Dynamical Many-body Localization and Delocalization in Periodically Driven Closed Quantum Systems
FFebruary 14, 2017
Dynamical Many-body Localization and Delocalizationin Periodically Driven Closed Quantum Systems
Asmi Haldar ∗ and Arnab Das Quantum interference lies at the heart of several sur-prising equilibrium and non-equilibrium phenomenain many-body Physics. Here we discuss two recentlyexplored non-equilibrium scenarios where external peri-odic drive applied to closed (i.e., not attached to any ex-ternal bath) quantum many-body systems have appar-ently opposite effects in respective cases. In one caseit freezes/localizes a disorder free system dynamically,while in the other it delocalizes a disordered many-bodylocalized system, and quantum interference is respon-sible for both the effects. We review these in the per-spective of more general questions of ergodicity, energyabsorption, asymptotic behavior, and finally the essen-tial role of quantum mechanics in understanding theseissues in periodically driven closed many-body systems.In this article we intend to deliver a non-technical ac-count of some recent developments in this field in amanner accessible to a broad readership.
Non-equilibrium dynamics of periodically driven closedquantum systems has gained significant attention re-cently, both because of its fundamental importance as apotential host of new quantum phenomena, and its recentexperimental realizations (see, e.g., [1–4]). Few centralissues, like energy absorption, thermalization, characteri-zation of the asymptotic behavior of the system and roleof quantum mechanics in qualitatively understanding ofthese issues is overarching theme of this review. Morespecifically, we focus on two different settings where ad-dressing these issues have lead to interesting phenom-ena and novel physical scenarios. Interest in periodicallydriven systems also stems from the possibility of gener-ating topologically non-trivial phases (see, e.g., [5] for areview) but here we restrain ourselves from discussingthose very interesting aspects. First, we review the phenomenon of dynamical many-body freezing (DMF) [6] in presence of disorder [7]. DMFis observed in a large class of translationally invariant in-tegrable systems under strong and rapid periodic drive:the drive induces destructive quantum interference ina massive scale (i.e., affecting almost all degrees of free-dom), and observables remain frozen close to their initialvalues for all time and for any arbitrary initial state. Thepicture here is, translational invariance and integrabilityallows one to map the many-body dynamics of these mod-els to the population dynamics of a set of independenttwo-level systems, where it is possible to tune the driveparameters (frequency and amplitude) in such a way thatstrong destructive interference simultaneously affects allthe modes. Introduction of disorder breaks translationalinvariance, rendering the above mentioned fine-tuningimpossible, and observables eventually decays with time.Here it is worth noting that disorder is usually associatedwith localization and consequent freezing of dynamics,hence one needs to choose suitable observables and ini-tial states in order to see the dynamics induced by dis-order. Interestingly, however, even in presence of strongdisorder, dramatic reminiscence of DMF still manifestsitself: an enormous enhancement of decay-timescale (or-ders of magnitude longer compared to the undriven case)is observed under the drive conditions corresponding tomaximal freezing in the disorder-free systems. Thus inthis case, periodic drive leads to freezing/localization inuniform systems and the disorder leads to unfreezing.Second, we review the effect of time-periodic drive onmany-body localized systems [8, 9]. Here, disorder andinteraction localizes a many-body system, and a periodicdrive (unlike in the case of dynamical localization), delo-calizes the system. Though the phenomenology soundsmore intuitive than that of DMF, its mechanism is subtlyquantum mechanical. ∗ Corresponding author E-mail: [email protected] Indian Association for the Cultivation of ScienceDepartment of Theoretical Physics2A & 2B Raja S. C. Mullick Road, Kolkata - 700032, India
Copyright line will be provided by the publisher a r X i v : . [ c ond - m a t . o t h e r] F e b . Haldar and A. Das: Dynamical Many-body Localization and Delocalization in Periodically Driven Closed Quantum Systems Figure 1
Dynamical many-body freezing of transverse magnetization in homogeneous Ising chain in periodically driven transversefield. (a) m z vs t (synchronization has not yet been attained within the t timescale shown). (b) Q (long-time average of m z ) vs ω -numerical and analytical results under rotating wave approximation are compared. (Fig. 1(b) taken from [6]) It is known since long that periodically driven quantumsystems with a single degree of freedom can undergodynamical localization (or freezing) due to strong de-structive quantum interference under certain drive condi-tions [10–16]. In some cases this happens even though theundriven system is quantum chaotic (i.e., the quantumsystem is obtained via quantization of a classically chaoticsystem). However, the physical picture that representsbest the role of quantum interference in such localizationphenomena can be quite diverse in different settings. Forexample, in case of a quantum kick-rotator, the mecha-nism of dynamical localization was identified [16,17] withthat of Anderson localization [18], while in the case ofa single particle moving on a plane [12] or in a double-well potential (coherent destruction of tunneling) [13, 