Mobility edge and multifractality in a periodically driven Aubry-André model
MMobility edge and multifractality in a periodically driven Aubry-Andr´e model
Madhumita Sarkar, Roopayan Ghosh, Arnab Sen, and K. Sengupta
School of Physical Sciences, Indian Association for the Cultivation of Science,2A and 2B Raja S. C. Mullick Road, Jadavpur 700032, India (Dated: February 25, 2021)We study the localization-delocalization transition of Floquet eigenstates in a driven fermionicchain with an incommensurate Aubry-Andr´e potential and a hopping amplitude which is variedperiodically in time. Our analysis shows the presence of a mobility edge separating single-particledelocalized states from localized and multifractal states in the Floquet spectrum. Such a mobilityedge does not have any counterpart in the static Aubry-Andr´e model and exists for a range ofdrive frequencies near the critical frequency at which the transition occurs. The presence of themobility edge is shown to leave a distinct imprint on fermion transport in the driven chain; italso influences the Shannon entropy and the survival probability of the fermions at long times. Inaddition, we find the presence of CAT states in the Floquet spectrum with weights centered arounda few nearby sites of the chain. This is shown to be tied to the flattening of Floquet bands over arange of quasienergies. We support our numerical studies with a semi-analytic expression for theFloquet Hamiltonian ( H F ) computed within a Floquet perturbation theory. The eigenspectra of theperturbative H F so obtained exhibit qualitatively identical properties to the exact eigenstates of H F obtained numerically. Our results thus constitute an analytic expression of a H F whose spectrumsupports multifractal and CAT states. We suggest experiments which can test our theory. I. INTRODUCTION
Localization phenomenon in an one-dimensional (1D)fermion chain with quasiperiodic potentials has beenstudied extensively in the past . These studies havereceived a new impetus in recent times due to experi-mental realization of such potentials in ultracold atomchains . In contrast to the more conventional mod-els with uncorrelated disorder which exhibits such local-ization for any disorder strength in 1D , fermion chainswith non-random but quasiperiodic potentials harbor alocalization-delocalization transition . The simplest ofsuch models termed as Aubry-Andr´e (AA) model hasan Hamiltonian given by H = H k + H p where H k = − J (cid:88) j c † j ( c j +1 + c j − ) H p = (cid:88) j V cos(2 πηj + φ ) c † j c j . (1)Here j is the site index of the chain, c j denotes fermionicannihilation operator at site j , J is the hopping am-plitude, η is an irrational number usually chosen to bethe golden ratio ( √ − / V is the amplitude of thepotential, and φ is an arbitrary global phase. The AAHamiltonian can be shown to be self-dual and hosts alocalization-delocalization transition at V c = 2 J . For V > ( < ) V c , all the single-particle states in the spectrumfor the model are localized (delocalized). Such transitionsalso occur in models with a more general class of suchquasiperiodic potentials (termed as generalized Aubry-Andr´e (GAA) potentials). These GAA Hamiltoniansmay have several forms; for example, they may be givenby Eq. 1 with a different form of the quasiperiodic poten-tial [cos(2 πηj + φ ) → cos(2 πηj + φ ) / (1 − α cos(2 πηj + φ ))] or with longer range hopping J → J ij = J / | i − j | a , where a is an exponent . One of the key aspects ofthese GAA Hamiltonians which is absent in the AAmodel is the presence of a mobility edge in the spec-trum. Moreover, the latter class of GAA models alsohost band of multifractal eigenstates in the delocalizedphase. These states, unlike their delocalized counterpart,are non-ergodic; thus their presence change the ergodic-ity properties of these models .The study of non-equilibrium dynamics of closed quan-tum systems has gained tremendous impetus in the lastdecade . More recently, it was realized that periodically(or quasiperiodically) driven systems host a wide rangeof interesting phenomena that have no analog in theirundriven counterparts. These include topological tran-sitions in driven systems , dynamical transitions ,dynamical freezing , realization of time crystals ,and weak ergodicity breaking behavior . Moreover suchdriven systems are known to lead to novel steady stateswhich have no analog in non-driven systems . Forperiodically driven systems, most of these phenomenacan be understood by analyzing its Floquet Hamilto-nian H F which is related to the evolution operator via U ( T,
0) = exp[ − iH F T / (cid:126) ] , where T = 2 π/ω D is thedrive period, ω D is the drive frequency, and (cid:126) is thePlanck’s constant.In this work, we study the properties of the Floqueteigenstates of a driven fermionic chain in the presenceof an AA potential. The Hamiltonian of the chain thatwe study is given by Eq. 1 with J → J ( t ), where J ( t )is a periodic function of time characterized by a drivefrequency ω D . In our study, we choose two distinct pro-tocols for J ( t ). The first is the square pulse protocolwhere J ( t ) = J ( −J ) for t ≤ ( > ) T / J ( t ) = J cos ω D t .For both these protocols, we choose J (cid:29) V . In theregime, of large drive frequency, H F (cid:39) H p , so that all a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b the Floquet states are localized. In contrast, for qua-sistatic drive ω D (cid:39)
0, all the states are expected to bedelocalized since J (cid:29) V . This features ensures thepresence of a localization-delocalization transition in theFloquet spectrum; the aim of the present work is to un-derstand the nature of the Floquet eigenstates near thetransition. We carry out this analysis numerically usingexact diagonalization of the fermionic chain followed bynumerical computation of U ( T, H F using a Floquet perturbation theory(FPT) .The central results that we obtain from this analysisare as follows. First, we find that for a range of drive fre-quencies around the localization-delocalization transition(occurring at a critical value of the drive frequency ω c ),the Floquet spectrum of the driven AA model supports amobility edge. This mobility edge, which has no analogin the static AA model, occurs for ω D ≤ ω c and separatesthe delocalized states from either localized or multifractalband of states. We chart out the drive frequency rangefor which these multifractal states are present for boththe square-pulse and continuous drive protocols. Second,we unravel the presence of single-particle CAT states inthe Floquet spectrum for ω D ≥ ω c . These states oc-cur at two specific quasienergies in the Floquet spectrumand have wavefunctions which are localized around two orthree next-nearest neighbor sites in the chain. We tie thepresence of these states to the presence of near-flat banddispersion in the Floquet spectrum at these quasiener-gies and provide a semi-analytic understanding of theirexistence. Third, we study the transport in such drivenchain by tracking the steady state value of the fermiondensity as a function of drive frequency starting from adomain-wall initial state. This initial state constitutesa many-body state where all sites to the left (right) ofthe chain center are occupied (empty). The density ofthe fermions in the steady state stays close to its initialprofile in the localized phase; in contrast it evolves toan uniform density profile for the delocalized phase. Inbetween, near the transition where the mobility edge ex-ists, it shows an intermediate behavior which arises fromthe presence of both delocalized and localized (or multi-fractal) states in the Floquet spectrum. Analogous fea-tures are found in the Shannon entropy, and the returnprobability of a single-particle fermion wavefunction (ini-tially localized at the center of the chain) measured in thesteady state. Fourth, we construct a semi-analytic, albeitperturbative, expression of the Floquet Hamiltonian H F using a FPT. We show that this semi-analytic Hamil-tonian qualitatively captures the physics of the drivensystem and use it to explain the presence of the mobilityedge and the multifractal states in the Floquet spectrum.Finally, we discuss possible experiments which can testour theory.The plan of the rest of the paper is as follows. In Sec.II, we provide a detailed numerical study of the drivenchain charting out the phase diagram, demonstrating the existence of the mobility edge, and determining the loca-tion of the multifractal and CAT states. This is followedby Sec. III where we construct a semi-analytic FloquetHamiltonian using FPT. Finally, we summarize our mainresults and suggest experiments which can test our the-ory in Sec. IV. Some details of the calculation of H F anda discussion of the approach of the driven chain to thesteady state are presented in the Appendices. II. NUMERICAL RESULTS
In this section, we present exact numerical result on thedriven fermion chain for both square-pulse (Sec. II A) andsinusoidal drive protocol (Sec. II B). Henceforth, we setthe global phase φ = 0 in Eq. 1 without loss of generality. A. Square pulse protocol
For the square-pulse protocol, we vary the hopping am-plitude of the AA model (Eq. 1) as J ( t ) = −J , t ≤ T / J , t > T / H F (cid:39) (cid:82) T H ( t ) dt/T = H p , the Flo-quet Hamiltonian represents a localized phase. To nu-merically find out the Floquet spectrum at arbitrary fre-quency, we first find the eigenspectrum of H ± = H [ ±J ](Eq. 1) using exact diagonalization (ED). We denotethese eigenvalues and eigenvectors as (cid:15) ± m and | ψ ± m (cid:105) re-spectively. Next, we note that for the protocol given byEq. 2, the evolution operator at t = T can be written as U ( T,
0) = e − iH + T/ (2 (cid:126) ) e − iH − T/ (2 (cid:126) ) (3)= (cid:88) p + ,q − e i ( (cid:15) + p − (cid:15) − q ) T/ (2 (cid:126) ) c p + q − | ψ + p (cid:105)(cid:104) ψ − q | where the coefficients c p + q − = (cid:104) ψ + p | ψ − q (cid:105) denote overlapbetween the two eigenbasis. Next, we numerically diag-onalize U ( T,
0) and obtain its eigenvalues λ m and | ψ m (cid:105) .The corresponding eigenstectrum of H F is then obtainedusing the relation U ( T,
0) = exp[ − iH F T / (cid:126) ] which iden-tifies the eigenvectors of U ( T,
0) and H F and yields U ( T,
0) = (cid:88) m λ m | ψ m (cid:105)(cid:104) ψ m | , λ m = e − i(cid:15) Fm T/ (cid:126) (4)where (cid:15) Fm are the quasienergies which satisfy H F | ψ m (cid:105) = (cid:15) Fm | ψ m (cid:105) . In this section, we shall use the propertiesof these Floquet eigenvalues and eigenvectors to studyphase diagram of the driven chain along with multifrac-tality of Floquet eigenstates and transport of fermions.
