Effects of local periodic driving on transport and generation of bound states
EEffects of local periodic driving on transport and generation of bound states
Adhip Agarwala and Diptiman Sen Department of Physics, Indian Institute of Science, Bengaluru 560012, India Centre for High Energy Physics, Indian Institute of Science, Bengaluru 560012, India (Dated: September 28, 2018)We periodically kick a local region in a one-dimensional lattice and demonstrate, by studyingwave packet dynamics, that the strength and the time period of the kicking can be used as tuningparameters to control the transmission probability across the region. Interestingly, we can tune thetransmission to zero which is otherwise impossible to do in a time-independent system. We adaptthe non-equilibrium Green’s function method to take into account the effects of periodic driving; theresults obtained by this method agree with those found by wave packet dynamics if the time periodis small. We discover that Floquet bound states can exist in certain ranges of parameters; when thedriving frequency is decreased, these states get delocalized and turn into resonances by mixing withthe Floquet bulk states. We extend these results to incorporate the effects of local interactions atthe driven site, and we find some interesting features in the transmission and the bound states.
I. INTRODUCTION
Periodically driven quantum systems have attractedan immense amount of interest for many years. A largevariety of interesting phenomena resulting from periodicdriving have been discovered including the coherent de-struction of tunneling , the generation of defects , dy-namical freezing , dynamical saturation and localiza-tion , dynamical fidelity , edge singularity in the prob-ability distribution of work and thermalization (for areview see Ref. 12). There have also been studies of peri-odic driving of graphene by the application of electromag-netic radiation , Floquet topological phases of mat-ter and the generation of topologically protected statesat the boundaries . Some of these aspects have beenexperimentally studied .In addition, there have been several studies of theeffects of interactions between electrons in periodicallydriven systems . The effects of interactions in Flo-quet topological insulators have been studied in Ref. 57.It is known that interactions can lead to a variety of topo-logical phases (some of which have elementary excitationswith fractional charges) in driven Rashba nanowires ,and to a chaotic and topologically trivial phase in theperiodically driven Kitaev model . The effects of peri-odic driving on the stability of a bosonic fractional Cherninsulator has been investigated . Interestingly some ofthese systems have been realized experimentally demon-strating correlated hopping in the Bose Hubbard model and many-body localization , and realizing bound statesfor two particles in driven photonic systems .Periodic driving can lead to an interesting phenomenoncalled dynamical localization. Here the particles becomeperfectly localized in space due to periodic driving ofsome parameter in the Hamiltonian. Systems exhibitingdynamical localization include driven two-level systems ,classical and quantum kicked rotors , the Kapitzapendulum , and bosons in an optical lattice . It hasbeen shown that remnants of dynamical localization maysurvive even in the presence of strong disorder .In an earlier paper, it was shown that a combination of interactions and periodic δ -function kicks with a partic-ular strength on all the sites on one sublattice of a one-dimensional system can lead to the formation of multi-particle bound states in three different models . Thesebound states are labeled by a momentum which is a goodquantum number since the system is translation invari-ant. This naturally leads us to ask if periodic kicks ap-plied to only one site in a system can also lead to the for-mation of a bound state which is localized near that par-ticular site. Further, it would be interesting to the effectof such a localized periodic kicking on the transmissionacross the site; a similar analysis for localized harmonicdriving has been carried out in Refs. 75,76. One can alsostudy what happens if there is both a time-independenton-site potential (which can produce a bound state andaffect the transmission on its own) and periodic kicking atthe same site. Finally, one can study what the combinedeffect is of an interaction (between, say, a spin-up anda spin-down electron) and periodic kicking at the samesite. We will study all these problems in this paper.In one dimension it is known that periodic driving in alocal region can lead to charge pumping; see Refs. 77,78and references therein. This is a phenomenon in which anet charge moves in each time period between two leadswhich are connected to the left and right sides of theregion which is subjected to the driving. Charge pumpingcan happen even when no voltage bias is applied betweenthe leads; however, this requires a breaking of left-rightsymmetry which can only occur if the periodic driving isapplied to more than one site. In this paper, we will studythe effect of driving at only site; this cannot producecharge pumping.The plan of this paper is as follows. In Sec. II, wewill introduce the basic model. We will consider a tight-binding model with spinless electrons in one dimensionwhere periodic δ -function kicks are applied to the poten-tial at one particular site. The strength and time periodof the kicks will be denoted by α and T respectively. InSec. III, we will discuss wave packet dynamics and howthis can be used to compute the reflection and trans-mission probabilities across the site which is subjected a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p to the periodic kicks. In Sec. IV, we will discuss whythere is perfect reflection from the kicked site for a par-ticular value of α and how this is related to dynamicallocalization. In Sec. V, we will show how an effectiveHamiltonian can be defined and will use this to calculatethe zero temperature differential conductance (which isrelated to the transmission probability) using the non-equilibrium Green’s function method . We will see thatthis matches the result obtained by the wave packet dy-namics if T is less than some value. In Sec. VI, we willdiscuss how the periodic kicking can lead to the forma-tion of a state which is localized near the kicking site.If T is small enough, this is a bound state, while if T is large, this is a resonance in the continuum of bulkstates as we will discuss. In Sec. VII, we will seehow a time-independent potential at one site affects thetransmission and how periodic kicking at that site canlead to an increase in the transmission. In Sec. VIII, wewill extend the model to include spin and will introduce aHubbard like interaction between spin-up and spin-downelectrons at the same site which is subjected to periodickicks. We again study the effects of the interaction onthe transmission of a two-particle wave packet which isin a spin singlet state. We will also study the possibilityof bound states in this system. We will end in Sec. IXwith a summary of our results and some directions forfuture work. II. THE MODEL
