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Dynamical transitions in aperiodically kicked tight-binding models

Vikram Ravindranath∗

Department of Physics and Astronomy, CUNY College of Staten Island, Staten Island, NY 10314,

Physics Program, The Graduate Center, CUNY, New York, NY 10016.

M. S. Santhanam†

Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411 008, India.

If a localized quantum state in a tight-binding model with structural aperiodicity is subject tonoisy evolution, then it is generally expected to result in diffusion and delocalization. In this work,it is shown that the localized phase of the kicked Aubry-Andre-Harper (AAH) model is robust to theeffects of noisy evolution, for long times, provided that some kick is delivered once every time period.However, if strong noisy perturbations are applied by randomly missing kicks, a sharp dynamicaltransition from a ballistic growth phase at initial times to a diffusive growth phase for longer times isobserved. Such sharp transitions are seen even in translationally invariant models. These transitionsare related to the existence of flat bands, and using a 2-band model we obtain analytical support forthese observations. The diffusive evolution at long times has a mechanism similar to that of a randomwalk. The time scale at which the sharp transition takes place is related to the characteristics ofnoise. Remarkably, the wavepacket evolution scales with the noise parameters. Further, using kicksequence modulated by a ‘coin toss’, it is argued that the correlations in the noise are crucial to theobserved sharp transitions.

I. INTRODUCTION

It is by now well established through theoretical stud-ies and experiments that periodic forcing imparted toquantum systems can lead to novel states of matter thatdid not exist in its time-independent counterpart [1–4].For the periodically driven Hamiltonian systems, Floquetstates represent the natural generalization of the sta-tionary eigenstates for time-independent systems. Thesestates are interesting objects of study because they canexhibit more complex dynamics and allow more controlthrough external driving in comparison to their staticcounterparts. Many phenomena in static systems such asthe topology of band structures [5] and transport proper-ties [6, 7] can be tuned by an appropriate driving mech-anism. A class of such techniques now known as Flo-quet engineering [1, 8, 9] attempt to create novel Floquetstates with desired topological properties [10, 11] by de-signing an effective time-independent Hamiltonian for thedriven systems. In the last decade, experiments have im-plemented a variety of approaches based on such Floquetmanipulations – effective Hofstadter Hamiltonian usingcold atoms in optical lattices [12], control over directionand interference of phonon flow in 2D array of trappedions [13], the implementation of Haldane model in ultra-cold fermions to realize topological insulators [14] andcontrol over heating due to periodic drive in pre-thermalphase of Bose-Hubbard system [15]. Recently, principlesof Floquet engineering have been extended to quantumdissipative systems as well thus providing a handle tounderstand non-equilibrium steady states [16].

∗ vravindranath@gradcenter.cuny.edu† santh@iiserpune.ac.in

In this broader context, of particular interest in a va-riety of condensed matter systems is the observation ofquantum localization of eigenstates in time-independentsystems and of Floquet states in time-dependent systems[17, 18]. Quantum kicked rotor, representing the dynam-ics of a periodically kicked pendulum, is a well studiedexample of a classically non-integrable system. The clas-sical limit of kicked rotor, for strong kick strengths, dis-plays chaotic dynamics and energy diffusion [19]. Thecorresponding quantum system exhibits stark differencesfrom classical regime, and localization of all its Floquetstates is a notable feature amongst them [19]. The quan-tum kicked rotor has been mapped to the Anderson tightbinding model for a particle in a crystalline lattice withrandom on-site potentials [20]. These connections havebeen experimentally explored using cold atoms in opticallattices over the last three decades [21]. Thus, far frombeing theoretical constructs, tight-binding models havebecome popular partly owing to their ease of experimen-tal realisation in optical lattices [22].

Floquet systems are characterised by periodic driv-ing with periodicity T such that drive term f(t) satis-fies f(t + T ) = f(t). In typical experimental situations,there are bound to be imperfections and this providesa motivation to consider noise in T and other parame-ters related to the drive term. More importantly, thereis a strong possibility of novel effects arising due to pres-ence of aperiodicity in Floquet systems. For instance,all the Floquet states of the quantum kicked rotor inone and two dimensions are localized, and do not dis-play a localization to delocalization transition. However,noisy kick sequences can induce such a transition. If thekick period is perturbed by stationary noise, dynamicallocalization is destroyed through an exponentially decay-ing decoherence process resulting in a delocalized phase.This aspect has been extensively studied in kicked rotors

2

[23–25]. Interestingly, recent theoretical proposals [26]and experimental realizations show that the localizationto delocalization transition can be non-exponential uponintroducing nonstationary noise in the kicking sequence[27]. In general, without noisy kicks, the localization todelocalization transition is absent in the one-dimensionalkicked rotor.

Though the localization-delocalization transition is ab-sent in kicked rotors (and also in Anderson model) of oneand two dimensions, it is known that quasi-periodic lat-tice models such as the Aubry-Andre model do displaysuch a transition even in one dimension [28]. Dependingon the presence or absence of interactions and the para-metric regime under consideration, particular effects ofperiodically driving the Aubry-Andre system vary frompreserving the localization transition [29], inducing delo-calization [30] to the appearance of a Griffiths-like phasemanifesting as slow dynamical spreading [31]. Generally,the phase boundaries depend on the amplitude and fre-quency of the driving field [6].

Then, the question arises as to how such systems withbuilt-in structural aperiodicity react to noisy drivingfields or kick sequences imparted to them. In this paper,the kicked Aubry-Andre system, belonging to a class ofaperiodic lattice models, is studied to examine the effectsof noisy kicks in which noise manipulates the periodickick sequence in three different ways. In this context, itis of interest to distinguish between milder forms of noisykicks in which only the amplitude of kicks are modulatedas opposed to the stronger forms in which entire kickscan be missed. If a tight-binding model that supportsa localized regime is irregularly kicked, we generally ex-pect to observe a localization to delocalization transitionanalogous to the behaviour of the quantum kicked rotorsunder milder forms of noisy kicks [23, 26, 27]. For in-stance, such a scenario also unfolds in the case of Floquettopological chains treated analytically using the Floquetsuperoperator formalism [32]. In this case, strictly pe-riodic kicks preserve topologically protected end states,and noisy kicks lead to a decay of these topologically pro-tected modes. It is also known that localized regime inthe kicked Aubry-Andre model survives even if a milderform of noise is present [33].

In contrast, in this study, the focus is on the effectsof stronger forms of noisy kick sequences imparted toan initially localized state. It is shown that a dynam-ical transition from ballistic to diffusive growth results.The characteristics of this transition, whether smooth orabrupt, and the timescale at which this transition takesplace depend on the noise properties. Remarkably, bytuning the characteristics of noise, the timescale for thetransition from ballistic to diffusive can be tuned as well.It is shown that the wavepacket growth quantified by itssecond moment scales with a noise parameter. Rest ofthe article is structured as follows – in sections II and III,we describe the AAH model and noisy driving protocols,in sections IV and V the simulation results are presented,and in sections VI and VII the results are extended to

translationally invariant models and analytical supportis provided for the main features of the results. SectionVIII provides a brief summary of the results.

