24
Time0123456 S A = 1 . Bond 16Bond 18Bond 22Bond 24
Figure 3: Left Column: Time-evolved density profile of spinful fermions for the initial staggered density-wave state underharmonic potential with no disorder. Middle column: Time dynamics of entanglement entropy on di ff erent bonds. Blackdashed lines gives the physical border of localization as predicted by the Stark localization with F c ≈ . with l ∈ [ − L / , L / J = U =
1. Notethat we assume the same curvature A for both ↑ , ↓ spin components.Fig. 3 shows the time dynamics of the density profile defined via S zl = ˆ n l ↑ − ˆ n l ↓ for twodi ff erent curvatures of the harmonic trap. We show results at “quarter filling” with the initialseparable state has a single fermion (with the spin alternatively pointing up and down) on oddsites and empty even sites. The observed dynamics has the same general features as observed forspinless fermions. In the center of the trap apparently we observe a fast “delocalization”, goingaway from the center we observe transition to left and right localized regions. As before, we maydefine the e ff ective local electric field as a derivative of the chemical potential . As shown bydashed lines in the top row of Fig. 3 the local field matches the estimate of the threshold given as F c = . ff erent bonds. Linear-in-time increase of the entropy in the central delocal-ized region changes into the slow logarithmic-like behavior in the localized parts – that may beconsidered as an another evidence for localization in the outer regions.Observe, however, that the dynamics in spinful case, as visualized in Fig. 3 looks moreviolent. One clearly observes a nontrivial density dynamics in the localized regime, similarlywhile the entropy growth seems confined in space for spinless case (as seen in Fig. 1), dynamicsof spinful fermions show signs of the fact that the high entropy region grows in time. Whilewe do not have data for longer times one might speculate that for much longer times the systemdestabilizes. This is especially plausible if we recall that the spin dynamics in the system seemsdelocalized [59, 18, 60, 61] - we concentrate here on density time dynamics only.6 . Bosons in the harmonic trap Let us now discuss the fate of the bosons in the harmonic trap [46]. The standard Bose-Hubbard Hamiltonian reads:ˆ H = − J L / − (cid:88) k = − L / (ˆ b † k ˆ b k + + h . c . ) + U L / (cid:88) k = − L / ˆ n k (ˆ n k − + (cid:88) k µ k ˆ n k (3)where ˆ b k (ˆ b † k ) denote bosonic annihilator (creator) operators obeying standard commutation rela-tion [ˆ b k , ˆ b † t ] = δ kt , ˆ n k = ˆ b † k ˆ b k and µ k = A ( k − L / corresponds to the harmonic binding (forstandard MBL studies [17] one assumes µ k to be random). We consider the system at half filling.Are bosons bringing some new twist to the problem? Figure 4: Top row: Time-evolved density profiles for bosons for the initial density-wave state under harmonic potentialwith no disorder for di ff erent curvatures A and interaction strength U as indicated in the panels. Black dashed lines givethe border of localization as predicted by the Stark localization with F c ≈ . A = U as shown in three bottom panels - for a discussion see text. Fig. 4 shows the evolution of the density profiles for a number of cases considered. Againthe time dynamics is accessed by TDVP algorithm, the number of bosons per site is limited to n max = ffi cient for the average density 1 /
2. The initial state has a single bo-son on each even site with odd sites being empty ( | ψ (0) (cid:105) = | , , , , ... (cid:105) ). To alert the reader wenow plot the time vertically while the sites are presented horizontally, opposite to the fermioniccases discussed previously. The top row shows a, by now, typical situation with the middle regionapparently delocalized. Dashed lines give the local field estimate coming from the harmonic po-tential which yield reasonable estimates of the borders between extended and localized regimes.The critical field is taken to be F c = . ff erent for cases for which A = U . Apart from thedelocalized region in the middle, one observes a parabolically shaped emission of particles intolocalized regions. This phenomenon, observed first in [46] may be explained by consideringthe degenerate in energy subset of states | ψ j (cid:105) that are also resonant with the initial state | ψ (cid:105) = , , , , , , , , , ... (cid:105) : | ψ (cid:105) = | , , , , , , , , , ... (cid:105)| ψ (cid:105) = | , , , , , , , , , ... (cid:105)| ψ (cid:105) = | , , , , , , , , , ... (cid:105) ... (4)The energy di ff erence between states | ψ j (cid:105) and state | ψ (cid:105) is ∆ E , j = µ j − + µ j + − µ j − U = A − U .Within this degenerate subspace the states are coupled by second order hopping terms with ratesdepending on the position. The rate dependence in j fully explains the parabolic spreadingobserved in Fig. 4 - for details see [46].
