Localisation transition in the driven Aubry-Andre model

D. Romito and C. LoboUniversity of Southampton

A. RecatiUniversita degli Studi di Trento

A recent experiment by P. Bordia et al. [1] has demonstrated that periodically modulating thepotential of a localised many-body quantum system described by the Aubry-Andre Hamiltonianwith on-site interactions can lead to a many-body localisation-delocalisation transition, providedthe modulation amplitude is big enough. Here, we consider the noninteracting counterpart of thatmodel in order to explore its phase diagram as a function of the strength of the disordered potential,the driving frequency and its amplitude. We will first of all mimic the experimental procedure of [1]and use the even-odd sites imbalance as a parameter in order to discern between different phases.Then we compute the Floquet eigenstates and relate the localisation-delocalisation transition totheir Inverse Participation Ratio. Both these approaches show that the delocalisation transitionoccurs for frequencies that are low compared to the bandwidth of the time independent model.Moreover, in agreement with [1] there is an amplitude threshold below which no delocalisationtransition occurs. We estimate both the critical values for the frequency and the amplitude.

I. INTRODUCTION

The study of periodically driven quantum systems hasgained interest in the last years. The most importanttool in this context is the Floquet theorem. One of itsmost interesting consequences is that it allows to writethe evolution over multiples of the driving period in termsof a time independent effective Hamiltonian [2], [3]. Theexistence of such an effective Hamiltonian can open theway to the so called Floquet engineering, that is, the pos-sibility to realise non trivial time independent models byperiodically modulating a quantum system with a suit-able protocol. This concept has been employed very suc-cessfully in various experiments with ultracold atoms indriven optical lattices [4]. This includes dynamic locali-sation ([5], [6], [7], [8]), “photon”-assisted tunneling ([9],[10]), control of the bosonic superfluid-to-Mott-insulatortransition ([11], [12]) and the realisation of artificial mag-netic fields ([13], [14], [15], [16]).

A recent seminal experiment [1] has focused insteadon the case of a driven many-body localised system, de-scribed by the Aubry-Andre model [17], showing thata properly tuned periodic driving can lead the systemacross the localisation transition. The interplay betweena many-body localised quantum system and a periodicdriving has drawn a lot of attention and recent theo-retical works have put forward the possibility that thiscombination can generate symmetry protected topologi-cal phases which have no equilibrium analogues [18] [19][20] [21] [22]. Motivated by the particular case of theexperiment reported in [1] we consider its single parti-cle counterpart, i.e. the periodically driven Aubry-Andremodel, in order to understand which kind of effects can bedisentangled from a strict many-body description. Thesame model was recently studied in [23], with which ourresults show good agreement, particularly in estimatingthe critical amplitude.

II. THE MODEL

We consider the Aubry-Andre Hamiltonian H0, withperiodically modulated potential V (t):

H(t) = H0 + V (t) (1)

where

H0 = J

N∑i

(|i〉 〈i+ 1|+ |i+ 1〉 〈i|)+λ∑i

cos(2πβi+φ) |i〉 〈i|

(2)

V (t) = A cos(ωt)∑i

cos(2πβi+ φ) |i〉 〈i| (3)

Here β is an irrational number, |i〉 is the Wannier stateon site i and λ is the disorder strength. We choose peri-odic boundary conditions for the lattice. The term V (t)satisfies V (t+ T ) = V (t), where T = 2π/ω.