II. DYNAMICAL PROPERTIES

A. Model

The Hamiltonian of the model considered in this workis given by

H = H0 + V∑

n

δ(t− Tn)

L∑

i=1

Vi c†i ci, (1)

where the integrable part of the Hamiltonian representingthe kinetic energy term is

H0 = −JL∑

i=1

(c†ici+1 + c†i+1ci

). (2)

In this, c†i and ci are the creation and annihilation op-erators at i-th site. This system has L lattice sites withperiodic boundary conditions applied to it. The onsitepotential is Vi ≡ cos(2παi) and α is a real parameterthat modulates the aperiodicity of the lattice. The tem-poral dependence in the form of Dirac delta function inEq. 1 ensures that the onsite potential kicks-in when-ever Tn = nτ , where n is an integer and τ is the kickingperiod. The parameter V represents the amplitude ofperiodic kicks and J is the amplitude for the strength ofnearest-neghbour hopping.Without the train of δ-kicks and if α is an irrational

number, then Eq. 1 is the well-known Aubry-Andremodel [34] for electron transport in a lattice with struc-tural disorder. Extensive investigation of this model [35]has shown that if α is a strongly incommensurable num-ber such as the Golden ratio, then all the eigenstatesdisplay a transition from extended to localized statesat the critical point V/J = 2. The transition to lo-calized states has been experimentally observed in thetest-bed of non-interacting ultracold atoms in quasiperi-odic optical lattices [36], as well as in the interactingcase [37]. Numerical evidence too points to the existenceof a many-body localization transition in the interact-ing Aubry-Andre model [38]. The corresponding asymp-totic dynamics of an initially localized wavepacket withwidth σ is ballistic, and the wavepacket width grows asσ2t ∼ t2, for V/J < 2. However, if V/J > 2 the spread

of the wavepacket is completely suppressed [39], σ2t ∼ t0.

This transition is accompanied by multifractality in thewavefunction [39], diffusive dynamics [40] and a fractalspectrum at the transition point [41, 42]. The dynamicscan even be strongly hyperballistic when a quasiperiodicsection is embedded in a tight-binding lattice [43].If the periodic kicks are applied at times Tn = nτ ,

(n ∈ Z+), the critical point for localization transition in

3

Type of Noise DescriptionA Timing noise

(Aperiodicity)[33]

The position of kick (in time axis) isa uniformly distributed random num-ber drawn from the interval [T −

δT, T + δT ].B Amplitude noise

[26]Kicks are delivered periodically, butafter time intervals randomly chosenfrom Yule-Simon distribution, theamplitude V is altered by a randomamount δV .

C Timing noise(Missed kicks)[27]

The spacing between kicks is a ran-dom number drawn from uniform,normal and Poisson distributions andhence some kicks are missed.

TABLE I. Descriptions of three noisy kicking protocols,labeled A, B and C, used in this work. See Fig. 1 for a

visual schematic of these protocols.

the limit of high-frequency of kicks is found to be VJT ∼ 2

[44]. However, if this critical point is approached from thedelocalized phase, the system exhibits a range of growthrates; i.e., σ2

t ∼ tγ , 1 ≤ γ ≤ 2, with a sharp fall-off toσ2t ∼ t0 for V > 2JT [33].

B. Classical limit

In the classical limit, the Hamiltonian in Eq. 1 ex-hibits chaotic dynamics since kicking eliminates energyas a conserved quantity. This can be seen in the classicalHamiltonian obtained by taking a continuum limit of thekicked tight-binding model as follows :

∑

x

|x〉〈x+ 1| → exp(−ιp)∑

x

|x+ 1〉〈x| → exp(ιp). (3)

−JL∑

i=1

(c†ici+1 + c†i+1ci

)= −J

∑

x

(|x〉〈x+ 1|+ |x+ 1〉〈x|) ,

= −2J cos(p) (4)

Hcl = −2J cos(p) + V∑

n

δ(t− Tn) cos(2παx). (5)

Starting from Hcl we can derive the so-called kickedHarper map [45],

pn+1 = pn + 2παV sin(2παxn),

xn+1 = xn + 2Jτ sin(pn+1). (6)

Numerical investigations have shown that such a model ischaotic regardless of the value of α, leading one to positthat all kicked tight-binding models ought to be quan-tum chaotic. Numerous studies on this map [45–47] haveeffectively established what is known about the Aubry-Andre-Harper model to be true even in the presence ofkicking, with the addition of intriguing dynamics such as

FIG. 1. Schematic of noisy kicking protocols. Kick ampli-tudes are shown as a function of integer time n. Grey colorvertical lines are the positions of kicks had the kick sequencebeen periodic. Red colour lines indicate the actual kicks im-parted for the given type of noise protocol. Labels A, B andC refer to three kicking protocols (see Table I).

ballistic transport and quantum localization, dependingon the location in (semi-classical) phase space [45]. Thesehave been corroborated by studies on their tight-bindinganalogues which have begun to be explored only recently[33, 44].

III. NOISY DRIVING : PROTOCOLS ANDMETHODS

In this section, results related to the dynamical effectsthat arise from a stochastic perturbation to the drivingterm in Hamiltonian in Eq. 1 are presented. Indeed,some effects of aperiodicty in the kicking have been stud-ied previously [33], where it was found that localizationpersists even in the presence of strong aperiodicity forsufficiently long time. The kicking protocol used wassuch that the time interval between kicks was drawn uni-formly from an interval (T − δT, T + δT ). Against thisbackdrop, in this paper, the AAH model has been subjectto stronger forms of noise protocols in order to probe therange of phenomena that driving, periodic or noisy, canelicit. In this paper, noise protocols were implementedby modifying the periodic potential in Eq. 1 to

∑

n

(V + δVn) gn δ(t− nτ)

L∑

i=1

Vi c†ici. (7)

These noisy protocols are listed in Table I. The case ofthe periodic kick sequence corresponds to gn = 1 andδVn = 0 for all n. Protocol A, shown in Fig. 1(A),corresponds to gn = 1 and δVn > 0 for all n, and itsoutcome has been reported before [33]. In this protocol,white noise is superimposed on kick times [33] such that,on an average, there is still one kick during every timeperiod τ though the actual positions of kicks are noisy.

4

950 1000 1050x

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

ρ(x)

100100010000

101

102

103

104

n

1

σ2

(a) (b)

FIG. 2. Density profile ρ(x) and spread of the wavefunctionunder protocol B. (a) ρ(x) at times n = 100, 1000 and 10000.(b) Width of the time evolving wavefunction (indicated byσ2). In this, timing noise is drawn from a Yule-Simon distri-bution with exponent β = 0.75 and amplitude noise is uni-formly distributed in the range [−V/2, V/2].

In protocol B, visualized in Fig. 1(B), a kick is alwaysdelivered at periods of τ and hence gn = 1. However,after time intervals drawn from Yule-Simon distributiongiven by

f(k) = βB(k, β + 1)k→∞=⇒ k−β−1,

the amplitude is made noisy (δVn > 0) and is drawn froma uniform distribution. In this, B is the Beta functionand β is a parameter. In protocol C, δVn = 0 and gntakes either 0 or 1. If gn = 0 (gn = 1), kicks are missed(present). The time interval between successive occur-rences of 1 is Knτ with Kn being an integer drawn fromuniform, Poisson or normal distributions.In this work, the evolution of a wavepacket, initially

localized at the center of the lattice, is characterized bytwo quantities – its (noise-averaged) spread σ2(t) and aninstantaneous, locally averaged diffusion exponent γm,defined as follows :

σ2(t) =

N∑

i=1

(i −N/2)2|Ci(t)|2; Ci(t) ≡ 〈xi|ψ(t)〉 , (8)

γm(t) ≡ 1

m

m2∑′

t=−m2

log(σ2 (t+ t)

)− log

(σ2(t)

)

log(1 + t

t

) . (9)

The utility of γm lies in its ability to expose local varia-tions in the dynamics. The caveat, then, is that genuinelocal fluctuations need to be aptly distinguished fromnoise, especially in the cases where the system is ape-riodically driven.The numerical results shown in this paper were im-

plemented using the Armadillo linear algebra library[48, 49]. The initial wavefunction, placed at the middle ofthe lattice of size L, 〈x|ψ(t = 0)〉 = δx,L/2 is evolved un-der the action of noisy kicks applied to the Hamiltonian

400 500 600x

0

0.01

0.02

0.03

ρ(x)

