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Floquet Engineering of Lie Algebraic Quantum Systems
Jayendra N. Bandyopadhyay1, 2, ∗ and Juzar Thingna1, 3, †
1Center for Theoretical Physics of Complex Systems,Institute for Basic Science (IBS), Daejeon 34126, Republic of Korea
2Department of Physics, Birla Institute of Technology and Science, Pilani 333031, India3Basic Science Program, University of Science and Technology (UST), Daejeon 34113, Republic of Korea
We propose a ‘Floquet engineering’ formalism to systematically design a periodic driving protocolin order to stroboscopically realize the desired system starting from a given static Hamiltonian. Theformalism is applicable to quantum systems which have an underlying closed Lie-algebraic structure,for example, solid-state systems with noninteracting particles moving on a lattice or its variantdescribed by the ultra-cold atoms moving on an optical lattice. Unlike previous attempts at Floquetengineering, our method produces the desired Floquet Hamiltonian at any driving frequency and isnot restricted to the fast or slow driving regimes. The approach is based on Wei-Norman ansatz,which was originally proposed to construct a time-evolution operator for any arbitrary driving. Here,we apply this ansatz to the micro-motion dynamics, defined within one period of the driving, andobtain the driving protocol by fixing the gauge of the micro-motion. To illustrate our idea, we usea two-band system or the systems consisting of two sub-lattices as a testbed. Particularly, we focuson engineering the cross-stitched lattice model that has been a paradigmatic flat-band model.
Introduction.– Floquet formalism [1] has been instrumen-tal to study the dynamic evolution of a system sub-jected to a periodic driving. The dynamics is decomposedinto two parts a time-periodic part describing the micro-motion of the system within a period, and an effectivestroboscopic part governed by a static ‘Floquet Hamil-tonian’. The problem of reverse engineering the drivingprotocol in order to obtain a desired Floquet Hamilto-nian is known as Floquet engineering. It has garnereda lot of attention over the past several years and hasbeen applied in different experimental paradigms [2–6].Floquet engineered solid-state materials have been dis-cussed extensively to develop “quantum matter on de-mand” by controlling post-semiconductor materials [7–9]and several exotic properties like unconventional super-conductivity [10, 11], topologically nontrivial band struc-tures [12], etc. have been realized. Moreover, the ef-fect of periodic driving has been studied on a variety oftimely solid-state systems such as Luttinger liquid [13],superconducting circuit [14], bilayer graphene [15], andstrongly correlated electrons (Mott materials) [16].
Most of these studies investigated the effect of periodicdriving on a given system in some particular driving fre-quency regime, namely, high and low frequency. In thehigh-frequency limit, the effect of driving is consideredperturbatively by creating a series in inverse frequency[17–21], whereas a perturbative series in the frequency isused to treat the low frequency regime [22, 23]. However,a systematic theory of designing a driving protocol suchthat the desired Floquet Hamiltonian is obtained exactlyat any driving frequency is still missing in the literature.
In this Letter, we propose to bridge this gap by for-mulating a theory of Floquet engineering for a class ofsystems whose Hamiltonians have underlying closed Lie-algebraic structure. Our formalism is based on the Wei-Norman ansatz, which was originally proposed to obtain
the dynamics for any time-dependent system [24, 25].Since the form of the long-time evolution part is alreadyknown from the Floquet theory, we massage the Wei-Norman ansatz to the micro-motion part of the dynam-ics. We focus on systems having an underlying SU(2)algebra that covers a wide range of noninteracting solidstate systems on a lattice with multiple sub-lattices orultra-cold atoms hopping on an optical lattice. We an-alytically design the driving protocol for two-band sys-tems based only on the Wei-Norman ansatz and the de-sired Floquet Hamiltonian. Unfortunately, this does notfix the gauge of the micro-motion and hence we pro-vide physically motivated guiding principles to fix thegauge. Finally, we apply our formulation to design adriving protocol to realize cross-stitched lattice, given anautonomous site Hamiltonian, an interesting two-bandsystem with one band dispersive and the other one aflat [26, 27].Formalism.– The Hamiltonian of a generic periodicallydriven quantum system reads
H(t) = H0 + V (t), V (t+ T ) = V (t), (1)
where H0 and V (t) are the static Hamiltonian andthe driving potential with time-period T , respectively.The corresponding time-dependent Schrodinger equation(TDSE) is
idU(t)
dt= H(t)U(t), (~ = 1). (2)
The operator U(t) is the unitary time-evolution operator.According to the Floquet theorem, the solution of theTDSE can always be expressed as
U(t) = P (t) e−iHeff t, (3)
where the micro-motion operator P (t + T ) = P (t) de-scribes the dynamics of the system within one period
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[t, t + T ] and Heff is a static Hamiltonian that gov-erns the long-time dynamics of the system. The initialcondition U(0) = 1 imposes the condition P (0) = 1,whereas the time-periodicity gives P (nT ) = 1 for ev-ery n ∈ Z
+ (positive integers). Consequently, we have
U(nT ) = e−iHeffnT = [e−iH
effT ]n = [U(T )]n. If thedynamics of the system is observed stroboscopically att = nT , then it is governed by the effective static Hamil-tonian Heff .
