Phase transition in a Aubry-Andre system with rapidly oscillating magnetic field

Tridev Mishra,1, ∗ Rajath Shashidhara,1, † Tapomoy Guha Sarkar,1, ‡ and Jayendra N. Bandyopadhyay1, §

1Department of Physics, Birla Institute of Technology and Science, Pilani 333031, India.

We investigate a variant of the Aubry-Andre-Harper (AAH) model corresponding to a bosonicoptical lattice of ultra cold atoms under an effective oscillatory magnetic field. In the limit of highfrequency oscillation, the system maybe approximated by an effective time independent Hamiltonian.We have studied localization/delocalization transition exhibited by the effective Hamiltonian. Theeffective Hamiltonian is found to retain the tight binding tri-diagonal form in position space. In astriking contrast to the usual AAH model, this non-dual system shows an energy dependent mobilityedge - a feature which is usually reminiscent of Hamiltonians with beyond the nearest neighbourhoppings in real space. Finally, we discuss possibilities of experimentally realizing this system inoptical lattices.

PACS numbers: 03.75.-b, 72.20.Ee, 37.10.Jk, 05.60.Gg

I. INTRODUCTION

Ever since its proposal by Anderson [1], localization,and transitions between localized and extended states,have been studied in a variety of systems [2]. Exten-sive analysis has been undertaken to understand vari-ous aspects of metal-insulator transitions, localization aswell as existence of mobility edges in quasi-periodic ordisordered 1D lattices using scaling and renormalizationtechniques [3–10]. A system which has served as a richprototype for such studies is the Hamiltonian, originallydue to Harper [11], and investigated for phase transitionsby Aubry and Andre [12].

An important feature of the Harper Hamiltonian isthe existence of a metal-insulator transition [12] remi-niscent of Anderson transition. However, a notable dif-ference is the absence of an energy dependent mobilityedge separating the localized and extended states, whichis a distinguishing feature of the Anderson transition in3D. The Aubry-Andre-Harper (AAH) Hamiltonian ex-hibits a sharp, duality driven, transition at a uniquecritical value of the lattice modulation strength for allenergies [13–16]. An ensuing trend in recent works hasbeen to develop variations on the model which manifesta Anderson-transition like mobility edge [17, 18].

The quest is legitimized further by the substantialprogress made by the cold atom community in repro-ducing complex condensed matter phenomena includingAnderson localization [19, 20].

The experimental investigation of localization in 1Dsystems, especially of the quasiperiodic/incommensuratecrystalline variety has witnessed a sustained interest eversince such lattices could be realized using ultra coldatoms in a bichromatic optical potential, or photonic qua-sicrystals [20–27]. These studies have ranged from direct

∗ tridev.mishra@pilani.bits-pilani.ac.in† rajath.shashidhara@gmail.com‡ tapomoy1@gmail.com§ jnbandyo@gmail.com

experimental demonstration [20, 22, 28–31] to numericalcalculations [23–25] with accompanying proposals for ob-serving the appearance of localized phases and the Metalto Insulator or superfluid to Mott insulator transition (inthe presence of interactions). Here, the control on the de-gree of commensurability has helped to identify the pointof transition which, in the AAH model is the self dualityinduced critical point [27]. The Hofstadter variant andthe AAH model in a 2D optical lattice have also beensuccessfully realized [32, 33], in the context of simulat-ing homogeneous magnetic fields in optical lattices. Theeffects of periodic driving on localization phenomena in1D disordered systems as a possible means of weakeninglocalization and arriving at extended or non local stateshas shown encouraging results [34, 35]. Similar pursuitsin AAH systems with a view to analyzing diffusive trans-port behaviour and wavepacket dynamics in the presenceof driving, have been promising in terms of appearanceof delocalized states [36–38].

The technique of ’shaking’ of ultracold atoms in opticallattices has risen to prominence as a flexible means of gen-erating new effective Hamiltonians which may replicatethe effects of disorder, curvature, stresses and strains,and several other phenomena as synthetic gauge fieldsboth Abelian or non-Abelian [21, 39–41]. Some recentstudies in driven cold atom setups have looked at inducedresonant couplings between localized states thereby mak-ing them extended [42] or at localization through incom-mensurate periodic kicks to an optical lattice [43]. Inthese models, the phase transition instead of being drivenby disorder, is a consequence of deliberate incommensu-rate periodicity. Demonstration of this behaviour hasalso been sought in a phase space analysis of the transi-tion [44, 45].