20304060200

100

101

102

n

1

10

102

103

104

σ2

(a) (b)

FIG. 3. Density profile ρ(x) and its evolution under protocolC for parameters V = 1.5 and A = 50. (a) ρ(x) at timesn = 20, 30, 40, 60 and 200. (b) Width of the time evolvingwavefunction as a function of time. The dashed lines haveslopes 1 and 2. See text for details of noise imposed on kicksequence.

in Eq. 1. If the kicks are periodically applied (noiselesscase), then the wavefunction at every time step is actedupon by the unitary operator

U(τ−, 0−) = e−ιτH0 e−ιV∆t, (10)

with ∆t = τ only for the strictly periodic case. Thesuperscript ‘−’ indicates the mapping of the systembetween two instants immediately before kicks. In

noisy cases, the operator e−ιτH0 is applied at every

time step while the kicking potential e−ιV , V =

V∑L

j=1 cos (2παj + φ) c†jcj is applied at times tn and thetime interval ∆t will be a random variable as per the pro-tocols listed in Table I. Apart from these noisy protocols,

the case where the kick operator e−ιτ V is applied basedon the result of a biased coin-toss provides an insight intothe role played by the correlations in the kicks. This isdiscussed in section VII(D). All these are implemented

by direct matrix multiplication, since H0 is tridiagonalin the position basis, and can be efficiently diagonalisedand hence exponentiated.

IV. EVOLUTION UNDER NOISY KICKS

A. Noisy kicks versus missing kicks

For the purposes of the numerical results presented inthis section, the parameters for the noise-free AAH modelare set at τ = 0.5 and J = 1 resulting in Vc ∼ 2Jτ =1 at which the delocalization to localization transitiontakes place. To motivate the main results, two broadscenarios arising from the application of protocols B andC are displayed in Fig. 2 and 3. The density profileof the evolving wavepacket and σ2(n) are shown in Fig.

5

1 10 100n

100

101

102

103

104

105

σ20.250.50.950.91.051.11.51

1 10 100n

0.250.50.950.91.051.11.51

V V

(a) (b)

FIG. 4. Wavefunction evolution under the effect of missedkicks (protocol C), with the time intervals without kicks aredrawn from uniform distribution. Width of the evolving stateσ vs. time (symbols) shown for several values of kick strengthV with (a) A = 10 and (b) A = 100. For V > Vc, a sharpbreak in the dynamics is observed at n = nc ∼ A. The dashedlines have slopes 1 and 2.

2(a,b) for protocol B. Remarkably, when the model is inthe localized phase (V = 1.5), despite strong amplitudenoise with a width equal to V , the density profile nearlyretains its shape. The corresponding growth of σ2(n) isso strongly muted that it is of O(1) even after 104 timesteps. A similar scenario of robust localization emerges ifprotocol A is applied (not shown here) and this has beenreported before in Ref. [33].

Hence, in any protocol that ensures that a kick is con-sistently delivered at multiples of τ , or at least in a smalltime window around it [33], the wavepacket spread isweakly subdiffusive, indicating the robustness of the lo-calized phase to noise. Even if the amplitude is alteredat a random time, localization persists for an extensivenumber of kicks, as evident in Fig. 2(a,b). This picturechanges when a kick is not guaranteed at every timestepas in the case of protocol C displayed in Fig. Fig. 1(c). IfTN is the time at which the N -th kick is delivered, thenKN ≡ (TN+1 − TN ) /τ , where KN > 0 is an integer ran-dom variable drawn from a discrete distribution F (K).In the rest of the paper, we shall call F (K) the wait-ing time distribution since it represents the time intervalbetween successive appearance of kicks.

As we show in the next section, the character of diffu-sion undergoes a sharp change at a timescale decided bythe specifics of the noise process. This is observed acrossdifferent types of noise superimposed on the standardkick sequence. Figure 3(a) shows the evolution of densityprofile due to noisy protocol C. A global picture is seen inFig. 3(b). At short timescales, the dynamics can displaysubdiffusive to ballistic growths, and tends asymptot-ically towards diffusive dynamics. This crossover fromgrowth with slope 2 (in log-log plot) to slope 1 is observedin Fig. 3(b). If nc represents this crossover timescale,then superdiffusive growth takes place for times n < nc,and becomes diffusive for n > nc. The numerical value

1 10 100n

100

101

102

103

104

σ2

0.250.50.950.91.051.11.515

1 10 100n

0.250.50.950.91.051.11.515

V V

(a) (b)

FIG. 5. Wavefunction evolution under the effect of missedkicks (protocol C), with waiting times drawn from Poissondistribution. Width of the evolving state σ vs. time (symbols)shown for several values of V for (a) λ = 0.5 and (b) λ = 10.Note the sharp transition point for λ = 10. Dashed lines haveslopes 1 and 2.

of nc is dependent on the properties of superimposednoise. Given that localization is robust even if noisykicks are imparted once every kicking period, in the sub-sequent sections, we focus on the wavefunction evolutionunder the effect missed kicks (protocol C). In the limit oft→ ∞, the unbounded diffusive growth continues thoughin practice it is interrupted by the finite size of the lattice.

V. DYNAMICS UNDER MISSING KICKS

In this section, the diffusive effects induced by the miss-ing kicks, protocol C, are studied. To begin with, sinceK ∈ Z, we consider the case of F (K) being a discreteuniform distribution U(K;A), with K ∈ [1, A] definingthe support for the distribution. In this case, a ballis-tic growth at initial times and an asymptotic diffusivegrowth is generically observed. The simulations that re-sult from the Hamiltonian in Eq. 1, shown in Fig. 4,reveal the existence of a crossover timescale nc. In Fig.4(a), σ2 is shown as a function of n for several valuesof kick strength V for A = 10. At short time scales ofn < nc, the wavefunction spread is ballistic as the in-crease in σ2 is parallel to dashed line with slope 2. Forn > nc, wavefunction spread becomes diffusive, as evi-dent in σ2(n) becoming parallel to a line with slope 1(Fig. 4(a)). This scenario of transition from ballisticgrowth at short times to asymptotic diffusive growth re-peats for A = 100 as well, as shown in Fig. 4(b).When the kicking is in the delocalized regime, V < Vc,

then the transition is smooth and is corroborated in Fig.4(a). However, in the cases when V & Vc, the transitionsuddenly sharpens, and one can notice distinct regimesin the dynamics. In this case, a distinct crossover pointbetween the two regimes is observed. A common featureis that regardless of the number of distinct regimes, tran-sitions between them are always sharp and hence we can

6

1 10 100n

100

101

102

103

104

105

σ210010505

1 10 100n

1001050575

AA

(a) (b)

FIG. 6. Growth of density profile in the delocalized and local-ized regime as the parameter A of the waiting time distribu-tion U(K;A) varies. In this, (a) V = 0.5 and, (b) V = 1.5 arefixed, but A is varied. Abrupt transitions from superdiffusiveto diffusive growth is seen only in the localized regime withV > Vc. Dashed lines have slopes 1 and 2.

label them as being distinct. In particular, for V > Vc, the transition to asymptotic diffusive behaviour alwaysoccurs at n = nc ≈ A. This is borne out by the nu-merical results in Fig. 4(a,b). The asymptotic diffusivebehaviour is expected as a generic feature of decoherence[50], which takes place when noise is introduced into asystem [24, 51].

If the waiting times are Poisson distributed, i.e.,F (K) = P (K;λ), where λ is the mean of the distribu-tion, a scenario similar to that of uniform distributionemerges. For λ < 1, which physically implies that kicksare rarely missed, the dynamics of the almost periodicallykicked Hamiltonian gives rise to approximately ballisticinitial growth until asymptotic diffusive growth sets in atlater times. The timescale at which this sets in is muchlonger when V < 1 than if V > 1. These features areobserved in the simulation results in Fig. 5(a) for whichλ = 0.5. In all the cases, asymptotic diffusive growth setsin as n >> 1. For large λ, as shown in Fig. 5(b), thereis an initial ballistic growth and it crosses over to normaldiffusion, as in the case of uniform noise. The transitionpoint nc from superdiffusive to diffusive growth is sharpif V > 1, and it is smooth if V < 1.