14],the mechanism of freezing seems to be viewed best asan effective dressing of the couplings/mass term in theHamiltonian by the drive. An interesting connection be-tween dynamical localization and coherent destructionof tunneling has been revealed in [15], and what hap-pens to the phenomena in presence of interactions is aninteresting open question.A generalization of dynamical localization, namely dy-namical many-body freezing (DMF) has been observedrelatively recently in integrable translationally invariantquantum many-body systems [5, 6, 19–23] and has beenrealized experimentally [3]. DMF is a generalization of the conventional dynamical localization in the followingsenses. First, in DMF the effect of more than one (evenmutually non-commuting) terms in the Hamiltonian canbe simultaneously muted down via destructive interfer-ence induced by the drive. This is unlike the conventionaldynamical localization where only one term (e.g., the ki-netic energy) is suppressed. In simple cases, this mightrender strong freezing of certain observables for any ar-bitrary initial state , as the effective Hamiltonian respon-sible for the dynamics vanishes entirely [6, 19]. Second,DMF is a generalization of the conventional dynamicallocalization (observed for a single degree of freedom) to(infinitely) many-body systems (see, however, [21, 24–27]for conventional dynamical localization in many-bodysystems studied more recently). It is to be noted thatDMF is observed so far only in systems of non-interactingparticles, or in those which can be mapped to one such.However, it is “many-body" in the sense that it surviveseven in the case when the dynamics is not factorizable tosingle-particle sectors, i.e., it does not conserve the parti-cle number. Moreover, DMF can be observed in presenceof superconducting-like pair-creation/annihilation pro-cesses, which induces correlation between the particles(though not via a non-integrable interactions). Such inter-particle correlations are sufficient to drive long-range or-dering and quantum phase transitions in many of thesesystems. Here we review the fate of DMF in presence ofdisorder [7].In [7], the following disordered one-dimensional Isingchain subjected to a sinusoidal transverse field has been Copyright line will be provided by the publisherebruary 14, 2017 considered. The Hamiltonian is H ( t ) = − α J L (cid:88) i J i σ xi σ xi + − L (cid:88) i { h sin ( ω t ) + α h i } σ zi , (1)where σ α ’s ( α = x , y , z ) are components of Pauli spins, J i ’s and h i are respectively the (quenched) interactionsand on-site fields - both drawn randomly from a uni-form distribution between ( − +
1) . The transverse fieldis subjected to an external drive of frequency ω (period T = π / ω ) and amplitude h ( ħ =
1, and periodic bound-ary condition). The study focuses on the regime of strong ( h (cid:192) α J J i , α h i ) and fast ( ω (cid:192) α J J i , α h i ) drive. Onestarts with the ground state at the initial Hamiltonian H ( t = m z ( t ) as the response. DMF in the Homogeneous Chain (Fig. 1:
First we reca-pitulate the phenomenon of DMF in absence of disorder( α J J i = h i = ∀ i ) [6]. In this case m z ( t ) settles to a T − periodic state oscillating around a non-zero averagevalue Q = lim τ →∞ τ (cid:82) τ m z ( t ) d t for ever. Clearly, the sys-tem does not absorb enough energy to destroy the orderin the initial state however long one might drive (Fig. 1a). For dynamics with a fully polarized initial state, i.e., m z (0) =
1, the magnitude of Q can be used as a measureof freezing ( Q = Q = Q turns out to be a highly non-monotonic func-tion of the drive frequency ω : Q ≈ + J (4 h / ω )) (here J is the Bessel function of first kind of order 0, “ ≈ " de-notes the rotating wave approximation). Thus under thecondition J (4 h / ω ) =
0, (2)the system freezes maximally, and one gets Q ≈ P , P etc in Fig. 1 b (we will refer to thecondition in Eq. (2) as freezing peaks in the rest of thereview). Emergence of Stroboscopic Conserved Quantities andRole of Quantum Interference:
Existence of DMF clearlyindicates absence of ergodicity and breakdown of Fermi-Golden Rule type scenario even within the Hilbert spaceallowed by the inherent integrable structure of the model.To be precise, integrability leads to decoupling of the de-grees of freedom into independent two-level systems inmomentum space. If the dynamics was ergodic for thesetwo-level systems, each would keep on absorbing energyuntil it reaches an infinite temperature like state in thesense of having equal occupation probability for both en-ergy levels, and one would have lim t →∞ m z ( t ) =
0. The non-zero value of Q or freezing is a consequence of re-peated coherent interference of the amplitudes of thefundamental fermionic excitations in momentum space.If the interference effect is not taken into account, andthe drive is assumed to change the population only ac-cording to the transition probabilities (neglecting the in-terference between the transition amplitudes ) after eachcycle, the system asymptotically approaches m z = Q = Unfreezing by Disorder and Strong Remnants of DMF:The Phenomenology
In presence of disorder, m z alwaysdecays to zero regardless of the drive parameters at infi-nite time and DMF is eventually destroyed. The decay canbe fitted well with the exponential decay form (Fig. 2(a)) 〈 m z ( t ) 〉 = m z e − t / τ , (3)where the overbar denotes average over disorder real-izations. However, the decay time-scale τ depends dra-matically on the drive parameters ( h , ω ), and exhibits aspectacular reminiscence of DMF. As shown in Fig. 2(b),the relaxation time scale τ shoots up by several orders ofmagnitude when the freezing condition for the uniformchain (Eq. 2) is satisfied by the drive. Interestingly, if η is kept fixed to a value such that J ( η ) =
