Dynamical Many-body Localization and Delocalization in Periodically Driven Closed Quantum Systems

TL;DR

This paper reviews how periodic driving can either localize or delocalize quantum many-body systems, highlighting quantum interference's role in these contrasting phenomena and discussing implications for ergodicity and energy absorption.

Contribution

It provides a non-technical overview of recent developments in understanding dynamical many-body localization and delocalization under periodic driving.

Findings

Periodic drive can induce localization in disorder-free systems.

Periodic drive can delocalize disordered many-body localized systems.

Quantum interference is key to both localization and delocalization effects.

Abstract

Quantum interference lies at the heart of several surprising equilibrium and non-equilibrium phenomena in many-body Physics. Here we discuss two recently explored non-equilibrium scenarios where external periodic drive applied to closed (i.e., not attached to any external bath) quantum many-body systems have apparently opposite effects in respective cases. In one case it freezes/localizes a disorder free system dynamically, while in the other it delocalizes a disordered many-body localized system, and quantum interference is responsible for both the effects. We review these in the perspective of more general questions of ergodicity, energy absorption, asymptotic behavior, and finally the essential role of quantum mechanics in understanding these issues in periodically driven closed many-body systems. In this article we intend to deliver a non-technical account of some recent developments in this field in a manner accessible to a broad readership.

Tables

Table 1. Table 1: Effect of driving frequency in the presence and absence of a mobility edge

Mobility edge	low frequency	high frequency
present	delocalized	delocalized
absent	delocalized	localized

Equations

$$H(t)=-\alpha J\sum _i^L{J}_i{\sigma }_i^x{\sigma }_{i+1}^x-\sum _i^L\left\{h_0\sin (\omega t)+\alpha h_i\right\}{\sigma }_i^z,$$

$${\mathcal{J}}_0(4h_0/\omega )=0,$$

$$\overline{⟨m^z(t)⟩}=m_0^z{e}^{-t/\tau },$$

$$U(ϵ)=e^{-i{H}_{eff}(ϵ)T}.$$

$$\begin{gathered} H(t)=-\alpha J\sum _i^L{J}_i\left(c_i^{†}{c}_{i+1}^{†}+c_i^{†}{c}_{i+1}+h.c.\right) \\ -2\sum _i^L\left\{h(t)+\alpha h_i\right\}{c}_i^{†}{c}_i, \end{gathered}$$

$$\begin{gathered} H_{\mathrm{eff}}\approx -J\sum _i{j}_i^{(0)}\left(c_i^{†}{c}_{i+1}^{†}+h.c.\right) \\ -J\sum _i{j}_i^{(1)}\left(c_i^{†}{c}_{i+1}^{}+h.c\right)-\mu \sum _i{j}_i^{(2)}{c}_i^{†}{c}_i^{}, \end{gathered}$$

$$j_i^{(0)}$$

$$j_i^{(1)}$$

$$j_i^{(2)}$$

$$\mu$$

$$H(t)=H_0+H_D(t),$$

$$\exp \left(-i{H}_{eff}\left(ϵ\right)T\right)=\mathcal{T}\exp \left(-i{\int }_{ϵ}^{ϵ+T}dtH(t)\right),$$

$$|\psi (0)⟩=\sum _i{c}_i|{\mu }_i⟩$$

$$\hat{\mathcal{O}}=\sum _{i,j}{\mathcal{O}}_{ij}|{\mu }_i⟩⟨{\mu }_j|.$$

$$⟨\psi (nT+ϵ)|\hat{\mathcal{O}}|\psi (nT+ϵ)⟩=\sum _{i,j}{c}_i{c}_j^{\ast }{\mathcal{O}}_{ij}{e}^{-i({\mu }_i-{\mu }_j)(nT+ϵ)}.$$

$${\hat{\rho }}_{{}_{Diag}}=\sum _i|c_i{|}^2|{\mu }_i⟩⟨{\mu }_i|.$$

$$H_0=H_{hop}+\sum _{r=1}^2{V}_r\sum _{i=1}^{L-1}{n}_i{n}_{i+r}+\sum _{i=1}^L{U}_i{n}_i$$

$$H_D\left(t\right)=\delta \tilde{\delta }(t)H_{hop}$$

$$H_{eff}$$

$$H^{(0)}$$

$$H^{(1)}$$

$$H^{(2)}$$

$$H(t)$$

$$H_0$$

$$H_D(t)=-\delta \cos \omega t\sum _{i=0}^L(-1{)}^i{S}_i^z.$$

$$E_{ex}(nT)=\frac{⟨\psi |H(nT)|\psi ⟩-E_{\min }}{\overline{E}-E_{\min }},$$

Full text

\shortabstract

1 Introduction

Non-equilibrium dynamics of periodically driven closed quantum systems has gained significant attention recently, both because of its fundamental importance as a potential host of new quantum phenomena, and its recent experimental realizations (see, e.g.,  [1, 2, 3, 4]). Few central issues, like energy absorption, thermalization, characterization of the asymptotic behavior of the system and role of quantum mechanics in qualitatively understanding of these issues is overarching theme of this review. More specifically, we focus on two different settings where addressing these issues have lead to interesting phenomena and novel physical scenarios. Interest in periodically driven systems also stems from the possibility of generating topologically non-trivial phases (see, e.g., [5] for a review) but here we restrain ourselves from discussing those very interesting aspects.