What is the nature of transition if the parameter char-acterising the waiting time distribution is varied ? Thequalitative picture portrayed in Fig 4 does not changeeven if the parameter A in the uniform waiting timedistribution U(K;A) is varied. As displayed in Fig.6, varying A does not modify the initial superdiffusiveregime and the asymptotic diffusive regime. Its effectis to simply shift the time at which the sharp transi-tion is observed for V > Vc (Fig. 6(a)), and to changethe timescale on which asymptotic diffusion sets in forV < Vc (Fig. 6(b)). The distinct dynamical behaviourupon variation of λ is evident in Fig. 7. No sharp transi-tions are observed for V < 1, irrespective of the value of

1 10 100n

100

101

102

103

104

105

σ2

0.010.10.5101505

1 10 100n

0.010.10.5101505

λ λ

(a) (b)

FIG. 7. Growth of density profile in the delocalized and local-ized regime with the change in the parameter λ of the wait-ing time distribution P (K; λ). In this, (a) V = 0.5 and, (b)V = 1.5 are fixed, but λ is varied. Sharp transitions fromsuperdiffusive to diffusive growth is seen only in the localizedregime with V > Vc and λ >> 1. Dashed lines have slopes 1and 2.

λ. In this case, all the transitions are smooth as seen inFig. 7(a). For large λ, longer mean waiting times ensurethat the ballistic regime stays on for longer timescales.On the other hand, for V > 1 sharp ballistic to diffusivetransitions are observed if λ > 1 (Fig. 7(b)). In sectionVII, we provide a justification for these observations andalso for the qualitative similarity of the dynamics withPoisson noise (λ & 2) to that of uniform noise. Thus,the parameters A and λ effectively tune the separationof timescales with distinct growth regimes. Further, sim-ilar results as in Figs. 4-7 were also observed (not shownhere) when Kn were chosen as the integer parts of a nor-mally distributed variable.

VI. TRANSLATIONALLY INVARIANTMODELS

In the light of the results presented above, one signifi-cant question of interest would be the dependence of theobserved phenomena on the lack of translational invari-ance in the model. To address this, the parameter α inthe onsite potential Vi in Eq. 1 is tuned from rationalto an irrational number. A standard way to do this isto consider a sequence αi such that αi = Fi/Fi+1, where

Fi is the ith Fibonacci number. Then, limi→∞

αi =√5−12 .

When α = pq , where p and q are integers, the Hamilto-

nian in Eq. 1 becomes translationally invariant and willhave q-bands under periodic driving. In this section, wesimulate the wavefunction evolution for the q-band modeland obtain σ2 as a function of time n.Firstly, as Figs. 8 and 9 show, a transition from bal-

listic to diffusive growth of σ persists even in the trans-lationally invariant models. Further, the transition issharp under identical parametric regimes, i.e., V > Vc,

7

100

102

104

σ2

1001020050575

1001050575

1 10 100n

100

102

104

σ2

0.250.50.91.11.55

1 10 100n

0.250.50.91.11.55

(a) (b)

(c) (d)

FIG. 8. Diffusion in 2-band model with uniformly distributedwaiting times between kicks. The parameters are (a) V = 0.5and (b) V = 1.1 for various values of A, and (c) A = 10 and(d) A = 100, for various values of kicking strength V . Dashedlines have slopes 1 and 2.

considered earlier in Figs. 6 and 7. As the system isswitched from a free model to a q-band model, whereq is a small integer, it is surprising to observe a sharpbreak connecting these two regimes. In the simulationswith 2-band model and noisy protocol C with uniformwaiting time distributions, sharp transition is absent ifV < Vc whereas it is clearly visible if V > Vc as seenin Figs. 8(a,b,c,d). In the case of Poisson distributedwaiting times shown in Fig. 9 a sharp transition is seenif λ >> 1. The asymptotic diffusive dynamics in allthe cases arises due to decoherence. Such dynamics isgenerically seen across all the q-band structures we havesimulated for 2 ≤ q ≤ 10. Hence, the results shown inFigs. 4 to 7 are also valid for models with translationalinvariance.In general, what determines if the nature of transition

will be sharp or smooth? Further, in certain windows ofthe strength of kicks V , a drastic slowing down in thegrowth of σ at the transition point is observed. This isaccompanied by the diffusion exponent almost reachingzero and, in certain cases, even falling further. To explainthe observed features in Figs. 4 to 9, we further studythe band structure of the 2-band model in greater detail.

A. Band Structures : kicking induced flat bands

A first striking observation from the band structuresis that, for the model in Eq. 2, kicking alone can beused to engineer flat-bands and hence, the localization ofexcitations. Lattice models that exhibit flat bands havebeen of considerable interest in recent times [52–54]. Aneffective Hamiltonian which gives rise to the same dy-namics as that of the periodically driven model over onetime period can be obtained from the Floquet opera-

100

102

104

σ2

0.010.10.5101505

0.010.10.5101505

1 10 100n

100

102

104

σ2

0.250.50.91.55

1 10 100n

0.250.50.91.55

(a) (b)

(c) (d)

FIG. 9. Diffusion in 2-band model with Poisson distributedwaiting times between kicks. The parameters are (a) V = 0.5and (b) V = 1.1, for various λ, and (c) λ = 0.5 and (d) λ = 10,for various V . Dashed lines have slopes 1 and 2.

tor using the Baker-Campbell-Hausdorff expansion. Ina naıve weak-potential approximation at high kick fre-quencies, one simply obtains an effective Hamiltonian

Heff = H0 + V /τ , leading to a tight-binding model withthe potential rescaled by τ . However, for no on-site po-tential of finite strength can there be flat bands in staticq-band tight-binding models. Analytically, this can beseen for the simple cases of the 2, 3 and 4 band models.Concretely, the dispersion relation for a 2-band modelwith onsite potentials ±U is given by [55]

E(k) = ±√U2 + 4J2 cos2

(ka

2

), (11)

where a is the lattice spacing, and J is the hopping ampli-tude. Only in the limit where the hopping is completelysuppressed and there are on-site potential terms alone, isthe spectrum flat.Evidently, this is not the case for the kicked AAH

model in Eq. 1. Figure 10 shows the band structurefor the 2-band model. As seen in Fig. 10(d), flat bandscan indeed be realised with finite on-site potential in thepresence of kicking. By diagonalising the Floquet opera-tor, the quasi-energies ǫ(k) can be expressed as

ǫ(k) = ± cos−1 {cos (2Jτ cos k) cosV } . (12)

Thus, flat-bands exist for V = (2l+ 1) π2 , l ∈ Z and

the kicking has shown a relatively straightforward mech-anism to construct flat bands. This is confirmed in Fig.10(d) where flat bands appear for V = π/2 ≈ 1.55. Atsuch parameter values, where the allowed bands are ex-actly flat, an abrtupt transition point is indeed observed(Fig. 8(c,d)).However, as simulation results reveal, perfectly flat

bands are not required in order to observe either a

8

-2

-1

0

1

2 (a) V=0.50

ε(k)

(b) V=0.90

-2

-1

0

1

2

-1.3 -0.7 0.0 0.7 1.3

(c) V=1.10

ε(k)

k

-1.3 -0.7 0.0 0.7 1.3

(d) V=1.55

k

FIG. 10. Band structures obtained from simulations for the2-band model. Note that the flat bands are obtained for V =π/2 ≈ 1.55.

sharp break or a slowing down in the dynamics. For in-stance, sharp transitions begin to appear approximatelyfor V > Vc as seen in Fig. 8(c,d). Hence, in the next sec-tion, we provide analytical support to the observations insections III to VI using 2-band model. In particular, wealso analyse the question, how “flat” is “flat enough” ?Moreover, are there other transitions that one can expectto observe that accompany the appearance of a break inthe dynamics ?