0, (correspond-ing to peak freezing in the homogeneous system) and ω is increased, then τ increases exponentially with it (seeFig. 2(c)). Thus, though introduction of disorder eventu-ally kills the freezing of m z , the timescale of decay still Copyright line will be provided by the publisher . Haldar and A. Das: Dynamical Many-body Localization and Delocalization in Periodically Driven Closed Quantum Systems Figure 2
Fate of dynamical localization in presence of random interactions without random fields. (a)
Exponential relaxationof the expectation value of m z with time for different values of the drive frequencies ω . Unless otherwise indicated, the driveamplitude h is fixed at . For certain specific values of ω (e.g., ω = ), the relaxation is tremendously slowed down.The relaxation in the absence of the drive is labeled separately for comparison. The inset compares a representative sample ofthe numerical data (shown as points) to the curves fitted to them using Eq. 3. (b) τ vs ω for fixed h : The sharp peaks indicatedramatic enhancements of τ for certain values of ω . Three of the most prominent peaks are identified as P − . The values of ω at these peaks are P ≈ , P ≈ , and P ≈ . Those values are identified to be the ones for which the effectiveHamiltonian H e f f vanishes. The red dot pointed by the arrow-head represents the case in absence of the drive. (c) τ vs ω atfixed η : Comparison of enhancement of τ as ω is increased keeping η = h ω fixed for two cases - under the freezing condition J ( η ) = ( η ≈ ), and away from it J ( η ) ≈ ( η = ) as marked in the Fig. Exponential enhancement of τ with ω is observed (numerical data fitted with the τ ( ω ) (cid:175)(cid:175) J ( η ) = = τ + τ s e ω / ω s form) under the freezing condition, while no noticeablevariation of τ is observed away from the freezing condition. Results are for L = averaged over > disorder realizations ofthe bonds J i . The error-bars due to disorder-induced fluctuations are about the point size, hence omitted. Qualitatively similarresults are observed with random fields. (Fig taken from [7]) bears a very strong signature of the extreme freezing ob-served in the absence of disorder. The Points of Maximal Freezing from Floquet Flow Equa-tion Approach:
It seems difficult to find analytical solu-tion of time-dependent Schrödinger equation with dis-ordered Hamiltonians. Hence one can resort to the fol-lowing Floquet analysis and determine the effective Flo-quet Hamiltonian approximately using a flow equationapproach [29]. For the present purpose, we adopt the sim-plest formulation of Floquet theory and define the effec-tive time-independent Hamiltonian that describes theevolution of the stroboscopically observed wave-functionas follows. Let us denote the time evolution operatorevolving a state through a period from t = (cid:178) to t = (cid:178) + T (0 ≤ (cid:178) < T ) by U ( (cid:178) ). Since U ( (cid:178) ) is unitary, it can always beexpressed in terms of a hermitian operator H e f f as U ( (cid:178) ) = e − i H e f f ( (cid:178) ) T . (4) Clearly, if observed in a “stroboscopic" fashion at in-stants t = (cid:178) , (cid:178) + T , . . . , (cid:178) + nT ( n is an integer), the dynam-ics can be considered to be effectively governed as if bya time-independent Hamiltonian H e f f . With H e f f onegets the same wave-function as that with H ( t ) at theinstants t = (cid:178) + nT , because the time-evolution opera-tor is same in both cases for evolution to those instants([ e − i H e f f T ] n = e − i H e f f nT ). This of course holds for every (cid:178) , hence we get different stroboscopic series for eachof them (actually choice of (cid:178) is equivalent to choosinga gauge, as shown in [5]). For characterizing the long-time behaviour of the system under rapid drive, it is suffi-cient to observe the system strobocopically, since nothingmuch happens within a single cycle. Hence it is sufficientto follow the dynamics governed by H e f f ( (cid:178) =
0) at t = nT .Moreover, the set of all (i.e., for all values of (cid:178) ) strobo-scopic observations are sufficient to construct the entire Copyright line will be provided by the publisherebruary 14, 2017 time evolution (see [5] for an elegant and efficient way ofextracting this information).The Hamiltonian in Eq. 1 can be mapped to the follow-ing non-interacting Hamiltonian using standard prescrip-tion (see, e.g., [22]). H ( t ) = − α J L (cid:88) i J i (cid:179) c † i c † i + + c † i c i + + h.c. (cid:180) − L (cid:88) i { h ( t ) + α h i } c † i c i , (5)with hard-core bosons created (annihilated) by c † i ( c i )These bosons satisfy { c † j , c j } =