We consider a chain of length L on which spinless elec-trons hop between neighboring sites with the Hamilto-nian H T B = − γ L − (cid:88) n =1 ( c † n c n +1 + H.c. ) , (1)where γ is the hopping integral, and c † n and c n are thefermion creation and annihilation operators at site n re-spectively. (We will set γ = 1 in all our numerical cal-culations. We will also set the lattice spacing and (cid:126) to 1 in this paper). The energy-momentum dispersionfor this Hamiltonian is given by E k = − γ cos k , where k lies in the range [ − π, π ]; hence the group velocity is v k = | γ sin k | . We now apply periodic δ -function kicksat a single site labeled as L c lying in the middle of thesystem; the kicks are described by the time-dependentpotential H K = α ∞ (cid:88) m = −∞ δ ( t − mT ) c † L c c L c . (2)Hence the complete Hamiltonian (see Fig. 1) is H = H T B + H K . (3)We are interested in studying the properties of this sys-tem as we tune parameters such as the strength α andthe time period T of the kicking. L c L α − γ FIG. 1: Schematic figure of a one-dimensional lattice where afermion can hop between nearest-neighbor sites with ampli-tude − γ . A periodic δ -function kick is applied at the centralsite L c of a lattice of length L . The kicking strength is α . III. WAVE PACKET DYNAMICS ANDTRANSPORT
We will first investigate the effect of the kicking on thetransport properties. To this end, we first construct aninitial wave packet at time t = 0 given by ψ i ( r ) = 1(2 πσ ) / exp (cid:16) − ( r − L o ) σ + ik c r (cid:17) , (4)which satisfies (cid:82) dr | ψ i ( r ) | = (cid:80) n | ψ i ( n ) | = 1. Here σ denotes the width of the wave packet in real space, k c isthe central value of the wave vector of the wave packet,and L o is the position in real space where the wave packetis initially centered. Since the wave packet is centeredat the momentum k c we know that the effective groupvelocity of the packet will be | k c | . We evolve thesystem for a time ( L − L o ) / | k c | ; this allows thewave packet the time to travel a distance L/ − L o whenit reaches the site where the periodic kicks are appliedand then allows the transmitted part of the wave packetto travel further by an equal distance L/ − L o . At theend of that time, we have a wave function ψ f ; we thendefine the transmission and reflection probabilities T and R as R = L c (cid:88) n =1 | ψ f ( n ) | , T = L (cid:88) n = L c +1 | ψ f ( n ) | . (5)These definitions ensure that R + T = 1. (For spinlesselectrons, the transmission probability T at an energy E is related to the zero temperature differential conduc-tance G = dI/dV as G ( E ) = ( e /h ) T ( E ). In our fig-ures, we will plot G rather than T , since G is a directlymeasurable physical quantity). The numerical results for k c = π/ k c = π/ . In one dimension, it is known that thewidth of a wave packet spreads in time at a rate whichis proportional to ( ∂ E k /∂k ) k = k c = 2 γ cos k c ; this van-ishes at k x = π/ R goes to 1 and G = ( e /h ) T goes to zero as α approaches π . Hence there is perfect reflection at aparticular value of α . This can be seen more clearly bydirectly observing the evolution of a wave packet in thepresence of the periodic kicking. Some representativecases are shown in Fig. 3. Here T and R are calculatedfor a wave packet which is centered at the site L o = 50with width σ = 5 on a lattice with L = 801 sites. Thekicking is done at the L c = 401-th site (denoted by avertical blue line). The kicking time period is taken tobe T = 1, and the central momentum of the wave packetis taken to be k c = π/
2. The wave packet is shown atdifferent intervals of time. From top to bottom, the dif-ferent cases correspond to kicking strengths α = 0 , . α is close to π , one finds that wave packet gets completelyreflects from the central site.It is also interesting to see what happens when α isfixed at a particular value and T is varied. This is shownin Fig. 2. We notice that at small T , the transmission isextremely small, a feature which we find to be generic inmost cases for non-zero α . IV. PERFECT REFLECTION ANDDYNAMICAL LOCALIZATION
A curious feature noted in the last section is that thewave packet completely reflects when the kicking strength α is close to π . This is intimately related to dynamicallocalization. We will make this connection clear in thissection. It has been shown in previous work that a pe-riodic kicks of strength π on one sublattice of a bipartitesystem can lead to the phenomena of dynamical local-ization where a wave packet remains localized in space;this holds even when there is no disorder present in thesystem.In the present context, the time evolution operator fora single time period T can be written as U = exp( − iαc † L c c L c ) exp[ iγT L − (cid:88) n =1 ( c † n c n +1 + H.c. )] . (6)It is particularly instructive to look at U which evolvesthe system for a period 2 T . We rewrite H T B in Eq. (1)as H T B = H r − γ ( c † L c c L c +1 + c † L c c L c − + H.c. ) , (7)where H r denote the rest of the terms. Then U = e − iαc † Lc c Lc exp( − iH T B T ) × e − iαc † Lc c Lc exp( − iH T B T ) . (8)We can evaluate this for α = π by noting that e − iπc † Lc c Lc = e iπc † Lc c Lc (since c † L c c L c can only take the α RG G ( e / h ) T FIG. 2: (Top) Reflection probability R (red circles) and dif-ferent conductance G = ( e /h ) T (blue squares) vs α of a wavepacket which is centered at the site L o = 50 with width σ = 5on a lattice with L = 401 sites. The kicking is done at the L c = 201-th site. The kicking time period is T = 0 .