VII. ANALYSIS

In this section, analytical justification is presented,mainly, for the case 2-band model. Firstly, γ4, as de-fined in Eq. (9), is computed to obtain nc, the time atwhich sharp transition takes place in the dynamics. Sec-ondly, as a further evidence for sharp transtion at timen = nc, a naıve scaling of log

(σ2)is presented in the case

of uniform noise. Finally, the role of noise correlations inthe sharpness of the transition is elucidated.

A. Calculation of γ4

The time nc at which the dynamics sharply transitionsfrom superdiffusive to diffusive is obtained by calculatingthe averaged local diffusion exponent γm, and observingthe time at which sharpest drop occurs. As is evidentfrom Figs. 6-7, the transition from initially superdiffusiveto diffusive growth is smooth in the delocalized regime ofkick strength V < Vc. This transition becomes sharpas one approaches and crosses into the localized regime.In our investigations, there does not appear to be anyqualitatively different behaviour at the critical point V =Vc.In Fig. 11, for the case of AAH model (Fig. 11(a,b))

and 2-band model (Fig. 11(c,d)), γ4(n) is shown for uni-formly distributed waiting times U(K) with K ∈ [1, A].Clearly, a sharp dip in γ4 occurs exactly at n = A when

-0.5

0

0.5

1

1.5

2

(a)

g4

A

1050100

(b)

-0.5

0

0.5

1

1.5

2

20 40 60 80 100

(c)

g4

20 40 60 80 100

(d)

FIG. 11. γ4 vs. n for noisy models with waiting times drawnfrom a uniform distribution U(K), with K ∈ [1, A]. (a,b)AAH model, and (c,d) 2-band model. The parameters are(a,c) V = 0.5, (b,d) V = 1.5.

V > Vc. In contrast, for V < Vc, a smooth transitionoccurs from the superdiffusive and diffusive regime.

In fact, the dips in γ4 in general seem to be far moresteep in the translationally invariant case than in theAAH model, across different values of V . As expected,the flatter the bands, the sharper the dips in γ4. Thisprovides a quantitative evidence to associate flat bandswith sharp transition in diffusive regimes.

B. Scaling of σ2(t)

The scaling arguments are presented in this section tocorroborate the fact that the time of transition is nc ∼ A,when K is drawn from a uniform distribution [1, A].

It can be posited that σ2(n;A) can be expressed interms of a function σ2

0 , defined as

σ20(n) =

{n2 0 < n < 1

n n > 1, (13)

σ2(n;A) = A2σ20

( nA

). (14)

In order to demonstrate this scaling, (the logarithm of)the function fA(n) ≡ 1

A2σ2(An) is plotted for different

values of A in Fig. 12. It can be seen from this figurethat fA(t) ∼ σ2

0(t) for all A. Taken together, for uni-formly distributed waiting times, Figs. 10-12 display thecentral result that at parametric regimes at which flatbands appear, the transition from superdiffusive to diffu-sive growth of evolving states is sharp, and the transitionpoint follows a excellent scaling with respect to a param-eter A that controls the strength of noise.

9

C. Mechanisms for time of transition and diffusion

In this section, analytical support is provided for as-sociating flat bands with the sharp superdiffusive to dif-fusive transition in the case of 2-band model. To begin,the following three probabilities are defined:

P(n) ≡ the probability of a kick occurring at the nth

time stepPN (n) ≡ the probability of the N th kick occurring at

the nth time stepQ(n;N) ≡ the probability that exactly N kicks have

been received before the nth time stepThese quantities are related to each other as

P(n) =

n∑

N=1

PN (n) (15)

Q(n;N) =n∑

n′=N

(PN (n′)− PN+1(n′)) . (16)

If ti, i = 1, 2, . . .N , denote the time intervals betweensuccessive kicks on the time axis, then the calculation ofPN (n) amounts to finding the number of ways one can

choose {t1, t2, . . . , tN} such thatN∑i=1

ti = n. Thus

PN (n) = PN

(N∑

i=1

ti = n

)(17)

For the specific case of F (K) = U(K;A), note thatPN (n) = 0 if N <

[nA

]+ 1, and by construction,

PN (n) = 0 if N > n for all cases of F (K). The nor-malisation condition is

NA∑

n=N

PN (n) = 1 (18)

from which one can see that

Q(n ≥ (N + 1)A;N) =

n∑

n′=N

(PN (n′)− PN+1(n′))

=

NA∑

n′=N

PN (n′)−(N+1)A∑

n′=N+1

PN+1(n′)

= 1− 1 = 0. (19)

The last form uses the normalization in Eq. 18. In allcases, P(n) does depend on n, at least for small n.This discussion begins by considering what happens

to an initial state ψ0 under the dynamics of the 2-bandmodel after the system receives exactly one kick. Thequantity of interest, then, is

σ2(t1, t2, V ) ≡ 〈ψ0|eιt1H0eιV eιt2H0 x2e−ιt2H0e−ιV e−ιt1H0 |ψ0〉 ;

|ψ0〉 ≡1√L

∑

k

|k〉 . (20)

10-5

10-4

10-3

10-2

10-1

100

101

102

10-1

100

101

102

(a)

s2 /A2

10-1

100

101

102

(b)

51050

100

FIG. 12. Scaling of σ2 with the A for the case of uniformlydistributed waiting times with V = 1.5. (a) AAH model,and (b) 2-band model. Note the excellent data collapse afterscaling using Eq. 14.

where |k〉 are the momentum states. In the mo-mentum basis, the evolution operator decomposes into

2 × 2 subspaces, in which, H0 is diagonal and x2 →∑k1,k2

∂k1∂k2

δ(k1 − k2). The Floquet matrix in a single2× 2 k-subspace is

Fk(V, T ) =

(eι2JT cos k cosV −ιeι2JT cos k sinV

−ιe−ι2JT cos k sinV e−ι2JT cos k cosV

).

(21)

After some calculations whose details are given in Ap-pendix A, we obtain

σ2(t1, t2, V ) = 2 (Jτ)2 (t21 + t22 + 2 cos(2V )t1t2

). (22)

For perfectly flat bands V = π2 , and this reduces to

σ2(t1, t2, π/2) = 2 (Jτ)2(t1 − t2)

2. (23)

This can be interpreted as follows. An impulse with aflat band Hamiltonian leads to a reversal of the velocityof the wavepacket. Concretely, consider the case whereF (K) = U(K;A). The probability that the system hasreceived only one kick by the time n = A steps haveelapsed is

Q(A; 1) =

(1

2+

1

2A

)>

1

2. (24)

Under the assumption that the system is faced with this,the most probable scenario for σ would be

σ2(Aτ) ∼ (Jτ)2

3A(A+ 2) ∼ 1

6σ2(A, 0, 0). (25)

The leading order term here shows that while the coef-ficient has reduced, the dynamics is still ballistic. How-ever, when one considers time steps n > A, it is most

10

10 20n

0.0

0.5

1.0

Q(n;N

≥2)

A = 10

0 50 100n

A = 50

FIG. 13. Cumulative mass function Q(n;N ≥ 2) for F (K) =U(K;A). The transition happens at time nc such thatQ(n;N ≥ 2) = 1

2. This figure shows that nc ∼ A.

likely that the system will have seen at least 2 kicks (de-tails of the calculation in Appendix B) and its probabilitycan be expressed as

Q(n > A;N ≥ 2) = 1−Q(n > A; 1)−Q(n > A; 0) >1

2.