5. Domain wall physics
Localization properties may also be studied looking at domain wall melting as shown in [15].This issue was addressed recently for the tilted Heisenberg chain problem [51]. The authorsconsidered a single domain wall initial state, | D (cid:105) , (with initially left half of the spin pointingup and right half pointing down) as well as a double domain walls state, | D (cid:105) , in which, forthe system of length L (with open boundary conditions) the spins l ∈ [ − L / , L /
4] are pointingup with the remaining ones pointing down. Both initial states belong to total S z = F , values below theSMBL border as estimated in [41, 42]. This is quite interesting providing, strictly speaking, acounterexample for the eigenstate thermalization hypothesis in disorder free system. The resultsare obtained using TDVP for lattice size up to L =
48. We believe that this claim has to becarefully verified analyzing di ff erent system sizes – this is beyond the scope of this work. Here,rather we would like to discuss features not exposed in [51] and, in particular, consider the e ff ectof adding a harmonic potential to the tilted lattice. -10 -5 0 5 1000.050.10.15 -10 0 10 2000.050.10.15 Figure 5: The density of state histogram for the Heisenberg chain with length L =
16, under the static field F = . F = . | D (cid:105) in energy spectrum, and the red stardenotes that of two-domain-wall state | D (cid:105) . It is important to realize where, in the energy spectrum of the system, the domain wall statesare located. This is discussed qualitatively in [51] we show the example in Fig. 5 where we plotthe density of states (DOS) for a smaller system with L =
16, amenable to exact diagonalization.8
500 1000 1500 2000Time t . . . . . (a) S at the left wall S at the center S at the right wall t . . . . (b) S N at the left wall S C at the left wall t . . . . (c) S N at the right wall S C at the right wall i − . − . . . . h ˆ S z i t = 1 t = 10 t = 20 t = 50 t = 100 t = 500 (d) i . . . . . . S N (e) t = 1 t = 10 t = 20 t = 50 t = 100 t = 500 i . . . . . . S C (f) t = 1 t = 10 t = 20 t = 50 t = 100 t = 500 Figure 6: The time evolution of the double domain wall state. Top row represents the entropy dynamics. After a short ini-tial transient the entropy saturates showing strong fast oscillations.The total entanglement entropy changes significantlyat left and right wall only (a). At left wall we have mainly classical single particle tunneling while di ff erent configura-tions contribute to the entropy at the right well. Averaging the fast oscillations (bottom row) reveals both the ling timesin distribution as well as the fact that the entropy, both classical S N and quantum S C , is produced mainly at the rightwall. It is apparent that the single domain wall state probes the low energy tail of DOS close to theground state. Then it seems not so interesting for “high temperature” limit, i.e. , evolution in theregion of maximal DOS. On the other hand the | D (cid:105) state lies close to the center of DOS beingseemingly much more relevant for a generic dynamics in the system. Let us note that a closeresemblance in time dynamics between | D (cid:105) and | D (cid:105) states were observed in [51] apparentlycontradicting the energy argument. That, in our opinion, is due to the fact that long stretchesof spins pointing in the same direction in domain wall states are e ff ectively immobile. Thus theinteresting dynamic is restricted to domain walls only. Those regions carry only the portion ofthe energy - so the energy argument involving the full system is to some extend irrelevant. Stillwe shall consider in the following | D (cid:105) states only for simplicity.While [51] consider the spin profile and the time dependence of the spin correlation functionand the entropy on the domain wall we find it worthwhile to consider enriched set of observables.Let us recall that the entanglement entropy may be divided into two parts, a classical like “numberentropy” and the inherently quantum “configuration entropy”: S ( t ) = − N (cid:88) n = p n log( p n ) − N (cid:88) n = p n (cid:88) i ρ ( n ) ii log( ρ ( n ) ii ) ≡ S N ( t ) + S C ( t ) , (5)where for N particles p n is the probability to have n particles in, say, left subsystem. The numberentropy S N describes a real exchange of particles between the two subsystems while S C quanti-fies the reshu ffl ing of particles among di ff erent configurations in left and right part of the system.Both quantities are plotted in Fig. 6. Here we assume F = . ff erent configurations ofspin occurs. The fast oscillations visible in the time domain may be removed by a high frequencyfilter (equivalent to an average over some range of final times). That allows us to get the longtime averages for the spin configuration as well as for the entropies presented in the bottom row.We stress that that the entropy production is quasi-local and is concentrated to the domainwall regions. Entropy production saturates fast, reaching quasi-stationary values already at sev-eral tens of (cid:126) / J . Similarly the long time spin profile is reached after a few tunnelings only.While we cannot claim that the observed behavior persists for very long times (yet we reachedmaximally 10000 tunneling times without changes in the picture observed), it seems that thequasi stationary state obtained for the spin profile and entropies reveals the equilibrium dynam-ics. If this is so that indeed, the double domain wall initial state breaks the ergodicity of thedelocalized regime and avoids thermalization. This claim, however, needs, in our opinion a moredetailed analysis. i − . − . . . . h ˆ S z i t = 1 , A = 1 / t = 500 , A = 1 / t = 1 , A = 1 / t = 500 , A = 1 / (a) i . . . . . . . S (b) t = 1 , A = 1 / t = 500 , A = 1 / t = 1 , A = 1 / t = 500 , A = 1 / i − . − . . . . h ˆ S z i t = 1 , A = − / t = 500 , A = − / t = 1 , A = − / t = 500 , A = − / (c) i S (d) t = 1 , A = − / t = 500 , A = − / t = 1 , A = − / t = 500 , A = − / Figure 7: The time dynamics of the double wall initial state in the combined static ( F = . A the e ff ective local field adds to the static field making meltingsmaller than for F = .
5. The opposite e ff ect occurs for negative A which enhances the melting of the right wall. – seethe discussion in the text. Let us now add a weak harmonic confinement to the static field. The results are presented10n Fig 7 for two sets of values of curvature A . for positive A the e ff ective field increases at thedomain wall. The A = /
120 corresponds to the additional local field at the wall position of F loc = . F tot = F + F loc = . A = / F = F tot indicating that the e ff ective local field notion works well in this case. Somewhatsurprizingly, at first, it is not so for a negative curvature A when local field weakens the e ff ectivefield. In particular A = − /
60 case would lead to F tot = . ff ect on the right wing of thedomain wall is quite dramatic - the melting reaches the edge of the chain. This is due to the factthat the global field F tot decreases fast to the right of the wall as a harmonic negative confinementterm becomes stronger with the distance from the center. In e ff ect the local static field at the rightedge of the system is merely F tot = .
1. This example shows that one has to carefully apply thepredictions of the local e ff ective field induced by the harmonic trap.Finally for this part, consider now the extreme case with no static tilt, i.e. F = A > ff ective field is negative at the leftand positive at the right wall. The harmonic trap acts apparently as the source of the local staticfield (tilt) symmetric for the left and for the right wall. t . . . . . . . (a) S at the left wall S at the center S at the right wall t . . . . . . (b) S N at the left wall S C at the left wall t . . . . . . (c) S N at the right wall S C at the right wall i − . − . . . . h ˆ S z i t = 1 t = 10 t = 20 t = 50 t = 100 t = 500 (d) i . . . . . S N (e) t = 1 t = 10 t = 20 t = 50 t = 100 t = 500 i . . . . . . S C (f) t = 1 t = 10 t = 20 t = 50 t = 100 t = 500 Figure 8: The fate of the double wall initial state in the purely harmonic potential. While at the left domain wallthe e ff ective field is negative, at the right wall it is positive (with the same magnitude). The dynamics seems to bequite symmetric at both walls. A ≈ .