0. 0.25 0.5 0.75 1. 1.251.00.80.60.40.20.0 ω D /( π ) m / L I m FIG. 1: Plot of I m as a function of the normalizedeigenfunction index m/L and ω D / ( π J ) showing the local-ized/delocalized nature of the Floquet eigenstates | ψ m (cid:105) . Here L = 2048 and we have set J = 1, V / J = 0 .
05, and scaledall energies(frequencies) in units of J ( J / (cid:126) ).
1. Phase diagram and CAT states
Having obtained the eigenspectrum of H F , we firstanalyze the localization properties of normalized single-particle eigenstates | ψ m (cid:105) as a function of the drive fre-quency. To this end, we compute the inverse participa-tion ratio (IPR) of these states given by I m = L (cid:88) j =1 | ψ m ( j ) | , ψ m ( j ) = (cid:104) j | ψ m (cid:105) (5)where j denotes the coordinate of the lattice sites of thechain of length L . The IPR I m ∼ L − in d = 1 for adelocalized (localized) state and thus acts as a measureof localization of a quantum state.This analysis leads to the phase diagram shown in Fig.1, where we present I m as a function of the drive fre-quency ω D and for V / J = 0 .
05. As expected, the Flo-quet eigenstates stay localized at large drive frequency; incontrast, they are delocalized at low drive frequencies. Inbetween, around (cid:126) ω D / J (cid:39) . π , we find a localization-delocalization transition. Near the transition, for drivefrequencies 0 . π ≤ (cid:126) ω D / J ≤ . π , we find the exis-tence of a mobility edge separating a delocalized bandhosting states with I m (cid:39) /L ≤ − from those withfinite I m > .
1. The nature of the states separated bythis mobility edge will be analyzed in detail in the nextsubsection.In addition to the mobility edge near the transition,we also find a two narrow bands of states which retaina smaller value of I m (cid:39) . | ψ m ( j ) | at a fixed frequency ω D ,shown in the top panels of Fig. 2, reveals that these stateshave their weights spread between few lattice sites evendeep inside the localized phase. This behavior is to be ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● / L | ψ m ( j ) | ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● / L | ψ m ( j ) | ●●●●●●●●●●●●●●●●● / L I m - - / L ϵ m F FIG. 2: Top Panels: Spatial distribution of the CAT statesat m/L = 0 .
24 (left panel) and m/L = 0 .
76 (right panel) and ω D / ( π J ) = 1. Bottom left panel: Plot of I m as a functionof m/L showing the dip in I m for the CAT states. Bottomright panel: Plot of (cid:15) Fm as a function of m/L at the samefrequency showing flattening of Floquet bands before the gapnear m/L (cid:39) . , .
76. All other parameters are as in Fig. 1.See text for details. contrasted with that of a canonical localized state where | ψ m ( j ) | is finite only on a single site. This feature makesthem perfect examples of CAT states whose wavefunc-tions are localized over more than one site. These statescan also be distinguished from either localized or delo-calized states via I m . This can be clearly seen in thebottom left panel of Fig. 2 where one sees a clear dip in I m for these states. The reason for the existence of suchstates can be understood from the structure of the Flo-quet eigenenergies (cid:15) Fm shown in the bottom right panel ofFig. 2 for (cid:126) ω D / J = π deep inside the localized regime.The Floquet energy dispersion becomes flat near the gapsin the spectrum; we have checked that this characteristicpersists for all ω ≥ ω c for large enough L . We find thatthe CAT states reside in these flat band regimes of theFloquet spectrum.The existence of such CAT states and its relation tothe flat regions in the Floquet band can be understood, inthe high frequency regime, as follows. We first note thatin this regime, from the first order Magnus expansion H F (cid:39) H p ; thus H F is almost diagonal in the positionbasis; each of its eigenstates is localized on one of thesites and these eigenstates can be approximately labeledby site indices of the chain. The off-diagonal terms aregenerated at higher order in the Magnus expansion andare therefore typically small in this region. Usually if thequasienergies are well-separated from each other, theseoff-diagonal terms do not change the nature of the Flo-quet spectrum. However, we note that this assumptionbreaks down around m/L (cid:39) . , .
76; the quasienergyspacing between the states localized in this regime ap-proaches zero as can be seen from flattening of the Flo-quet band. Consequently, the presence of off-diagonalterm in H F arising from H K , however small, becomes L = = = = = = / L I m L L = = = = = = / L I m L . FIG. 3: Left Panel: Plot of I m L as a function of m/L (sortedin increasing order of I m ) for ω D / ( π J ) = 0 .
245 for several L showing collapse of the delocalized states below the mobilityedge around the middle of the spectrum. Right panel: Similarplot for I m L . showing collapse of the multifractal statesabove the mobility edge. All other parameters are same as inFig. 1. important and leads to hybridization of the states local-ized on nearby sites. This leads to a pair of CAT statesin the spectrum. We note that these states persists onlyfor frequencies where the Floquet spectrum has a flat re-gion; for ω D ≤ ω c , this feature is absent and one doesnot find the CAT states in this regime.
2. Multifractal states
The multifractal nature of a quantum state can not beascertained from the IPR alone. To this end, we nowpresent computation of a generalized IPR defined as I ( q ) m = L (cid:88) j =1 | ψ m ( j ) | q (6)where I m ≡ I (2) m . It is well-known that I ( q ) m ∼ L − τ q ,where the fractal dimension of the state is given by D q = τ q / ( q − D q = 1 forall q , while for localized states D q = τ q = 0. Multifractalstates typically yield 0 < D q < ω D (cid:29) ω c ; in contrast, they are delo-calized for ω D (cid:39)
0. Thus it is evident that the presenceof multifractal states, if any, would be near the transitionwhere the mobility edge separates delocalized states froma bunch of states for which 0 < τ <
1. With this expec-tation, we first plot I m for all Floquet eigenstates nearthe transition corresponding to (cid:126) ω D / J = 0 . π in Fig.3 after sorting the eigenstates in terms of increasing IPR,which clearly shows the presence of delocalized states andmultifractal states separated by a mobility edge. The leftpanel shows a plot of I m L for all states; we find that forthe states with m/L < .
6, this quantity collapses forthe different system sizes indicating that I m ∼ L − andhence, the delocalized nature of these states. In con-trast, the right panel of the Fig. 3 indicates that I m for all states with m/L > . L − . indicating τ = D = 0 .