However, conspicuous by their absence have beenworks which look at the AAH model with a rapidly os-cillating magnetic field, using the extensive tunability ofcold atom setups. The existing study of AAH systemsassumes the magnetic field to be static in time. If themagnetic field is periodic, then one may find a pertur-bative solution in the limit of high frequency driving.We address this neglected aspect by employing a formal-

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ism based on Floquet analysis to obtain an approximateeffective time independent Hamiltonian for the system[46–51]. A pertinent enquiry about the effective systemwould be to look for a metal to insulator phase transitionwith an energy dependent mobility edge.

In this work, we consider a high frequency, sinusoidaleffective magnetic field which couples minimally to theAAH Hamiltonian. The effective Hamiltonian is obtainedfor this system and its localization characteristics arecompared with the usual self-dual AAH model in real andFourier space. An energy dependent mobility edge hasalready been studied in the context of an AAH Hamil-tonian with an exponentially decaying strength of hop-ping parameters (beyond nearest neighbor coupling) [17].We explore the possibility of a similar mobility edge inour physically motivated effective Hamiltonian with onlynearest neighbor hopping. The non self-dual nature ofour model is analysed and some general features are in-vestigated. In the section on Discussions, we attempt toreconcile our findings for the specific case with genericfeatures of such non-dual models thereby putting the re-sults in perspective. Finally, we discuss some possibleexperimental techniques that could be adapted to realizea version of the model presented here. Here, we high-light the difficulties involved in doing so and compareour model to some other driven cold atom AAH modelsin the literature.

II. FORMALISM

Recent successes in synthesizing tunable, possiblytime-dependent, artificial gauge fields for systems of ultracold neutral atoms in optical lattices [52, 53] has openeda gateway to the strong field regime required for Hofs-tadter like systems [54]. The system to be studied heremay also be realized as an incommensurate superposi-tion of two 1D optical lattices [27], with the laser beamsfor one of them undergoing a time-dependent frequencymodulation. This shall be discussed in detail later, underExperimental Aspects.

We consider a tight binding Hamiltonian with nearestneighbor coupling that can be expressed as a time de-pendent Aubry-Andre-Harper Hamiltonian of the formH(t) = H0 + V (t), where

H0 =∑n

|n〉〈n+ 1|+ |n〉〈n− 1|

V (t) =V0

∑n

cos[2πα0n cos(ωt) + θ

]|n〉〈n|.

(1)

The summation here runs over all lattice sites. The time-dependent parameter α(t) = α0 cos(ωt) denotes the fluxquanta per unit cell. An irrational value of α0 shall ren-der the on-site potential to be quasi-periodic. The har-monic time dependence of α(t) owes its origin to a timedependent magnetic field B = B0 cos(ωt)z. The otherparameter θ is an arbitrary phase. The |n〉’s are the

Wannier states pinned to the lattice sites which are usedas the basis for representing the Hamiltonian and V0 de-notes the strength of the on-site potential. The time de-pendence in the argument of the cosine modulation of theon-site potential is different from usual time dependentAAH models where it is in the overall magnitude of theon-site potential. The periodic time-dependent operatorV (t) can be expanded in a Fourier series as

V (t) = V0 +∑

1≤j<∞

Vjeijωt +

∑1≤j<∞

V−je−ijωt. (2)

In order to obtain the effective time independent Hamil-tonian one writes the time evolution operator as

U(ti, tf ) = e−iF (tf )e−iHeff (tf−ti)eiF (ti), (3)

where, one introduces a time dependent Hermitian oper-ator F . The idea is to push all the time dependence to theinitial and final “kick” terms and render the main timeevolution to be dictated by a time independent Hamil-tonian. The systematic formalism yields in the limit oflarge ω the following perturbative expansion for the ef-fective time independent Hamiltonian given by [50]

Heff = H0 + V0 +1

ω

∞∑j=1

1

j[Vj , V−j ]

+1

2ω2

∞∑j=1

1

j2

([[Vj , H0], V−j

]+ h.c.