First, we review the phenomenon of dynamical many-body freezing (DMF)  [6] in presence of disorder [7]. DMF is observed in a large class of translationally invariant integrable systems under strong and rapid periodic drive: the drive induces destructive quantum interference in a massive scale (i.e., affecting almost all degrees of freedom), and observables remain frozen close to their initial values for all time and for any arbitrary initial state. The picture here is, translational invariance and integrability allows one to map the many-body dynamics of these models to the population dynamics of a set of independent two-level systems, where it is possible to tune the drive parameters (frequency and amplitude) in such a way that strong destructive interference simultaneously affects all the modes. Introduction of disorder breaks translational invariance, rendering the above mentioned fine-tuning impossible, and observables eventually decays with time. Here it is worth noting that disorder is usually associated with localization and consequent freezing of dynamics, hence one needs to choose suitable observables and initial states in order to see the dynamics induced by disorder. Interestingly, however, even in presence of strong disorder, dramatic reminiscence of DMF still manifests itself: an enormous enhancement of decay-timescale (orders of magnitude longer compared to the undriven case) is observed under the drive conditions corresponding to maximal freezing in the disorder-free systems. Thus in this case, periodic drive leads to freezing/localization in uniform systems and the disorder leads to unfreezing.

Second, we review the effect of time-periodic drive on many-body localized systems [8, 9]. Here, disorder and interaction localizes a many-body system, and a periodic drive (unlike in the case of dynamical localization), delocalizes the system. Though the phenomenology sounds more intuitive than that of DMF, its mechanism is subtly quantum mechanical.

2 Fate of Dynamical Many-body Freezing/Localization in Presence of Disorder

It is known since long that periodically driven quantum systems with a single degree of freedom can undergo dynamical localization (or freezing) due to strong destructive quantum interference under certain drive conditions [10, 11, 16, 12, 13, 14, 15]. In some cases this happens even though the undriven system is quantum chaotic (i.e., the quantum system is obtained via quantization of a classically chaotic system). However, the physical picture that represents best the role of quantum interference in such localization phenomena can be quite diverse in different settings. For example, in case of a quantum kick-rotator, the mechanism of dynamical localization was identified [16, 17] with that of Anderson localization [18], while in the case of a 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 as an effective dressing of the couplings/mass term in the Hamiltonian by the drive. An interesting connection between dynamical localization and coherent destruction of tunneling has been revealed in  [15], and what happens to the phenomena in presence of interactions is an interesting open question.

A generalization of dynamical localization, namely dynamical many-body freezing (DMF) has been observed relatively recently in integrable translationally invariant quantum many-body systems [6, 19, 23, 20, 21, 5, 22] and has been realized experimentally [3]. DMF is a generalization of the conventional dynamical localization in the following senses. First, in DMF the effect of more than one (even mutually non-commuting) terms in the Hamiltonian can be simultaneously muted down via destructive interference induced by the drive. This is unlike the conventional dynamical localization where only one term (e.g., the kinetic energy) is suppressed. In simple cases, this might render strong freezing of certain observables for any arbitrary initial state, as the effective Hamiltonian responsible for the dynamics vanishes entirely [6, 19]. Second, DMF is a generalization of the conventional dynamical localization (observed for a single degree of freedom) to (infinitely) many-body systems (see, however,  [24, 21, 25, 26, 27] for conventional dynamical localization in many-body systems studied more recently). It is to be noted that DMF is observed so far only in systems of non-interacting particles, or in those which can be mapped to one such. However, it is “many-body” in the sense that it survives even in the case when the dynamics is not factorizable to single-particle sectors, i.e., it does not conserve the particle number. Moreover, DMF can be observed in presence of superconducting-like pair-creation/annihilation processes, which induces correlation between the particles (though not via a non-integrable interactions). Such inter-particle correlations are sufficient to drive long-range ordering and quantum phase transitions in many of these systems. Here we review the fate of DMF in presence of disorder [7].