58 and a multifractal nature. To confirmthis, we plot I ( q ) m for these states for q = 3 and q = 4 as
0. 0.25 0.5 0.75 1. 1.251.00.80.60.40.20.0 ω D /( π ) m / L I m ( )
0. 0.25 0.5 0.75 1. 1.251.00.80.60.40.20.0 ω D /( π ) m / L I m ( ) ● ● ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆◆◆ ● ω D = π ■ ω D = π ◆ ω D = π - - -
20 ln L l n I ( L / ) ω D = πω D = πω D = π - τ q FIG. 4: Top Panels: Plot of I (3) m (left panel) and I (4) m (rightpanel) as a function of m/L and ω D indicating multifractalstates at intermediate frequency. Bottom left panel: Plot forln I m vs ln L used for extracting τ for several representativefrequencies for the state corresponding to m/L = 0 .
75. Thebehavior of perfectly delocalized (blue dots at ω D / ( π J ) =0 . ω D / ( π J ) = 0 .
5) can bedistinguished from that of a multifractal states (red dots ω D / ( π J ) = 0 . τ q as a func-tion of q for a delocalized (blue dots at ω D / ( π J ) = 0 . ω D / ( π J ) = 0 .
5) and multifractal(red dots at ω D / ( π J ) = 0 . shown in top panels of Fig. 4. The value of τ q is extractedfrom a plot of ln I ( q ) m vs ln L for several L as shown in thebottom left panel of Fig. 4, where m/L ≈ / I m for each L . A plot of τ q obtained using this procedure is shown as a function q for representative drive frequencies in the right bottompanel of Fig. 4. We find that τ q ∼ D q ( q −
1) for allplots; D q ∼ (cid:126) ω D / J = 0 . . π while 0 < D q < (cid:126) ω D / J = 0 . π .Numerically we find the presence of multifractal statesfor a wide range of frequencies below ω c , till (cid:126) ω D / J =0 . π . This is shown in the left panel of Fig. 5 where weplot τ q for all states in the Hilbert space after sorting inincreasing order of I m as a function of ω D . This clearlyshows the presence of multifractal states with quasiener-gies higher than the mobility edge for 0 . π ≤ (cid:126) ω D / J ≤ . π . Our analysis indicates that the multifractal di-mension D q is a non-monotonic function of ω D . This isshown in the right panel of Fig. 5 for a randomly chosenstate corresponding to m/L = 0 .
75. The dip in the plotaround (cid:126) ω D / J = 0 . π corresponds to the narrow fre-quency region where we find localized, rather than mul-tifractal, states above the mobility edge.Another test of multifractality of a given state is thedistribution of the energy difference between the odd-even ( s o − e m = (cid:15) F m +1 − (cid:15) F m ) and the even-odd ( s e − o m = ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ● q = ■ q = ◆ q = ω D /( π ) D q FIG. 5: Left: Plot of τ as a function of m/L (after sorting inincreasing order of I m ) and ω D / ( π J ) showing the mobilityedge separating delocalized and multifractal states for 0 . ≤ ω D / ( π J ) ≤ .
25. The system sizes used for extracting τ are L = 500 , · · · , D q as afunction of ω D / ( π J ) for m/L = 0 .
75. All other parametersare same as in Fig. 1. See text for details. (cid:15) F m − (cid:15) F m − ) energies . For delocalized states these twogaps are different due to almost doubly degenerate spec-trum leading to s o − e m (cid:39) s o − e m and s e − o m show a scattered be-havior. Thus one can distinguish between different set ofstates by studying these energy gaps.In our case, due to the drive, we study the difference ofquasi-energies. Since our spectrum has a mobility edge,the quasi-energy spectrum is folded. There is no generalway to unfold the spectrum in such a case; consequently,the identification of odd and even energies cannot be doneuniquely at low and intermediate drive frequencies. How-ever, the distribution of quasi-energy differences wouldstill show the same features as discussed in the last para-graph. Hence to highlight the expected behavior, wedefine two new quantities, s minm = Min[ s o − e m , s e − o m ] and s maxm = Max[ s o − e m , s e − o m ], which would allow us to sepa-rate the two gaps properly in the delocalized region ofthe spectrum.A plot of ln s minm and ln s maxm is shown in Fig. 6 as afunction of m/L for several representative frequencies.For (cid:126) ω D / J = 0 . π , where all states are delocalized,the plot shows clear separation of these two quantities forall m/L ; we find, in accordance to standard expectation,that s minm (cid:39) m . In contrast for (cid:126) ω D / J = 0 . π ,where all states are localized we find regular distributionof both energy gaps as shown in the bottom right panelof Fig. 6. The small difference between s minm and s maxm in this regime is a finite size effect and reduces with in-creasing L . In contrast, in the intermediate frequencyregime at (cid:126) ω D / J = 0 . π (top right panel of Fig.6), we find clear signature of a mobility edge separat-ing delocalized and multifractal states; the latter classof states can be recognized by strong scattering in dis-tribution of both ln s minm and ln s maxm . The presence ofa mobility edge separating the localized and delocalizedat (cid:126) ω D / J = 0 . π is shown in the bottom left panelof Fig. 6. We find that the presence of localized states s m max s m min - - - - - - -
50 m / L l n ( s mm ax , m i n ) - - - - - -
50 m / L l n ( s mm ax , m i n ) - - - - -
50 m / L l n ( s mm ax , m i n ) - - -
50 m / L l n ( s mm ax , m i n ) FIG. 6: Top Panels: Plot of ln s minm (red dots) and ln s maxm (blue dots) as a function of m/L for ω D / ( π J ) = 0 .
025 (topleft panel), 0 .
175 (top right panel), 0 . . L = 4181, since L needs to be a Fibonacci number. All other parameters aresame as in Fig. 1. above the mobility edge can be clearly distinguished fromthat of multifractal states because here there is a overlapof ln s minm and ln s maxm unlike the scattered distributionfound in the latter states.Before ending this subsection, we would like to pointout that our analysis shows that the driven AA model, atintermediate frequencies, shows mobility edge and multi-fractal states even when the parent Hamiltonian (Eq. 1)does not host either of these features. This distinguishesthis phenomenon from earlier studies of driven GAAmodel where the drive, in the high frequency regime,creates a multifractal state by superposing localized anddelocalized states across the mobility edge of the staticGAA Hamiltonian . For completeness, we note herethat Ref. 27 showed that periodic modulations of thephase of the hopping amplitude (e.g., by applying a timedependent gauge field) in the AA model also exhibiteda mobility edge and multifractal states. However, in ourcase, the time dependent hopping amplitudes are real-valued. The mechanism leading to the multifractal statesfor the driven AA model shall be discussed in Sec. III.