)+O(ω−3),

(4)

where, ω−1 is the small perturbation parameter, and theseries is truncated at O(ω−2). In order to find the effec-tive approximate Hamiltonian representing our system inthe large ω limit, one needs to compute the Fourier co-efficients in Eq.(2). This is done by using the followingcommonly valid expansions [55]

cos(r cosx) = J0(r) + 2

∞∑p=1

(−1)pJ2p

(r) cos(2px)

sin(r cosx) = 2

∞∑p=1

(−1)p−1J2p−1

(r) cos[(2p− 1)x],

(5)

where, Jn(r) ’s are Bessel functions of order n. The

Fourier coefficients of V (t) may be obtained by invertingEq.(2) using these expansions. We obtain

Vj =(−1)j2V0 cos θ

∑n

Jj(2πα0n) |n〉〈n|; j = ±2, 4, 6...

Vj =(−1)j+12 V0 sin θ

∑n

Jj(2πα0n) |n〉〈n|; j = ±1, 3, 5...

V0 =V0 cos θ∑n

J0(2πα

0n) |n〉〈n|. (6)

We find that [Vj , V−j ] = 0 owing to the symmetric na-ture of the Fourier coefficients (for real V). Therefore the

3

O(ω−1) correction to the effective Hamiltonian vanishesand the first non-trivial correction is at O(ω−2). The

O(ω−2) term of the effective Hamiltonian would require

the commutator bracket

[[Vj , H0], V−j

], which on evalu-

ation yields

[[Vj , H0], V−j

]=

∑n

−V 20 cos2 θ

[(J

j[2πα0(n+ 1)]− J

j(2πα0n)

)2

|n〉〈n+ 1|

+

(Jj [2πα0(n− 1)]− Jj (2πα0n)

)2

|n〉〈n− 1|] if j = ±2, 4, 6...

∑n

−V 20 sin2 θ

[(Jj [2πα0(n+ 1)]− Jj (2πα0n)

)2

|n〉〈n+ 1|

+

(J

j[2πα0(n− 1)]− J

j(2πα0n)

)2

|n〉〈n− 1|] if j = ±1, 3, 5...

(7)

Using the above expression in Eq. (4) we obtain the effec-tive Hamiltonian, Heff , for our system. We find that up toO(ω−2), the effective Hamiltonian yields a nearest neigh-bor tight binding model with a zeroth order Bessel func-tion modulating the site energies, and higher order Besselfunctions make their appearance in the hopping terms.There have been works which have looked at inhomogen-ities in the hopping of the AAH model, arising not fromdriving but from the choice of next nearest neigbour hop-pings in the corresponding 2D quantum hall model[56–59]. However, these models consider situations where theoff-diagonal modulations are quasiperiodic through in-commensurate modifications of cosine kind of terms. Inour case above, the incommensurability is embedded inhigher order Bessel functions, thereby variations in hop-ping strength are far more erratic and with signaturesbordering on those of disorder. This is expected to haveramifications for the localization/extended behaviour ofthe eigenstates. This has been discussed in the next sec-tion and illustrated through localization phase plots.

III. RESULTS

The simple AAH model has a well studied transitionfrom localized (insulating) to delocalized (metallic) phasewhich occurs at a critical value V0 = 2. To quantifythe localization property we use the inverse participation

ratio (IPR) defined as IPR =

L∑n=1

|an|4/

(L∑n=1

|an|2)2

where an’s are the expansion coefficients of the energyeigenstates in a local discrete site basis and L the num-ber of lattice sites [60, 61]. The IPR takes a value inthe range 1 to 1/L with 1 indicating a perfectly local-ized state and 1/L for completely extended states. Fig-ure 1(a) shows the transition in the real space IPR forthe ground state of the AAH system with choice of ir-

1 10

V0

10-4

10-3

10-2

10-1

100

IPR

1 10

V0

0 1 2 3 4V0

0

1

D2

0 1 2 3 4V0

0

1

D2

(a) (b)

1 10

V0

10-4

10-3

10-2

10-1

100

IPR

0.1 1 10

V0

0 2 4 6V0

0

1

D2

0 2 4 6V0

0

1

D2

(c) (d)

FIG. 1. (Upper Panel)The metal to insulator transitionof the AAH Hamiltonian for the system’s ground state.Plot (a) shows the IPR versus V0 in real space for L =144, 1597 and 10946 (top to bottom). The inset shows thevariation of D2 with V0 which also exhibits a transition. Plot(b) exhibits the mirror behaviour in the dual space. (LowerPanel) The transition seen in the IPR versus V0 curve for theHeff . Plots (c) and (d) are the real and dual space plots forthe ground state with phase θ = 0.

rational α0 as inverse of the golden mean (√

5 − 1)/2and L = 144, 1597 and 10946. The inset in this plot in-dicates variation in the magnitude of the quantity |D2|with V0, where IPR ∝ L−D2 . For a given V0, D2 is ob-tained by fitting a linear regression line between log IPRand logL and thereby obtaining the slope. The regres-sion fitting is done using several values of L, taken tobe large Fibonacci numbers. In all the plots discussedhere the transitions depicted are for some finite choiceof system size and hence not exactly ’step’ changes butramp up or down over some finite range of V0 values.

