Distinctive response of many-body localized systems to strong electric field
DDistinctive response of many-body localized systems to strong electric field
Maciej Kozarzewski, Peter Prelovˇsek,
2, 3 and Marcin Mierzejewski Institute of Physics, University of Silesia, 40-007 Katowice, Poland J. Stefan Institute, SI-1000 Ljubljana, Slovenia Arnold Sommerfeld Center for Theoretical Physics,Ludwig-Maximilians-Universit¨at, D-80333 M¨unchen, Germany
We study systems which are close to or within the many-body localized (MBL) regime and aredriven by strong electric field. In the ergodic regime, the disorder extends applicability of the equi-librium linear–response theory to stronger drivings, whereas the response of the MBL systems is verydistinctive, revealing currents with damped oscillations. The oscillation frequency is independent ofdriving and the damping is not due to heating but rather due to dephasing. The details of dampingdepend on the system’s history reflecting nonergodicity of the MBL phase, while the frequency of theoscillations remains a robust hallmark of localization. We show that the distinctive characteristic ofthe driven MBL phase is also a logarithmic increase of the energy and the polarization with time.
PACS numbers: 71.23.An, 75.10.Pq, 75.10.Jm, 05.60.Gg
Introduction.–
The many–body localized (MBL) sys-tems together with the Anderson insulator [1–4] mightrepresent the only generic solid–state systems which donot thermalize [5–14] in the thermodynamic limit andmay be used to store quantum information [15, 16].Among most characteristic features of the MBL systemsare the absense of d.c. transport at any temperature, T , [17–19] and the entanglement entropy growing onlyas a logarithmic function of time [20–23]. The MBL hasrecently been identified in optical lattices [24] by measur-ing the relaxation dynamics of a particular initial statewith charge–density–waves (see also [25, 26]). In orderto stimulate further experimental studies, it is essentialto specify which hallmarks of the MBL [27–35] could bedetected with well established experimental techniques.Several theoretical studies have reported unusual proper-ties of the optical conductivity, σ ( ω ), obtained from thelinear response (LR) theory [17–19, 29, 36–40]. The cru-cial observations concern the low–frequency part of σ ( ω )that goes as | ω | α with the exponent 1 ≤ α < σ ( ω ) can be measured via the opti-cal spectroscopy. However, it is still unclear when (orwhether) the equilibrium LR theory itself is applicablein such systems. It has recently been found that sub-ject to nonzero driving they go nonlinearly and display ahighly nonlocal response at low enough frequencies [41].Moreover, these systems are expected to posses extensivenumber of local conserved quantities [22, 42–47] hence,due to these conservation laws, they do not thermalize[16], whereas it is common for the LR studies to startfrom a thermal initial state. A first step in clarifyingthese essential problems is to study the MBL systemsdriven by a non-zero electric field. Since strong fieldsdrive the system out of equilibrium, such studies allowto test not only the linearity of the response but also theconsequences of nonthermal initial states. The evolution of the MBL under strong fields is impor-tant also for a general understanding of driven lattice sys-tems. Typically, the particle and energy currents Blochoscillate with a frequency that is proportional to the field[48], as confirmed for the Falicov–Kimball model [49], in-tegrable [50] and