The qualitative features of the undriven system in themany-body and the noninteracting cases are quite simi-lar [24]. In the noninteracting case the Hamiltonian H0

is that of a particle moving in a lattice with two lenghtsL1 and L2 which are incommensurate, with β = L1/L2.This produces a pseudorandom potential and the differ-ent realisations of this randomness are obtained chang-ing the value of the phase φ. Such a Hamiltonian is thesimplest realisation of a quasiperiodic crystal, or qua-sicrystal (see [25] for a review on the topic). When βis an irrational number the time independent model isproven to undergo a metal to insulator transition: aboveλ = λc = 2J the eigenstates of H0 are localised, whereasbelow they are delocalised ([17], [26], [27])

In the many-body case the transition is controlled bythe parameters J , λ and U , where U is the intensity of

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the repulsive on-site interaction. There is a critical dis-order strength which depends on J and U above whichthe system becomes localised [28]. More precisely, untilU ≈ 2λ the interaction decreases the degree of localisa-tion, while for large |U |, increasing U helps to make thesystem more localised. This is understood as a conse-quence of the formation of repulsive stable bound atompairs in optical lattices described by a Hubbard Hamil-tonian ([29]).For the first realization of this effect withcold atoms see [30] These pairs have a reduced effectivetunneling rate of Jeff ≈ J2/|U | which thus increases thedegree of localisation . Both above and below 2λ for eachvalue of U there is a definite value of λ for which the tran-sition occurs. It is thus interesting to see whether theseanalogies are retained in presence of the time periodicmodulation.

III. SETUP

A. Imbalance

As a first step to explore the phase diagram of themodel we mimic as closely as possible the procedure de-scribed in [1] but in a single particle context. The initialstate there is chosen as a density-wave pattern in whichfermions occupy the even sites of the lattice. The pa-rameter which discerns between a localised and a non-localised phase is the asymptotic imbalance:

I = limt→∞

1

t

∫ t

0

dt′Ne(t

′)−No(t′)Ne(t′) +No(t′)

(4)

where Ne and No are the number of particles in the evenand odd sites respectively. A persistent imbalance in-dicates a localised phase, while it obviously drops to 0in absence of localisation, indicating that the system isergodic as it does not retain the memory of its initialconditions.

To properly imitate the experiment we consider differ-ent realisations of the system each initially localised ona single even site and let them evolve separately underthe Hamiltonian H(t). The initial state in the Wannierstates basis for each realisation m = 1, ..., N/2 reads:

ψ(2m)(i, t = 0) = 〈i|ψ(2m)(t = 0)〉 = δi,2m (5)

After a long evolution time we sum the modulus squaredamplitudes of all the realisations to obtain the final den-sity, namely:

n(i, t) =∑

m=1,...N/2

|ψ(2m)(i, t)|2 (6)

The above definition is justified by the fact that the one-body density of a noninteracting many-body system isthe sum of the densities of the occupied single particlestates [31]. The analogues to the occupation numbers Ne

FIG. 1: Imbalance in the time independent case as a func-tion of the disorder strength λ, for lattice size N = 50, withperiodic boundary conditions. We see a clear critical valueat λc = 2J indicating the localisation transition. The timeof integration is τ = 1000h/J , at which the imbalance hasreached its asymptotic value

and No are then calculated by simply using this densityfunction as a weight in the following sum:

Ne(t) =

N/2∑i=1

n(2i, t) (7)

and similarly

No(t) =

N/2−1∑i=0

n(2i+ 1, t) (8)

With these definitions we can simply calculate the im-balance as defined in equation 4.

Before moving to the results for the driven lattice weshow how the Imbalance behaves around the phase tran-sition for the time independent model. Figure 1 was ob-tained considering a lattice made of N = 50 sites andcalculating the asymptotic Imbalance for different valuesof the disorder strength λ. It shows how the transition ismarked by a nonzero value of the imbalance as a functionof λ (all energies are in units of J), at the critical valueλc = 2J .

The imbalance is thus able to signal in a clear way thetransition from the localised to the delocalised phase.