4

Further, the references to such transitions as abrupt oroccuring at a critical value have to be interpreted withinsuch numerical constraints. D2 values shows an abrupttransition from 1 to 0 at the critical value irrespectiveof lattice size. This establishes the transition to be a in-tegral feature of the model even in the thermodynamiclimit of infinite lattice size and the critical point is pro-tected in this limit. In order to switch to states in theFourier domain, i.e, |m〉 ’s from the position space kets|n〉 we use the transformation

|m〉 =1√L

∑n

exp(−i2πmα0n)|n〉. (8)

This enables one to write the AAH Hamiltonian inFourier space and compute the IPR in this space. Fig-ure 1(b) shows the transition in Fourier space for a setof parameters identical to those in plot (a). Here, againthe characteristic transition occurs at the critical pointV0 = 2 and the curves in plot (a) are a mirror reflectionof the curves in plot (b) about V0 = 2. This is due to theexactly self dual nature of the AAH Hamiltonian. Thus,an extended regime in real space implies a localized one inFourier space and vice versa. The inset for D2 in Fourierspace accordingly mirrors its real space counterpart.

In the case of our effective model, the IPR for theground state exhibits a similar trend, as shown inFig.1(c). The transition in this case for this state occursat a new critical value V0 ≈ 4.6, all parameters beingkept the same as in former plots. Another distinguishingfeature of this transition for the driven case is the mannerin which the IPR values approach the critical value anddepart from it, relative to the behaviour in the standardAAH model. In the extended regime, instead of the per-fectly flat value of IPR ≈ 0, a slow positive gradient isobserved indicating weak localization that progressivelygets stronger until a sharp surge occurs at the criticalpoint. Even beyond the transition there is a fall in theIPR due to the still imperfect nature of the localization.This unique behavior can be partially attributed to thenon self-dual nature of the effective Hamiltonian whichshall be discussed later. D2, in the inset, continues toretain its scale invariant attributes and manifests the im-print of the transition. The difference from the simpleAAH model, lies in the fall in its value from 1 to a lowerplateau before the transition. An indication of the ex-istence of a parameter regime where the state is neitherpurely localized nor extended but a sort of composite i.e.critical and possibly multifractal. This is due to the un-usual behavior of the wavefunction in the case of α0 beinga Liouville irrational number whereby, for some latticemodulations, no finite localization length may be foundover which the state could be said to appreciably decay[15, 16]. Plot (d) which shows the Fourier space IPR forour model exhibits reciprocal behavior of the kind seen inthe simple AAH model but with the major difference thatplots (c) and (d) are not mirror reflected about the samecritical value. This deviation is expected on grounds ofthe non self dual nature of our system’s Hamiltonian.

In Fourier space, D2 analogously is not an exact mirrorimage of its real space version but all other qualitativecharacteristics remain the same. There are features inthe driven system’s Fourier space IPR which stand incontrast from the AAH model, as seen in plots (b) and(d) of Fig.1. Most notably, the driven model shows a dis-crimination between the different lengths as the curves inplot (d) transition from localized to extended regimes atdifferent rates. This is not the case in the ordinary AAHmodel, where all lengths transition together, as seen inplot (b). This difference is an indicator of non-nearestneighbor couplings in our dual space effective Hamilto-nian and the accompanying anomalous behavior of thewavefunctions in a certain parameter range.

In order to understand how the properties of the tran-sition are related to the normal modes of the effectivedriven system, we look at the localization phase diagram,i.e. the variation of IPR with V0 and the low energy re-gion of the spectrum at each V0 . Figure 2 shows theIPR in the V0 − E plane. We consider three such plotsfor values of the phase, θ = 0 , θ = π/4 and θ = π/2 inEq.(1) and lattice size L = 4181. The nature of the vari-ation of IPR reveals a sharp energy dependent mobilityedge for our model. The portion of the energy spectrumfor which the IPR variation has been illustrated is cho-sen to clearly indicate the appearance of localized states. The choice of values for the phase is intended to isolateand compare the relative effects of the modified onsiteterm and the site dependent hopping terms. In all threeplots the sector corresponding to low V0 values and nearto E = 0 shows a dense region of extended states which,(see eq.(1)), reflects the bare hopping structure.