In  [7], the following disordered one-dimensional Ising chain subjected to a sinusoidal transverse field has been considered. The Hamiltonian is

[TABLE]

where $\sigma^{\alpha}$’s ($\alpha=x,y,z$) are components of Pauli spins, $J_{i}$’s and $h_{i}$ are respectively the (quenched) interactions and on-site fields - both drawn randomly from a uniform distribution between $\left(-1,+1\right).$ The transverse field is subjected to an external drive of frequency $\omega$ (period $T=2\pi/\omega$) and amplitude $h_{0}$ ($\hbar=1$, and periodic boundary condition). The study focuses on the regime of strong ($h_{0}\gg\alpha JJ_{i},\alpha h_{i}$) and fast ($\omega\gg\alpha JJ_{i},\alpha h_{i}$) drive. One starts with the ground state at the initial Hamiltonian $H(t=0),$ drives the transverse field sinusoidally, and measures the transverse magnetization $m^{z}(t)$ as the response.

DMF in the Homogeneous Chain (Fig. 1: First we recapitulate the phenomenon of DMF in absence of disorder ($\alpha JJ_{i}=1$ and $h_{i}=0$ $\forall i$)  [6]. In this case $m^{z}(t)$ settles to a $T-$periodic state oscillating around a non-zero average value $Q=\lim_{\tau\to\infty}\frac{1}{\tau}\int_{0}^{\tau}m^{z}(t)dt$ for ever. Clearly, the system does not absorb enough energy to destroy the order in the initial state however long one might drive (Fig. 1 a). For dynamics with a fully polarized initial state, i.e., $m^{z}(0)=1$, the magnitude of $Q$ can be used as a measure of freezing ($Q=0$ corresponds to the adiabatic drive and $Q=1$ corresponds to infinitely rapid drive). As shown in Fig 1(b), $Q$ turns out to be a highly non-monotonic function of the drive frequency $\omega$: $Q\approx 1/(1+{\cal J}_{0}(4h_{0}/\omega))$ (here ${\cal J}_{0}$ is the Bessel function of first kind of order 0, “$\approx$” denotes the rotating wave approximation). Thus under the condition

[TABLE]

the system freezes maximally, and one gets $Q\approx 1$ (freezing peaks; labeled as $P_{1},P_{2}$ etc in Fig. 1 b (we will refer to the condition in Eq. (2) as freezing peaks in the rest of the review).

Emergence of Stroboscopic Conserved Quantities and Role of Quantum Interference: Existence of DMF clearly indicates absence of ergodicity and breakdown of Fermi-Golden Rule type scenario even within the Hilbert space allowed by the inherent integrable structure of the model. To be precise, integrability leads to decoupling of the degrees of freedom into independent two-level systems in momentum space. If the dynamics was ergodic for these two-level systems, each would keep on absorbing energy until it reaches an infinite temperature like state in the sense of having equal occupation probability for both energy levels, and one would have $\lim_{t\to\infty}m^{z}(t)=0.$ The non-zero value of $Q$ or freezing is a consequence of repeated coherent interference of the amplitudes of the fundamental fermionic excitations in momentum space. If the interference effect is not taken into account, and the drive is assumed to change the population only according to the transition probabilities (neglecting the interference between the transition amplitudes) after each cycle, the system asymptotically approaches $m^{z}=0$ state exponentially rapidly, rendering $Q=0$ regardless of the drive condition and initial state (illustrated in Suppl. Mat. of  [3]).

The lack of thermalization (see, e.g., [6, 19, 23, 20]) can also be attributed to emergence of stroboscopic conserved quantities which leads to a periodic generalized Gibbs’ ensemble (PGE) description of the system at long times [28]. The ensemble description of course involves concept of maximization of Von Neumann entropy in the Floquet space, which is equivalent to making the drastic assumption of equal apriori probability taking into account only the relevant (stroboscopic) conservation laws.

The above two standpoints of viewing the drive-dependent asymptotic states indicate the role of quantum interference - it determines the statistical ensemble to which the system approaches asymptotically by fixing the relevant stroboscopic conserved quantities.

Quantum interference of course continues to play the overarching role also in the case of disordered systems (since the above argument continues to hold), but the simples physical pictures necessary for intuitive understanding of the scenarios under different circumstances are quite diverse as we will see in the rest of the review.

Unfreezing by Disorder and Strong Remnants of DMF: The Phenomenology In presence of disorder, $m^{z}$ always decays to zero regardless of the drive parameters at infinite time and DMF is eventually destroyed. The decay can be fitted well with the exponential decay form (Fig.  2(a))

[TABLE]

where the overbar denotes average over disorder realizations. However, the decay time-scale $\tau$ depends dramatically on the drive parameters $(h_{0},\omega),$ and exhibits a spectacular reminiscence of DMF. As shown in Fig.  2(b), the relaxation time scale $\tau$ shoots up by several orders of magnitude when the freezing condition for the uniform chain (Eq. 2) is satisfied by the drive. Interestingly, if $\eta$ is kept fixed to a value such that ${\cal J}_{0}(\eta)=0,$ (corresponding to peak freezing in the homogeneous system) and $\omega$ is increased, then $\tau$ increases exponentially with it (see Fig. 2(c)). Thus, though introduction of disorder eventually kills the freezing of $m^{z},$ the timescale of decay still bears a very strong signature of the extreme freezing observed in the absence of disorder.