5, andthe momentum is centered at k c = π/
2. The wave packet isevolved up to a time ( L − L o ) / | k c | . (Bottom) Differen-tial conductance G = ( e /h ) T vs T when α = 1 is kept fixed.Other parameters are the same as in the top panel. values 0 and 1), and using the identities e iπc † Lc c Lc c L c e − iπc † Lc c Lc = − c L c ,e iπc † Lc c Lc c † L c e − iπc † Lc c Lc = − c † L c . (9)We then find U = exp[ − iT { H r + γ ( c † L c c L c +1 + c † L c c L c − + H.c. ) } ] × exp[ − iT { H r − γ ( c † L c c L c +1 + c † L c c L c − + H.c. ) } ] . (10)Using the Baker-Campbell-Hausdorff formula e X e Y = e X + Y + [ X,Y ]+ ··· , (11) x t=0
0 500 x t=100T
0 500 x t=150T
0 500 x t=200T
0 500 x t=250T x t=0
0 500 x t=100T
0 500 x t=150T
0 500 x t=200T
0 500 x t=250T x t=0
0 500 x t=100T
0 500 x t=150T
0 500 x t=200T
0 500 x t=250T FIG. 3: Transmission and reflection of a wave packet whichis centered at the site L o = 50 with width σ = 5 on a latticewith L = 801 sites. The kicking is done at the L c = 401-th site (denoted by a vertical blue line). The kicking timeperiod is T = 1, and the momentum is centered at k c = π/
2. The wave packet is shown at various intervals of time.From top to bottom, the different cases correspond to kickingstrengths α = 0 , . and assuming that γT (cid:28)
1, we can evaluate Eq. (10) tofirst order in T ; we obtain U = exp( − i H r T ) . (12)We now examine the form of H r . We see that H r isthe part of the tight-binding Hamiltonian in which thehoppings to the central site are removed, i.e., H r is ef-fectively described by two disconnected chains. This isthe underlying reason why a wave packet completely re-flects back at α = π . Interestingly, this is also the regimewhich leads to dynamical localization in translationallyinvariant systems where the periodic kicking is appliedto all the sites on one sublattice of a bipartite lattice .We note here that the parameter α appearing in Eq. (6)is really a periodic variable, namely, α and α + 2 π givethe same results since c † L c c L c can only take the values 0and 1. In particular, α equal to any integer multiple of2 π will have no effect on the time evolution.For later purposes, it is convenient to consider the Flo-quet eigenstates ψ j and eigenvalues e − i(cid:15) j T of the uni-tary operator U defined in Eq. (6). The (cid:15) j ’s are called quasienergies; since they are only defined modulo 2 π/T ,we can take them to lie in the range [ − π/T, π/T ]. V. NON-EQUILIBRIUM GREEN’S FUNCTIONMETHOD
The non-equilibrium Green’s function (NEGF) methodis one of the most robust methods for evaluating the con-ductance of a time-independent Hamiltonian . Here, weextend it to a periodically driven system and show thatsuch a formalism appears to work for large driving fre-quencies or small time periods T .The time evolution operator for a single time period T can be written as U = exp( − iαc † L c c L c ) exp[ iγT L − (cid:88) n =1 ( c † n c n +1 + H.c. )] ≡ exp( − iH eff T ) , (13)where H eff can be found exactly by a numerical calcula-tion. We now propose to use H eff as a time-independentHamiltonian and implement the NEGF method. Namely,we use the Hamiltonian H eff , along with the self-energiesΣ ( E c ) and Σ ( E c ) at the left and right ends of the sys-tem (here E c = − γ cos k c is the energy of a particle withmomentum k c ), to compute the zero temperature differ-ential conductance G at the energy E c . (See Ref. 83 fordetails of the procedure).The comparison of the differential conductance G ( E c )obtained using the NEGF method and the exact valueusing wave packet dynamics is shown in Fig. 4. (An ana-lytical expression for G ( E c ) will be presented in Eq. (16)below for the case when T is small). It is clear that theNEGF method using H eff works well for small T , butdeviates significantly as T becomes large. It is naturalto ask what determines the crossover time scale betweenthe two regimes. Another observation from Fig. 4 is that,even when α ∼ π , the wave packet dynamics shows thatthe transmission T is quite far from zero when the timeperiod T is large. Both of these observations can be un-derstood by the following argument. Since a wave packetwith width σ and centered at a momentum k c has a ve-locity | γ sin k c | , it will take a time ∆ t = σ/ | γ sin k c | to cross any particular site on the lattice. If the kickingtime period T is larger than this ∆ t , one expects that thewave packet may not sample the kick and will thereforepass right through the site where the kicking is beingapplied. Therefore the kicking can properly affect thetransmission only when T (cid:46) σ | γ sin k c | . (14)In Fig. 4, we have chosen σ = 5 and k c = π/
2; this gives T (cid:46) . T = 0 . , . .