B. Inverse Participation Ratio

In this subsection we link the localisation properties ofthe model to the localisation of its Floquet modes. Thetime periodicity of the full Hamiltonian H(t) allows us tomake use of the Floquet theorem, which states that we

3

can write the time evolution of an arbitrary initial stateas [2], [3]:

|ψ(t)〉 =∑n

cne−iεn(t)/h |un(t)〉 (9)

with cn = 〈un(0)|ψ(0)〉 and εn are the so-calledquasienergies. We emphasize the fact that these coef-ficients do not depend on time. The states |un(t)〉 arecalled Floquet modes. They have the same time pe-riodicity of the original Hamiltonian and can be foundas eigenstates of the propagator over one period, whichreads:

U(T, 0) := T exp(−i

h

∫ T

0

H(τ)dτ)

(10)

where T denotes the time ordering operator.We expand the Floquet modes at t = 0 in the Wannier

state basis yielding:

|un(0)〉 =∑i

b(n)i |i〉 . (11)

We define the averaged Inverse Participation Ratio (IPR)as the average of the IPRs of all the Floquet eigenmodeson the Wannier states, namely:

IPR =1

N

∑i,n

|b(n)i |

4 (12)

where N represents the number of Floquet modes whichcoincides with the number of sites of the lattice.

If each one of the Floquet eigenmodes is localised on asingle Wannier state then for any n there exists an i such

that |b(n)i | ≈ 1 and the sum approaches 1. If instead the

eigenmodes are distributed among many Wannier states

then |b(n)i | ≈ 1/

√N for all n, i and the averaged IPR

goes to 0 as 1/N .Thanks to the form of equation (9) we can expect a

localised dynamics when very few Floquet modes partic-ipate in the time evolution of an initial Wannier state.This would resemble in the context of time-periodic sys-tems the phenomenon of Anderson localisation where thelocalisation of the eigenstates of the Hamiltonian impliesnon-ergodic dynamics [32]. This is not however the onlymechanism that is in play, as localisation can occur asa consequence of the degeneracy of energy levels e.g.when the renormalized hopping parameter J becomesvery small due to the driving. This particular mecha-nism is often referred to as dynamic localisation or bandcollapse ([33], [34]). In [33], in particular, the authorspropose to observe a dynamic localisation effect in a re-alization of the Aubry-Andre Hamiltonian by tuning theamplitude and frequency to a value for which the renor-malized hopping would vanish. The periodic modulationthat we are considering here is however different fromtheirs and doesn’t allow the same tunability of the renor-malized hopping. Confronting the Inverse Participation

FIG. 2: Imbalance as a function of frequency for A = λ,normalized to its value in the absence of driving. While forlow frequencies the imbalance is vanishing, it approaches thevalue for A = 0 for high frequencies. This plot can be com-pared with the ones present in [1] and confirms the fact thatthe single particle picture is able to capture the qualitativefeatures of the many-body experiment.

Ratio with the Imbalance allows us to discern which ef-fects are due to collapse of the bandwidth of quasiener-gies and which ones are due to an Anderson localisationtransition.

IV. RESULTS

In what follows we will indicate the disorder strength,λ, and the amplitude of the modulation, A, in units of Jand times in units of 1/J . The calculations below weredone considering a lattice made of N = 50 sites, aver-aging over 20 different realisations of the disorder, whichare obtained by varying the value of the phase φ in equa-tion (2). In choosing β we decided to follow as close aspossible the choice of the experiment in reference [1], sowe chose β = 532/738.2. The simulations were madeusing the standard Matlab toolbox, solving the time evo-lution with the ode45 function in order to compute theimbalance and exactly diagonalizing the propagator overone period to find the Floquet modes.

As a first step to outline the behaviour of this model wecalculated the imbalance for strong driving, i.e. A = λ,for a broad range of frequencies, keeping the disorderstrength at a fixed value above the critical one λ = 3 = A.The results are shown in figure 2, which highlights a de-localised regime for low frequencies while for high fre-quencies the imbalance approaches that of the model inabsence of driving. The similarity in the main featuresbetween this figure and the ones in [1] is striking. In par-ticular the dip appearing after the imbalance has startedto rise, around hω = 2λ, is present also in the many-bodyexperiment.