The Points of Maximal Freezing from Floquet Flow Equation Approach: It seems difficult to find analytical solution of time-dependent Schrödinger equation with disordered Hamiltonians. Hence one can resort to the following Floquet analysis and determine the effective Floquet Hamiltonian approximately using a flow equation approach [29]. For the present purpose, we adopt the simplest formulation of Floquet theory and define the effective time-independent Hamiltonian that describes the evolution of the stroboscopically observed wave-function as follows. Let us denote the time evolution operator evolving a state through a period from $t=\epsilon$ to $t=\epsilon+T$ ($0\leq\epsilon<T$) by $U(\epsilon)$. Since $U(\epsilon)$ is unitary, it can always be expressed in terms of a hermitian operator $H_{eff}$ as

[TABLE]

Clearly, if observed in a “stroboscopic” fashion at instants $t=\epsilon,\epsilon+T,\dots,\epsilon+nT$ ($n$ is an integer), the dynamics can be considered to be effectively governed as if by a time-independent Hamiltonian $H_{eff}.$ With $H_{eff}$ one gets the same wave-function as that with $H(t)$ at the instants $t=\epsilon+nT,$ because the time-evolution operator is same in both cases for evolution to those instants ($[e^{-iH_{eff}T}]^{n}=e^{-iH_{eff}nT}$). This of course holds for every $\epsilon,$ hence we get different stroboscopic series for each of them (actually choice of $\epsilon$ is equivalent to choosing a gauge, as shown in [5]). For characterizing the long-time behaviour of the system under rapid drive, it is sufficient to observe the system strobocopically, since nothing much happens within a single cycle. Hence it is sufficient to follow the dynamics governed by $H_{eff}(\epsilon=0)$ at $t=nT.$ Moreover, the set of all (i.e., for all values of $\epsilon$) stroboscopic observations are sufficient to construct the entire time evolution (see  [5] for an elegant and efficient way of extracting this information).

The Hamiltonian in Eq. 1 can be mapped to the following non-interacting Hamiltonian using standard prescription (see, e.g., [22]).

[TABLE]

with hard-core bosons created (annihilated) by $c^{\dagger}_{i}$ ($c^{\;}_{i}$) These bosons satisfy $\{c_{j}^{\dagger},c_{j}\}=0$, and the usual bosonic commutation relations for $i\neq j$. Also, $h(t)=h_{0}\sin{\omega t}$. Using Floquet-flow equation technique, one can construct the following analytical expression for $H_{eff}$ for the evolution from $t=0$ to $t=T$ governed by the Hamiltonian in Eq. 5 [7].

[TABLE]

with $\eta\equiv 4h_{0}/\omega$, and the constants $j^{(s)}_{i},\mu$ defined as follows.

[TABLE]

Here, ${\cal J}_{n}(\eta)$ denote Bessel function of the first kind of order $n,$ and $\beta(\eta)\equiv\sum_{n\neq 0}{{\cal J}_{n}(\eta)}/{n}$. The above is obtained under a rotating-wave approximation (RWA) which holds for $\omega\gg J,\alpha.$ This effective Hamiltonian accurately reproduces the dynamics of the full system stroboscopically to the leading order in $\alpha/\omega.$

The above expression of $H_{eff}$ shows that when ${\cal J}_{0}(\eta)=0$ (freezing peaks) and $h_{i}=0,$ $H_{eff}$ vanishes, implying complete freezing of the dynamics. Though this does not happen here and $m^{z}$ decays to zero due to the effect of the higher order terms in $\alpha/\omega$ (unlike in the case of the homogeneous chain), the time-scale $\tau$ gets enormous jumps at these points (see Fig. 2 a,b). For $h_{i}\neq 0$ the picture is qualitatively similar, though the enhancements of $\tau$ are smaller due to the presence of the $\beta$-term. Note that the huge 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, $\tau$ can also be enhanced exponentially by tuning $\omega,$ (Fig. 2c), demonstrating a great control achievable on the disordered induced decays via periodic drive.