(26)As more kicks are imparted, the particle is subject to anincreasing number of velocity reversals, and it executes arandom walk, leading to diffusive behaviour.Based on this argument, we posit that the break-time

nc is the time at which Q(n;N ≥ 2) – a monotonic, in-creasing function – exceeds 1

2 . Then, for n > nc, thesystem has most likely received more than one kick. Thequantity of interest Q(n;N ≥ 2) is shown to be theCumulative Mass Function (CMF) for the distributionP2(n) in Appendix B. Thus, the posited break-time isthe median of the distribution P2(n) which is A, for thecase of uniform noise and ∼ λ for λ >> 1, in the Poissoncase.The CMF for the case of uniformly distributed waiting

times is shown in Fig. 13, for two different values of A.Using its expression (obtained in Eq. B7 in AppendixB), we obtain nc = A, the value at which CMF becomeslarger than 1

2 .This estimate for nc agrees with the timeat which sharp transition is observed in Fig. 6(b).Similarly, for the case of Poisson waiting time distribu-

tion, the CMF is shown in Fig. 14 for various values ofmean waiting time λ. In this case, the median of P2(n)approximately corresponds to the time at which transi-tion to diffusive dynamics takes place. This can be clearlyseen for the cases of λ = 5, 10 upon comparing the me-dian with the actual transition observed in Fig. 7(b).Further, the median occurs at n > 1 only when λ ' 0.7,and at n ≥ 3 only when λ ' 1.25, which explains whythe break is not observable in Fig. 7(b) for λ ≤ 1.At large times n >> 1, regardless of the form of the

F (K), the contribution to P(n) comes from PN (n) where

0.0

0.5

1.0

Q(n;N

≥2)

λ = 0.1 λ = 0.5

0 20 40n

0.0

0.5

1.0

Q(n;N

≥2)

λ = 1.5

0 20 40n

λ = 10

FIG. 14. Cumulative mass function Q(n;N ≥ 2) for F (K) =P (K;λ). Notice that nc is barely greater than 1 for λ ≤ 1,explaining why the break is quite abrupt for λ > 1.5

N is large. By the central limit theorem and Eq. 17,PN (n) tends to a normal distribution that depends onlyon the properties of F (K), and thus, P (n) is no longerdependent on n. This implies that the velocity of the par-ticle is reversed with a fixed probability P∞ = P (n→ ∞)at each time step. Since the wavepacket evolves with thatvelocity for one time step, this is akin to a 1-dimensionalclassical discrete-time random walk. In this case, it isknown that

⟨x2(t)

⟩∼ t, and hence a linear increase in

the square of the width of the wavepacket for n >> 1can be expected.

D. Role of n dependence of P(n)

From the arguments made in the previous sub-section,it is seems that the time dependence of P(n) at shorttimes plays a crucial role in the appearance of a sharptransition time nc. In order to test this, the AAH modelin Eq. 1 is evolved and kicks are applied depending onthe outcome of a biased coin toss, with various (time-independent) biases. P(n) is now constrained to be in-dependent of n, as a result, and is denoted by pk. Inthe 2-band case, the quantity σ2 has an exact solution interms of a 4×4 “disorder matrix”, as shown in AppendixC. The final result turns out to be

σ2(N) =1

L

∑

i,j

[A(N) +B(N) + C(N)]i,j . (27)

where A(N), B(N) and C(N) are defined in the Ap-pendix C. Note that the indices in brackets are not thetrue indices of the 4× 4 matrix D. The recipe to switchbetween the two type of indices is by employing the Kro-necker product and is given in Appendix C. The nu-merically simulated results for the coin-toss model ap-plied to the Hamiltonian in Eq. 1 and for the 2-bandmodel are, respectively, in Fig 15(a) and 15(b). Notice

11

the absence of sharp transitions even for V = 1.57 (atwhich sharp transitions were observed in the case of uni-form and Poisson distributed waiting times). This il-lustrates another facet of wavefunction spreading – thesmooth nature of the crossover from the superdiffusiveto the diffusive regime, even at a value of V that pro-duces sharp break-points for other kick sequences withtime-dependent P(n) seems to suggest that this time de-pendence is crucial to the sharp change observed in thedynamics. This time dependence manifests in form ofcorrelation for the times at which kicks are imparted,in that P(n|n′) 6= P(n)P(n′). Indeed, there has beenwork on continuously and stochastically perturbed mod-els where the has played a non-trivial role in engineeringsuch a transition [56].

VIII. CONCLUSIONS

This work had focussed on how an initially localizedwavepacket evolves under the action of noisy kick se-quences applied to a lattice model with quasi-periodiconsite potential, namely, the Aubry-Andre model. Thissystem is well known to display localization to delocal-ization transition even in the one-dimensional case, incontrast to the Anderson model. Periodically drivingsuch systems is known to either preserve localization orinduce delocalization depending on the choice of parame-ters. This work presents an extensive study of effects dueto noisy kicks and we place it in the context of the cur-rent interest in a variety of Floquet engineering schemesin quantum systems.In summary, three distinct noisy kick sequences are

considered as shown in Fig. 1. The first two sequencesdeliver one kick on an average in every period, in whichcase the localized phase appears to be sufficiently robustto application of noise. However, if the kick sequence is

10-1

100

101

102

103

104

105

106

101

102

(a)

g=1

g=2

s2

101

102

(b)

g=1

g=2

0.050.200.350.500.650.800.95

FIG. 15. σ2 vs. t/τ at V = 1.57 for coin-toss noise in the (a)AAH model and (b) 2-band model

chosen such as to miss certain kicks entirely, then theresults are quite dramatic. Most of this paper discussesthe effects due to randomly missed kicks. Under theseconditions, it is found that the dynamics of an initiallylocalized wavepacket can be made ballistic for arbitrar-ily long times by tuning a parameter characterising thenoise process. Following the ballistic regime, there is asharp transition to diffusive behaviour. In particular, thetransition is not always smooth. This was observed bothin translationally invariant as well as disordered models;the former required some degree of flatness to the bands.We argue and show, using a 2-band model, that the emer-gence of diffusive behaviour in the long time limit is in-trinsically linked to velocity reversals which leads to clas-sical random walk type behaviour. More importantly, byusing an uncorrelated kick sequence for comparison, it isshown that correlations in the noise play a crucial rolein facilitating the sharpness of this transition, which isdetermined by the parameters characterising the noiserealizations.

It would be interesting to obtain a more rigorous an-alytic backing for the criterion to find the ballistic todiffusive transition timescale nc. Moreover, the observa-tion of these phenomena in the presence of interactions isyet to be explored in this model, but results in countin-uously (stochastically) driven models that could serve asguides to future studies [56].

ACKNOWLEDGMENTS

We are grateful to G. J. Sreejith for helpful discus-sions and comments on the thesis that led to this work.One of the authors (MSS) would like to acknowledgethe financial support from Science and Engineering Re-search Board, Govt. of India, through MATRICS grantMTR/2019/001111.

Appendix A: Calculation of σ2 (t1, t2,V)

Let Uk1 ≡ Fk(V, t2)Fk(0, t1), where Fk(V, T ) is thesame as in Eq. (21). σ2 (t1, t2, V ), as defined in Eq. (20),can equivalently be written as

1

L

∑

{in,jn}k1,k2

〈i0|U †k2|i1〉 〈j1|Uk1

|j0〉∂2δ (k1 − k2)

∂k1∂k2

=1

L

∑

{in,jn}k1,k2

∂

∂k2〈i0|U †

k2|i1〉

∂

∂k1〈j1|Uk1

|j0〉 δ (k1 − k2) δi1,j1

(A1)

12

{in, jn} = 1, 2 and k1, k2 ∈ [0, π).