nonintegrable [48] models of spinlessfermions as well as for the Hubbard model [51, 52]. Dueto finite d.c. conductivity, driving causes also the Jouleheating that damps Bloch oscillations.In this work, we show that the strong-field responseof the MBL systems is very different from the responseof standard tight–binding models. Although the particlecurrent undergoes damped oscillations, the frequency isfield-independent and the damping is not due to heat-ing but mainly due to dephasing. The magnitude of thecurrent as well as the damping depend on the initial con-ditions, i.e., response within the MBL phase reveals pro-nounced memory effects, but on the other hand the os-cillation frequency is independent of the initial (possiblynonthermal) state and is a very robust hallmark of thelocalization. We show furtheron that the oscillations canbe well attributed to the local physics within the MBLregime and explained with a local toy-model. But beyondthat our result clearly display also a logarithmic increaseof energy under constant driving which is evidently anonlocal effect, having an analogy with a similar slowbut steady increase of the entanglement entropy withinthe MBL phase [20–23]. Model–
We study interacting spinless fermions on aone–dimensional lattice with periodic boundary condi-tions. The system is threaded by a time–dependent mag-netic flux, φ ( t ), which induces the electric field F ( t ) = − ˙ φ ( t ). The time–dependent Hamiltonian reads, H ( t ) = − t h (cid:88) j (cid:104) e iφ ( t ) c † j +1 c j + H .c. (cid:105) + (cid:88) j ε j ˆ n j + V (cid:88) j ˆ n j ˆ n j +1 + V (cid:48) (cid:88) j ˆ n j ˆ n j +2 , (1) a r X i v : . [ c ond - m a t . s t r- e l ] F e b where ˆ n j = c † j c j . The hopping integral is taken as theenergy unit, t h = 1. The potentials ε j have uncorre-lated random values, uniformly distributed in the inter-val ( − W, W ). We have introduced the nearest–neighborrepulsion, V , as well as the next-nearest-neighbor repul-sion, V (cid:48) , so that the ballistic transport is avoided alsofor W → V (cid:48) = 0, first we check how thisinteraction affects the MBL transition. We repeat theanalysis of the energy-level statistics in Refs. [7, 28] anddetermine the ratio of two consecutive level spacings, δ n , δ n +1 . In Fig. 1a we show the average value of the ratio r = (cid:104) r n (cid:105) , r n = min { δ n , δ n +1 } / max { δ n , δ n +1 } for systemsof L = 10 , ,
14 sites. Upon increasing W , we observea change from r (cid:39) .
53, consistent with the Wigner-Dyson distribution for ergodic systems, to the result ofthe Poisson-distribution, r (cid:39) .
39, which is characteristicfor nonergodic (e.g., localized) systems [7]. In the ther-modynamics limit, the ergodic regime should extend atleast upto W ≤ W ≥ Time-evolution –
We assume that the field is switchedon at t = 0. For each set of { ε i } , the initial state | Ψ (cid:105) is chosen as a (thermal) microcanonical state [54] with N = L/ E , the latterrepresenting a high- T state. The relation between E and T can be then well estimated employing the high– T expansion for the model, Eq.(1), within the canonicalensemble [55], E = E ∞ − βL V + V (cid:48) − β (cid:88) i ε i , (2)where E ∞ (cid:39) L ( V + V (cid:48) ) / T → ∞ and β = 1 /T . If not specified otherwise, we choose β = 0 . L = 20 and V = V (cid:48) = 1. The time–evolutionof | Ψ t (cid:105) is obtained with the help of the short–iterativeLanczos method [56] and the Chebyshev polynomial ex-pansion of the time–propagator [57]. We calculate theenergy E t = (cid:104)(cid:104) Ψ t | H ( t ) | Ψ t (cid:105)(cid:105) c and the particle current I t = (cid:104)(cid:104) Ψ t | J ( t ) | Ψ t (cid:105)(cid:105) c , where J = − ddφ H ( t ) /L and (cid:104) ... (cid:105) c represents averaging over disorder configurations. Results –