To explain the origin of the above mentioned dip inthe value of the imbalance we first have to consider theresponse of the model to various values of frequency anddisorder. To this intent we computed the time averagedimbalance for different values of the disorder strength λand the angular frequency ω, setting the amplitude of

4

FIG. 3: Imbalance as a function of frequency and disorderstrength. On the vertical axis λ

hωis displayed in logarithmic

scale to allow comparison with the experiment in [1]. Thedashed line is for hω = 2λ, which is the approximate criticalvalue for the frequency.

FIG. 4: Inverse Participation Ratio as a function of frequencyand disorder strength. On the vertical axis λ

hωis displayed in

logarithmic scale to allow comparison with the experiment in[1]. The dashed line is for hω = 2λ, which is the approximatecritical value for the frequency.

the modulation in the strong driving regime i.e. A = λ.The evolution time is chosen to be 100 times the periodof the modulation. In 3 and 4 the vertical axis shows λ

hωto allow comparison with the experiment in [1]. In 5 theInverse Participation Ratio is displayed as a function ofλ and hω to more clearly show the relation between thefrequency response and the spectrum.

Figure 3 confirms that for very low frequencies the sys-tem is brought to a delocalised phase (marked by a van-ishing imbalance), while for high frequency the drivingis not able to bring the system to delocalisation any-more. Moreover it displays distinct analogy with thecorresponding phase diagram in [1].

As anticipated in the previous section we computed theInverse Participation Ratio for various values of ω and λ.Figures 5 and 4 distinctly show the separation betweenthe two phases. This also shows that the localisationproperties of the Floquet eigenmodes at initial time al-low us to discern in a broad sense the different localisa-tion properties of the system. This happens despite thefact that the initial Floquet modes carry no information

FIG. 5: Inverse Participation Ratio as a function of frequencyand disorder strength, for A = λ. The white dashed line isfor hω = 2λ, dividing the localised phase (yellow) from thedelocalised one (blue).

FIG. 6: Spectrum of the Aubry-Andre model as a function ofλ for N = 50 lattice sites.

on the structure of the quasienergy spectrum which cancontain accidental crossings of energy levels, causing thesystem to be partially localised. The relatively small sizeof the system implies that the Inverse Participation Ra-tio will display finite size effects in the delocalised phase,where it vanishes as 1/N . We run a simulation whichcomputes the IPR as a function of frequency for differ-ent system sizes, going from N = 50 to N = 500. Foreach system size the IPR goes to 0 with the correct scal-ing with respect to N , while in the localised phase itsbehaviour is largely unchanged.

There’s a transition line (white dashed line in 3, 4 and5) above which the system remains localised. This lineappears for hωc = 2λ which can be understood from thespectral properties of the Hamiltonian H0 of equation 2.To better illustrate this, in figure 6 we show the spectrumof the Aubry-Andre model as a function of the disorderstrength for N = 50 lattice sites. The bandwidth of theAubry-Andre Model is ≈ 2λ for any disorder strengthabove the transition point λc = 2J . Thus the transitionline in the time periodic case appears when the quanta ofenergy that the driving pumps into the system are too big

5

for the system to absorb. Above the transition line thesystem’s behaviour becomes that of the time independentmodel. This is because the period of the driving T = 2π

ωis now smaller than the fastest time scale present in theAubry-Andre Hamiltonian, making the system unable torespond to the driving.

Below the transition line there are other smaller re-vivals of the localised phase. We attribute this intri-cate structure again to the spectrum of the Aubry-Andremodel which is divided into smaller subbands divided byspectral gaps. In the intermediate range of frequencieswhere hω is comparable to the energy gaps present inthe spectrum, the presence of a localised phase has a nonmonotonic dependence on the frequency of the modula-tion. The study of the relevance of the single particlespectrum and the density of spectral lines for the fre-quency response of the model was studied in the thesisthat lead to this paper [35].

All these results confirm the qualitative picture out-lined in [1] and are well understood in terms of the singleparticle spectrum. This suggests that in this context thetime averaged imbalance, while providing a precise char-acterization of the phase diagram of the model, doesn’tseem able to disentangle the role of the interactions.