0, and the usual bosoniccommutation relations for i (cid:54)= j . Also, h ( t ) = h sin ω t . Us-ing Floquet-flow equation technique, one can constructthe following analytical expression for H e f f for the evolu-tion from t = t = T governed by the Hamiltonian inEq. 5 [7]. H eff ≈ − J (cid:88) i j (0) i (cid:179) c † i c † i + + h.c. (cid:180) − J (cid:88) i j (1) i (cid:179) c † i c i + + h.c (cid:180) − µ (cid:88) i j (2) i c † i c i , (6)with η ≡ h / ω , and the constants j ( s ) i , µ defined as fol-lows. j (0) i ≡ α J i (cid:110) J ( η ) − α h i ω β ( η ) (cid:111) , j (1) i ≡ α J i J ( η ), j (2) i ≡ h i J , µ ≡ α J . (7)Here, J n ( η ) denote Bessel function of the first kind of or-der n , and β ( η ) ≡ (cid:80) n (cid:54)= J n ( η )/ n . The above is obtainedunder a rotating-wave approximation (RWA) which holdsfor ω (cid:192) J , α . This effective Hamiltonian accurately repro-duces the dynamics of the full system stroboscopically tothe leading order in α / ω .The above expression of H e f f shows that when J ( η ) = h i = H e f f vanishes, implyingcomplete freezing of the dynamics. Though this does nothappen here and m z decays to zero due to the effect ofthe higher order terms in α / ω (unlike in the case of the ho-mogeneous chain), the time-scale τ gets enormous jumpsat these points (see Fig. 2 a,b). For h i (cid:54)= τ aresmaller due to the presence of the β -term. Note that thehuge enhancement of time-scale is achieved since the drive strongly suppresses three different mutually non-commuting terms in the Hamiltonian - a hallmark of DMF.At the freezing peaks, τ can also be enhanced exponen-tially by tuning ω , (Fig. 2c), demonstrating a great controlachievable on the disordered induced decays via periodicdrive.Before concluding this section a few words on RWAseems to be in order. In RWA, one essentially goes intoa “rotating frame" where the sinusoidal drive term in theHamiltonian can be expanded into sum of terms of theform e ( i Ω t ) , and drops out all the terms for which the fre-quency Ω is much larger than the characteristic frequen-cies of the undriven system. DMF occurs when the drivefrequency is high enough to be off-resonant with all thecharacteristic frequencies of the system (i.e., in the limit ofzero-photon process). In the present case, this conditiontranslates to ω (cid:192) J , α [7]. A detailed and critical review onthe domain of validity of RWA is given in [30]. In this part of the review we take a different standpoint:We consider interacting systems where disorder inducesmany-body localization (MBL) in the Fock space (see,e.g., [18, 31–34]), and the question is if one can delocalizethe system by applying an external periodic drive. Butbefore addressing this, we first make a small detour andbriefly review the application of the Floquet formalism instudying the asymptotic properties a periodically drivenmany-body systems in general.
Consider a static Hamiltonian H hosting a many-body lo-calized phase be driven by a time-periodic (non-commutingwith H ) part H D ( t ). The total Hamiltonian is thus H ( t ) = H + H D ( t ), (8)and H e f f ( (cid:178) ) be the corresponding Floquet Hamiltonian(see Eq. (4)). Thenexp (cid:161) − i H e f f ( (cid:178) ) T (cid:162) = T exp (cid:181) − i (cid:90) (cid:178) + T (cid:178) d t H ( t ) (cid:182) , (9)where T denotes time-ordering. Without loss of general-ity one can set (cid:178) =
0. Let | µ i 〉 denote the i -th eigenstate of H e f f corresponding to the eigenvalue µ i . Copyright line will be provided by the publisher . Haldar and A. Das: Dynamical Many-body Localization and Delocalization in Periodically Driven Closed Quantum Systems Under “generic" initial conditions and considering “generic"local operators as observables, the nature of the asymp-totic state can be understood from the following. In orderto consider the fate of the system at long times, we con-sider a initial state | ψ (0) 〉 = (cid:88) i c i | µ i 〉 and an observable ˆ O = (cid:88) i , j O i j | µ i 〉〈 µ j | . 〈 ψ ( nT + (cid:178) ) | ˆ O | ψ ( nT + (cid:178) ) 〉 = (cid:88) i , j c i c ∗ j O i j e − i ( µ i − µ j )( nT + (cid:178) ) . (10)For a many-body system (infinite-size limit), by “generic" | ψ (0) 〉 and by “generic" operator ˆ O , we mean that abovesum is extensive in the sense that there are sufficientlylarge number of quasi-energy states participating in thesum. In that case (see [28, 35] for further conditions),at long times ( n → ∞ ) the off-diagonal ( i (cid:54)= j ) terms inthe sum oscillates rapidly and their contributions addup almost randomly, summing up to a vanishingly smallquantity. Hence under above conditions, the state of thesystem can be described by an effective “diagonal ensem-ble" given by the mixed density matrix [36]ˆ ρ Diag = (cid:88) i | c i | | µ i 〉〈 µ i | .Thus, the asymptotic properties of a periodically drivensystem are effectively given by a statistical average overthe expectation values of the eigenstates of H e f f , andhence it is sufficient to study the nature of the eigenstatesand eigenvalues of H e f f in order to understand the long-time behaviour. Moreover, reduction of the stroboscopicdynamics to that due to a time-independent H e f f impliesthat quasi-energy plays similar role in the stroboscopicdynamics as energy plays in the dynamics governed by atime-independent Hamiltonian. These hold regardlessof whether the system is ergodic or many-body local-ized, see, e.g., [37]. Here it is worth noting that in spiteof this reducibility, nature of the dynamics due to periodicdrive can be different from a generic undriven case in fun-damental ways, since in the former case H e f f might behighly non-local. For example, dynamics under a generictime-independent local Hamiltonian leads to thermal-ization at a finite temperature, while evolution under an H e f f derived from a time-periodic generic Hamiltoniancan lead to heating up to an effectively infinite tempera-ture scenario [38, 39]. MBL is a thermodynamically stable non-ergodic phase ofmatter (see, e.g., [18, 31–34]) where an interacting many-body system remains localized in the Fock space due todisorder in absence of an external bath. Here we addressif periodic drive can destabilize such a phase and heat itup indefinitely. From the discussion in Sec 3.1 it is clearthat in order to distinguish between an MBL and an er-godic phase in a periodically driven system, it is sufficientto focus on the properties/statistics of eigenstates andeigenvalues of H e f f [8]. The general scenario depends onthe absence (presence) of many-body mobility edge assummarized below. Two mechanisms by which periodicdriving might destroy MBL are identified. The first, ratherrobust, mechanism is the mixing of undriven eigenstatesfrom everywhere in the spectrum by the driving; if there isa mobility edge, this results in delocalization of all statesof the effective Hamiltonian. The second mechanism ismore subtle and involves strong mixing of states [28]which causes a delocalization transition at a finite drivefrequency for a given disorder strength. The key findingsare summarised in Table 1. Mobility edge low frequency high frequencypresent delocalized delocalizedabsent delocalized localized