4, but not if T = 3 . T=0.40 G ( e / h ) α NEGFWavepacket
T=1.20 G ( e / h ) α NEGFWavepacket
T=2.40 G ( e / h ) α NEGFWavepacket
T=3.60 G ( e / h ) α NEGFWavepacket
FIG. 4: Differential conductance G ( E c ) = ( e /h ) T ( E c ) vs α . G is computed from the dynamics of a wave packet which iscentered at the site L o = 50 with width σ = 10 on a lattice with L = 401 sites. The kicking is done at the 201-th site, and thetime period takes the values 0 . , . , . . k c = π/
2, so that E c = 0. The transmissions obtained from the exact wave packet dynamics and the NEGF formalism arecompared. We see that the NEGF formalism matches the exact results for small T . We note that the use of an effective Hamiltonian isonly justified if 2 γT < π ; this can be seen as follows.We recall that the quasienergies (cid:15) j are only defined upto multiples of the driving frequency ω = 2 π/T . Sincethe (cid:15) j ’s are eigenvalues of H eff , this means that H eff is not uniquely defined to begin with. The eigenstatesof H T B in Eq. (1) lie in the range [ − γ, γ ]; hence if 2 γT < π , we can define the quasienergies of all the bulkstates to lie in the range [ − π/T, π/T ]. This will define H eff uniquely. On the other hand, the correspondencebetween the NEGF results and wave packet dynamics areexpected to hold if the condition in Eq. (14) holds; thiscondition depends on both the wave packet width σ andthe momentum k c . VI. FLOQUET BOUND STATES ANDRESONANCES
In the presence of kicking we can study if there areFloquet bound states in the system and explore the prop-erties of such bound states both analytically and usingnumerical techniques. At high frequencies (i.e., small values of T ), the ef-fective Hamiltonian prescription, as briefly discussed inSec. V, becomes more and more accurate. If both α and γT are small, we can use Eq. (11) to show that the effec-tive Hamiltonian is H eff = H T B + αT c † L c c L c (15)to lowest order in α and γT . This is effectively a time-independent system with a potential equal to α/T at thesite L c . It is known that such a potential on a latticegives a transmission probability T ( k c ) = 4 γ sin k c γ sin k c + α T (16)for a particle which is coming in with momentum k c andenergy E c . The form in Eq. (16) explains the shape ofthe first plot in Fig. 4 where T = 0 . α/T at one site also produces a bound state with energy (cid:15) b given by (cid:15) b = ± (cid:114) α T , (17)where the sign of (cid:15) b is the same as the sign of α/T .Numerically, given all the eigenstates of either a time-independent Hamiltonian H or a time evolution operator U , the bound states can be identified quickly by lookingat the values of the inverse participation ratio (IPR) ofall the states. The IPR of a state | ψ (cid:105) = (cid:80) Ln =1 ψ ( n ) | n (cid:105) is defined as (cid:80) Ln =1 | ψ ( n ) | . Typically, states which arespread over the entire system of length L have an IPR ofthe order of 1 /L , while a bound state with a decay length λ which is much smaller than L will have an IPR of theorder of 1 /λ which is much larger than 1 /L . Hence aplot of the IPR versus the eigenstate number will clearlyshow the bound states .The bound state with the energy given in Eq. (17) hasan exponentially decaying wave function of the form ψ ( n ) = N exp( −| n − L c | /λ ) if αT < , = N ( − n exp( −| n − L c | /λ ) if αT > , (18)where the normalization constant N = (cid:112) tanh(1 /λ ), andthe decay length λ is given by λ = (cid:32) arccosh (cid:114) α T (cid:33) − . (19)If λ (cid:29)
1, one can show that the Fourier transform ofthe wave function in Eq. (18) will have a peak at k = 0if α/T < k = ± π if α/T >
0. (The Fouriertransform of a wave function ψ ( n ) is defined as ˜ ψ ( k ) = √ L (cid:80) Ln =1 ψ ( n ) e − ikn ). The IPR of the wave function inEq. (18) is given byIPR = αT α T + 2 (cid:16) α T + 4 (cid:17) / . (20)The highest IPR and its corresponding quasienergycalculated numerically for the eigenstates of the time evo-lution operator U in Eq. (13) and their comparison with ε b α /T NumericalAnalytical I PR α /T NumericalAnalytical
FIG. 5: Bound state energy and IPR for α = 0 . T . The numerically calculated spectrum matcheswell the analytical expression as shown in Eqs. (17) and (20),for L = 401 and L c = 201. the analytical expressions in Eqs. (17) and (20) is shownin Fig. 5. We will see later that the highest IPR corre-sponds to a bound state in certain regions of the “phasediagram” in the α − T plane but to a resonance in thecontinuum in other regions.It is interesting to study the full phase diagram forthis system. This is shown in the left panel of Fig. 6when there is no time-independent on-site potential (i.e., V = 0, where V is defined in Eq. (21)). With increasing α one finds that the IPR increases, while increasing T reduces IPR. Both of these are expected results since theeffective potential due to the kicking is given by α/T .However we find that the bound state appears to vanishabruptly when T increases beyond π/