Before concluding this section a few words on RWA seems to be in order. In RWA, one essentially goes into a “rotating frame” where the sinusoidal drive term in the Hamiltonian can be expanded into sum of terms of the form $e^{(i\Omega t)},$ and drops out all the terms for which the frequency $\Omega$ is much larger than the characteristic frequencies of the undriven system. DMF occurs when the drive frequency is high enough to be off-resonant with all the characteristic frequencies of the system (i.e., in the limit of zero-photon process). In the present case, this condition translates to $\omega\gg J,\alpha$ [7]. A detailed and critical review on the domain of validity of RWA is given in [30].

3 Fate of Many-body Localization under Periodic Drive

In this part of the review we take a different standpoint: We consider interacting systems where disorder induces many-body localization (MBL) in the Fock space (see, e.g., [18, 31, 32, 33, 34]), and the question is if one can delocalize the system by applying an external periodic drive. But before addressing this, we first make a small detour and briefly review the application of the Floquet formalism in studying the asymptotic properties a periodically driven many-body systems in general.

3.1 Effective Floquet Hamiltonian and the Diagonal Ensemble

Consider a static Hamiltonian $H_{0}$ hosting a many-body localized phase be driven by a time-periodic (non-commuting with $H_{0}$) part $H_{D}(t).$ The total Hamiltonian is thus

[TABLE]

and $H_{eff}\left(\epsilon\right)$ be the corresponding Floquet Hamiltonian (see Eq. (4)). Then

[TABLE]

where ${\mathcal{T}}$ denotes time-ordering. Without loss of generality one can set $\epsilon=0.$ Let $|\mu_{i}\rangle$ denote the $i$-th eigenstate of $H_{eff}$ corresponding to the eigenvalue $\mu_{i}$.

Under “generic” initial conditions and considering “generic” local operators as observables, the nature of the asymptotic state can be understood from the following. In order to consider the fate of the system at long times, we consider a initial state

[TABLE]

and an observable

[TABLE]

[TABLE]

For a many-body system (infinite-size limit), by “generic” $|\psi(0)\rangle$ and by “generic” operator ${\hat{\cal O}},$ we mean that above sum is extensive in the sense that there are sufficiently large number of quasi-energy states participating in the sum. In that case (see  [35, 28] for further conditions), at long times ($n\to\infty$) the off-diagonal ($i\neq j$) terms in the sum oscillates rapidly and their contributions add up almost randomly, summing up to a vanishingly small quantity. Hence under above conditions, the state of the system can be described by an effective “diagonal ensemble” given by the mixed density matrix [36]

[TABLE]

Thus, the asymptotic properties of a periodically driven system are effectively given by a statistical average over the expectation values of the eigenstates of $H_{eff}$, and hence it is sufficient to study the nature of the eigenstates and eigenvalues of $H_{eff}$ in order to understand the long-time behaviour. Moreover, reduction of the stroboscopic dynamics to that due to a time-independent $H_{eff}$ implies that quasi-energy plays similar role in the stroboscopic dynamics as energy plays in the dynamics governed by a time-independent Hamiltonian. These hold regardless of whether the system is ergodic or many-body localized, see, e.g., [37]. Here it is worth noting that in spite of this reducibility, nature of the dynamics due to periodic drive can be different from a generic undriven case in fundamental ways, since in the former case $H_{eff}$ might be highly non-local. For example, dynamics under a generic time-independent local Hamiltonian leads to thermalization at a finite temperature, while evolution under an $H_{eff}$ derived from a time-periodic generic Hamiltonian can lead to heating up to an effectively infinite temperature scenario [38, 39].

3.2 Delocalization of MBL

MBL is a thermodynamically stable non-ergodic phase of matter (see, e.g.,  [18, 31, 32, 33, 34]) where an interacting many-body system remains localized in the Fock space due to disorder in absence of an external bath. Here we address if periodic drive can destabilize such a phase and heat it up indefinitely. From the discussion in Sec 3.1 it is clear that in order to distinguish between an MBL and an ergodic phase in a periodically driven system, it is sufficient to focus on the properties/statistics of eigenstates and eigenvalues of $H_{eff}$ [8]. The general scenario depends on the absence (presence) of many-body mobility edge as summarized below. Two mechanisms by which periodic driving might destroy MBL are identified. The first, rather robust, mechanism is the mixing of undriven eigenstates from everywhere in the spectrum by the driving; if there is a mobility edge, this results in delocalization of all states of the effective Hamiltonian. The second mechanism is more subtle and involves strong mixing of states [28] which causes a delocalization transition at a finite drive frequency for a given disorder strength. The key findings are summarised in Table 1.

This leads to a phase diagram outlined in Fig 3. For low enough $\omega$ and disorder strength, the system always delocalizes under the drive, while for high enough $\omega$ and disorder the system remains MBL. The blue line in Fig. 3 indicates a tentative boundary between these two phases, obtained by extrapolating the numerically determined transition points (red dots).