3. Transport, return probability and entropy
In this subsection, we address the effect of the presenceof mobility edge on fermion transport, survival probabil-ity of the fermion wavefunction in the steady state andtheir Shannon entropy .For studying transport property of the fermions westart from a domain-wall initial state defined, in thefermion number basis, by | ψ init (cid:105) = | n = 1 , ...n L/ = 1 , n L/ = 0 , ...n L = 0 (cid:105) (7)where we have taken L to be an even integer (chain witheven number of sites) and n j = (cid:104) ˆ n j (cid:105) denotes fermionoccupation number on the j th site and ˆ n j = c † j c j is thefermion number operator on that site. The wavefunctionafter n drive cycles is then given by | ψ (cid:48) (cid:105) = U ( nT, | ψ init (cid:105) = (cid:88) m c init m e − in(cid:15) Fm T/ (cid:126) | ψ m (cid:105) (8)where | ψ m (cid:105) denotes Floquet eigenstates with L/ c init m = (cid:104) ψ m | ψ init (cid:105) . Using this state, onemay compute the density profile of fermions in the steadystate. In what follows we study the quantities N j ( T ) = (cid:104) n j − / (cid:105) N av ( T ) = 4 L (cid:88) j =1 ..L (cid:104) ˆ(ˆ n j − / (cid:105) (9)where the average is taken with respect to the steadystate reached under a Floquet drive starting from | ψ init (cid:105) .In terms of the Floquet eigenfunctions | ψ m (cid:105) and the over-lap coefficients c init m (Eq. 8) these can be expressed as N j ( T ) = (cid:88) m | c init m | (cid:104) ψ m | n j − / | ψ m (cid:105) (10) N av ( T ) = 4 L (cid:88) j =1 ..L ( (cid:88) m | c init m | (cid:104) ψ m | ˆ(ˆ n j − / | ψ m (cid:105) ) We note that for the initial state |(cid:104) ψ init | ˆ2(ˆ n j − / | ψ init (cid:105)| = 0 and |(cid:104) ψ init | ˆ2(ˆ n j − / | ψ init (cid:105)| = 1while for free fermions, the ground state with J (cid:29) V , (cid:104) n j − / (cid:105) = 0. Thus N av ( T ) provides a measure ofdegree of delocalization of the driven chain. A similarreasoning shows that N j → N j = 1[ −
1] for j < [ > ] L/ N j takes values between 0 and 1 at differentsites.A plot of N j as a function of site index j/L and drivefrequency ω D for the steady state is shown in the leftpanel of Fig. 7. The density profile is seen to stay closeto that of the initial state confirming localization at highdrive frequency. In contrast, at low drive frequencies, itapproaches zero as expected for the delocalized regimewith J (cid:29) V . In between, N j indicates intermedi-ate behavior showing signature of partial transport suchthat 0 < | N j | <
1. The distribution of N j is muchmore spread out in the case where the mobility edge sep-arates delocalized and multifractal(as opposed to local-ized) states as can be clearly seen from the bottom panelsof Fig. 7. Thus we find that fermion number distributionin the steady state may provide a signature of presenceof the multifractal state in the driven AA model. A plotof N av ( T ) as a function of ω D , shown in the top rightpanel of Fig. 7, also confirms this behavior. We notethat an increased value of N av ( T ) (between 0 for per-fectly delocalized states and 1 perfectly localized states)for 0 . π < (cid:126) ω D / J < . π is a signature of presence ofboth localized (or multifractal) and delocalized states inthe Floquet spectrum and hence provides an indication ω D /( π ) N av ( T ) - - / L N j - - - - / L N j FIG. 7: Top Left Panel: Plot of N as a function of j/L and ω D / ( π J ) showing fermion density profile at all sites of thechain in the steady state as a function of ω D / ( π J ). TopRight Panel: Plot of N av ( T ) as a function of ω D / ( π J ) in thesteady state showing 0 ≤ N av ( T ) ≤ . ≤ ω D / ( π J ) ≤ .
25. Bottom Panels: Plot of N j as a function of j/L for ω D / ( π J ) = 0 . ω D / ( π J ) = 0 .
175 (right)where it separates the delocalized and multifractal states. Allother parameters are same as in Fig. 1. See text for details. ω D /( π ) S / l nL FIG. 8: Plot of the mean Shannon entropy S/ ln L as a func-tion of ω D / ( π J ). All other parameters are same as in Fig.1. See text for details. of the presence of mobility edge in the spectrum. More-over, the value of N av ( T ) seems to be larger in a narrowfrequency range around (cid:126) ω D / J = 0 . π where the mobil-ity edge separates delocalized and localized states. Thusour results show that the local fermion density in thesteady state starting from a domain-wall initial condi-tion in these chains may serve as a detector of mobilityedge in the Floquet spectrum.Next, we compute the Shannon entropy of the drivenchain. This is defined in terms of the overlap coef-ficients obtained by computing overlap of the single-particle Floquet eigenstates | ψ m (cid:105) with the eigenstates of H p = H F ( T = 0) | j (cid:105) : ψ m ( j ) = (cid:104) j | ψ m (cid:105) . The Floqueteigenstates can be written as | ψ m (cid:105) = (cid:80) j ψ m ( j ) | j (cid:105) . TheShannon entropy of the m th Floquet eigenstate is thengiven by S m = − (cid:88) j | ψ m ( j ) | ln | ψ m ( j ) | , S = 1 L (cid:88) m S m (11)where S is the mean entropy. We note that for highfrequency when H F (cid:39) H p , ψ m ( j ) (cid:39) δ mj leading to S m (cid:39) (cid:126) ω D / J (cid:29) H p are localized this means that S → (cid:126) ω D / J (cid:28) ψ m ( j ) (cid:39) / √ L for all m leading to maximum entropy of S (cid:39) ln L .A plot of S/ ln L as a function of the drive frequency,shown in Fig. 8, indicates this change. We find thatthe localization-delocalization transition is marked by asharp rise in S around ω D = ω c = 0 . π J / (cid:126) . The ap-pearance of the mobility edge just below the transitionleads to S/ ln L ≤
1; this value would have been closerto unity if all the Floquet eigenstates would be delocal-ized for ω D ≤ ω c . We note that S shows a narrow diparound (cid:126) ω D / J = 0 . π . This can be understood to bedue to the fact that around this frequency the mobilityedge separates localized, rather than multifractal, statesfrom the delocalized ones; the presence of these localizedstates in the spectrum leads to a lower value of S .Finally we compute the survival probability which isdefined as the probability of finding a fermion, initiallylocalized at a given site, within a neighborhood of length R around that site after n drive cycles. This is given by F n ( R ) = j + R/ (cid:88) j = j − R/ | ψ n ( j ) | ψ n ( j ) = U ( nT, ψ init ( j ) (12)where j denotes lattice sites, we shall consider the initialwavefunction to be localized at the center of the chain( j = L/
2) for the rest of this section. The limitingvalues of F n ( R ) can be easily deduced. For example, ifthe wavefunction remains localized F n ( R ) (cid:39) R and n ; in contrast if the drive leads to delocalization, F n ( R ) should linearly increase with R for large n . In thepresence of a mobility edge separating delocalized andmultifractal states, F n ( R ) should again increase with R ,but with a sublinear growth for large n . Moreover, thesteady state value of F n ( R ) can be obtained in terms ofFloquet eigenfunctions as F s ( R ) = j + R/ (cid:88) j = j − R/ (cid:88) m | ψ m ( j ) | (13)and is therefore controlled by the coefficients ψ m ( j ).A plot of F s ( R = L/
2) as a function of the drivefrequency ω D is shown in the left panel of Fig. 9. Wefind that F s ( L/
2) shows a sharp dip at the localization-delocalization transition. Below the transition, the decay ω D /( π ) F s ( R = L / ) ω D = πω D = πω D = πω D = π / L F s ( R ) FIG. 9: Left Panel: Plot of the survival probability F s ( L/ ω D / ( π J ). Right panel: Plot of F s ( R ) asa function of R for several representative value of ω D / ( π J ).All other parameters are same as in Fig. 1. See text for details. of F s ( L/
2) is gradual and non-monotonic; this seems tobe a direct consequence of the presence of the mobilityedge. The right panel of Fig. 9 shows the R dependenceof F s ( R ) for several representative drive frequencies. Wefind that at high drive frequencies (cid:126) ω D / J = 0 . π , thesystem remains localized leading to F s (cid:39) R ; in contrast it linearly decreases to zero as R isdecreased in the low frequency limit (cid:126) ω D / J = 0 . π .In between, in the regime where the mobility edge sep-arates delocalized states from multifractal or localizedstates in the Floquet eigenspectrum, we find sublineardecay of F s ( R ) as a function of R ; this decay is fasterif states with quasienergies above the mobility edge arelocalized ( (cid:126) ω D / J = 0 . π ). Thus F s ( R ) distinguishesbetween mobility edge separating delocalized states withmultifractal or localized states.Finally, we study the V dependence of our results. Inparticular we concentrate on obtaining an estimate of therange of V / J over which the multifractal states exist.To this end, we plot the mean Shannon entropy of theFloquet eigenstates as a function of V and ω D . This plot,shown in the left panel of Fig. 10, indicates that a mo-bility edge separating delocalized and multifractal states(indicated by blue in the plot) are present of over a rangeof frequency whose width tend to be maximal around V (cid:28) J . For V ≥ J , the Floquet states are either alllocalized (red region) or display a mobility edge separat-ing delocalized and localized states (green regions). For V , (cid:126) ω D (cid:28) J , the Floquet states are all delocalized (vi-olet regions). The right panel shows a plot of S/ ln L asa function of V for (cid:126) ω D / J = 0 . π . The plot shows in-dication of a mobility edge separating delocalized and lo-calized states for 0 . ≤ V / J ≤ .