Obtaining the analytic quantum evolution for anyHamiltonian is highly nontrivial and hence we restrictourselves to those Hamiltonians which can be written asa linear combination of operators which form a closedLie-algebra, i.e.,
H0 = h01+
N∑α=1
hαAα = h01+ h ·A, (4)
V (t) = f0(t)1+
N∑β=1
fβ(t)Aβ = f0(t)1+ f(t) ·A, (5)
where · denotes the standard scalar product. Above,h0 and h (column vector with elements hα and dimen-sion N) are time-independent parameters, whereas f0(t)and f(t) are time-dependent functions due to the ex-ternal field. The column vector of the linear operatorsA = {Aα} forms a finite N -dimensional simple Lie-algebra LN , which satisfies
[Aα, Aβ ] =
N∑γ=1
ΛγαβAγ , (6)
where Λ’s are the structure constants of the algebra LN .From the Floquet engineering perspective, the underly-ing Lie-algebraic structure will be exploited to design adriving scheme {f0(t),f(t)} to achieve a desired effectiveHamiltonian Heff for a given initial static HamiltonianH0.
The Wei-Norman ansatz [24, 25], i.e., expressing thefull evolution operator as a product of exponentials, hasbeen successfully applied to solve the TDSE for a drivenquantum system. In our case, since we are particularlyinterested in Floquet engineering wherein Heff is known,the natural choice is to apply the ansatz to the micro-motion operator,
P (t) = e−im0(t)
(N∏α=1
e−imα(t)Aα
). (7)
The initial condition and the time-periodic property ofP (t) imposes following conditions: m0(nT ) = 2νnπ ande−imα(nT )Aα = 1 for all α = 1, · · · , N , n = 0, 1, 2, · · · ,and any integer ν. Besides, we have a gauge free-dom to choose any time-dependent functional form ofmα(t). Using the above form of P (t), if we substitute
U(t) in the TDSE [Eq. (21)], we get the relations be-tween the driving protocols {f0(t),f(t)} and the func-tions {m0(t),m(t)} as
[h0 + f0(t)] + [h+ f(t)] ·A= m0(t) + ζ(m, m) ·A+ P (t)Heff P †(t).
(8)
Here, the components of the column vector ζ are lin-ear functions of m(t) = {dmα(t)/dt} and nonlinearfunctions of m(t). Therefore, we can always expressζ(m, m) =M1(t) · m, whereM1(t) is a N ×N matrixwhose elements are nonlinear functions of m. This non-linearity is decided by the underlying Lie-algebra. Con-sider the general form Heff = heff
0 1+heff ·A, the last termon the right hand side of Eq. (8) can also be representedin terms of the operators A as
P (t)Heff P †(t) = heff0 1+ ξ(m,heff) ·A. (9)
The vector ξ(m,heff) is a linear function of heff , buta nonlinear function of m(t). Therefore, we also writeξ(m,heff) =M2(t) ·heff . From Eqs. (8) and (9), equat-ing the coefficients of the identity operator 1 and thelinear operators A = {Aα}, we get
h0 + f0(t) = m0(t) + heff0 , (10)
h+ f(t) =M1(t) · m(t) +M2(t) · heff ,
where M1(nT ) = M2(nT ) = 1 for n = 0, 1, 2, · · · .The gauge freedom in the micro-motion operator makesM1(t) and M2(t) non-unique, but it can be fixed atany arbitrary time t 6= nT by choosing an appropriategauge. According to Wei-Norman, if LN is not a solv-able algebra, the transformation matrices M1(t) andM2(t) could be ill-defined for an arbitrary representa-tion. Therefore, unless we find a representation which isglobally well-defined, we cannot apply the Wei-Normanansatz to design the driving protocol.
Our Lie-algebraic Floquet engineering protocol can beapplied to any system having an underlying finite dimen-sional closed algebra. We now apply this formalism toan arbitrary 2-bands systems. In principle, this formal-ism can also be applied to multi-bands systems, but thecomplexity of the problem increases with the number ofbands. In the supplementary material [28], we shed lighton the three-band case.Two-bands systems.– In the momentum space (k-space),the Hamiltonian of the periodically driven 2-band sys-tems can be written in terms of the Nambu’s spinorsΨk = (ak, bk)
Tas
H(t) =∑k
Ψ†kHk(t)Ψk, (11)
where Hk(t) = Hk0 + Vk(t) and Vk(t+ T ) = Vk(t). The
components of the Nambu spinors ak(a†k) and bk(b†k) arerespectively representing the annihilation (creation) op-erators corresponding to the valence band and the con-duction band. The Hamiltonian kernel Hk(t) for each
3
mode k has a time-independent part Hk0 and a time-periodic part Vk(t) with periodicity T , which can be ex-pressed as,
Hk0 = hk01+ hk · SVk(t) = fk0(t)1+ fk(t) · S. (12)
The operators 2S = σ follow SU(2) algebra, where thecomponents of σ are the Pauli matrices. This finite di-mensional algebra facilitates the application of the Wei-Norman formalism to study the dynamics of two-bandsystems.