In the following two different models (with/without the mobility edge) are considered separately in order to illustrate the phenomenology above.

3.2.1 No mobility Edge

A model of interacting hard-core bosons is considered, which is described by a driven, local Hamiltonian (Eq. 8) with

[TABLE]

where $H_{hop}=\left(-\frac{1}{2}J\sum_{i=1}^{L-1}\left(b_{i}^{\dagger}b_{i+1}+b_{i+1}^{\dagger}b_{i}\right)\right)$ is a hopping operator, the $b$ are hard-core bosonic operators, $U_{i}$ an on-site random potential uniformly distributed between $-w$ and $+w$ and $H_{D}\left(t\right)$ a time-periodic hopping term

[TABLE]

with $\delta$ a dimensionless constant, $\tilde{\delta}(t)=-1(+1)$ in the first (second) half of each period $T=2\pi/\omega$. Via Jordan-Wigner transformations this model is related to a fermionic interacting system as well as to a spin-1/2 chain. The results are presented for $V_{1}/J=V_{2}/J=1$, although the qualitative conclusions are not sensitive to this.

Non-ergodicity of the MBL Phase from Eigenstate Expectation Values (EEV): It has been shown that generic interacting systems, when driven periodically in time at low frequencies (low compared to the bandwidth of the undriven system), the system keeps on absorbing energy without bounds, ending up in a state which is indistinguishable from an infinite temperature state as far the expectation values of local observables are concerned [38, 39]. This infinite temperature like scenario is reflected in the fact that the eigenstate expectation values (EEV) of any local operator $\langle{\hat{\cal O}}\rangle_{i}=\langle\mu_{i}|{\hat{\cal O}}|\mu_{i}\rangle$ over all eigenstates of $H_{eff}$ are almost equal to each other (i.e., when EEV is plotted with respect to $\mu_{i}$ it is almost flat for any given ordering of $i$). On the other hand, in the localized phase EEV fluctuates wildly. This is illustrated in Fig. 4.

Locating the MBL-Delocalization Transition: To accurately locate the localization-delocalization transition for the driven system, the level statistics of the eigenvalues of $H_{eff}$ has been calculated. That is, after obtaining the quasi-energies $\epsilon_{n}$, one calculates the following ratio involving adjacent level spacings $\delta_{n}=\epsilon_{n}-\epsilon_{n+1}$: $r_{n}=\min\left(\delta_{n},\delta_{n-1}\right)/\max\left(\delta_{n},\delta_{n-1}\right)$. The mean $\eta=\int_{0}^{1}dr\,rP(r)$ distinguishes between Circular Unitary Ensemble (ergodic) and Poissonian statistics (non-ergodic). One calculates $\eta$ for a sequence of system sizes and extrapolate the limit of $\eta$ as $L\rightarrow\infty,$ and consider the statistics of the extrapolated values of $\eta.$

To obtain the frequency $\omega_{c}$ above which delocalization sets in for a driven system, one plots $\eta$ for several values of disorder amplitude $w$, averaged over $\sim 10^{4}$ disorder realisations and for several system sizes. Typical results are shown in Fig. 5. The transition is located at the crossing point of the lines for different system sizes: if increasing system size results in larger $\eta$ then we conclude that the system is delocalized, since $\eta=\eta_{CUE}$ for a delocalized system and $\eta=\eta_{P}$ for a localized system with $\eta_{P}<\eta_{CUE}$. Here $\eta_{CUE}$ is the value for the CUE ensemble [39].

For the results to be applicable in the thermodynamic limit, it is necessary (though might not be sufficient) to take the drive frequency $\omega$ much lower than the width of the energy spectrum of the undriven Hamiltonian. The main practical problem is the following: with decreasing disorder amplitude $w$ and for fixed system size, the value $\omega_{c}$ increases while the energetic width of the DOS decreases. Since $\omega$ must be small compared to the width in order for the extrapolation to the thermodynamic limit to be meaningful, the $\omega_{c}$ for values of the disorder close to $w_{c}$ are inaccessible for the system sizes for which the numerics could be done. The width of the DOS is indicated in Fig. 5 by vertical lines; the crossing point of the curves cannot lie to the right of this line, since otherwise the finite size of the system would be important (and thus the results would not be reliable in the thermodynamic limit).

Fig. 5 reveals the following features: for $w/J\leq 6$ (where the undriven system is delocalized) the lines for successive, increasing $L$ do not cross for values of $\omega$ below the bandwidth, indicating that the thermodynamic limit is delocalized, as expected. For $w/J>6$, there is a clear crossing point, which indicates the position of the transition. The crossing value of $\omega$ determined by this method is plotted as a function of $w/J$ in Fig. 3.