2; in contrast, the mo-bility edge separates delocalized and multifractal statesfor V / J ≤ .
2. Thus our results show that the multi-fractal states are indeed present in the Floquet spectrumfor a wide region in the ( V / J , (cid:126) ω D / J ) plane. B. Sinusoidal Protocol
In this section, we study the properties of AA modelin the presence of a continuous drive. Such a drive is V / S / l nL FIG. 10: Left panel: Distribution of S/ ln L as a function ofthe Aubrey Andre Strength V / J and the drive frequency ω D / ( π J ). The regions with all localized states are indicatedby red, delocalized states by violet. The blue regions denotemixture of delocalized and multifractal states and green mix-ture of delocalized and localized states. Right Panel: Plot of S/ ln L as a function of V / J for (cid:126) ω D / ( π J ) = 0 .
22. . Allother parameters are same as in Fig. 1. implemented by choosing J ( t ) = J cos ω D t (14)A numerical study of the AA model in the presence ofsuch a continuous drive involves decomposition of theevolution operator into N Trotter steps such H ( t ) doesnot change significantly in the interval t j and t j + T /N for any time instant t j . One can define the eigenval-ues and eigenfunctions of the instantaneous Hamiltonian H j = H [ t j + T / (2 N )] as (cid:15) jn and | ψ jn (cid:105) ; these are obtainednumerically by exact diagonalization of H j on a lattice ofsize L . One can then construct the evolution operator as U ( T,
0) = N (cid:89) j =1 (cid:88) n e − i(cid:15) jn T/N | ψ jn (cid:105)(cid:104) ψ jn | (15)We note this procedure requires numerical diagonaliza-tion of N instantaneous Hamiltonians; this make numeri-cal study of continuous protocols significantly more costlycompared to their discrete counterparts. Having con-structed U ( T, I m as afunction of ω D and m in the top left panel of Fig. 11,we find that the transition shifts to (cid:126) ω D / J (cid:39) . π ;the mobility edge exists over narrower regions (one near0 . π ≤ (cid:126) ω D / J ≤ . π and another near 0 . π ≤ (cid:126) ω D / J ≤ . π ) as can be seen from the plot of τ as a function of m and ω D in the top right panel. Aplot of D q shown in the middle left panel confirms thepresence of multifractal states in these regions. In the ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ● q = ■ q = ◆ q = ω D /( π ) D q ω D /( π ) S / l nL ω D /( π ) N av ( T ) FIG. 11: Top Left Panel: Plot of I m as a function of m/L and ω D / ( π J ). Top Right Panel: Plot of τ as a function of m/L and ω D / ( π J ). Middle Left Panel: Plot of D q as a function of ω D / ( π J ) for m/L = 0 .
75. Middle Right Panel: A plot of S asa function of ω D / ( π J ) showing the signature of localization-delocalization transition at (cid:126) ω D / ( π J ) (cid:39) .
25. Bottom leftPanel: Plot of N j as a function of j/L and ω D / ( π J ) showingfermion density profile at all sites of the chain in the steadystate as a function of ω D / ( π J ). Bottom right Panel:Plot of N av as a function of ω D / ( π J ). The system sizes used tocalculate τ and D q is L = 200 ∼ L = 1024.Here V / J = 0 .
025 ,all other parameters are same as in Fig. 1. See text for details. middle right panel, we show the plot of the mean Shan-non entropy S as a function of ω D . We find that S also bears the signature of the localization-delocalizationtransition. The bottom panels show steady state dis-tribution of particles in this system starting from thedomain wall state. The bottom left panel shows thedistribution of N j over lattice sites (scaled by systemsize( L )) as a function of the drive frequency. The plotdemonstrates the non-monotonic behavior of N j as afunction of ω D just below the transition. Finally thebottom right panel shows a plot of N av as a function of ω D in the steady state; we find that it displays signa-ture of the localization-delocalization transition around (cid:126) ω D / J = 0 . π and also shows peaks at intermediatefrequency where the mobility edge appears in the spec-trum. The height of these peaks are less than unity; thisis a consequence of the fact that the entire spectrum isnot localized at these frequencies.Finally, we plot the mean Shannon entropy S/ ln L as V / S / l nL FIG. 12: Left: Plot of S/ ln L as a function of V / J and ω D / ( π J ) for the sinusoidal protocol with L = 610. Right:Plot of S/ ln L as a function of V / J for a cut taken at (cid:126) ω D / ( π J ) = 0 .
22. All other parameters are same as in Fig.1. a function of V and ω D . We find that for the sinu-soidal protocol the presence of multifractal states occursin a reduced area of parameter space compared to thatfor square pulse protocol studied earlier. Moreover, weget localized states at a lower frequency as compared tosquare pulse protocol for the same value of V . Fromboth the graphs we observe that as we increase V thefrequency at which all states becomes localized decreases.This expected since the off-diagonal hopping terms be-comes small compared to the diagonal AA potential termleading to dynamical localization.Our results therefore indicate that the localization-delocalization transition in these systems along with thepresence of the CAT and multifractal state exists for bothdiscreet and continuous protocols. However, the range of ω D for which the multifractal states exists is significantlyreduced in the latter case. III. FLOQUET PERTURBATION THEORY
In this section, we aim to obtain an analytic, albeit per-turbative, understanding of several features of the drivenAA model found via exact numerics using FPT which isknown to provide accurate results in the large drive am-plitude limit . The square pulse protocol will betreated in Sec. III A while the continuous drive protocolwill be addressed in Sec. III B.
A. FPT for square pulse protocol
In this section, we shall focus on the square pulse pro-tocol given by Eq. 2 in the large drive amplitude limit J (cid:29) V . In this limit, we consider the contributionfrom H p to the evolution operator U ( T,
0) as perturba-tion and develop a systematic expansion for U followingRefs. 17,21,22. To this end, we first note that the firstterm in such an expansion is given by U which can be written as U = (cid:81) k U k where U k ( t,
0) = e it J cos kc † k c k , t ≤ T / e i ( T − t ) J cos kc † k c k , T / ≤ t ≤ T Here and in the rest of this section, we have set (cid:126) tounity. This leads to U ( T,
0) = I (where I denotes theidentity matrix) and H F = 0. The vanishing H F canbe seen to be the consequence of the symmetric natureof the drive protocol.The first order perturbative correction of U , withinFPT, is given by U = − i (cid:90) T dtU † ( t, H p U ( t,
0) (17)To evaluate this, we use the number basis in momentumspace, | k (cid:105) ≡ | n k (cid:105) , since U is diagonal in this basis. Thematrix element of U in this basis is then given by (cid:104) k | U | k (cid:105) = 4 V ( k − k ) J f ( k , k ) sin[ J f ( k , k ) T / × e iT J f ( k ,k ) / (18) V ( k ) = V (cid:88) j exp[ ikj ] cos(2 πηj ) = V ( − k ) ,f ( k , k ) = cos k − cos k = − f ( k , k )where we have set φ = 0 without loss of generality. Thisindicates that the Floquet Hamiltonian to first order inperturbation theory is given by H F = i (cid:88) k ,k V ( k − k ) J T f ( k , k ) sin[ J f ( k , k ) T / × e iT J f ( k ,k ) / c † k c k (19)A similar procedure for U ( T,
0) yields the relation U ( T,
0) = U ( T, / H F = 0. Thedetails of this calculation is similar to that carried outin Ref. 17 and is not presented here. In what follows,we shall analyze H F (Eq. 19) with the aim of obtainingqualitative understanding of the presence of multifractalstates in the Floquet spectrum.A straightforward numerical diagonalization of H F yields the Floquet eigenstates and eigenvalues. To studythe nature of these Floquet eigenstates as a function ofdrive frequency, we plot the IPR I m and τ correspondingto these states in Fig. 13. The top left panel of this plotshows the plot of I m as a function of m/L and ω D . I m ob-tained from the eigenstates of H F retains all qualitativefeature of the Floquet eigenstates obtained from exactnumerics. In particular, the plot shows a localization-delocalization transition around (cid:126) ω D / J (cid:39) . π whichis close to the exact value (cid:126) ω c / J = 0 . π . More-over, the spectrum indicates the presence of the CATstates in the spectrum; we have checked that the ori-gin of these state can be tracked back to the flat regionsin the Floquet spectrum as can be seen from the top0 - - / L ϵ m F ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ● q = ■ q = ◆ q = ω D /( π ) D q FIG. 13: Top Left Panel: Plot of I m as a function of m/L and ω D / ( π J ). Top Right Panel: Plot of eigenstates (cid:15) Fm obtainedby diagonalizing H F as a function of m/L for (cid:126) ω D / ( π J ) =1. Bottom Left Panel: Plot of τ as a function of m/L and ω D / ( π J ). Bottom right Panel: Plot of D q as a function of ω D / ( π J ) for m/L = 0 .