The non-solvable SU(2) algebra has two well-knownrepresentations: XY Z representation with SXY Z =(Sx, Sy, Sz)
T and ±Z representation with S±Z =(S+, S−, Sz)
T, where S± = (Sx ± iSy). For an arbi-trary choice of A, e.g., A = SXY Z , it is not guaranteedthat the time-dependent functions mα(t) appearing inthe micro-motion operator, Eq. (7), are smooth contin-uous functions for all time t [24, 25]. However, followingRef. [[29]], we later show that the (±Z) representationgives a globally well-defined M1(t) matrix. Therefore,for the two-bands case, Wei-Norman ansatz along witha proper choice of representation S ≡ S±Z can be ap-plied to design a driving scheme, with arbitrary drivingfrequency, to achieve the desired effective Hamiltonianfrom a static Hamiltonian.
Floquet engineering protocol.– We now provide the basicsteps to Floquet engineer a two-band system, where thedesired stroboscopic Hamiltonian is Heff
k = heffk01+heff
k ·S.The protocol is divided into three essential steps:
1. Wei-Norman Ansatz : Use Uk(t) = Pk(t)e−iHeffk t via
Floquet theorem, and apply the Wei-Norman ansatz toconstruct the micro-motion operator
Pk(t) = e−imk0e−imk+S+e−imk−S−e−imkzSz . (13)
The function mk0 is real, but the other functions(mk±,mkz) are complex. The explicit time-dependenceof the functions m has been suppressed for notationalsimplicity. Also note that, the last three terms of theabove expression are not individually unitary, but theirproduct is unitary which imposes
Im[mkz] = ln(1 + |mk+|2
)mk− =
m∗k+
1 + |mk+|2. (14)
The above condition reduces the seven independent pa-rameters (real mk0 and real and imaginary parts ofmk±,z) to four. We choose
(mk0,mk+,m
∗k+,m
Rkz
), where
mRkz = Re[mkz], as the independent variables and the
micro-motion operator reads
Pk(t) =e−imk0√
1 + |mk+|2
(e−
i2m
Rkz −imk+e
i2m
Rkz
−im∗k+e− i
2mRkz e
i2m
Rkz
).
2. Transformation matrices: Consider heffk =
{heffk−, h
effk+, h
effkz} in (±Z) representation. Substituting
Uk(t) in the TDSE and using Eq. (10), we obtain M1
andM2 for a given k as
Mk1 =1
1 + |mk+|2
1 0 imk+
0 1 −im∗k+
im∗k+ −imk+ 1− |mk+|2
(15)
Mk2 =1
1 + |mk+|2
e−imRkz −iqkmk+ imk+
iq∗km∗k+ eim
Rkz −im∗k+
−2q∗k −2qk 1− |mk+|2
,
with qk = imk+eimR
kz . The above form ensures that thesematrices are identity at t = nT for n = 0, 1, 2, · · · .3. Driving protocol : For the Floquet engineering pro-tocol, a bare minimal Hamiltonian is considered as theinitial static Hamiltonian H0. For example, here we seth = 0, i.e. H0 = h01. Therefore, using the previous twosteps, the driving functions are
h0 + fk0(t) = mk0 + heffk0,
fk(t) =Mk1 · mk +Mk2 · heffk .
(16)
The transformation matrices are well-defined at all timest (implicit dependence in mk) and not just stroboscop-ically [see from Eq. (15)]. Moreover, the globally well-defined Mk ∀k ensures the driving protocol is well-defined for all times. It is worth emphasizing our Flo-quet engineering protocol is exact in the driving fre-quency ω and does not require frequency-based per-turbative expansions that lead to non-convergent series[17, 18, 20, 21].Guiding principle to fix the gauge of the micro-motionoperator.– The gauge for the micro-motion is fixed bychoosing {mk0,mk} satisfying the boundary conditionsat t = nT : e−imk01 = e−imk+S± = e−imkzSz = 1,∀n.This can be achieved in various ways and here we il-lustrate a physically motivated gauge choice. The ap-proach outlined below can be used as a guiding prin-ciple to fix gauge for any other systems. First, anatural choice to consider that each mk(t) has a sep-arable form as a product of momentum and time-dependent functions, such that {mk0(t),mk(t)} ={φk0µ0(t), φk+µ+(t), φ∗k+µ
∗+(t), φR
kzµRz (t)}. Further-
more, we set φRkz = 1 and φk+ = eik. Physically, these
assumptions respectively suggests that during the micro-motion intra sub-lattice hopping is suppressed and onlyinter sub-lattices hopping is allowed. Consequently, Eq.(16) becomes
fk0(t) = φk0µ0(t) + heffk0
fk(t) = Mk1 · µk(t) + Mk2 · heffk , (17)
where
Mk1 =1
1 + |µ+(t)|2
eik 0 iµ+(t)eik
0 e−ik iµ∗+(t)e−ik
iµ∗+(t) −iµ+(t) 1− |µ+(t)|2
,
4
and Mk2 → Mk2, defined in Eq. (15), with mRkz →
µRz (t) and mk+ → µ+(t)eik. We now set µ0(t) =a0 sinωt, µ+(t) = a+e
iθ sinωt, and µRz (t) = pωt where
p is any integer. This choice respects the boundaryconditions and ensures the frequency of all the time-dependent functions equals ω = 2π/T . The real am-plitudes {a0, a+}, the phase factor eiθ, and the integerp are arbitrary that depend on the physical system asshown below with a specific example.Application.– We now apply our Floquet engineeringprotocol to realize a desired Hamiltonian Heff
k of two-bands system whose one band is flat or dispersionlessand the other band is dispersive. This property is ob-served in a cross-stitch lattice structure and later, ob-served in some other variants of this model [26, 27]. Inthe momentum-space, the Hamiltonian of this system isHeffk = heff
k01+heffk (S+ +S−), where heff
k0 = −2α cos k andheffk = −(2α cos k + ∆). The energy of the flat band is
∆ and the dispersive band is given as −4α cos k−∆ [seeFig. 1(a)].