∂

∂k1

∑

j0

〈j1|Uk1|j0〉 = (−ι2J sin(k1))×

{

δj1,1

((t1 + t2) cos(V )eι2J(t1+t2) cos(k1) +

ι(t1 − t2) sin(V )e−ι2J(t1−t2) cos(k1))−

δj1,2

(−(t1 + t2) cos(V )e−ι2J(t1+t2) cos(k1) +

ι(t1 − t2) sin(V )eι2J(t1−t2) cos(k1))}

(A2)

∂

∂k2

∑

i0

〈i0|U †k2|i1〉 = (−ι2J sin(k2))×

{

δi1,1

(−(t1 + t2) cos(V )e−ι2J(t1+t2) cos(k2) +

ι(t1 − t2) sin(V )eι2J(t1−t2) cos(k2))+

δi1,2

((t1 + t2) cos(V )eι2J(t1+t2) cos(k2) −

ι(t1 − t2) sin(V )e−ι2J(t1−t2) cos(k2))}

(A3)

Using these results, redefining ti → ti/τ and somestraightforward calculations, one arrives at

σ2(t1, t2, V ) =8

L

(∑

k

sin2(k)

)×

(Jτ)2((t1 + t2)

2 − 4 sin2(V )t1t2

).

(A4)

Replacing 1L

∑k sin

2(k) → 12π

∫ π

0 dk sin2(k) = 14 , we

have

σ2(t1, t2, V ) = 2 (Jτ)2 (t21 + t22 + 2 cos(2V )t1t2

)(A5)

Appendix B: Calculation of Q(n;N ≥ 2)

1. General form of Q(n;N ≥ 2)

We begin by considering Q(n;N = 0), which is theprobability that no kick has been imparted to the systemuntil step n. From the definition,

Q(n; 0) =∑

n′=n+1

P1(n′) (B1)

= 1−n∑

n′=1

P1(n′) (B2)

Q(n; 1) =

n∑

n′=1

(P1(n′)− P2(n

′))

= 1−Q(n; 0)−n∑

n′=1

P2(n′)

n∑

n′=1

P2(n′) = 1−Q(n; 1)−Q(n; 0)

Q(n;N ≥ 2) =

n∑

n′=1

P2(n′) (B3)

2. Q(n;N ≥ 2) for Specific Cases

We first consider the case of F (K) = U(K;A). Webegin by calculating P1(n) and P2(n).The first kick can take place at any 1 ≤ n ≤ A, with

equal probability.

P1(n) =

{1A , 1 ≤ n ≤ A

0, otherwise(B4)

From Eq. 17, P2(n) = P (K0 +K1 = n), with {Ki} ≤A. For n ≤ A+ 1, there are n− 1 ways of picking {Ki},since choosing K0 from 1 to n − 1 decides K1. For n >A+1, the upper limit on {Ki} requires that K0 can onlytake (2A− n+ 1) values from n−A to A.

P2(n) =1

A2

n− 1, 2 ≤ n ≤ A

2A− n+ 1, A < n ≤ 2A

0, otherwise

(B5)

If n ≤ A,

Q(n;N ≥ 2) =

n∑

n′=1

P2(n′)

=1

A2

n∑

n′=2

(n′ − 1) (by Eq.18)

Q(n; 1) =n(n− 1)

2A2(B6)

If n > A

Q(n;N ≥ 2) =1

A2

A∑

n′=2

(n′ − 1) +1

A2

n∑

n′=A+1

(2A− n′ + 1)

=2n−A− 1

2A− (n−A− 1)(n−A)

2A2(B7)

Putting these together,

Q(n;N ≥ 2) =

n(n−1)2A2 , n ≤ A

2n−A−12A − (n−A−1)(n−A)

2A2 , A ≤ n ≤ 2A

1, n > 2A

.

(B8)

13

Next, we consider F (K) = P (K;λ), for which it isstraightforward to calculate PN (n) for any n and N . Byconstruction, more than 1 kick at a time step is not al-lowed and thus we have

P (K;λ) ≡ e−λ λK−1

(K − 1)!θ(K − 1), (B9)

where θ(x) is the Heaviside Step-Function with the con-vention that θ(0) = 1. Then, we get

PN (n) =∑

{nj}

′

N∏

j=1

P (K = nj ;λ) (B10)

where prime indicates that the summation over {nj} is

performed subject to the constraintN∑j=1

nj = n.

PN (n) = θ(n−N)∑

{nj}

′

e−NλN∏

j=1

λnj−1

(nj − 1)!(B11)

= θ(n−N)e−Nλλn−N∑

{nj}

′ 1

(n1 − 1)! . . . (nN − 1)!

Under the redefinition of nj − 1 → nj , such that thecondition under the sum becomes

∑nj = n − N , and

the multinomial expansion, we have

PN (n) = θ(n−N)e−Nλ (Nλ)n−N

(n−N)!(B12)

The actual form of Q(n;N ≥ 2) is not particularlyuseful, and we require just the median of P2(n), which isknown to be ≈ 2λ for large λ.

Appendix C: σ2(N) for the 2-band coin-toss model

In the 2-band case, the quantity σ2 has an exact solu-tion in terms of a 4× 4 “disorder matrix”, introduced in[57]. We begin by defining σ2(N) using the noisy opera-tors,

σ2(N) =∑

{in,jn}k1,k2

〈ψ0|i0〉 〈j0|ψ0〉(

N∏

n=1

〈in−1|F †k1|in〉 〈jn−1|FT

k1|jn〉

)∂2

∂k1∂k2δ (k1 − k2) (C1)

〈in−1|F †k1|in〉 〈jn−1|FT

k1|jn〉 = pk 〈in−1|F †

k1(λ)|in〉 〈jn−1|FT

k1(λ)|jn〉

+ (1 − pk) 〈in−1|F †k1(0)|in〉 〈jn−1|FT

k1(0)|jn〉 . (C2)

One can define the following matrices

D1 := pk

[F †k (λ) ⊗ FT

k (λ)]+ (1− pk)

[F †k (0)⊗ FT

k (0)]

D2 := pk

[F †k (λ) ⊗ ∂k

(FTk (λ)

)]+ (1 − pk)

[F †k (0)⊗ ∂k(F

Tk (0))

]

D3 := pk

[∂k(F

†k (λ)) ⊗ FT

k (λ)]+ (1− pk)

[∂k(F

†k (0))⊗ FT

k (0)]

D4 := pk

[∂k(F

†k (λ)) ⊗ ∂k(F

Tk (λ))

]+ (1− pk)

[∂k(F

†k (0))⊗ ∂k(F

Tk (0))

], (C3)

in terms of which one can express σ2(N) as

σ2(N) =

1

L

∑

i0,j0iN ,jN

N∑

n=1

δiN ,jN

[Dn−1

1 D4DN−n1 +

n−1∑

m=1

(Dm−1

1 D3Dn−m−11 D2D

N−n1 +Dm−1

1 D2Dn−m−11 D3D

N−n1

)]

(i0,j0,iN ,jN )

,

(C4)

where the δiN ,jN follows from applying δ (k1 − k2) in thediscrete limit. Certain recursion relations prove helpful in

reducing the computational complexity of this evaluation

14

to O(N). These are

A(N) = A(N − 1)D1 +DN−11 D4; A(0) := 0,

B(N) = B(N − 1)D1 + B(N − 1)D3; B(1) := 0,

B(N) = B(N − 1)D1 +DN−11 D2; B(0) := 0

C(N) = C(N − 1)D1 + C(N − 1)D2; C(1) := 0

C(N) = C(N − 1)D1 +DN−11 D3; C(0) := 0,

σ2(N) =1

L

∑

i,j

[A(N) +B(N) + C(N)]i,j .