Table 1
Effect of driving frequency in the presence and ab-sence of a mobility edge
This leads to a phase diagram outlined in Fig 3. Forlow enough ω and disorder strength, the system alwaysdelocalizes under the drive, while for high enough ω anddisorder the system remains MBL. The blue line in Fig. 3indicates a tentative boundary between these two phases,obtained by extrapolating the numerically determinedtransition points (red dots). In the following two differ-ent models (with/without the mobility edge) are consid-ered separately in order to illustrate the phenomenologyabove. A model of interacting hard-core bosons is considered,which is described by a driven, local Hamiltonian (Eq. 8) Copyright line will be provided by the publisherebruary 14, 2017
Figure 3
Driven MBL with no mobility edge: Plot of drivingfrequency ω c below which the system delocalizes as a functionof disorder amplitude w . The shaded areas correspond to de-localization. The red dots are obtained from finite-size studiesof the level statistics of the system. The disorder amplitude w c is the value below which the undriven system is delocalized inthe absence of driving. The blue line is a guide to the eye. (Fig.taken from [8]) with H = H hop + (cid:88) r = V r L − (cid:88) i = n i n i + r + L (cid:88) i = U i n i (11)where H hop = (cid:179) − J (cid:80) L − i = (cid:179) b † i b i + + b † i + b i (cid:180)(cid:180) is a hoppingoperator, the b are hard-core bosonic operators, U i anon-site random potential uniformly distributed between − w and + w and H D ( t ) a time-periodic hopping term H D ( t ) = δ ˜ δ ( t ) H hop (12)with δ a dimensionless constant, ˜ δ ( t ) = − +
1) in the first(second) half of each period T = π / ω . Via Jordan-Wignertransformations this model is related to a fermionic inter-acting system as well as to a spin-1/2 chain. The resultsare presented for V / J = V / J =
1, although the qualitativeconclusions are not sensitive to this.
Non-ergodicity of the MBL Phase from Eigenstate Ex-pectation Values (EEV):
It has been shown that genericinteracting systems, when driven periodically in timeat low frequencies (low compared to the bandwidth ofthe undriven system), the system keeps on absorbingenergy without bounds, ending up in a state which isindistinguishable from an infinite temperature state asfar the expectation values of local observables are con-cerned [38, 39]. This infinite temperature like scenario isreflected in the fact that the eigenstate expectation values
Figure 4
Plots of eigenstate expectation values (EEV) of thedensity at a single arbitrarily chosen site in all the eigenstatesof H e f f for a system with w / J = , size L = for aHilbert space dimension of D H = and driving ampli-tude δ / J = . For driving frequency above the blue line inFig. 3, ω / J = (left), the EEVs fluctuate wildly between dif-ferent eigenstates of H e f f . In contrast, for a driving frequencybelow the blue line, ω / J = (right), there is markedly lesseigenstate-to-eigenstate variation, consistent with all statesbeing fully mixed. This is the expected behaviour of the EEVsfor clean (therefore delocalized) driven systems (see Ref. [38]).In the undriven system the EEVs appear qualitatively similar tothose in the left panel. (Fig. taken from [8]) (EEV) of any local operator 〈 ˆ O 〉 i = 〈 µ i | ˆ O | µ i 〉 over all eigen-states of H e f f are almost equal to each other (i.e., whenEEV is plotted with respect to µ i it is almost flat for anygiven ordering of i ). On the other hand, in the localizedphase EEV fluctuates wildly. This is illustrated in Fig. 4. Locating the MBL-Delocalization Transition:
To accu-rately locate the localization-delocalization transition forthe driven system, the level statistics of the eigenval-ues of H e f f has been calculated. That is, after obtain-ing the quasi-energies (cid:178) n , one calculates the followingratio involving adjacent level spacings δ n = (cid:178) n − (cid:178) n + : r n = min ( δ n , δ n − ) / max ( δ n , δ n − ). The mean η = (cid:82) dr r P ( r )distinguishes between Circular Unitary Ensemble (er-godic) and Poissonian statistics (non-ergodic). One cal-culates η for a sequence of system sizes and extrapolatethe limit of η as L → ∞ , and consider the statistics of theextrapolated values of η .To obtain the frequency ω c above which delocalizationsets in for a driven system, one plots η for several valuesof disorder amplitude w , averaged over ∼ disorder re-alisations and for several system sizes. Typical results areshown in Fig. 5. The transition is located at the crossingpoint of the lines for different system sizes: if increasingsystem size results in larger η then we conclude that thesystem is delocalized, since η = η CUE for a delocalizedsystem and η = η P for a localized system with η P < η CUE .Here η CUE is the value for the CUE ensemble [39].