This effective Hamiltonian is obtained via a suitabledriving protocol designed for the initial bare static Hamil-tonian Hk0 = hk01, where hk0 = −2α cos k. This Hamil-tonian describes two uncoupled sub-lattices, where eachsub-lattice is a 1D chain with zero onsite energy. Theparameter α determines the nearest neighbor hoppingstrength in each of the sub-lattices. The choice of thestatic Hamiltonian reduces the complexity of the expres-sions and can be easily realized experimentally. Conse-quently, we obtain heff
k0 = 0 implying fk0(t) = φk0µ0(t)using Eq. (17). As mentioned earlier, the gauge can befully set with a physical model and hence in this case wehave a freedom to set φ0(k) = 0 and θ = 0. Thus, wehave fk0(t) = 0∀t and the function µ+(t) becomes real.
Using Eq. (17), and from the relations fkx(t) =2 Re[fk−(t)] and fky(t) = −2 Im[fk−(t)], we obtain thedriving functions in the XY Z representation as
fkx(t) = fe(t)[a+ωCωtCk + heff
k Cpωt − a+pω SωtSk
+a2+h
effk C2k+pωtS
2ωt
],
fky(t) = −fe(t)[a+ωCωtSk − heff
k Spωt + a+pω SωtCk
+a2+h
effk S2k+pωtS
2ωt
],
fkz(t) = fe(t)
[pω
(1− a2
+
2
)+pωa2
+
2C2ωt
+4a+heffk Sk+pωtSωt
], (18)
where fe(t) =(1 + a2
+ S2ωt
)−1, Cw = cosw and Sw =
sinw. In the above expression, we have two free param-eters: a real parameter a+ and an integer p. We shallset these two parameters in such a way that each of thedriving function should not have any static (DC) part.First, we set a2
+ = 2. This removes the first term of thedriving function fkz(t). Next we set p = 3, this is theminimal integer that ensures absence of any static term
−π −π
20 π
2
π−6
−4
−2
0
2
(a)
k
ε k
0 5 10 15 20
10−5
10−4
10−3
10−2
10−1
100101
(b)
n
c 2n
ω = 4 + 2∆ω = 2∆
FIG. 1. (a) Band diagram of the cross-stitch lattice for α =1.0 and ∆ = 2.0. (b) Fourier coefficients of the envelopefunction for a2+ = 2.0.
in either driving protocol. For p = 1, fkz develops astatic contribution; whereas for p = 2, fkx and fky havestatic parts.
This time-periodic envelope function is an even func-tion. Therefore, its Fourier expansion will be a cosineseries with one constant coefficient c0. We consider twomoderate (same order of the band gap) cases of the driv-ing frequency: ω = 4 + 2∆ = 8 and ω = 2∆ = 4. Forthese two frequencies, the Fourier coefficients are shownin Fig. 1(b) as a function of the coefficient indices. Forboth frequencies, the odd coefficients c2n+1 are zero andthe even coefficients c2n fall exponentially with n. There-fore, the envelope function can be realized with high ac-curacy considering only a few even harmonics. In Fig. 2,
(a)
−π
−π
2
0
π
2
π
−1
0
1(d)
−1
0
1
(b)
−π
−π
2
0
π
2
π
k
−2
−1
0
1
2 (e)
−1
0
1
(c)
0 0.2 0.4 0.6 0.8 1−π
−π
2
0
π
2
π
t (T )
−5
0
5(f)
0 0.2 0.4 0.6 0.8 1t (T )
−5
0
5
FIG. 2. Density plot of the driving functions fkx(t) (a,d),fky(t) (b,e), fkz(t) (c,f) are plotted as a function of momen-tum k and time t for ω = 4 + 2∆ = 8 (a-c) and ω = 2∆ = 4(d-f).
5
the density plot of all the driving functions are plottedon the plane of time t and momentum k that stroboscop-ically give us Heff with band diagram given in Fig. 1(a).Figures 2(a)-(c) show the driving functions fkx, fky andfkz, respectively for ω = 4+2∆ = 8, whereas Figs. 2(d)-(f) show the same for ω = 2∆ = 4.Conclusion.– We have introduced a Floquet engineeringprotocol applicable to systems whose Hamiltonians havean underlying Lie-algebraic structure. A large number ofphysically relevant systems, like noninteracting particlesmoving on a lattice that show multiple energy bands,are prominent examples of this class of systems. In thisformalism, we have applied the Wei-Norman ansatz [25]to the micro-motion part of the Floquet dynamics, andfrom that, we have prescribed how to design a drivingprotocol to reach the desired system starting from a givenstatic Hamiltonian. The underlying Lie-algebra of two-bands systems is SU(2), which is not solvable. Therefore,to ensure that the driving protocol is well defined we usedthe representation studied in Ref. [29]. Importantly, ourformulation does not rely on any perturbative expansionsand is exact.