The features in the these phase diagrams owe their ori-gin to the relative dominance of different terms the driveneffective Hamiltonian for a given θ. The anisotropy of thezero-order Bessel modulated on-site energy adds impu-rity/disorder like effects on top of the inherent quasiperi-odicity of the model. The fall off of this on-site energywith lattice sites diminishes the actual system size to areduced one. The modified hopping strengths are alsosite dependent and vary in an oscillatory manner, witha damping with increasing site index. When the onsitepotential damps out, the hopping terms from H0 sur-vive. This is expected to contribute to an increase inthe IPR as the behaviour tends to one of a lattice withdisorder. These factors collectively influence the spectralspread and density alongwith the loclalized/delocalizedbehaviour of the various eigenstates.

Plot 2 (a), for θ = 0, depicts the appearance of quasi-localized states (yellow fringes) at the boundaries of theextended (deep blue) region. As V0 increases, the denseregion of extended states near E = 0 begins to mani-fest traces of localization in IPR values, introducing themobility edges. This can be noted from the bifurca-tions of the phase boundary that begin to show up inwith increasing V0 with localized eigenstates piercing intoportions of the spectrum which at lower V0 were domi-nated by extended states. For higher energy the IPR

5

FIG. 2. (Color online) The localization phase diagram with IPR in the (E,V0) phase plane, for lattice size L = 4181 threevalues of the variable θ (a) θ = 0 , (b) θ = π/4, (c) θ = π/2.

values vary primarily between critical and extended be-haviours. This is notably absent in the plots for θ = π/4and θ = π/2 where critical behaviour is hardly observedand that too in a very narrow region around the phaseboundary. The wider gapping in the eigenvalues as com-pared to the other two cases can be accorded to the over-all stronger influence of the on-site term as compared tothe hoppings.

Plot (b), for θ = π/4, includes the effects of all theterms of the effective Hamiltonian. The appearance oflocalized states around E = 0 takes place as before.However, there is a notable lack of appearance of welllocalized states in the higher energy regions as comparedto plot (a). This indicates a closer competetion betweenthe on-site and hopping terms of the driven model. Thesignificant localization effects, hence mobility edges, ap-pear distinctly in a band around E = 0 and that too athigher V0. This may be accounted for by the fact that forθ = π/4 both the sine and cosine modulations are presentin the modified hopping (see eq.(7)), unlike the previousθ = 0 case. Apart from contributing to the predominanceof extended states in most of the spectrum, this more in-fluential hopping part also makes the spectrum relativelyless gapped.

Plot (c), for θ = π/2 has been specifically shown to il-lustrate how the hopping terms in the driven AAH modelbehave in the absence of any on-site term. For this choice

of θ the V0 in eq.(6) goes to zero. As expected in thepresence of just the hopping, the IPR values show anextended behaviour everywhere in the phase plot. How-ever, one can still note a phase boundary differentiatingthe region of the purely extended states from somewhatless extended ones. The appearance and nature of the bi-furcations in the boundary of this dense part of the phaseplot with changing V0, indicates the qualitative effects ofthe inhomogeneities in the hopping terms. Looking atplots (a) (b) and (c) it is clear that the role of the mod-ulated onsite has the effect of enhancing the localizationof states as well as creating a sharper mobility edge.

This indicates that the driven model shows a strong

sensitivity to the phase θ in terms of the localizatonbehaviour and the appearance of mobility edges. Thenumber of these edges, as can be seen, is more for theθ = π/4 case and is almost absent in the phase plot forθ = π/2. A recent work [62], looks at a topological classi-fication of AAH models with cosine modulated hoppingswhich differ by a phase factor from the onsite modula-tion. This helps to realize topologically distinct familiesof AAH Hamiltonians with the possibility of topologicalphase transitions between the different classes via a mod-ification of the lattice modulations. Similar behaviourwould be interesting to study in our driven context.