75. All other parameters are same asin Fig. 1 and Fig. 5. See text for details. right panel of Fig. 13. The bottom left panel showsa plot of τ as a function of m/L and ω D . We findthat just below the transition, we find a wide range of frequency 0 . π ≤ (cid:126) ω D / J ≤ . π where we findstates with 0 < τ < . π ≤ (cid:126) ω D / J ≤ . π ; for 0 . π ≤ (cid:126) ω D / J ≤ . π we do not find delocalized states in the spectrum whichis contrast with that obtained in exact numerics. Thepresence of multifractal states in the spectrum of H F is further confirmed by plotting D q for q = 2 , , ω D for m/L = 0 .
75 in the bottom right panelof Fig. 13; the plot shows clear signature of multifrac-tality for 0 . π ≤ (cid:126) ω D / J ≤ . π . Our results indicatethat H F constitutes semi-analytic expression of a Flo-quet Hamiltonian which support multifractal states in itseigenspectrum.To understand the origin of these multifractal states,we obtain a real space representation of H F . A Fouriertransform of Eq. 19 yields H F = (cid:88) j,j (cid:48) H jj (cid:48) c † j c j (cid:48) H jj (cid:48) = (cid:90) π − π dk dk π e i ( k j − k j (cid:48) ) iV ( k − k ) J T f ( k , k ) × sin[ J f ( k , k ) T / e iT J f ( k ,k ) / (20)A straightforward calculation outlined in the Appendixleads to an analytic expression for H jj (cid:48) given by H jj (cid:48) = i √ ∞ (cid:88) w = | j − j (cid:48) | / ( T J ) w (2 w + 1) √ πκ (cid:48) ( j, j (cid:48) , w ) (cid:20) w − (cid:18) ww (cid:19) V ( j + j (cid:48) ) / − w w − w − (cid:88) z =0 ( − z (cid:18) wz (cid:19) ( V [( w − z ) + ( j + j (cid:48) ) /
2] + V [( j + j (cid:48) ) / − ( w − z )]) (cid:35) j − j (cid:48) = 2 n = − √ ∞ (cid:88) w =( | j − j (cid:48) |− / ( T J ) w +1 (2 w + 2) √ πκ (cid:48) ( j, j (cid:48) , w + 1) ( − w w w (cid:88) z =0 ( − w (cid:18) w + 1 z (cid:19) × ( V [( j + j (cid:48) ) / − (2 w + 1 − z ) / − V [( j + j (cid:48) ) / w + 1 − z ) / , j − j (cid:48) = 2 n + 1 (21)where n is an integer and the function κ (cid:48) is given by κ (cid:48) ( m, n, p ) = 1 , p = 0 (22)= | m − n | p/ (cid:89) s =1 ,s (cid:54) = | m − n | / [( m − n ) − (2 s ) ] , p = 2 k = ( p − / (cid:89) s =0 ,s (cid:54) =( | m − n |− / [( m − n ) − (2 s + 1) ] , p = 2 k + 1for integer k . From the expression of H jj (cid:48) we clearlyfind that the Floquet Hamiltonian corresponds to a hop- ping Hamiltonian whose range increases with decreas-ing drive frequency. At very high frequencies, only the j = j (cid:48) (on-site) term survives and we get back the Mag-nus result. As the frequency is decreased, the amplitudeof terms for which j (cid:54) = j (cid:48) (hopping terms with range | j − j (cid:48) | ) increases. Thus at intermediate frequencies, thiscorresponds to a Hamiltonian with on-site quasiperiodicterm (corresponding to j = j (cid:48) ) and intermediate rangehopping terms (for both odd and even | j − j (cid:48) | ) whoseamplitude depends on the drive frequency. It is wellknown that similar Hamiltonians, for specific range ofhopping amplitudes, supports multifractal states in their1 FIG. 14: Left Panel: Plot of I m as a function of m/L and ω D / ( π J ). Right Panel: Plot of τ as a function of m/L and ω D / ( π J ). All other parameters are same as in Fig. 1 andFig. 5. See text for details. spectrum . We point out that here the drive frequencymay be used to engineer these amplitudes. Our resultthus constitutes an example of analytic form of a Flo-quet Hamiltonian which supports multifractal states. B. FPT for continuous protocol
For the continuous protocol, we choose J ( t ) = J cos( ω D t ) so that for J (cid:29) V , U is given by U ( t,
0) = exp (cid:34) − i J ω D sin( ω D t ) (cid:88) n c † k c k (cid:35) (23) where we have set (cid:126) = 1. This leads to U ( T,
0) = I and H F = 0. We note that the eigenbasis for U is still givenby | k (cid:105) ≡ | n k (cid:105) The perturbative contribution to the first order term inthe Floquet Hamiltonian is then by Eq. 17 with U ( t, U (cid:104) k | U ( T, | k (cid:105) = − iT V ( k − k ) J ( x ) (24)where J denotes Bessel functions and x = J f ( k , k ) /ω D . Using this, we find that the first orderFloquet Hamiltonian is given by H F = (cid:88) k ,k V ( k − k ) J ( x ) c † k c k (25)We note that for ω D → ∞ , J → H F → H p which reproduces the Magnus results. The second orderterms can be computed in an analogous fashion. Thecomputation procedure is same as charted out in Ref. 22and yields H F = (cid:88) k ,k ,k ,k ∞ (cid:88) n =0 V ( k − k ) V ( k − k )(2 n + 1) ω D [ J ( x ) J n +1 ( x ) − J ( x ) J n +1 ( x )] c † k c k c † k c k . (26)We note that H F → ω D → ∞ which is consis-tent with the Magnus expansion results which yields avanishing second order contribution to H F .Next, we obtain the Floquet eigenstates and corre-sponding quasienergies via numerical diagonalization of H F = H F + H F . The results are shown in Fig. 14.Again, qualitative features like the presence of CATstates, mobility edge and multifractal states are all cap-tured by the perturbative H F obtained from FPT. TheFPT results also show a significant reduction in the rangeof drive frequencies ω D that give rise to multifractal states consistent with the exact numerics for the sinu-soidal protocol.Finally we obtain a representation of H F in real spacefollowing an analysis which is identical to that carriedout in the previous section for the square-pulse proto-col. For this purpose, we consider H F and obtain itsanalytic form in real space. The details of the cal-culation in charted out in the appendix. This yields H F = (cid:80) jj (cid:48) H jj (cid:48) c † j c j (cid:48) where H jj (cid:48) = 0 for | j − j (cid:48) | = 2 m +1.For | j − j (cid:48) | = 2 m , it is given by2 H jj (cid:48) = ∞ (cid:88) p =( j − j (cid:48) ) / , ( − p p )!( J T / (2 π )) p √ πp !2 p − p + 1) κ ( j, j (cid:48) , p ) (cid:20)(cid:18) pp/ (cid:19) V (( j + j (cid:48) ) / − p/ p/ − (cid:88) z =0 ( − z (cid:18) pz (cid:19) ( V [( p/ − z ) + ( j + j (cid:48) ) /
2] + V [( j + j (cid:48) ) / − ( p/ − z )]) κ ( m, n, p ) = 1 , for p = 0 , κ ( m, n, p ) = ( m − n ) p (cid:89) s =1 ,s (cid:54) =( m − n ) / [( m − n ) − (2 s ) ] , otherwise (27)We note that similar to the square pulse protocol, weget a real-space Floquet whose range increases with de-creasing frequency. The high-frequency limit leads to acompletely local Hamiltonian consistent with the Mag-nus result. However, for the continuous drive proto-col discussed in this subsection, H F only induces next-nearest neighbor couplings. The coupling between oddand higher neighboring sites which differ by an odd num-ber of lattice sites is induced by H F . We do not computethis terms here but merely observe that their contribu-tion would be smaller by at least a factor of V /ω D . Thedifference in coupling strength between sites differing byodd and even number of lattice sites also explains thereason for the structure of the CAT states. We find thatthey are distributed between a site and its next-nearestneighbor (rather than the expected nearest one). This isclearly a consequence of having larger H jj (cid:48) between thenext-nearest neighbor sites compared to the nearest ones. IV. DISCUSSION