We have particularly applied the formalism to studysystems having two energy bands, which can be easilyadapted for any two-state system (k =constant) with a
generalized driving protocol than what was studied inRef. [30], [31] and [32]. For the two-band system, wehave also described in detail a guiding principle that fixesthe gauge of the micro-motion and hence fixes the driv-ing protocol. An inappropriate gauge fixing can lead toa complicated driving protocol, which may not be veryeasy to implement experimentally. We have applied ourformalism to realize the cross-stitched model’s band dia-gram [26, 27] from a static Hamiltonian with only near-est neighbor hopping in each sub-lattice. In other words,we showed how Floquet engineering on a simple near-est neighbour lattice with a complex driving protocolcould help replicate the cross-stitch model, which re-quires a complicated nearest and next-nearest neighbourcouplings that could be hard to engineer.
In principle, the formalism presented here can be ap-plied to multi-band systems (see supplementary mate-rial [28] for a discussion on three-bands systems) andcould be applied to Floquet engineer technologically rel-evant materials like higher-order topological insulator(HOTI) [33–35] or reproduce Z2 lattice gauge theory incold atom setup [36].
Acknowledgments.– This research was supported by theInstitute for Basic Science in Korea (IBS-R024-Y2).
Supplementary Information
General formalism
In this section we present the details of the general formalism presented in the main text. We begin with a generalform of a time-dependent Hamiltonian having a period T that reads,
H(t) = H0 + V (t), V (t+ T ) = V (t), (19)
where H0 is the time-independent static part and V (t) is the periodic driving. Utilizing operators Aα that form aclosed Lie algebra of dimension N , we can express the Hamiltonian as
H0 = h01+
N∑α=1
hαAα = h01+ h ·A and V (t) = f0(t)1+
N∑α=1
fα(t)Aα = f0(t)1+ f(t)·A, (20)
where h’s are functions of the system parameters, the driving functions f(t)’s are functions of both time t and systemparameters. The algebraic structure of A’s is governed by the structure constants Λγαβ which are defined in the maintext via Eq. (6). The time-periodic condition of V (t) implies that all the driving functions are also time-periodic.If U(t) is the corresponding time-evolution operator, then this operator will satisfy the time-dependent Schrodingerequation (TDSE) with ~ = 1,
idU(t)
dt= H(t)U(t), (21)
whose solution is
U(t) = T exp
{−i∫ t
0
H(t′) dt′}, (22)
6
where T is the time-ordering operator. The Floquet theorem suggests that the above time-evolution operator canalways be written as a product two unitary operators: operator P (t) that describes the short-time dynamics withinone period with P (t+ T ) = P (t); the other operator describes the long time dynamics governed by an effective staticHamiltonian HF called the ‘Floquet Hamiltonian’. Therefore, we have
U(t) = P (t) e−iHF t. (23)
Following the initial condition U(0) = 1, we get P (0) = 1. Moreover, the time-periodic property of P (t) suggeststhat P (nT ) = 1 for any arbitrary positive integer n. If one observes the dynamics of a particle stroboscopically atevery time interval nT , then that dynamics is effectively governed by the static Hamiltonian HF . From the Floquetengineering perspective, this effective static Hamiltonian is also the desired one which Floquet engineers want to obtainby designing a driving protocol. Therefore, in the remaining part of this supplementary information, we denote theFloquet Hamiltonian HF by the effective Hamiltonian Heff . Since this Hamiltonian is obtained from a time-dependentHamiltonian with underlying Lie algebraic structure, the general form of Heff will also be a linear combination of{1, Aα} of the form
Heff = heff0 1+
N∑α=1
heffα Aα = heff
0 1+ heff ·A . (24)
Substituting the Floquet form of the time-evolution operator in the TDSE, given by Eq. (21), we obtain[idP (t)
dt+ P (t)Heff
]e−iH
eff t = H(t)U(t). (25)
The idea of Floquet engineering implies that the form of the evolution operator corresponding to the long timedynamics is known, but the micro-motion part is unknown. Since the Hamiltonian has an underlying Lie algebraicstructure, we can apply the Wei-Norman ansatz [24, 25] to the micro-motion operator. According to this ansatz, wecan write
P (t) = e−im0(t)N∏α=1
e−imα(t)Aα . (26)
From the condition P (nT ) = 1, we get the following conditions
e−im0(nT ) = 1 and e−imα(nT )Aα = 1 ∀ α and n. (27)
Substituting the above Wei-Norman form of P (t) in Eq. (25), then the first term at the left hand side will become
idP (t)
dte−iH
eff t =
m0(t)P (t) + m1(t) e−im0(t)A1
{N∏α=1
e−imα(t)Aα
}︸ ︷︷ ︸
=A1P (t)
+ m2(t) e−im0(t) e−im1(t)A1A2︸ ︷︷ ︸applying BCH
{N∏α=2
e−imα(t)Aα
}+ · · ·
+ mβ(t) e−im0(t)
{β−1∏α=1
e−imα(t)Aα
}Aβ︸ ︷︷ ︸
applying BCH
N∏
α′=β
e−imα′ (t)Aα′
+ · · ·
+ mN (t) e−im0(t)
{N−1∏α=1
e−imα(t)Aα
}AN︸ ︷︷ ︸
applying BCH
e−imN (t)AN
e−iHeff t.