IV. DISCUSSION

In order to qualitatively analyze some of our resultswe consider a simplified model comprising of a trivialconstant hopping term and an on-site potential T whichis aperiodic or quasi-periodic. The Schrodinger equationis given by

an+1 + an−1 + ΛT (α0n+ φ)an = Ean (9)

where, Λ is the strength of on-site energy and E are theenergy eigenvalues. One can go to the dual space for theabove system by defining an expansion, as follows

an =eikn√L

∑m

ameim(α0n+φ) (10)

where, the am’s are the dual space amplitudes, and k is awave vector from the Bloch wave expansion ansatz. Thisallows T to be expressed as

T (α0n+ φ) =1√L

∑m

Tmei m(α0n+φ). (11)

Equations (10) and (11) yield an on-site term in thedual space from the the hopping terms of Eq.(9) as

an−1+an+1 =eikn√L

∑m

ameim(α0n+φ) cos(α0m+k) (12)

6

with a cosine modulation of the on-site energy (as seenin the Aubry and Andre model). Interestingly, the realspace on-site energy term transforms as

ΛT (α0n+ φ)an =Λeikn

L

∑m

∑m

Tmamei (m+m)(α0n+φ).

(13)The RHS can be slightly rearranged to give

Λeikn

L

∑m 6=±1

∑m

Tmam+meim(α0n+φ)+

Λeikn

L

∑m

(T1am−1 + T−1am+1)eim(α0n+φ)

(14)

In the above form, the second term clearly indicates theapparent nearest neighbor hopping terms in the dualspace whose strength is modulated by the Fourier com-ponents of T . It is the first term in the above expressionwhich explicitly breaks the exact duality. The form of Tmdetermines the extent to which different m values in thedual space are coupled. It is well known that for decay-ing oscillatory functions like Sinc and Bessel function ofthe zeroth order Tm is a rectangular function, with pos-sibly a m dependent modulation, symmetric about theorigin. Thus, in our case, we expect a truncation effectin dual space which restricts the range of couplings. Thisdeviation from exact duality is expected to have some im-pact on the probability of an “analytic accident” alongthe lines of [12]. The appearance of localized states (realeigenfunctions) happens when there are superpositionsof counter-propagating plane waves with wave vectors ofnear-commensurate magnitude. This would mean, in ourmodel, some harmonics from the expansion of T shallscatter the wave with wave vector k by an amount com-mensurate with 2nπ. This has to be considered togetherwith the fact that for a rational approximation of α0 asa ratio of two large successive Fibonacci numbers, thetrue momentum(Fourier) space eigenvalues κ are relatedto m as κ = mFi−1modFi, where Fi−1 and Fi are suc-cessive Fibonacci numbers [5, 44]. Thus, what appear tobe close neighbors in m could possibly be well separatedin the actual wave vector space. Further, the range of mvalues that shall remain coupled in the dual space will bedictated by the extent of T in the real lattice for exam-ple the first zero in the Bessel function. The set of m’swhich conspires with a given k value to yield a localizedstate shall be dictated by V0 and E(k). This explains theenergy dependent mobility edge in Fig.2.

In the dual space, where k acts as a phase (see Eq.(12)),a state localized at few m values could be shifted by largeamounts for a small change in k. This allows for theinterpretation that a small change in φ could in effectcause a state localized around some lattice site to lo-calize about a far off site. In terms of symmetry, theabsence of translational invariance in Euclidean space ofquasi-periodic structures with two incommensurate peri-odicities can be restored in an extended space using the φ

dimension [63, 64]. This effect of φ on localization prop-erties leads to the differences between the three plots inFig.2.

V. EXPERIMENTAL ASPECTS

The experimental realization of our system may beachieved in several different ways, with ease and fea-sibility of implementation being the guiding criteria inthe choice of method. We will explore two options here,from recent literature, which are promising. One way isto begin with a 2D optical lattice and then proceed inthe manner described in some recent realizations of theHarper-Hofstadter Hamiltonian [32, 33]. The notion ofsimulating a synthetic magnetic field by means of gener-ating effective flux per plaquette of the lattice is a genericfeature. However, the true appeal of these methods com-pared to others for generating artificial magnetic fields forultracold neutral alkali metal atoms in optical lattices, isthe absence of coupling between different hyperfine statesof the atoms. It is possible therefore to proceed with asingle internal state and far detuned lasers to achievehomogeneous magnetic fields by a laser assisted hoppingprocess. A pair of far detuned Raman lasers is employed,while tunneling in a particular direction is obstructed bymeans of a gradient/ ramp in the site energies using grav-ity or magnetic fields, to restore resonant tunneling be-tween sites. The AAH Hamiltonian is obtained in a timeindependent effective way by averaging over the high fre-quency terms and the hopping energy is modified by acomplex position dependent phase.