A Physical Picture: Though the physical picture consistently fitting the full phenomenology described above is not entirely clear, significant efforts have been made in this direction [8, 40, 41]. We summarize the gist in the following.

Delocalization Under Low-frequency Drive: In the MBL phase and in the absence of driving, the system is effectively integrable in that there exist extensively many local integrals of motion [33, 42, 43, 44, 45]. The system may thus be thought of as a set of local subsystems, of finite spatial extent. The matrix elements connecting these local systems are suppressed in much the same way hopping amplitude between sites is suppressed in Anderson localization. Under slow periodic drive, the MBL Hamiltonian is replaced by an $H_{eff},$ which is not just the time-average of the periodic Hamiltonian over a period, but also have significant other components which consists of effective long-range hopping and interactions. These terms might introduce matrix elements (tunneling, say) between different localized systems, resulting in delocalized eigenstates of $H_{eff}.$ From the numerical investigations discussed above, this appears to be the case in general, but it is not obvious to what extent this argument should work, since in case of simple non-integrable models like kick-rotor (see Sec. 2), periodic drive actually induces localization by producing random destructive quantum interference (hence killing matrix elements), and the problem can be exactly mapped to Anderson localization. Other interesting scenarios are also being suggested and investigated recently, see. e.g., [46, 47]

Stability of MBL towards High-frequency Drive: This can be understood as follows. For fast drive one can do Magnus expansion [48] for $H_{eff}$ as follows.

[TABLE]

For fast enough drive (small enough $T$), the series might converge, and one can keep only the leading order term $H^{(0)}=\frac{1}{T}\int_{0}^{T}H(t)dt$ [39]. In that case of course $H_{eff}$ is trivially an MBL Hamiltonian, particularly when $\int_{0}^{T}H_{D}(t)dt=0$ (which is the case considered in the above studies).

3.3 In Presence of Mobility Edge

The QREM as a case study: We now turn to the case in which a mobility edge is present in the undriven spectrum. The central result is based on the observation [38] that a periodic perturbation acting on a system couples each undriven state to states spread uniformly throughout the spectrum of $H_{0}$. As a result, if part of the spectrum corresponds to delocalized eigenstates then all eigenstates of $H_{eff}$ will necessarily be delocalized. Whether there exists mobility edge in local models of MBL is debatable [49], but delocalization for all values of $\omega$ has been numerically confirmed by studying the Quantum Random Energy Model (QREM), recently studied in Ref. [50] where it was shown to have a mobility edge. The model is defined for $N$ Ising spins with the Hamiltonian $H=E\left(\left\{\sigma_{j}^{z}\right\}\right)-\Gamma\sum_{j}\sigma_{j}^{x}$, where $E$ is a random operator diagonal in the $\sigma^{z}$ basis (that is, it assigns a random energy to each spin configuration) and $\Gamma$ a transverse field. Extensivity of the many-body spectrum is satisfied if the random energies are drawn from a distribution $P\left(E\right)=\frac{1}{\sqrt{\pi N}}\exp\left(-E^{2}/N\right)$.

The diagnostic of localization used here is the participation ratio (PR), defined for the state $|\psi\rangle$ with respect to the Fock basis $\{|n\rangle\}$ as \phi=\sum_{n}\mbox{\left|\left\langle n\right|\left.\psi\right\rangle\right|}^{4} with $n$ enumerating Fock states. $\phi$ approaches unity for a state localized on a single Fock state and $2^{-N}$ for one fully delocalized in Fock space. The leftmost panel in Fig. 6 shows the average $\phi$ versus energy (scaled with system size) of the 256 eigenstates of an undriven $N=8$ system averaged over 1000 disorder realisations, demonstrating the existence of a mobility edge.

The system is also driven by modulating $\Gamma\left(t\right)=\Gamma_{0}\left(1+\delta\tilde{\delta}\left(t\right)\right)$, $\tilde{\delta}\left(t\right)=+1(-1)$ for the first (second) half of the period with an amplitude $\delta=0.2$ and frequency $\omega=2\pi/T=0.1$. The PR of the eigenstates of $H_{eff}$ are shown in the second panel of Fig. 6. As expected, periodic driving causes delocalization of the entire spectrum so long as part of the undriven spectrum at the same $\Gamma_{0}$ is delocalized.