(28)
7
In the above expression, the first and the second terms at the right side are equal to m0(t)U(t) and m1(t)A1 U(t),respectively. If we apply Baker-Campbell-Hausdorff (BCH) formula [25] to the expression e−im1(t)A1A2 of the thirdterm, then we can push the exponential operator to the right and the third term will take the form
third term = m2(t)
{N∑γ=1
M(3)1γ (m1)Aγ
}U(t),
where the coefficients M(3)1γ (m1) are in general nonlinear functions of m1(t). The term is a linear function of the
operators Aγ ’s because these operators form a closed Lie algebra. Similarly, for the general (β + 1)th term of theabove expression, we can push all the exponential operators in the following expression{
β−1∏α=1
e−imα(t)Aα
}Aβ
to the right by applying the BCH formula (β−1) times. Then the general (β+1)th term will be of the following form
(β + 1) term = mβ(t)
{N∑γ=1
M(β+1)1γ (m1,m2, · · · ,mβ−1)Aγ
}U(t).
Here again the coefficientsM(β+1)1γ (m1,m2, · · · ,mβ−1) are nonlinear functions of (m1(t),m2(t), · · · ,mβ−1(t)) and the
linearity of the operators Aγ ’s is due to the closed Lie algebraic structure as mentioned above. Therefore, using allabove relations in Eq. (28) we obtain,
idP (t)
dte−iH
eff t =
m0(t)1+ m(t) ·M1(t)T︸ ︷︷ ︸= ζ(m, m)
·A
U(t), (29)
where m = {mα(t)} is a N -dimensional vector whose components are time-dependent functions mα(t), A = {Aα} isa vector whose each element is the operator Aα, the elements of the matrixM1(t) are in general nonlinear functionsof m(t).
We now consider the second term on the left-hand side of Eq. (25),
P (t)Heff e−iHeff t =
[P (t)Heff P †(t)
]U(t) =
[heff
0 1+
N∑α=1
heffα
{P (t)Aα P
†(t)}]
U(t). (30)
Again due to the closed Lie algebra of {Aα}, applying the BCH formula, we have
N∑α=1
heffα
{P (t)Aα P
†(t)}
= heff ·M2(t)T︸ ︷︷ ︸= ξ(heff ,m)
·A. (31)
Therefore, Eq. (25) becomes[m0(t)1+ m ·M1(t)T ·A+ heff
0 1+ heff ·MT2 (t) ·A
]U(t) = H(t)U(t). (32)
Using the Hamiltonian H(t) given by Eqs. (19)-(20) and equating the coefficients of 1 and Aα’s from both sides, weobtain
h0 + f0(t) = m0(t) + heff0 and h+ f(t) =M1(t) · m+M2(t) · heff . (33)
which matches Eq. (16) from the main text.
8
Formalism for the two-bands case: Floquet Engineering Protocol
The two-band Hamiltonians are represented by the operators that follow SU(2) algebra. Here we represent theHamiltonian in terms of the operators (1, S±, Sz), where S± = Sx ± iSy and 1 is a 2 × 2 identity matrix. Here,2S = σ where σ = (σx, σy, σz) are spin-1/2 Pauli matrices. Following the Wei-Norman ansatz, the correspondingmicro-motion operator for each k can be written as
Pk(t) = e−imk0(t)1e−imk+(t)S+e−imk−(t)S−e−imkz(t)Sz . (34)
Since, S± are not Hermitian, then mk±(t) are complex and e−imk±(t)S± is not unitary, but over all the operator P (t)is unitary. Moreover, even though Sz is a Hermitian operator, we still have to consider mkz(t) to be complex. Thisis because, for the real mkz(t), we cannot find any pair of complex numbers mk±(t) except zeros, such that P (t)will be unitary. From the unitary property of P (t), we have shown in the main text that mk− and Im[mkz] are notindependent functions and both of these can be expressed in terms of mk+ (see Eq. (14) of the main text).
We now have to construct M1(t) and M2(t) matrices for each momentum value k to determine the drivingprotocol. We denote these matrices as Mk1(t) and Mk2(t), respectively. The matrix Mk1(t) will be constructedfrom the time derivative of Pk(t), Eq. (29), using the Wei-Norman form given in Eq. (34). Therefore, we obtain
idPk(t)
dt=
mk01+{mk+ +m2
k+mk− + imk+(1−mk+mk−)mkz}︸ ︷︷ ︸
=C+
S+
+ (mk− − imk−)︸ ︷︷ ︸=C−
S− + {mkz(1− 2mk+mk−)− 2imk+mk−}︸ ︷︷ ︸=Cz
Sz
Pk(t).
(35)
Above, just like the main text we suppress the explicit time dependence in the functions m for notational simplicity.We now express above equation in terms of two independent variables mk+ and the real part of mkz, i.e., mR
kz.Therefore, we replace mk− and the imaginary part of mkz, i.e., mI
kz, by the following expression given in the maintext (see Eq. (14) in the main text),
mk− =m∗k+
1 + |mk+|2and mI
kz = ln(1 + |mk+|2), (36)
where m∗k+ is the complex conjugate of mk+. Subsequently, the time derivative of these functions are
mk− =m∗k+ −m∗2k+mk+
(1 + |mk+|2)2 and mI
kz =mk+m
∗k+ +m∗k+mk+
1 + |mk+|2. (37)
Using the relations obtained in Eqs. (36) and (37), we obtain the coefficients of Eq. (35) as
C+ =1
1 + |mk+|2(mk+ + imk+m
Rkz), C− = C∗+, and
Cz =1
1 + |mk+|2[im∗k+mk+ − imk+m
∗k+ + (1− |mk+|2) mR
kz
].