2. This value of T corresponds to the driving frequency ω = 4 which isalso the band width of the tight-binding model with γ =1. Since the quasienergies of the bulk states (namely,the states which are extended throughout the system)form a continuum going from − γ to 2 γ . Hence, for T < π/
2, the quasienergies do not cover the full range -3-2-1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 α TIPR α TIPR
FIG. 6: The maximum IPR value of the eigenstates of the time evolution operator as a function of T and α for V = 0 (leftpanel) and V = − L = 401 sites and the central site is kicked periodically. For V = 0, the IPRincreases as α increases, while increasing T reduces the IPR. When T crosses π/ [ − π/T, π/T ]; this makes it possible for a bound stateto appear with a quasienergy which does not lie in therange of the bulk quasienergies; hence the bound andbulk states do not mix. However, for T > π/
2, the bulkquasienergies cover the full range; hence any bound statesmust have a quasienergy which lies in the continuum ofthe bulk quasienergies. Such a situation is generally notpossible except in special cases where the bound and bulkstates cannot mix due to some symmetry or topological reasons; see Ref. 80 and references therein. Thus thedisappearance of bound states above a certain value of T is a unique feature of the Floquet system, since in atime-independent system in one dimension, a non-zeropotential will always produce a bound state. Althoughthere are no bound states for T > π/
2, we will now seethat there can be a resonance in the continuum; such astate is a superposition of a state which is localized nearone point and some of the bulk states.In Fig. 7, we show the Floquet eigenvalues (since thetime evolution operator is unitary, its eigenvalues lie on aunit circle in the complex plane), the probabilities | ψ ( n ) | at different sites of a bound state, and the square of themodulus of the Fourier transform of the bound state fora system with 401 sites in which periodic δ -function kicksare applied at the 201-th site with strength α = 0 . T = 1. The bound state is easily identifiedbecause it has the largest IPR equal to 0 . − . − . i which is shown by a large red dot lying just outside the contin-uum of the eigenvalues of the bulk states; this eigenvalueagrees well with exp( − i(cid:15) b T ) = − . − . i , where (cid:15) b = (cid:112) α /T is the bound state energy given inEq. (17). According to Eq. (19), the decay length of thisstate is equal to λ = 5. The IPR equal to 0 . . | ˜ ψ ( k ) | ,of the bound state is found to have peaks at k = ± π .Figure 8 shows the Floquet eigenvalues, the proba-bilities | ψ ( n ) | at different sites of a resonance state,and the square of the modulus of the Fourier transformof the resonance state for a system with 401 sites inwhich δ -function kicks are applied at the 201-th site with α = 0 . T = 2. The resonance state has the largestIPR equal to 0 . − . . i which is shown by a large red dot lyingwithin the continuum of the bulk eigenvalues; this valueagain agrees well with exp( − i(cid:15) b T ) = − . . i ,where (cid:15) b is the bound state energy given in Eq. (17) (thebound state has turned into a resonance here due to mix-ing with the bulk states). According to Eq. (19), thedecay length of this state is equal to λ = 10. The IPR -1-0.5 0 0.5 1-1 -0.5 0 0.5 1 I m ( E ) Re(E) | Ψ ( x ) | x x | Ψ ~ ( k ) | k FIG. 7: (Left) Eigenvalues of time evolution operator for a system with 401 sites in which δ -function kicks are applied atthe 201-th site with α = 0 . T = 1. There is a bound state with IPR equal to 0 . − . − . i shown by a large red dot. (Middle) Probability | ψ ( n ) | of the bound state. (Right) Square of the modulusof the Fourier transform, | ˜ ψ ( k ) | , of the bound state. It has peaks at k = ± π . equal to 0 . . k = ± .
967 (foundfrom the peaks in the Fourier transform). Accordingto Eq. (19), the decay length of this state is equal to λ = 10. The square of the modulus of the Fourier trans-form, | ˜ ψ ( k ) | , of the bound state is found to have peaksat both k = ± π and k = ± . k = ± .
967 as follows: we note that thereare bulk states at these values of k with a Floquet eigen-value equal to exp( i γT cos k ) = − . . i . Thisis close to the Floquet eigenvalue of the resonance statewhich can therefore mix easily with these bulk states.To see how the IPR of a bound or resonance statevaries with the system size, we study the maximum IPRversus L , taking L to be odd, the kicking site to be at themiddle, L c = ( L + 1) /
2, and open boundary conditions.For α = 0 . T = 1, we find that the maximum IPRis equal to 0 . ≤ L ≤ L is much larger than the decay length λ of the central part of the state). This size independence is asignature of a bound state. On the other hand, for α =0 . T = 2, we find that the maximum IPR fluctuatessignificantly for small changes in L but on the averagedecreases as L increases. This is shown in Fig. 9; thefluctuations demonstrate a sensitivity to the system sizeand confirm that it is a resonance rather than a boundstate. We have chosen a fit of the form IP R = a/L b ; wefind that the best fit is given by the exponent b = 0 . α = 0 . , T = 2, and 0 .