058 corresponding to e ff ective local fields at the positions of the walls to be F e ff = ± ∼ . Finally let us consider bosons in a combined static tilt and the harmonic potential The Hamil-tonian of the system considered is:ˆ H = − J (cid:88) (cid:104) l , k (cid:105) ˆ a † l ˆ a k + U (cid:88) l ˆ n l (ˆ n l − + L / (cid:88) l = − L / ( Fl + Al n l . (6)11gain the tunneling J is set to be 1, and we fix interactions at U = J . The time evolution isstudied as, before, in unit of tunneling time (cid:126) / J . Following the dynamics with TDVP we againtake a 1D lattice of L =
48 sites. We consider mean half filling case with left L / L / | D (cid:105) )state. The double domain ( | D (cid:105) ) wall state has occupied sites in [ − L / , L /
4] interval. Note thatin this subsection the harmonic potential is not symmetric with respect to the center of the trap,the e ff ective local tilt is positive and adds to the constant tilt F – compare (6).Time propagation for bosons is more demanding numerically as one has to restrict localHilbert space. For the mean half feeling case we assume the maximal atom number on site to be n max =
4. The evolution is carried out as before restricting the auxiliary maximal bond dimensionto χ max = χ = χ max showing that the resultsare at the edge of the convergence. Still we believe, basing on a limited tests, that the resultspresented are converged to within the width of the line in the plots. InitialF = 0.5F = 1F = 2F = 3
InitialF = 0.5F = 1F = 2F = 3
Figure 9: The occupation profile for time-evolved states under di ff erent values of the static field as indicated in thefigure. The blue square denotes initial profile, | D (cid:105) on the left and | D (cid:105) on the right. The results are time averaged for t ∈ [400 , (cid:126) / J interval (due to fast oscillations). Fig. 9 shows the final profiles obtained for both | D (cid:105) (left) and | D (cid:105) (right) initial statesat di ff erent values of the static tilt F being well below the critical field F c estimated for thetypical transition to SMBL [46]. The results are averaged over a final time window to wash outthe e ff ect of rapid oscillations due to Bloch-like oscillations. While significant deformations ofinitial profiles are seen resulting in the wall melting the memory of the initial state is clearlypreserved up to the final times. Note that the wall itself does not prohibit transport as in thepreviously discussed spinless fermion case as multiple occupation of sites is permitted. For asingle domain wall a slight accumulation of particles in visible in the left part of the lattice.Similar excess occupation is visible for the | D (cid:105) in the middle of the chainThe melting dynamics may be put on a bit more quantitative basis by calculating the finalmelting range ξ . We define it as the distance between the site at which the initial populationchanges from unity to 0.95 and the position (on the other side of the domain wall) where thepopulation changes from 0 by 0.05. The resulting ξ ( F ) dependence is shown in Fig. 10. Observethat for both single and double domain states two sets of points overlap with very good accuracy.The blue squares correspond to a pure linear tilt with a given F value. The red symbols areobtained at field values corrected by the local tilt ∆ F = Al d due to the presence of an additional12armonic potential with AL =
10 – compare (6). The excellent agreement shows that the localfield approximation works very well for bosons.
Figure 10: The melting range ξ calculated from di ff erent domain wall: (a) The central domain wall of | D (cid:105) states, (b) Theleft domain wall of | D (cid:105) state, and (c) The right domain wall of | D (cid:105) state. The blue squares denote ranges calculatedfor a pure static force case while the red squares are for AL =
10 harmonic trap with a horizontal shift ∆ F . The ∆ F is determined by the domain wall position l d : ∆ F = Al d and the collapse of di ff erent data points shows the goodperformance of local field approximation.
6. Conclusions
We discussed the time dynamics of 1D lattice systems in a disorder-free potential combingthe static field (a constant in space tilt of the lattice) with the harmonic potential. Realistic lat-tice sizes of the order of L =
50 were studied using TDVP numerical routines. In particularwe reviewed the case of pure harmonic binding that may lead to the phase separation and thecoexistence of extended and localized regions. The border between the two regions can be wellestimated using the notion of the local electric field obtained from the derivative of the harmonicpotential – such an approach works quite well for su ffi ciently large lattices. This may be un-derstood by the fact that the localization, when it sets in the many particle system has a shortlocalization length of few physical sites only.We considered spinless or (briefly) spinful fermions, as well as bosonic systems. Manipu-lating with static and harmonic trap one can shift the localized regions as presented for spinlessfermions. We have noted that the presence of the localized regions slows down considerably theentanglement entropy growth also on the delocalized size.We have studied in detail in combined static and harmonic potential the dynamics of domainwall initial states. Their very slow and apparently nonergodic (on a relatively short time scaleof few thousands of the tunneling times studied) dynamics is related to the stretches of spinspointing in the same direction - such a situation prevents any real transport for spin 1 / ff ect the growth of entropy (being it the quantum one due to configuration change or thenumber entropy due to real particle transport over a studied bond) is limited to the very vicinityof walls separating domains (for spins). When no “frozen spin” stretches occurs, as for bosons,the melting occurs more e ff ectively although it is still a quite slow process and the memory ofthe initial state shape persists on the time scale of few hundreds of tunneling times. Again the13ocal e ff ective field due to a derivative of the harmonic potential describes very well the physicsstudied. It suggests that such a local field approach may work quite well for arbitrary smoothpotentials. Acknowledgments
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