In this work, we have charted out the phase diagramof the driven AA model using both square pulse and si-nusoidal drive protocols. Our numerical studies, carriedout using exact diagonalization of the fermionic system,reveals the presence of localization-delocalization transi-tion in this system occurring at a critical drive frequency ω c . Moreover, below ω c , for a range of drive frequencies,we find the existence of a mobility edge which separatesdelocalized Floquet eigenstates with quasienergies belowthe edge from localized or multifractal eigenstates aboveit. Our analysis shows the presence of multifractal statesin the Floquet eigenspectrum over a wide range of drivefrequencies. We show that the presence of the mobilityedge leaves its imprint on the transport of the systemand on survival probability and Shannon entropy of thedriven fermions. Moreover, the fermion transport start-ing from a domain wall state where all the fermions arelocalized to the left-half of the chain can discern the pres-ence of multifractal states in the Floquet eiegnspectrum.We note that the non-driven AA model does not supportmobility edge or multifractal states in its spectrum; thusour results constitute dynamical signatures which have no analog in the non-driven model.The numerical results that we find can be semi-analytically understood within FPT. Our results re-garding this constitutes derivation of semi-analytic, al-beit perturbative, Floquet Hamiltonians for both squarepulse and sinusoidal drive protocols. We show thatthese perturbative, semi-analytic Hamiltonians repro-duce the localization-delocalization transition obtainednumerically; moreover, they support CAT and multifrac-tal states in their eigenspectrum. The reason for thepresence of such states can be understood by obtain-ing real-space representation of these Hamiltonians. Inreal-space, these Floquet Hamiltonians contain on-sitequasiperiodic terms along with hopping terms which con-nects between fermions at different sites. We find thatthe range of these latter class of terms increase with de-creasing drive frequency. Consequently, these FloquetHamiltonians belong to a class of Hamiltonians withAubrey-Andr´e interactions and quasi-long range hoppingterms. It was shown in Ref. 3 that these Hamiltoniansupport multifractal states.Our results indicate that the signature of thelocalization-delocalization transition can be obtained bystudying fermionic transport. This allows us to suggestrealistic experiment which can test our theory. We sug-gest realization of the Aubrey-Andr´e potential in an opti-cal lattice as done recently in Ref. 6. The drive of the hop-ping term may be induced by tuning the laser strengthcreating the optical lattice using either of the periodicprotocols discussed. In addition, one can start from aconfiguration where the fermions in the lattice are con-fined to the left-half of the chain. Our prediction is thatthere will be critical drive frequency ω c below which thesystem will eventually delocalize. This will be reflectedin a sharp drop in the value of N as sketched in Fig. 7.Moreover, for a range of frequencies below ω c , N willremain between its values for localized ( N = 1) and de-localized ( N = 0) states signifying the presence of themobility edge.In conclusion, we have studied the driven Aubrey-Andr´e model and showed the presence of a drive-inducedlocalization-delocalization transition. Our results indi-cate the presence of mobility edge and multifractal statesin the Floquet eigenstates; their existence can be seen3from analytic, perturbative form of H F which we deriveusing a Floquet perturbation theory which represents aresummation of an infinite class of terms in the Magnusexpansion. We show that the presence of this mobilityedge is reflected in fermionic transport and suggest ex-periments which can test our theory. Acknowledgments
The authors acknowledges related discussions atICTS, Bengaluru during the program Thermalization,Many body localization and Hydrodynamics (Code:ICTS/hydrodynamics2019/11). The work of A.S. ispartly supported through the Max Planck Partner Groupprogram between the Indian Association for the Cultiva-tion of Science (Kolkata) and the Max Planck Institutefor the Physics of Complex Systems (Dresden).
Appendix A: Real space representation of H F
1. Square pulse
We start from Eq. 20 from the main text, H F = (cid:88) j,j (cid:48) H jj (cid:48) c † j c j (cid:48) H jj (cid:48) = (cid:90) π − π dk dk π e i ( k j − k j (cid:48) ) iV ( k − k ) J T f ( k , k ) × sin[ J f ( k , k ) T / e iT J f ( k ,k ) / (A1)We shift to the center of momentum coordinates, q = k − k and r = k + k where dqdr = dk dk and write H jj (cid:48) = 12 π (cid:90) π − π (cid:90) π − π dqdre i [ q ( j + j (cid:48) )+ r ( j − j (cid:48) )] iV (2 q ) J T g ( q, r ) × sin[ J g ( q, r ) T / e iT J g ( q,r ) / (A2)Now expanding the oscillatory part, we writesin[ J g ( q, r ) T / e iT J g ( q,r ) / = T ∞ (cid:88) p =0 ( iT J g ( q, r ) / p ( p + 1)! (A3)Hence Eq. A2 can be written as, H jj (cid:48) = 12 π ∞ (cid:88) p =0 ( iT J / p ( p + 1)! (cid:90) π − π (cid:90) π − π dqdr × e i [ q ( j + j (cid:48) )+ r ( j − j (cid:48) )] iV (2 q )(2 sin q sin r ) p (A4) The next task is to perform the integrals. Performing theintegral over r first, we get H jj (cid:48) = 12 π ∞ (cid:88) p =0 ( iT J / p ( p + 1)! (cid:90) π − π dqe i [ q ( j + j (cid:48) )] (A5) × iV (2 q )(sin q ) p p +1 p !( − i ) p sin[( j − j (cid:48) ) π ] κ ( j, j (cid:48) , p )where κ ( j, j (cid:48) , p ) = ( j − j (cid:48) ) p/ (cid:89) s =1 [( j − j (cid:48) ) − (2 s ) ] , p = 2 k = ( p − / (cid:89) s =0 [( j − j (cid:48) ) − (2 s + 1) ] , p = 2 k + 1(A6)for any integer k . It is to be noted that only when s = | j − j (cid:48) | / p even and s = ( | j − j (cid:48) | − / p odd, theintegrals give a finite contribution. Hence the summationover p must start from p = | j − j (cid:48) | . This gives, H jj (cid:48) = i π ∞ (cid:88) p = | j − j (cid:48) | ( iT J / p ( p + 1)! (cid:90) π − π dqe i [ q ( j + j (cid:48) )] × V (2 q )(sin q ) p p +1 p !( − i ) p πκ (cid:48) ( j, j (cid:48) , p ) (A7)where , κ (cid:48) ( j, j (cid:48) , p ) = 1 , p=0 (A8)= | j − j (cid:48) | p/ (cid:89) s =1 ,s (cid:54) = | j − j (cid:48) | / [( j − j (cid:48) ) − (2 s ) ] , p = 2 k = ( p − / (cid:89) s =0 ,s (cid:54) =( | j − j (cid:48) |− / [( j − j (cid:48) ) − (2 s + 1) ] , p = 2 k + 1Next, we perform the integral over q using standardtrigonometric identities. First one should separate outthe even and odd parts of the integral, and note thatwhen j − j (cid:48) is even, p necessarily is always even as therest of the terms integrate to 0 and similarly for j − j (cid:48) odd. Hence for j − j (cid:48) even, assuming p = 2 w , we get4 H jj (cid:48) = i π ∞ (cid:88) w = | j − j (cid:48) | / ( T J ) w (2 w + 1) πκ (cid:48) ( j, j (cid:48) , w ) (cid:90) π − π dq cos[ q ( j + j (cid:48) )] V (2 q )[ 12 w (cid:18) ww (cid:19) + ( − w w − w − (cid:88) z =0 ( − z (cid:18) wz (cid:19) cos[2( w − z ) q ]](A9)For odd j − j (cid:48) , we consider p = 2 w + 1 and obtain H jj (cid:48) = − π ∞ (cid:88) w =( | j − j (cid:48) |− / ( T J ) w +1 (2 w + 2) πκ (cid:48) ( j, j (cid:48) , w + 1) (cid:90) π − π dq sin[ q ( j + j (cid:48) )] V (2 q ) × ( − w w w (cid:88) z =0 ( − w (cid:18) w + 1 z (cid:19) sin[(2 w + 1 − z ) q ] (A10)Using the inverse Fourier transform √ π (cid:82) π − π V (2 q ) e i qx = V ( x ) and integrating over q ,we get Eq. 21 of the main text.