83 for α = 1 , T = 2 .
5. This isto be compared with the IPRs of the bulk states whichdecrease as 1 /L . Thus although the peak value of thewave function goes to zero as L increases, the ratio ofthe peak value to the value of the wave function far fromthe peak grows with L . The value of the exponent b isnot universal; we find that it depends on the values of α and T . However, it is smaller than 1 over a wide range ofparameters and reasonably large system sizes, implyingthat although the IPR of the resonance state decreases,the IPRs of the bulk states decrease even faster as L increases.To summarize, we find that a bound state differs froma resonance in several ways.(i) The Floquet eigenvalue of a bound state lies out-side the continuum of the Floquet eigenvalues of the bulkstates, while the Floquet eigenvalue of a resonance lieswithin the continuum of the bulk eigenvalues.(ii) The wave function of a bound state is peaked atsome point, decays rapidly away from that point, and es-sentially becomes zero beyond some distance. The wavefunction of a resonance is also peaked at some point anddecays away from that point, but it does not becomecompletely zero no matter how far we go; this is becauseit contains a non-zero superposition of some plane wavesand therefore remains non-zero even far away from thepeak. (iii) If the system size is large enough, the propertiesof a bound state, such as its IPR and the peak value ofits wave function, become independent of the system size L and the boundary conditions (for instance, whetherwe have periodic, anti-periodic or open boundary condi-tions). For a resonance, however, the IPR and peak valueof the wave function depend sensitively on the boundaryconditions and the value of L , and on the average theykeep decreasing as L is increased. This is because such astate contains some plane waves which sample the entiresystem, and the quasienergies of the plane waves is sensi-tively dependent on the boundary conditions and L . (Werecall that if periodic boundary conditions are imposed,the momentum of the plane wave states is quantized inunits of 2 π/L . Hence the values of the momentum and -1-0.5 0 0.5 1-1 -0.5 0 0.5 1 I m ( E ) Re(E) | Ψ ( x ) | x x | Ψ ~ ( k ) | k FIG. 8: (Left) Eigenvalues of time evolution operator for a system with 401 sites in which δ -function kicks are applied to the201-th site with strength α = 0 . T = 2. (There are more eigenvalues on the left side than on the rightside; hence the plot looks solid on the left and dotted on the right). There is a resonance state with Floquet eigenvalue equalto − . . i shown by a large red dot. (Middle) Probability | ψ ( n ) | of the resonance state. (Right) Square of themodulus of the Fourier transform, | ˜ ψ ( k ) | , of the resonance state. It has peaks at both k = ± π and k = ± . I PR L III
FIG. 9: The maximum IPR value of the eigenstates of thetime evolution operator as a function of the system size L for (I) α = 0 . T = 2 and (II) α = 1 . T = 2 . α has much largerfluctuations than curve II. The dotted lines show a fit of theform IP R = a/L b for the average IPR; for (I) the values a = 0 .
31 and b = 0 .
41 give the best fit; for (II) a = 2 .
65 and b = 0 . therefore the quasienergies − γ cos k depend on L ).Next, we introduce a time-independent on-site poten-tial at the same site where the periodic kicking is beingapplied; this potential is given by H V = V c † L c c L c . (21)To investigate the effects of kicking, we again plot themaximum value of the IPR of all the eigenstates of thetime evolution operator as a function of T and α . Thisis shown in the right panel of Fig. 6 for V = −
4. We seea number of features in this plot including some straightlines; we now provide a qualitative understanding of thesefeatures. The bold lines within which the IPR is close to1 is basically determined by whether the bound state mixes with the continuum states or not. Since the bulkquasienergies lie between − γ and 2 γ , the bound statedoes not mix with the continuum states and thereforeexists in the regions (cid:15) b + αT > γ, (22) (cid:15) b + αT < − γ, (23)where (cid:15) b is the bound state energy (cid:15) b = − (cid:112) V + 4 γ (24)produced by an on-site potential V <
0. Similar to thecondition in Eq. (22) and using the fact that quasiener-gies are only defined modulo 2 π/T , we see that anotherline for the existence of a bound state is given by (cid:15) b + αT > − πT + 2 γ. (25)Between the two lines given in Eqs. (22) and (23) andbelow the line given in Eq. (25), the bound state mixeswith the bulk quasienergies and therefore turns into aresonance in the continuum. VII. INCREASE IN CONDUCTANCE DUE TOPERIODIC KICKING