First, we recall how generic systems respondto constant driving with F ( t >
0) = F . In Fig. 1b weplot I t as a function of flux φ ( t ) = F t for a weak dis-order. Here, I t vanishes for t → ∞ due to the Jouleheating, ˙ E t = F LI t [50]. For weak F < I t . On the other hand, for F ≥ φ ( t )], with afrequency ω B ∼ F . Such field–dependence of ω B is thecharacteristic feature of the Bloch oscillations. Fig. 1dshows seemingly similar behavior for MBL systems inthat I t also undergoes damped oscillations. However,the frequency is clearly field–independent, whereas theamplitude is roughly proportional to F . Ergodic regime –
Before explaining the latter result, we W r (a) L = 10L = 12L = 14 0 10 20 30 φ −0.06−0.030.000.030.060.09 I t (b) F = 0.3F = 0.5F = 1.0 F = 1.5F = 2.0F = 3.04 5 6 7 8 9 E t −0.020.000.020.04 I t F (c) F = 0.3F = 0.5 F = 1.0F = 1.5 0 5 10 15 20 t −0.0050.0000.0050.0100.0150.0200.025 I t F (d) F = 0.5F = 1.0F = 1.5 F = 2.0F = 2.5F = 4.0
FIG. 1. (Color online) (a) The level spacing ratio r vs. dis-order W ; (b) current I t vs. φ ( t ) for weak disorder W = 1; (c)conductivity I t /F vs. instantaneous energy E t for intermedi-ate W = 3, and (d) I t /F vs. t for W = 6 within the MBLphase. All results are for the V = V (cid:48) = 1. briefly discuss the case of intermediate disorder, 2 (cid:46) W (cid:46)
4, which is too weak to cause MBL but strong enough toproduce anomalous optical response [18, 19, 36, 38, 40].Such systems are ergodic and relax towards the thermalstate, hence the only concern related to the applicabilityof the equilibrium LR theory is whether the response isindeed linear in F . Strictly speaking, even a slow Jouleheating is a nonlinear effect which, however, can be easilyaccounted for within a simple extension of the LR theory[50]. A convenient way to filter out the heating effect isto plot the observables as a function of the instantaneousenergy E t (see Fig.1c). For modest driving the systemthen undergoes a quasi-thermal evolution, i.e., the time-dependent expectation values of all local operators areexpected to be determined solely by E t [55]. Fig. 1cshows that in the long–time regime, the effective conduc-tivity I t /F for intermediate W = 3 is indeed uniquelydetermined by E t and roughly F –independent. Compar-ing further results for moderate driving ( F ∼
1) one findsnon–oscillatory linear response for intermediate W = 3(Fig. 1c) and very clear Bloch oscillation for weak dis-order W = 1 (Fig. 1b). So our main conclusion for theergodic phase is that the LR theory is applicable to muchlarger fields in more disordered systems. MBL regime - memory effects–
A particularly interest-ing aspect of MBL are the memory effects. Since MBLsystems do not thermalize, their response may depend onthe history of the system, in particular, whether it was t −0.0250.0000.0250.050 I t (c) driving W = 3.0W = 4.0W = 5.0W = 7.0 0 10 20 30 t −0.030.000.03 I t (d) driving V=0, V’=0V=1, V’=1V=3, V’=10 10 20 30 40 50 t −0.010.000.010.020.030.04 I t (a) driving driving t −0.04−0.020.000.020.040.060.08 I t (b) driving driving FIG. 2. (Color online) Current I t vs. t for various parametersand driving protocols. Horizontal arrows mark time windowswhere F (cid:54) = 0: (a) weak W = 1 and F = 0 .