Copyright line will be provided by the publisher . Haldar and A. Das: Dynamical Many-body Localization and Delocalization in Periodically Driven Closed Quantum Systems Figure 5
Level statistics for various disorder amplitudes w / J as a function of driving frequency ω . The driving amplitude is δ / J = (cid:191) w / J , ω / J , and each point represents an average over 10000 disorder realisations. The dashed vertical lines indicatehalf the width of the energy spectrum; for ω greater than this, the results cannot be extrapolated to the thermodynamic limit. Thecolours correspond to system sizes L =
8, 10, 12 from bottom to top for the smallest ω . The values η CUE and η P correspond tothe presence and absence of level repulsion respectively, which in turn correspond to localized and delocalized phases. Thedotted vertical lines correspond to the typical spectral width of the system, for frequencies above which the results cannot beused to infer the thermodynamic limit. (Fig. taken from [8]). For the results to be applicable in the thermodynamiclimit, it is necessary (though might not be sufficient) totake the drive frequency ω much lower than the width ofthe energy spectrum of the undriven Hamiltonian. Themain practical problem is the following: with decreasingdisorder amplitude w and for fixed system size, the value ω c increases while the energetic width of the DOS de-creases. Since ω must be small compared to the width inorder for the extrapolation to the thermodynamic limitto be meaningful, the ω c for values of the disorder closeto w c are inaccessible for the system sizes for which thenumerics could be done. The width of the DOS is indi-cated in Fig. 5 by vertical lines; the crossing point of thecurves cannot lie to the right of this line, since otherwisethe finite size of the system would be important (and thusthe results would not be reliable in the thermodynamiclimit).Fig. 5 reveals the following features: for w / J ≤ L do not cross for values of ω below thebandwidth, indicating that the thermodynamic limit isdelocalized, as expected. For w / J >
6, there is a clear cross-ing point, which indicates the position of the transition. The crossing value of ω determined by this method is plot-ted as a function of w / J in Fig. 3. A Physical Picture:
Though the physical picture consis-tently fitting the full phenomenology described above isnot entirely clear, significant efforts have been made inthis direction [8, 40, 41]. We summarize the gist in the fol-lowing.
Delocalization Under Low-frequency Drive:
In the MBLphase and in the absence of driving, the system is effec-tively integrable in that there exist extensively many localintegrals of motion [33, 42–45]. The system may thus bethought of as a set of local subsystems, of finite spatial ex-tent. The matrix elements connecting these local systemsare suppressed in much the same way hopping amplitudebetween sites is suppressed in Anderson localization. Un-der slow periodic drive, the MBL Hamiltonian is replacedby an H e f f , which is not just the time-average of the peri-odic Hamiltonian over a period, but also have significantother components which consists of effective long-rangehopping and interactions. These terms might introducematrix elements (tunneling, say) between different local- Copyright line will be provided by the publisherebruary 14, 2017 ized systems, resulting in delocalized eigenstates of H e f f .From the numerical investigations discussed above, thisappears to be the case in general, but it is not obviousto what extent this argument should work, since in caseof simple non-integrable models like kick-rotor (see Sec.2), periodic drive actually induces localization by pro-ducing random destructive quantum interference (hencekilling matrix elements), and the problem can be exactlymapped to Anderson localization. Other interesting sce-narios are also being suggested and investigated recently,see. e.g., [46, 47] Stability of MBL towards High-frequency Drive:
Thiscan be understood as follows. For fast drive one can doMagnus expansion [48] for H e f f as follows. H e f f = ∞ (cid:88) n = H ( n ) e f f , where H (0) = T (cid:90) T H ( t ) d tH (1) = T ( i ħ ) (cid:90) (cid:178) + T (cid:178) d t (cid:90) t (cid:178) d t [ H ( t ), H ( t )] H (2) = T ( i ħ ) (cid:90) (cid:178) + T (cid:178) d t (cid:90) t (cid:178) d t ([ H ( t ), [ H ( t ), H ( t )]] + [ H ( t ), [ H ( t ), H ( t )]]) . . . (13)(14)For fast enough drive (small enough T ), the series mightconverge, and one can keep only the leading orderterm H (0) = T (cid:82) T H ( t ) d t [39]. In that case of course H e f f is trivially an MBL Hamiltonian, particularly when (cid:82) T H D ( t ) d t = The QREM as a case study:
We now turn to the case inwhich a mobility edge is present in the undriven spec-trum. The central result is based on the observation [38]that a periodic perturbation acting on a system coupleseach undriven state to states spread uniformly throughoutthe spectrum of H . As a result, if part of the spectrum cor-responds to delocalized eigenstates then all eigenstates of H e f f will necessarily be delocalized. Whether there existsmobility edge in local models of MBL is debatable [49],but delocalization for all values of ω has been numeri-cally confirmed by studying the Quantum Random EnergyModel (QREM), recently studied in Ref. [50] where it wasshown to have a mobility edge. The model is defined for N Figure 6
Driving the QREM. The top left figure shows the par-ticipation ratio φ for the eigenstates of the undriven model,showing (energy/quasienergy on the y axis) a mobile region(blue) surrounded by a localized region (red). Driving withfrequency ω / J = and amplitude δ / J = (top right)causes all states at a given Γ to become as delocalizedas the least localized state at that Γ in the undriven model.This is also shown in the bottom panel which shows φ for Γ = (red, blue and green line, from top to bot-tom) in the absence (presence) of driving with darker (lighter)colour. The driven points always lie below the undriven pointsfor the corresponding Γ . This is due to the strong mixing ofall undriven eigenstates by the driving. All data in this figure isfor spins and averaged over 1000 disorder realisations. (Fig.taken from [8]). Ising spins with the Hamiltonian H = E (cid:179)(cid:110) σ zj (cid:111)(cid:180) − Γ (cid:80) j σ xj ,where E is a random operator diagonal in the σ z basis(that is, it assigns a random energy to each spin configura-tion) and Γ a transverse field. Extensivity of the many-body spectrum is satisfied if the random energies aredrawn from a distribution P ( E ) = (cid:112) π N exp (cid:161) − E / N (cid:162) .The diagnostic of localization used here is the partici-pation ratio (PR), defined for the state | ψ 〉 with respect tothe Fock basis { | n 〉 } as φ = (cid:80) n (cid:175)(cid:175) 〈 n | ψ (cid:174)(cid:175)(cid:175) with n enumerat-ing Fock states. φ approaches unity for a state localizedon a single Fock state and 2 − N for one fully delocalized in Copyright line will be provided by the publisher . Haldar and A. Das: Dynamical Many-body Localization and Delocalization in Periodically Driven Closed Quantum Systems Figure 7
Proposed phase diagram for the long time state ofa driven strongly, disordered system as a function of the driv-ing frequency ω and strength δ . Red (I) and blue (III) indicateFloquet-ETH and Floquet-MBL behavior, where the system ap-proaches a fully-mixed, “infinite-temperature” state or remainslocalized, respectively. At (II), heating leads to energy growinglogarithmically slowly with time (Fig. 8(II)) over a broad timewindow. (Fig. taken from [9]). Fock space. The leftmost panel in Fig. 6 shows the aver-age φ versus energy (scaled with system size) of the 256eigenstates of an undriven N = Γ ( t ) = Γ (cid:161) + δ ˜ δ ( t ) (cid:162) , ˜ δ ( t ) = + −
1) for the first (second) halfof the period with an amplitude δ = ω = π / T = H e f f areshown in the second panel of Fig. 6. As expected, periodicdriving causes delocalization of the entire spectrum solong as part of the undriven spectrum at the same Γ isdelocalized. Energy absorption by an MBL system under periodic drivehas been studied extensively in [9]. The model studiedwas spin-1/2 XXZ chain in a disordered longitudinal field under monochromatic drive with period T = π / ω : H ( t ) = H + H D ( t ) (15) H = J ⊥ L (cid:88) i = ( S xi S xi + + S yi S yi + ) + J z L (cid:88) i = S zi S zi + (16) + L (cid:88) i = h zi S zi , (17)where h zi ∈ [ − κ , κ ], J ⊥ , J z ≥
0, and with driving H D ( t ) = − δ cos ω t L (cid:88) i = ( − i S zi . (18)The static part H is known to be MBL for J z (cid:54)= κ > κ c . Starting fromthe ground state of H at t =
0, real-time dynamics of“rescaled excess energy density" defined as E ex ( nT ) = 〈 ψ | H ( nT ) | ψ 〉 − E min E − E min , (19)with E = D − H tr[ H (0)] and D H being the Hilbert space di-mension, so that E ex = H (0), while E ex = H . This is the “infinite temperature" like sce-nario observed when a disorder free interacting system isdriven periodically [38].For strong disorder ( κ > κ c ), a qualitative phase-diagram depending on ω and δ (Fig. 7) has been proposed.Three regimes has been identified (marked in the figure)as follows. There are two regimes (I and III), which areknown from earlier works [8, 40] (see also Sec. 3.2.1). Inregime (I) an initially MBL state delocalizes and heats upto a state in which the system locally looks as if it is at infi-nite temperature (i.e., the expectation value of the localobservables over the state equals to that over an infinitetemperature ensemble). In (III), the drive fails to delocal-ize the system, but only pumps some energy into it. Theenergy of the system thus settles to some intermediate av-erage value. However, the numerics seems to suggest thereis another intermediate regime (II), where the system tendto reach the infinite temperature like scenario, but onlylogerithmically slowly. Sample of real time behaviour ofthese regimes are given in Fig. 8. Further technical details,particularly those relevant for validation of the numericalresults are given in detail in [9]. In this article we illustrate that periodic drive can induceboth freezing/localization and unfreezing/delocalization Copyright line will be provided by the publisherebruary 14, 2017