For very low frequencies the system is brought to de-localisation. We can define the following parameter:

λ(t)def= λ+A cos(ωt) (13)

If the frequency is very low the global parameter λ(t) ischanged adiabatically and sweeps through the transitionpoint λc = 2J , bringing the system to delocalisation.

This intuitive picture helps us understand the role ofthe amplitude of the driving A: even for arbitrarily lowfrequencies the system does not delocalise if the ampli-tude is not big enough to make λ(t) sweep through thecritical point λc = 2J . Following this reasoning we de-

fine the critical value for A to be such that mint

{λ(t)

}=

λc = 2J , namely:

Ac = λ− λc = λ− 2J (14)

This picture is clearly confirmed by the contourplotsof the imbalance and the Inverse Participation Ratio asfunctions of the disorder strength and the amplitude,which are shown on figure 7 and 8. For these plots weconsidered a frequency ν = ω/2π = 0.005(1/J). Thesame value for the critical amplitude was found in [23].

We stress that the very existence of a critical valueof A as determined here is valid only in the case of amodulation of the form considered in this work, whichcorresponds to a modulation of the disorder strength. Itis often stated in the literature (see [36], [37], [38]) thata modulation of arbitrarily small amplitude will alwaysdelocalise a many-body localised quantum system pro-vided that the driving frequency is small enough. Thisstatement however is not in contrast with our result asit refers to a driving of the form A cos(ωt)

∑i i |i〉 〈i|.

FIG. 7: Imbalance as a function of amplitude and disorderstrength, for ν = 0.005(1/J). There’s a clear line for A = λ−2separating the localised phase (yellow) to the delocalised one(blue).

FIG. 8: Inverse Participation Ratio of the Floquet eigenmodesas a function of amplitude and disorder strength, for ν =0.005(1/J). The result is consistent with the one in fig. 7,confirming that the localisation properties of the model aredue to the localisation of the Floquet eigenmodes

Last, we want to briefly comment on the role of the in-teraction. In presence of the interaction U , we expect toretain most of the qualitative results displayed, as com-parison with the many-body experiment seems to sug-gest. In particular, regarding the frequency response ofthe system we expect the description in terms of the spec-trum to be still relevant, with the critical value ωc to beshifted to be equal to the badwidth of the interactingmodel. According to [39] the many-body interaction willchange the size of the infinite number of spectral gapsof the noninteracting spectrum, without closing any ofthem. This makes the interacting model very similar tothe noninteracting one, except for the intermediate fre-quency regime where the effect of the interaction in cou-pling the energy levels is crucial, possibly explaining theless sharp peaks displayed in the experiment in [1].

6

V. CONCLUSIONS

Our work shows that the driven noninteracting Aubry-Andre model qualitatively reproduces many of the local-isation phenomena which are found in the experiment[1] such as the presence of a delocalised phase for lowfrequency, the persistence of localisation for a high fre-quency driving, and the existence of a critical value ofdriving amplitude for the onset of the localisation transi-tion. We were able to determine the critical values for thefrequency and the amplitude of the driving, and provide atheoretical explanation for their existence. Moreover werelated the phases of the model to the localisation of itsFloquet eigenmodes. Future theoretical studies shouldfocus on the role of interactions and the new qualitativeaspects they bring.

VI. AKNOWLEDGEMENTS

This project was supported by the University ofSouthampton as host of the Vice-Chancellor Fellowship

scheme.

D.R. would like to thank the Ludwig Maximilian Uni-versity (Munich) for the hospitality during the writing ofthe thesis that lead to this work.

VII. AUTHOR CONTRIBUTION

Starting from an idea by A. R., D. R. has carried outthe computations and produced the data in this work, aswell as working towards their interpretation with A. R.C. L. has contributed to detailing such interpretationsand helped D.R. in writing the paper.D. R., A. R., C. L. have contributed to the discussion ofthe results and the revision of the paper.

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