3, and (b) large W = 6 and F = 3 in marked time slots, respectively (Dashedlines in (a) and (b) show response to analogous second pulsebut for systems which up to t = t = 25 are in thermal states);(c) fixed driving F = 1 and various W ; (d) fixed W = 6 and F = 1 . V, V (cid:48) . previously driven out of equilibrium. It order to studythis effect, we turn off the driving for a time intervalsuch that the transient particle current relaxes I t ∼ t = t . Evolutionof I t under such specific driving is shown by continuouslines in Figs. 2a and b for the ergodic and MBL regimes,respectively. For comparison we present also the effectof the second pulse provided that the pulse excites thesystem within the thermal (microcanonical) state withthe same energy (see dashed curves). It clearly follows,that in contrast to the ergodic case, the MBL regime haspronounced memory effect, i.e. the response strongly de-pends on the initial conditions. Current oscillations–
Furtheron we return to oscilla-tions of I t close to or within the MBL regime. Resultspresented in Figs. 2c,d reveal that the frequency ω (cid:39) ω is independent of W as well as of inter-actions V, V (cid:48) , as seen in Figs. 2c,d. We note, that weaksignatures of damped oscillations are visible even in theergodic phase for W ∼ F ( t >
0) = F sin( ωt ). Other type of periodic driving has been stud-ied in Refs. [58, 59]. Fig. 3a shows that the strongestabsorption of energy is exactly for ω = ω . We find for t −6−303 E t (a) ω = 1.0ω = 2.0ω = 4.0ω = 6.0dc 10 −1 t − t −0.50.51.52.5 E t (d) t −0.010.000.010.020.030.040.05 I jt (b) t −0.02−0.010.000.010.02 I t (c) V=1, V’=1V=0, V’=0model
FIG. 3. (Color online) (a) Energy E t vs. t for a.c. driving F ( t >
0) = 3 sin( ωt ) including d.c. case F = 3, all for W = 6;(b) local currents I tj on consecutive bonds (shifted verticallyfor clarity) for d.c. F = 3 within the MBL regime, W = 6;(c) numerically obtained I t for F = 1, W = 8 within the in-teracting and noninteracting models, respectively, comparedwith result from the toy-model. (d) Long-time variation of E t for the same cases as in Fig. 2b. All results except c) arefor V = V (cid:48) = 1. such driving that E t increases and eventually approachesthe T = ∞ value ( E ∞ ≈
10 for parameters in Fig. 3a),whereas for the d.c. driving the energy apparently sat-urates at much lower values (see more detailed analysisfurtheron). This last observation can be reconciled withthe LR result for the MBL regime, σ dc ∼
0, which re-mains qualitatively valid even for strong fields
F >
1, asshown in Fig. 3a.A clearer picture of the oscillations arises from com-paring the currents flowing on individual bonds, I jt = (cid:104) Ψ t | ( i e iφ ( t ) c † j +1 c j + c . c . ) | Ψ t (cid:105) , (3)so that I t = (cid:80) j (cid:104) I jt (cid:105) c /L . In the ergodic phase, the cur-rents on the neighboring bonds, I jt and I j +1 t , are quitecorrelated with each other (not shown). However in theMBL regime, I jt and I j +1 t oscillate with very differentfrequencies and magnitudes, as shown in Fig. 3b. In con-trast to I t , the damping of currents on individual bondsis hardly visible. The latter result clearly indicates thatdamping of I t is actually due to destructive interferenceof various I jt . Toy model.
Within the MBL phase, the currents onneighbouring bonds appear to be independent of eachother. This suggests that results for decoupled two–siteclusters should capture the essential physics. Therefore,we briefly discuss a toy-model on two sites with the fol-lowing Hamiltonian and the current operator H ( t ) = (cid:32) (cid:15) + F ( t )2 − (cid:15) − F ( t )2 (cid:33) , J = (cid:18) i − i (cid:19) . (4)The distribution of random (cid:15) , depending on W , can alsoincorporate the many–body interaction between neigh-boring clusters. Here, we assume only that the probabil-ity density is even f (cid:15) = f − (cid:15) . An arbitrary initial statecan be written as ρ (0) = x | φ − (cid:105)(cid:104) φ − | +(1 − x ) | φ + (cid:105)(cid:104) φ + | +( α | φ − (cid:105)(cid:104) φ + | + H . c . ) , (5)where ( x − / + | α | ≤ /
4, while | φ ± (cid:105) are eigenstatesof H (0) with energies ±√ (cid:15) . In general, x and α may depend on (cid:15) , e.g., for the thermal state one obtains α = 0 and x (cid:15) = 1 / β √ (cid:15) ) /