2. Sinusoidal pulse
For this drive protocol we start from H jj (cid:48) = 12 π (cid:90) π − π (cid:90) π − π dk dk e i [ k j − k j (cid:48) ] V ( k − k ) × J ( J f ( k , k ) /ω D ) (A11)As in the case of square protocol, we switch to relativeand center of mass momenta and obtain H jj (cid:48) = 14 π (cid:90) π − π (cid:90) π − π dqdre i [ q ( j + j (cid:48) )+ r ( j − j (cid:48) )] V (2 q ) × J ( J g ( q, r ) /ω D ) (A12)where g ( q, r ) = − q sin r ). Next, we use the expan-sion of J ( x ), J ( x ) = ∞ (cid:88) p =0 ( − p p !Γ( p + 1) ( x/ p (A13)Substituting Eq. A13 in Eq. A12 we find, H jj (cid:48) = ∞ (cid:88) p =0 π (cid:90) π − π (cid:90) π − π dqdre i [ q ( j + j (cid:48) )+ r ( j − j (cid:48) )] V (2 q ) × ( − p p !Γ( p + 1) ( J sin q sin r/ω D ) p (A14)Integrating over r we get, H mn = − ∞ (cid:88) p =0 π (cid:90) π − π dqe i [ q ( j + j (cid:48) )] V (2 q ) (A15) × ( J sin q/ω D ) p ( − p p )! sin[( j − j (cid:48) ) π ] p !Γ( p + 1) κ ( j, j (cid:48) , p ) where, κ ( j, j (cid:48) , p ) = | j − j (cid:48) | p (cid:89) s =1 [( j − j (cid:48) ) − (2 s ) ] , And noting that the summation can only start from p =( m − n ) / H jj (cid:48) = − ∞ (cid:88) p = | j − j (cid:48) | / , | j − j (cid:48) | even π (cid:90) π − π dqe i [ q ( j + j (cid:48) )] V (2 q ) × ( J T sin q/ (2 π )) p ( − p p )! πp !Γ( p + 1) κ (cid:48) ( j, j (cid:48) , p ) (A16)where , κ (cid:48) ( j, j (cid:48) , p ) = 1 , p=0 (A17)= | j − j (cid:48) | p (cid:89) s =1 ,s (cid:54) =( j − j (cid:48) ) / [( j − j (cid:48) ) − (2 s ) ]otherwiseand we have replaced ω D by 2 π/T . One can imme-diately see if | j − j (cid:48) | is odd then no term contributesand H jj (cid:48) = 0. Then one can integrate over q aswell to get Eq. 27 of the main text. The expres-sions of the second order term in H F is quite compli-cated and we have not analyzed their form in positionspace. However, we note that these terms are of the form ∼ (cid:80) ∞ n =0 [ J n +1 ( x ) J ( y ) − J ( x ) J n +1 ( y )] / (2 n + 1). Thusfrom the expansion of J n ( x ), it can be seen that theseterms would actually give rise to odd powers of sin q .This means that here the terms of H jj (cid:48) where j − j (cid:48) oddwill be non-zero. The consequence of this is discussed inthe main text.5 n = n = n = n s - - - / L N j n = n = n = n s - - / L N j n = n = n = n s - - / L N j n = n = n = n s - - / L N j n = n = n = n s - - / L N j ω = πω = πω = πω = πω = π n N h / L FIG. 15: Top Left Panel: Plot of N j as a function of j/L for ω D / ( π J )=0.025 where the full spectrum is delocal-ized. Top Right Panel: Plot of N j as a function of j/L for ω D / ( π J )=0.175 at which there is a mobility edge betweendelocalized and multifractal states. Middle Left Panel: Plot of N j as a function of j/L for ω D / ( π J )=0.20 at which thereis a mobility edge between delocalized and localized states.Middle Right Panel: Plot of N j as a function of j/L for ω D / ( π J )=0.24 at which there is a mobility edge betweendelocalized and multifractal states. This drive frequency isnear the critical frequency ω c . Bottom left Panel: Plot of N j as a function of j/L for ω D / ( π J )=0.5 where the fullspectrum is localized. Bottom right Panel:Plot of the numberof particles present in the right half (beginning from domainwall initial state) as a function of number of drive cycles n forvarious representative drive frequencies. All other parametersare same as in Fig. 1. See text for details. Appendix B: Approach to Steady state
In this appendix we discuss, in brief, the approach ofour model subjected to square pulse drive, to the steadystates shown in the main text, starting from the domainwall initial state given by, | ψ init (cid:105) = | n = 1 , ...n L/ = 1 , n L/ = 0 , ...n L = 0 (cid:105) To this effect we study the distribution of fermion num-ber density N j = (cid:104) ψ ( n T ) | ˆ n j − / | ψ ( n T ) (cid:105) , where ψ ( n T ) = U ( n T, | ψ init (cid:105) and ˆ n j = c † j c j at differentnumber of cycles n . Fig. 15 shows the distributionsstudied for different drive frequencies. For low drivefrequencies ( ω D / ( π J ) = 0 . ω D /( π ) S v N / L ω = πω = πω = πω = πω = π n S v N / L FIG. 16: Left panel: Plot of half chain von-Neumann en-tanglement entropy( S vN ) in the steady state starting from adomain wall initial state scaled by the system length L as afunction of drive frequency ω D / ( π J ) showing the signatureof delocalized, localized and mixture of delocalized and multi-fractal and delocalized and localized states. Right Panel: Plotof half chain entanglement as a function of number of drivecycles n for various representative drive frequencies showinghow it reaches the steady state value. All other parametersare same as in Fig. 1. top right and middle left panels which show results for ω D / ( π J ) = 0 .
175 and ω D / ( π J ) = 0 .
20 respectively.However, it is seen that while 10 drive cycles is notenough to reach close to the steady states, 10 cycles isenough even for the drive frequency ( ω D / ( π J ) = 0 . ω D / ( π J ) = 0 . n = 10 gives the im-pression the system is localized from the distribution.Only at extremely large number of cycles n = 10 doesthe system give the expected behavior of the steady state.This is possibly due to the proximity of this drive fre-quency to the critical frequency ω D / ( π J ) = 0 .
30. Thebottom right panel of Fig. 15 shows the evolution of N h = (cid:80) Lj = L/ ˆ n j with time, i.e., the transport of par-ticles from the left-half of the system to the right-half. Italso shows that only after a sufficiently long time scaledoes N h for ω D / ( π J ) = 0 .
24 overtake ω D / ( π J ) = 0 . (cid:104) c † i c j (cid:105) ( t )) in the Heisenberg picture, and thenuse the technique outlined in Ref. 30 to extract the von-Neumann entropy between the left half of the system(between sites 1 and L/
2) and the right half ( L/ L ). To calculate the steady state entanglenment en-tropy, we utilize the steady state correlators calculatedusing the procedure outlined in the main text and thenuse the method of Ref. 30. The steady state entan-glement shows the expected features of a dip when thesystem’s eigenstates change from being fully delocalized6to delocalized and multifractal and then, to delocalizedand localized as the drive frequency is increased from ω D ≈
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