We suggest using Raman lasers of frequencies close tothose of the optical lattice lasers, as prescibed by the au-thors, and to use the tunability of the flux per plaquetteα0 offered by such choice, to set it to an irrational valueby adjusting the angle between the Raman beams. Intro-ducing the time dependence is admittedly tricky. This isdue to the fact that the static Harper Hamiltonian in theabove technique is itself achieved by time averaging andwe need it to have a further residual time dependence.For this one would have to vary α0 periodically by say,modifying the angle between the Raman lasers periodi-cally with time together with simultaneous time modu-lations of the detunings and the gradients in a fashionthat the overall effect is of a periodic change that is of arate slower than the oscillations to be averaged over whileresonant tunneling occurs, but fast enough to remain de-tuned from the energy gap between the ground and ex-cited bloch bands of the trapped atoms in the lattice po-tential. This yeilds a time scale which survives the firstaveraging and gives one a time dependent AAH Hamil-tonian effectively being driven by an oscillating magneticfield. There are some obstacles to be overcome here suchas arranging the time dependent detunings and the an-gular variation of the Raman lasers so as to vary α0 sinu-soidally as a function of time effectively, since there is agood chance of getting high frequency noisy components

7

that have to be averaged over. Another issue is that thescheme does not realize the simple Landau gauge for amagnetic field. We use this gauge in our analysis but theresults, essentially the nature and existence of the Metal-insualtor transition, are independent of any such choicethrough the gauge freedom embodied in the choice of θin the AAH Hamiltonian[12]. The analysis then wouldbe modified only upto a gauge transformation. It wouldindeed be interesting if the method could be modified toinclude the Landau gauge.

On account of the multiple time dependent modu-lations in the realization just discussed, heating andspontaneous photonic emission processes are a legiti-mate source of concern. We would like to outline an-other approach using a quasiperiodic 1D optical latticewhich may have better characteristics as regards dissi-pative processes. Here we suggest using the bichromatic1D optical lattice realization of the AAH model as de-scribed in [27] and suitably modifying it to implement ourmodel. Essentially, a bichromatic optical lattice setup isone with a pair of superposed standing waves whereinone provides the tight binding structure to the Hamilto-nian and the other, a weak secondary perturbing poten-tial which, through adjustable non-commensurability ofits wavelength with that of the first, offers a quasiperi-odic/pseudorandom potential for the ultracold gas ofatoms even to the extent of mimicking quasidisorder inthe lattice [20]. The two standing waves have wave-lengths in the ratio of two consecutive Fibonacci num-bers. This helps to realize a workable notion of incom-mensurability in a finite lattice system by tending thevalue of the α0 to near the inverse of the golden mean.As suggested in [27] this is the key requirement for theobservation of a transition from extended to localizedstates i.e. to keep a large number of lattice sites in asingle period of the on-site potential for finite systems.

The next step is to systematically introduce the driv-ing. This is done by introducing a time dependence in theratio of the wavelengths of the two standing wave lattices.More precisely, to do this we suggest generating the twostanding waves using beam splitting and retro reflectionby mirrors. If now the reflecting mirror corresponding tothe primary tight binding lattice is shaken according to aprotocol which mimics a sinusoidal drive say, by mount-ing it on a piezoelectric motor, it should be in principlepossible to generate a sinusoidal time dependence in theirrational flux term. It would be preferable to use actua-tors that move the mirrors so as to produce accelerationeffects on the lattice such that one may achieve time de-pendent Doppler shifts in the frequency and hence wave-lengths of the stationary waves (where one averages overthe fast oscillations in the amplitudes to get the hoppingterms) which could be controlled in a sinusoidal fashion.This may be technically demanding under the presentcapabilities of shaking in optical lattice systems but isdefinitely worth exploring as a powerful instrument forstudying effective Hamiltonians in a new time dependentregime.