In the presence of only a time-independent on-site po-tential V , we have a bound state and a transmission prob-ability T which is less than 1. We now ask if the trans-mission can be increased by periodic kicking at the samesite where the potential V is present. In Sec. II we sawthat the transmission can get reduced when we introducekicking. We now look at the opposite case where peri-odic driving can increase the transmission. In Fig. 10.the differential conductance is shown as a function of α for a system with T = 0 . V = − k c = π/
2. The0 G ( e / h ) α NEGFWavepacket
FIG. 10: Differential conductance G = ( e /h ) T vs α for asystem with V = −
4. We see that α can tune the conductanceall the way from zero to 1. Here L = 401, k c = π/ L o = 50, σ = 5, and T = 0 . maximum transmission should occur at α = − V T whichis equal to 2 for the parameters used in Fig. 10. Wesee in that figure that this is indeed true and the systembecomes “transparent” when α = − V T . VIII. EFFECTS OF INTERACTIONS
We now analyze the effects of interactions on the vari-ous aspects that we have discussed so far, namely, trans-port and the presence of bound states. We consider asystem containing two species of electrons with up anddown spins and a time-independent interacting term onthe site L c . The total Hamiltonian is H = − γ L − (cid:88) n =1 (cid:88) σ = ↑ , ↓ ( c † nσ c n +1 ,σ + H.c. )+ α ∞ (cid:88) m = −∞ δ ( t − mT ) c † L c σ c L c σ + U ˆ n L c ↑ ˆ n L c ↓ , (26)where ˆ n L c σ is the number operator for electrons withspin σ at site L c . In order to investigate the effect of theinteraction term, we begin with an initial wave packetwhich has two-particles in a singlet state of ψ ↑ ( r ) and ψ ↓ ( r ); the form of the wave packet is given in Eq. (4).We then study the effects of the interaction using exactdiagonalization and wave packet evolution. The effect of U is shown in Fig. 11 where the reflection probability foran electron with spin-up, R ↑ , is shown as a function of U . [Given an amplitude ψ ( n , n ) for a spin-up electronto be at n and a spin-down electron to be at n , wedefine the reflection and transmission probabilities for aspin-up electron to be R ↑ = (cid:80) L c n =1 (cid:80) Ln =1 | ψ ( n , n ) | and T ↑ = (cid:80) Ln = L c +1 (cid:80) Ln =1 | ψ ( n , n ) | , analogous to R U FIG. 11: Reflection probability R ↑ of a spin-up electron fora wave packet in a system with an on-site interaction U andkicking strengths α = 0 (lower curve) and 0 . α = 0 the wave packet is not completely trans-mitted in the presence of a finite U . This happens becausethere is a finite probability for the incoming wave packet toget trapped in a bound state which lives near the interactingsite. The effect of U acts asymmetrically in the presence ofa finite α . The effect of U reduces with increasing α and atlarge values of α , R ↑ is independent of U . We have taken L = 51, T = 0 . k c = π/ σ = 4, and L o = 6. Eq. (5). These satisfy R ↑ + T ↑ = 1. We can similarlydefine R ↓ and T ↓ ; our choice of the form of the wavepacket implies that R ↑ = R ↓ and T ↑ = T ↓ ]. Increasing α makes U less effective; this is because in the presenceof kicking, the wave packet is small at the site L c and istherefore unable to sample the interaction. Note that inthe presence of α , the effects of U and − U are different.This is because the driving produces an effective on-sitepotential equal to α/T ; hence the total quasienergy ofa state with two electrons at site L c is U + 2 α/T . Theminimum of this energy (and hence the minimum of thereflection probability) occurs at a non-zero value of U .As in the case of a non-interacting system, we find thatFloquet bound states can also appear in the presence ofinteractions. They have an interesting dependence onthe time period T . If U (cid:29) γ , there will be a boundstate in which both electrons are at the site L c , and thequasienergy of this state is U + 2 α/T in the presence ofkicking. Since the energy of the bulk states of the two-electron system goes − γ to 4 γ , the bound state will notmix with the bulk states if U + 2 αT > γ, (27) U + 2 αT < − γ. (28)We can calculate the IPR of a two-particle state andstudy its variation with U ; this is shown in Fig. 12. Inthe absence of kicking ( α = 0), we find that for | U | (cid:46) I PR U
0 20 40 0 20 40 0 0.01 | Ψ ( x , x ) | x x | Ψ ( x , x ) |
0 0.0005 0.001
FIG. 12: (Top) The value of the two-body IPR for the statewith the maximum IPR as a function of the Hubbard inter-action U , with and without periodic driving. We have taken L = 51, T = 0 . α = 0 . α = 0 (redsquares). (Bottom) A two-particle bound state wave functionfor L = 51, T = 0 . α = 0 .