(38)
Then from Eq. (35), equating the coefficients of {S±, Sz} from the both sides of the TDSE, we get Mk1 given byEq. (15) of the main text.
In order to derive the matrixMk2 we have to calculate Pk(t)Heffk P †k(t), Eq. (30), where Heff
k = Pk(t)Heffk P †k(t) +
heffk · S. In the (Sx, Sy, Sz) and (S±, Sz) representation, Heff
k is
Heffk = heff
k01+ hkxSx + hkySy + hkzSz = heffk01+ hk−S+ + hk+S− + hkzSz, (39)
where 2hk± = hkx ± ihky. Here we consider the ±Z representation to obtain,
Pk(t)Heffk P †k(t) = heff
k01+ Pk(t) (hk−S+ + hk+S− + hkzSz) P†k(t). (40)
9
We have found that
Pk(t)S+ P†k(t) =
e−imRkz
1 + |mk+|2(S+ +m∗2k+S− + 2im∗k+Sz
),
Pk(t)S− P†k(t) =
eimRkz
1 + |mk+|2(m2k+S+ + S− − 2imk+Sz
),
Pk(t)Sz P†k(t) =
1
2(1 + |mk+|2)
[2imk+S+ − 2im∗k+S− + 2(1− |mk+|2)Sz
].
(41)
The above relations can be derived in two ways: (1) Applying the BCH formula multiple times. This is a cumber-some approach. However, it is a very general method that can be applied for any closed algebra irrespective of itsrepresentation in any dimension. (2) A straightforward way is to explicitly write down the matrix representation ofPk(t) and S-matrices, then calculate matrix multiplication of three matrices to obtaining each of the three relations.Substituting the results obtained in Eqs. (35) and (41) in the TDSE idUk(t)/dt = Hk(t)Uk(t), and equating thecoefficients of (1, S±, Sz) from the both sides, we get the relations given by Eqs. (16) and (17) of the main text.
Three bands case
In order to represent any generic three bands tight-binding Hamiltonian, one needs the 3× 3 identity matrix 1 andeight linearly independent matrices. A natural choice for these is to consider the eight trace-less Hermitian Gell-Mannmatrices used in the standard description of SU(3) algebra [37, 38]. The Gell-Mann matrices are generalizations ofthe Pauli matrices for the 3× 3 case. In the standard basis, the Gell-Mann matrices are of the form:
λ1 =
0 1 01 0 00 0 0
, λ2 =
0 −i 0i 0 00 0 0
, λ3 =
1 0 00 −1 00 0 0
, λ4 =
0 0 10 0 01 0 0
,
λ5 =
0 0 −i0 0 0i 0 0
, λ6 =
0 0 00 0 10 1 0
, λ7 =
0 0 00 0 −i0 i 0
, λ8 =1√3
1 0 00 1 00 0 −2
.
(42)
In terms of the above matrices, one can write the Hamiltonian of any three-bands models such as [39]
Hk = hk1λ1 + hk4λ4 + hk6λ6, (Kagome Lattice) (43)
Hk = h′k4λ4 + h′k6λ6 (Lieb Lattice). (44)
If one wants to study Floquet version of the Kagome or Lieb lattice under the Wei-Norman formalism, then one hasto consider all the λ-matrices. Following the Wei-Norman ansatz, the micro-motion operator will take the form
Pk(t) = e−imk0(t)18∏
α=1
e−imkα(t)λα . (45)
Alternatively, one can construct the micro-motion operator using the following representation [40]:
a± =1
2(λ6 ± iλ7), b± =
1
2(λ1 ± iλ2), c± =
1
2(λ4 ± iλ5), a3 =
1
2(√
3λ8 − λ3), and c3 =1
2(√
3λ8 + λ3). (46)
In principle, one can follow our Floquet engineering protocol to realize any three-bands model using one of the above(or any other) representations of the SU(3) algebra. However, the matricesMk1(t) andMk2(t) which are crucial fordesigning the driving protocol will now be 8× 8 matrices. The large dimension of these matrices makes the Floquetengineering protocol for the three-bands case complicated.