(C5)

Note that the indices in brackets are not the true indicesof the 4 × 4 matrix D. The recipe to switch betweenthe two type of indices – by employing the KroneckerProduct – is

[DN

1

](i0,j0,iN ,jN )

:=[DN

1

]2(i0−1)+j0,2(iN−1)+jN

. (C6)

[1] T. Oka and S. Kitamura,Annual Review of Condensed Matter Physics 10, 387 (2019).

[2] D. Basov, R. Averitt, and D. Hsieh,Nature Materials 16, 1077 (2017).

[3] M. Bukov, L. D’Alessio, and A. Polkovnikov, Advancesin Physics 64, 139 (2015).

[4] A. Haldar and A. Das,Annalen der Physik 529, 1600333 (2017).

[5] L. Du, X. Zhou, and G. A. Fiete,Phys. Rev. B 95, 035136 (2017).

[6] C. M. Dai, W. Wang, and X. X. Yi,Phys. Rev. A 98, 013635 (2018).

[7] F. Grossmann, T. Dittrich, P. Jung, and P. Hanggi,Phys. Rev. Lett. 67, 516 (1991).

[8] F. Harper, R. Roy, M. S. Rudner, and S. Sondhi,Annual Review of Condensed Matter Physics 11, 345 (2020).

[9] M. J. Park, Y. Kim, G. Y. Cho, and S. Lee,Phys. Rev. Lett. 123, 216803 (2019).

[10] N. Goldman and J. Dalibard,Phys. Rev. X 4, 031027 (2014).

[11] T. Schuster, S. Gazit, J. E. Moore, and N. Y. Yao,Phys. Rev. Lett. 123, 266803 (2019).

[12] M. Aidelsburger, M. Atala, M. Lohse,J. T. Barreiro, B. Paredes, and I. Bloch,Phys. Rev. Lett. 111, 185301 (2013).

[13] P. Kiefer, F. Hakelberg, M. Wittemer, A. Bermudez,D. Porras, U. Warring, and T. Schaetz,Phys. Rev. Lett. 123, 213605 (2019).

[14] G. Jotzu, M. Messer, and R. D. et. al.,Nature 515, 237 (2014).

[15] A. Rubio-Abadal, M. Ippoliti, S. Hollerith, D. Wei,J. Rui, S. L. Sondhi, V. Khemani, C. Gross, and I. Bloch,Phys. Rev. X 10, 021044 (2020).

[16] T. N. Ikeda and M. Sato, Science Advances 6,10.1126/sciadv.abb4019 (2020).

[17] E. Abrahams, 50 years of Anderson Localization (WorldScientific, 2010).

[18] R. Nandkishore and D. A. Huse,Annual Review of Condensed Matter Physics 6, 15 (2015).

[19] L. Reichl, The Transition to Chaos: Conservative Classi-

cal Systems and Quantum Manifestations (Springer Sci-ence & Business Media, 2004).

[20] S. Fishman, D. R. Grempel, and R. E. Prange,Phys. Rev. Lett. 49, 509 (1982).

[21] F. L. Moore, J. C. Robinson, C. F.Bharucha, B. Sundaram, and M. G. Raizen,

Phys. Rev. Lett. 75, 4598 (1995).[22] A. Eckardt, Rev. Mod. Phys. 89, 011004 (2017).[23] B. G. Klappauf, W. H. Oskay, D. A. Steck, and M. G.

Raizen, Phys. Rev. Lett. 81, 1203 (1998).[24] E. Ott, T. M. Antonsen, and J. D. Hanson,

Phys. Rev. Lett. 53, 2187 (1984).[25] D. Cohen, Phys. Rev. A 44, 2292 (1991).[26] H. Schomerus and E. Lutz,

Phys. Rev. A 77, 062113 (2008).[27] S. Sarkar, S. Paul, C. Vishwakarma, S. Kumar, G. Verma,

M. Sainath, U. D. Rapol, and M. S. Santhanam,Phys. Rev. Lett. 118, 174101 (2017).

[28] D.-L. Deng, S. Ganeshan, X. Li,R. Modak, S. Mukerjee, and J. H. Pixley,Annalen der Physik 529, 1600399 (2017).

[29] P. Bordia, H. Luschen, U. Schneider, M. Knap, andI. Bloch, Nature Physics 13, 460 (2017).

[30] S. Ray, A. Ghosh, and S. Sinha,Phys. Rev. E 97, 010101 (2018).

[31] S. Xu, X. Li, Y.-T. Hsu, B. Swingle, and S. Das Sarma,Phys. Rev. Research 1, 032039 (2019).

[32] L. M. Sieberer, M.-T. Rieder, M. H. Fischer, and I. C.Fulga, Phys. Rev. B 98, 214301 (2018).

[33] T. Cadez, R. Mondaini, and P. D. Sacramento,Phys. Rev. B 96, 144301 (2017).

[34] S. Aubry and G. Andre, Ann. Israel Phys. Soc 3, 18(1980).

[35] S. Y. Jitomirskaya, Annals of Mathematics 150, 1159 (1999).[36] G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort,

M. Zaccanti, G. Modugno, M. Modugno, and M. Ingus-cio, Nature 453, 895 (2008).

[37] M. Schreiber, P. Hodgman, Sean S. and, H. P. Luschen,M. H. Fischer, R. Vosk, E. Altman, U. Schneider, andI. Bloch, Science 349, 842 (2015).

[38] S. Iyer, V. Oganesyan, G. Refael, and D. A. Huse,Phys. Rev. B 87, 134202 (2013).

[39] H. Hiramoto and M. Kohmoto,International Journal of Modern Physics B 06, 281 (1992).

[40] H. Hiramoto and S. Abe,Journal of the Physical Society of Japan 57, 1365 (1988).

[41] T. Geisel, R. Ketzmerick, and G. Petschel,Phys. Rev. Lett. 66, 1651 (1991).

[42] D. R. Hofstadter, Phys. Rev. B 14, 2239 (1976).[43] Z. Zhang, P. Tong, J. Gong, and B. Li,

Phys. Rev. Lett. 108, 070603 (2012).[44] P. Qin, C. Yin, and S. Chen,

15

Phys. Rev. B 90, 054303 (2014).[45] R. Lima and D. Shepelyansky,

Phys. Rev. Lett. 67, 1377 (1991).[46] R. Artuso, G. Casati, and D. Shepelyansky,

Phys. Rev. Lett. 68, 3826 (1992).[47] R. Artuso, G. Casati, F. Bor-

gonovi, L. Rebuzzini, and I. Guarneri,International Journal of Modern Physics B 08, 207 (1994).

[48] C. Sanderson and R. Curtin, Journal of Open SourceSoftware 1, 26 (2016).

[49] C. Sanderson and R. Curtin, in International Congress

on Mathematical Software (Springer, 2018) pp. 422–430.[50] W. H. Zurek, Decoherence and the transi-

tion from quantum to classical — revisited, inQuantum Decoherence: Poincare Seminar 2005 , editedby B. Duplantier, J.-M. Raimond, and V. Rivasseau

(Birkhauser Basel, Basel, 2007) pp. 1–31.[51] D. Cohen, Phys. Rev. Lett. 67, 1945 (1991).[52] J. T. Chalker, T. S. Pickles, and P. Shukla,

Phys. Rev. B 82, 104209 (2010).[53] L. Ge, Ann. Phys. (Berlin) 529, 1600182 (2017).[54] T. Mizoguchi and M. Udagawa,

Phys. Rev. B 99, 235118 (2019).[55] P. Markos and C. M. Soukoulis, Wave propagation: from

electrons to photonic crystals and left-handed materials

(Princeton University Press, 2008).[56] S. Gopalakrishnan, K. R. Islam, and M. Knap,

Phys. Rev. Lett. 119, 046601 (2017).[57] U. Bhattacharya, S. Maity, U. Banik, and A. Dutta,

Phys. Rev. B 97, 184308 (2018).