In the two approaches highlighted above, the respec-tive works [32] and [27] provide a clear map between theparameters of the simulated model and the experimentalparameters such as laser intensities, recoil energies of thetrapped neutral atoms and the energy gap between theground state and lowest excited bloch band in the lat-tice. This mapping translates readily to the formalism ofcalculating the effective Hamiltonian. For instance in thecase of the bichromatic construction the mapping of thecontinuous optical potential to a tight binding picturehas been carried out in [27] using a set of local Wannierbasis states. As per this construction our time depen-dent model, see eq.(1) would have V0 to be a ratio ofproduct of the height of the weak perturbing lattice withthe time dependent ratio of the wavelengths of the twostanding waves and an integral term, to the hopping ele-ment of the primary optical lattice. The term α0 is justthe ratio of the wavelengths of the two standing waves,made time dependent by shaking, which are two consec-utive Fibonacci numbers. From the expressions in eqs.(6) and (7) it can be readily seen how the experimentalparameters enter into the modified on-site and nearestneigbour hopping energy terms of the high frequency ef-fective static Hamiltonian for the driven system.

Thus the relation between parameters of the setupand the derived model Hamiltonian can be traced in astraightforward manner. It is true that this manner ofconstructing the system will make the strength of theon-site modulation (or its ratio with the hopping energy)also a sinusoidal function of time but this is not expectedto alter the system’s features studied here in any signifi-cant way. We propose studying this more general form oftime dependence, with the strength of the on-site to offsite energy also taken to be a function of time alongsidethe periodicity in α0, as a future line of work.

Briefly, we would like to survey how our work contrastswith other early and recent work, which has emphasizedperiodic driving through additional potentials [42] andshaking [21] in AAH systems. In [21], shaking introducesa time dependent phase in the cosine term of the on-siteenergy. This phase is seperate from the incommensurateposition dependence. The effect is a renormalization ofthe hopping energy so as to make it a function of thedriving amplitude. This enables one to tune across themetal-isulator transition by varying the amplitude of thedriving. Whereas in our system the driving is providedthrough an oscillatory effective magnetic field which man-ifests itself through the periodicity in α0, hence presentin the incommensurate position dependent term.

This is again different from [42] which employs a driv-ing that is a weak space quasiperiodic and time peri-odic perturbation onto the AAH system modeled as aquasiperiodic optical lattice. Here, the driving is a weakperturbation to the original AAH Hamiltonian. In ourcase, however, the manner in which the AAH model isdriven is non-perturbative by its very nature. An oscil-lating magnetic field, even of small amplitude, is in noway a weak perturbation and cannot be treated as such,

8

it has to be looked upon in the Floquet picture of peri-odic time dependent Hamiltonians. The high frequencynature of the driving permits a Floquet theoretic treat-ment of a slightly analytical variety through the highfrequency expansion available for the Floquet Hamilto-nian. Only here, in the parameter 1/ω, is one allowed touse a perturbative treatment. This is formally differentfrom [42] in that our modifications significantly alter theAAH model for which there is limited analytical footingin the high frequency regime. It would do well to regardthe newly obtained static effective Hamiltonian as an in-dependent system in its own right, with features that arenot to be expected in the undriven or weakly driven AAHmodels. This in fact merits looking into, as it would notbe unjustified to anticipate exotic modifications to thetraditional AAH Metal Insulator transition under thesecircumstances.

Also worth noting are the differences between themodel in [42] and our system from a reciprocal spacepoint of view. While our model is also non self dual ithas an exact 1D structure with couplings that are beyondthe nearest neighbour. In [42], the dual Hamiltonian isnot exactly 1D and the extended states appear due toresonant couplings of localized states that are drivinginduced. This differs considerably from the mechanism(discussed in the previous section)that causes localiza-tion/delocalization behaviour in our driven system. Infact, the non self duality of our model sets it apart evenfrom undriven variations on the AAH model (with mobil-ity edges) which are self dual by construction [17, 18], ir-

respective of the real space couplings being nearest neigh-bour or beyond it.

VI. CONCLUSION

In this work we have studied the Aubry- Andre-Harper problem with an oscillatory magnetic field inthe promising cold atom- optical lattice scenario. Theproblem is significantly simplified by going into aneffective Hamiltonian which approximately representsthe system in the limit of high frequency magnetic field.We find that this effective Hamiltonian is non-self dual,and though it exhibits a metal-insulator transition,it differs from the classic Aubry-Andre model in theemergence of an energy dependent mobility edge. Thenearest-neighbor coupling form of the effective Hamil-tonian yields this feature which is commonly observedin disordered 3D systems or Aubry-Andre like mod-els with hoppings extending beyond the nearest neighbor.

VII. ACKNOWLEDGEMENTS

T.M. would like to acknowledge Takashi Oka andAlessio Celi for valuable discussions on various theoreti-cal and experimental aspects of the problem.

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