Figure 8
Stroboscopic excess energy E ex ( nT ) in a strongly disordered system with η = and J z = , corresponding to threedifferent regimes. In (I), an initially MBL system delocalizes and heats up to a fully-mixed state. In the intermediate regime, (II),the system heats up to the fully-mixed state, but logarithmically slowly. This slow growth persists for longer times as we increase L . For Floquet-MBL, (III), driving does not delocalize the system, leading instead to a localized long-time state which has partiallyheated up to some intermediate energy. (Fig. taken from [9]). dynamically when applied to closed quantum systemswith many-degrees of freedom depending on the circum-stances. It has been argued that quantum interferenceplays an important role in all these phenomena, thoughthe intuitive pictures consistent with different circum-stances are quite varied. For example, while in the caseof DMF the simplest physical picture of freezing seemsto consist of renormalization of the Hamiltonian by theperiodic drive resulting in suppression of dynamics, dy-namical localization in a periodically kicked rotor canbe understood more easily by mapping it to the Ander-son localization problem, implying that the drive induceddynamical randomness has similar localizing effect asquenched disorder in static problem. Yet, in an inherentlylocalized many-body system with interaction, periodicdrive at low frequencies can cause delocalization . Therethe picture is, the drive generates matrix elements con-necting the spatially isolated localized parts of the un-driven MBL system. The system then eventually heats uptill it reaches a state which looks like an infinite tempera-ture state when expectation values of local observables aremeasured. Interestingly, it seems no noticeable dynamicallocalization is observed in periodically driven MBL in thelow ω regime (at least, not sufficiently strong to exhibitany noticeable freezing effect), unlike that observed insingle-body non-integrable quantum chaotic system likethe kick-rotator. The issue of energy absorption underexternal drive in system not attached to a bath is still abroad open question, and are being pursued under differ-ent drive protocols [51, 52].A number of interesting questions present themselves. For example, whether under high frequency drive, mech-anism of dynamical localization steps and lends an MBLphase greater stability?What happens if one drives an integrable models (whichcan be mapped to non-interacting fermions) with verylow frequencies, and the Magnus expansion breaks down?Moreover Does one achieve effective infinite tempera-ture thermalization scenario there? If yes, then how doesthe transition/crossover from PGE to thermal phase takesplace ? If not, then certainly interaction kills it (evidencesindicate that MBL gives way to ergodic phases whendriven with low enough frequencies). Then the behav-ior of MBL-ergodic crossover as a function of interactionstrength would be interesting.Can one expect Periodic (generalized) Gibbs’ Ensemble asa local description of the asymptotic states in periodicallydriven integrable systems which cannot be mapped tofree fermions? This seems plausible, but to our knowledgethere is no general proof of existence of extensive num-ber of stroboscopic conserved quantities necessary forthis. The same (open) question appears interesting in thecontext of DMF, in particular, whether extreme freezingpoints can occur in such integrable systems.One might wonder what happens when a bath is weaklycoupled to a periodically driven system. The effect ofquantum interference would be affected by external de-coherence and dissipation, and of course the asymptoticbehaviour cannot be expected to be determined by a sta-tistical average of the property of H e f f (system wouldnot go to a diagonal ensemble in the eigenbasis of H e f f Copyright line will be provided by the publisher . Haldar and A. Das: Dynamical Many-body Localization and Delocalization in Periodically Driven Closed Quantum Systems in general). The intuitive pictures developed here mighthave to be revised significantly, and emergence of funda-mentally new pictures are not unlikely. Acknowledgements:
AD acknowledges collaborationswith S. Bhattacharyya, S. Dasgupta, A. Lazarides, R. Moess-ner and A. Roy in various works covered in this article. AHand AD thankfully acknowledge support from DST-MPIpartner group program “Spin liquids: correlations, dynam-ics and disorder" between MPI-PKS (Dresden) and IACS(Kolkata), and the Visitor’s Program of MPI-PKS for a visitto MPI-PKS, during which many interesting discussionson the subject took place.
Key words.
Floquet system, thermalization, periodic Gibbs’ensemble, many-body localozation, dynamical many-bodyfreezing