3.4 Energy Absorption:

Energy absorption by an MBL system under periodic drive has been studied extensively in [9]. The model studied was spin-1/2 XXZ chain in a disordered longitudinal field under monochromatic drive with period $T=2\pi/\omega:$

[TABLE]

where $h^{z}_{i}\in[-\kappa,\kappa]$, $J_{\perp},J_{z}\geq 0$, and with driving

[TABLE]

The static part $H_{0}$ is known to be MBL for $J_{z}\neq 0$ and sufficiently strong disorder $\kappa>\kappa_{c}.$ Starting from the ground state of $H_{0}$ at $t=0,$ real-time dynamics of “rescaled excess energy density” defined as

[TABLE]

with $\overline{E}=D_{H}^{-1}\mathrm{tr}[H(0)]$ and $D_{H}$ being the Hilbert space dimension, so that $E_{ex}=0$ in the ground state of $H(0)$, while $E_{ex}=1$ for a state uniformly delocalized among all eigenstates of $H_{0}.$ This is the “infinite temperature” like scenario observed when a disorder free interacting system is driven periodically [38].

For strong disorder ($\kappa>\kappa_{c}$), a qualitative phase-diagram depending on $\omega$ and $\delta$ (Fig. 7) has been proposed. Three regimes has been identified (marked in the figure) as follows. There are two regimes (I and III), which are known from earlier works [8, 40] (see also Sec. 3.2.1). In regime (I) an initially MBL state delocalizes and heats up to a state in which the system locally looks as if it is at infinite temperature (i.e., the expectation value of the local observables over the state equals to that over an infinite temperature ensemble). In (III), the drive fails to delocalize the system, but only pumps some energy into it. The energy of the system thus settles to some intermediate average value. However, the numerics seems to suggest there is another intermediate regime (II), where the system tend to reach the infinite temperature like scenario, but only logerithmically slowly. Sample of real time behaviour of these regimes are given in Fig. 8. Further technical details, particularly those relevant for validation of the numerical results are given in detail in [9].

4 Summary and Outlook

In this article we illustrate that periodic drive can induce both freezing/localization and unfreezing/delocalization dynamically when applied to closed quantum systems with many-degrees of freedom depending on the circumstances. It has been argued that quantum interference plays an important role in all these phenomena, though the intuitive pictures consistent with different circumstances are quite varied. For example, while in the case of DMF the simplest physical picture of freezing seems to consist of renormalization of the Hamiltonian by the periodic drive resulting in suppression of dynamics, dynamical localization in a periodically kicked rotor can be understood more easily by mapping it to the Anderson localization problem, implying that the drive induced dynamical randomness has similar localizing effect as quenched disorder in static problem. Yet, in an inherently localized many-body system with interaction, periodic drive at low frequencies can cause delocalization. There the picture is, the drive generates matrix elements connecting the spatially isolated localized parts of the undriven MBL system. The system then eventually heats up till it reaches a state which looks like an infinite temperature state when expectation values of local observables are measured. Interestingly, it seems no noticeable dynamical localization is observed in periodically driven MBL in the low $\omega$ regime (at least, not sufficiently strong to exhibit any noticeable freezing effect), unlike that observed in single-body non-integrable quantum chaotic system like the kick-rotator. The issue of energy absorption under external drive in system not attached to a bath is still a broad open question, and are being pursued under different drive protocols [51, 52].

A number of interesting questions present themselves.

For example, whether under high frequency drive, mechanism of dynamical localization steps and lends an MBL phase greater stability?

What happens if one drives an integrable models (which can be mapped to non-interacting fermions) with very low frequencies, and the Magnus expansion breaks down? Moreover Does one achieve effective infinite temperature thermalization scenario there? If yes, then how does the transition/crossover from PGE to thermal phase takes place ? If not, then certainly interaction kills it (evidences indicate that MBL gives way to ergodic phases when driven with low enough frequencies). Then the behavior of MBL-ergodic crossover as a function of interaction strength would be interesting.

Can one expect Periodic (generalized) Gibbs’ Ensemble as a local description of the asymptotic states in periodically driven integrable systems which cannot be mapped to free fermions? This seems plausible, but to our knowledge there is no general proof of existence of extensive number of stroboscopic conserved quantities necessary for this. The same (open) question appears interesting in the context of DMF, in particular, whether extreme freezing points can occur in such integrable systems.

One might wonder what happens when a bath is weakly coupled to a periodically driven system. The effect of quantum interference would be affected by external decoherence and dissipation, and of course the asymptotic behaviour cannot be expected to be determined by a statistical average of the property of $H_{eff}$ (system would not go to a diagonal ensemble in the eigenbasis of $H_{eff}$ in general). The intuitive pictures developed here might have to be revised significantly, and emergence of fundamentally new pictures are not unlikely.

Acknowledgements: AD acknowledges collaborations with S. Bhattacharyya, S. Dasgupta, A. Lazarides, R. Moessner and A. Roy in various works covered in this article. AH and AD thankfully acknowledge support from DST-MPI partner group program “Spin liquids: correlations, dynamics and disorder” between MPI-PKS (Dresden) and IACS (Kolkata), and the Visitor’s Program of MPI-PKS for a visit to MPI-PKS, during which many interesting discussions on the subject took place.