2. We assumethat x (cid:15) = x − (cid:15) and α (cid:15) = α − (cid:15) . Then, straightforwardcalculations show that driving F ( t ) = F θ ( t ) induces thecurrent (given here only up to the linear term in F ), I t = (cid:104) Tr[ ρ ( t ) J ] (cid:105) c = (cid:104) I t + I Ft (cid:105) c + O ( F ) , (6) I t = − (cid:61) [ α exp( i (cid:112) (cid:15) t )] , (7) I Ft = (cid:18) x − (cid:19) sin(2 √ (cid:15) t )1 + (cid:15) F. (8) I t is independent of driving and arises solely due to non–steady initial conditions ( α (cid:54) = 0), whereas I Ft describesthe LR response. Eq. (8) explains, at least qualitatively,why the largest amplitudes of I jt shown in Fig. 3b areoscillating with the smallest frequencies. In the long-time regime, the disorder–averaged (cid:104) I Ft (cid:105) c can be obtainedanalytically for arbitrary x (cid:15) and f (cid:15) . The asymptotic formis then (cid:104) I Ft (cid:105) c = (cid:114) πt f (cid:15) =0 (cid:18) x (cid:15) =0 − (cid:19) F sin (cid:16) t + π (cid:17) . (9)The average current oscillates with the smallest possiblefrequency, ω = 2, and decays slowly in time as 1 / √ t due to destructive interference of oscillations with dif-ferent frequencies (as seen in Fig. 3b). Fig. 3c showsnumerical results for the original Hamiltonian (1) com-pared with (cid:104) I Ft (cid:105) c obtained from Eq. (9) for f = 1 /W and x = 1 / β ) /
2. The toy-model is too simple to de-scribe details of the damping which appears from Fig. 2dto be mostly determined by the many–body interactions.However, the toy-model correctly reproduces the specificfrequency ω of these oscillations. Most importantly, itexplains also why the same frequency is obtained for var-ious types of disorder ( f (cid:15) ) and various (also nonthermal)initial conditions ( α (cid:15) , x (cid:15) ). Logarithmic increase of energy and polarization–
Within the fully localized regime one expects that thedriving with constant field F would finally lead to the saturation (or oscillation) of various quantities. This isindeed the case for the noninteracting Anderson model.A more detailed analysis of the MBL results, however,reveals a deviation at long times. In Fig. 3d we present E t in the long–time window for exactly the same casesas in Fig. 2b, i.e. for thermal and non–thermal initialstates. In both cases, one observes a slow steady growthof E t even for t (cid:29)
1, which is consistent with the log-arithmic dependence, ∆ E t ∝ log( t ). Since the increaseof the energy is exactly related to the current [50] as˙ E t = LI t F ( t ), upon constant F ( t ) = F one can directlytest the plausible relation ∆ E t = L ∆ P t F , where P t isthe polarization of the system. Hence, the observed vari-ation implies as well the dependence ∆ P t (cid:29) ∝ log( t ).The growth without an upper bound opposes the plausi-ble picture that MBL insulator is a dielectric with a finitepolarizabiltity χ = P/F , still it is consistent with recentfindings [18, 36, 40] that the low-frequency dynamicalconductivity beyond the transition to the MBL phasebehaves as σ ( ω ) ∝ ω α with α ∼
1. Namely, at leastwithin the LR theory we would get χ ∝ (cid:82) dωσ ( ω ) /ω which for α = 1 diverges logarithmically. Similarly tothe particle current, also the polarization shows strongmemory effects, however the logarithmic character of itsgrowth seems to be independent of the initial conditions,as shown in Fig. 3d. Conclusions–
We have identified distinctive propertiesof the MBL systems driven by non–zero (strong) electricfield. They can be classified according to locality andlinearity of the underlying physics:(a) The oscillations of the particle current with afield-independent frequency are very pronounced alreadywithin the crossover to the MBL phase and even moredeeper within it. Our toy–model confirms that they arethe consequence of very local (two-site) physics, while de-cay of oscillations in the nonergodic (MBL phase) is dueto the dephasing. Such oscillations might be observedalso in recent experiments [25].(b) We find within the MBL phase very pronounced non-trivial memory effects which typically reveal physics be-yond the equilibrium LR.(c) In contrast to above phenomena, the observed log-arithmic increase of energy and polarization with timeunder constant driving clearly goes beyond the localphysics. 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