Interestingly, a careful observation reveals that λ1, λ2, and λ3 can be represented in terms of the Pauli matrices as
λα =
(σα 00 0
), where α = 1, 2, 3 or x, y, z and 0 =
(00
). (47)
These three matrices form an SU(2) sub-algebra: [λα, λβ ] = iεαβγλγ . Therefore, if one wants to design a Floquetengineering protocol for a system having three energy bands whose Heff
k can be expressed as a linear combination of
10
these three matrices and the 3×3 identity matrix, then the undriven Hamiltonian Hk0 and the periodic driving Vk(t)can also be expanded as a linear combination of the same. Instead of this representation, one can also use anotherrepresentation, which is equivalent to the ±Z representation of the main text, with
b± =1
2
(σx ± iσy 0
0 0
)=
(S± 00 0
)≡ Λ± and
1
2λ3 =
1
2
(σz 00 0
)=
(Sz 00 0
)≡ Λz. (48)
to express the Hamiltonian. Moreover, like S2± = 0, we now have Λ2
± = 0. Following the Wei-Norman ansatz, we canwrite down the form of the micro-motion operator for this case as
Pk(t) = e−imk0(t)1e−imk+(t)Λ+e−imk−(t)Λ−e−imkz(t)Λz . (49)
Here again, the operators Λ± are not Hermitian, and consequently the second and the third term in the aboveexpression are not unitary. The unitary property of the micro-motion operator Pk(t) once again gives the samerelation as given in Eq. (36). As a consequence, from the time-derivative of the above Pk(t), we construct Mk1(t)matrix for the SU(2) sub-algebra of the SU(3) algebra. ThisMk1(t) will be exactly identical to the expression givenin Eq. (16) of the main text. The form of theMk2(t) matrix is determined by the desired Hamiltonian Heff
k .We may consider one interesting case for this SU(2) sub-algebra. Consider a desired 3-bands Hamiltonian of the
form
Heffk = ηkxΛx + ηkyΛy + ηkzΛz = ηk−Λ+ + ηk+Λ− + ηkzΛz ≡ heff
k ·Λ, (50)
where
Λα =1
2λα, α = x, y, z and ηk± =
1
2(ηkx ± iηky)
and in ±Z representation heffk = (ηk−, ηk+, ηkz). For the above Hamiltonian, two bands will be dispersive
Ek = ±√|ηk+|2 +
η2kz
4= ±1
2
√η2kx + η2
ky + η2kz ≡ ±
1
2|ηk|
and the third band will be a flat-band at Ek = 0. If we add a term η01 to the above Hamiltonian, then the flat-bandwill be at energy Ek = η0 and the dispersive bands will be Ek = η0 ± 1
2 |ηk|.We now design the driving protocol to achieve the desired/effective Hamiltonian Heff
k . For simplicity, we areassuming the case when η0 = 0, that is the energy of the flat-band is zero. Therefore, for this case, we can assumethat there is no initial static Hamiltonian. We only need a pure time time-dependent Hamiltonian for any momentumk as
Hk(t) = Fkx(t) Λx + Fky(t) Λy + Fkz(t) Λz = Fk−(t) Λ+ + Fk+(t) Λ− + Fkz(t) Λz. (51)
The driving functions {Fkx(t), Fky(t), Fkz(t)} and {Fk±(t), Fkz(t)}, where Fk±(t) = 12 [Fkx(t)± iFky(t)], are all
time-periodic functions with period T .The next is to deriveMk1(t) andMk2(t) matrices for this case. Since we are considering only SU(2) sub-group of
the SU(3) group, here also we get the identicalMk1(t) andMk2(t) matrices as given by Eq. (15) of the main text.Moreover, if you consider the same gauge, then we shall also get identical Mk1(t) and Mk2(t) matrices as obtained inthe main text [just below Eq. (17)]. In order to respect the boundary condition, we also set µ+(t) = a+ sinωt = µ∗+(t)and µR
z (t) = pωt where p is any integer. We then obtain the driving functions in the XY Z representation, using therelations Fkx = 2Re[Fk−(t)] and Fky = −2Im[Fk−(t)], as:
Fkx(t) = 2Fe(t)[a+ωCωtCk − a+pωCωtSk + ηkxCpωt − ηkySpωt + a2
+S2ωt {C2k+pωtηkx − S2k+pωtηky} − a+SωtCkηkz
],
Fky(t) = −2Fe(t)[a+ωCωtSk + a+pωCωtCk − Cpωtηky − Spωtηkx + a+S
2ωt {S2k+pωtηkx − C2k+pωtηky}+ a+SωtCkηkz
],
Fkz(t) = Fe(t)
[2a+Sωtηky +
(1− a2
+
2
)pω +
1
2pωa2
+C2ωt +
{(1− a2
+
2
)+a2
+
2C2ωt
}ηkz
], (52)
where Fe(t) = (1 + a2+S
2ωt)−1, Cw = cosw, and Sw = sinw. Here again we set a2
+ = 2 and p = 3 to remove the staticpart from the driving. We have shown the driving functions for the one dimensional case k→ k with ηkz = 0 for all kand ηkx = −ηky = 2 cos k + ∆. Here we again consider two values of the frequency, ω = 4 + 2∆ = 8 and ω = 2∆ = 4.
In Fig. 3, we have shown the density plot of all the driving functions fkx, fky and fkz on the plane of time t andmomentum k.
11
(a)
−π
−π
2
0
π
2
πk
−50
0
50(b)
−50
0
50(c)
−10
0
10
20
(d)
0 0.2 0.4 0.6 0.8 1−π
−π
2
0
π
2
π
k
−40
−20
0
20
40 (e)
0 0.2 0.4 0.6 0.8 1t (T )
−40
−20
0
20
40 (f)
0 0.2 0.4 0.6 0.8 1
−5
0
5
10
FIG. 3. Density plot of the driving functions fkx(t) (a,d), fky(t) (b,e), fkz(t) (c,f) are plotted as a function of the time t andthe momentum k. Panels (a)-(c) represent ω = 4 + 2∆ = 8 and panels (d)-(f) are for ω = 2∆ = 4.
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