4, and U = 1 where a much morelocalized bound state is produced by the driving. strongly localized, while for | U | (cid:38)
2, the IPR is largewhich suggests a strongly localized state. Interestingly,we find that a finite kicking strength (such as α = 0 . U and T . A strongly localized two-particle bound state wavefunction is shown in Fig. 12. IX. CONCLUSIONS
In this paper we have studied the effects of periodicdriving at one site in some tight-binding lattice modelsin one dimension. We have taken the driving to be ofthe form of periodic δ -function kicks with strength α andtime period T . We have studied how the kicking affectstransmission across that site and whether it produces anybound states.The transmission (which is related to the differential conductance) has been calculated by constructing an in-coming wave packet which is centered around a particularmomentum and time evolving it numerically. The reflec-tion and transmission probabilities are found by comput-ing the total probabilities on the left and right sides ofthe kicking site after a sufficiently long time. The boundstate is found by computing all the eigenstates of the timeevolution operator U for one time period, and finding theeigenstate with the maximum value of the inverse partic-ipation ratio; we look at the corresponding wave functionto confirm that it is indeed peaked near the kicking site.When T is much smaller than the inverse of the hop-ping γ and | α | (cid:28)
1, the numerically obtained val-ues of both the transmission probability and the Flo-quet quasienergy of the bound state are consistent withthe fact that the kicking effectively acts like a time-independent potential equal to α/T at the special site.This is confirmed by calculating the effective Hamilto-nian and then using the non-equilibrium Green’s func-tion method to compute the transmission; this is found toagree well with the transmission found from wave packetdynamics if T is small. When T is small and α = π ,we find that the transmission probability is zero; this isbecause the effective hoppings between the kicked siteand its two neighboring sites become zero, and it is re-lated to the phenomenon of dynamical localization. Onthe other hand, when T becomes comparable to 1 /γ , theagreement between the transmissions found using wavepacket dynamics and the effective Hamiltonian breaksdown, showing that the effective Hamiltonian no longerprovides an accurate description of the system.We note that the effective breaking of a “bond” inthe Floquet description is a unique result and cannot befound in a static case. This feature is true even in higherdimensions and therefore provides a unique opportunityto experimentally simulate bond percolation problems in cold atom or photonic systems. In such systems local δ -function kicking can be implemented at randomly cho-sen sites in the system and their effect on the localizationphysics can be investigated.A bound state can appear only if its quasienergy doesnot lie the continuum of the quasienergies of the bulkstates going from − γ to 2 γ modulo 2 π/T . If the Floquetquasienergy of the would-be bound state lies within thecontinuum of the quasienergies of the bulk states (thisnecessarily happens if T > π/ (2 γ ) but it can also happenfor certain values of α if T < π/ (2 γ )), the bound stateceases to exist. However, we then find that in certainranges of values of α and T , there is a state which canbe described as a resonance in the continuum. The wavefunction of such a state consists of a superposition of astrongly peaked part which resembles a bound state anda plane wave part which does not decay even far awayfrom the kicking site. Further, the IPR of this state issensitively dependent on the system size and boundaryconditions and it gradually decays as the system size isincreased. This behavior is in contrast to a bound statewhose IPR becomes independent of the system size and2boundary conditions when the system size is larger thanthe decay length.Next, we have studied what happens if there is a time-independent potential V at a single site and periodic δ -function kicks are applied to the same site. Separatelyboth V and the kicks reduce the transmission from unityand can produce bound states. When both of them arepresent, we get a complex pattern of regions in the α − T plane where bound states are present. These regionscan be understood using a simple condition that the sumof the effective on-site potential α/T due to the kickingand the energy of the bound state produced by V aloneshould not lie within the continuum of the quasienergiesof the bulk states. Further, if V and α/T have oppositesigns, their effects can partially cancel each other and thetransmission probability can be higher than if only oneof them was present.Finally, we have studied a model with spin-1/2 elec-trons where there is a Hubbard interaction of strength U at a single site and periodic kicks are applied to thesame site. We numerically study wave packet dynamicsstarting with an initial wave packet which contains twoelectrons in a spin singlet state. In the absence of kick-ing, a state in which both particles are at the special sitehas an energy U . This has a similar effect as an on-sitepotential for the model of spinless electrons; the trans-mission probability is therefore reduced from 1 for anynon-zero value of U , and it has the same value for U and − U . When we introduce kicking, the effective potentialfor two particles at the special site is given by the sum of U and 2 α/T . Hence the transmission probability will behigher when U and 2 α/T have opposite signs, and willtherefore not be symmetric under U → − U . We also findthat a bound state can appear if its quasienergy does notlie within the continuum of bulk quasienergies. When U is non-zero, we find that kicking can convert stronglylocalized states to weakly localized ones and vice versa.We end by pointing out some directions for future stud-ies.(i) In this paper we have only examined systems withone or two particles. It may be interesting to study athermodynamic system with a finite filling fraction ofparticles. One can then investigate if, for example, themodel of spin-1/2 electrons with both an interaction U and periodic kicking at the same site can show a Kondo-like resonance . Related problems have been studiedin Refs. 87,88.(ii) It may be interesting to look at the effects of heat-ing. It is known that a system generally heats up to in-finite temperature when there are interactions and peri-odic driving at all the sites . However, if interactionsand periodic driving are both present in only a small re-gion as considered in this paper, it is not known if thesystem will heat up indefinitely at long times.(iii) The effects of periodic kicking at more than onesite, possibly with different strengths and phases, wouldbe interesting to study. It is known that harmonic drivingat two sites with a phase difference can pump charge(see Ref. 78 for references). We therefore expect thatthe application of δ -function kicks at two sites may alsopump charge. In addition, we can study what kinds ofbound states are generated in such a system.There has been an increasing interest in understand-ing the dynamics of a single impurity or of electrons ina quantum dot under the periodic modulation of someparameter. This is motivated both by theoretical con-siderations such as the effect of such a modulation onthe Kondo effect and by advances in cold atom ex-periments which allow for the imaging and modulationof systems up to single site resolution . The resultspresented in this manuscript describe many interestingphenomena which are realizable due to an interplay ofimpurity physics and dynamical modulation of some pa-rameter in the Hamiltonian. It will be interesting if sucheffects can indeed be observed in cold atom or mesoscopicsystems. Acknowledgments
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