The Floquet Engineer’s Handbook Mark S. Rudner 1 and Netanel H. Lindner 2 1 Niels Bohr International Academy and the Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark and 2 Physics Department, Technion, 320003, Haifa, Israel We provide a pedagogical technical guide to many of the key theoretical tools and ideas that underlie work in the field of Floquet engineering. We hope that this document will serve as a useful resource for new researchers aiming to enter the field, as well as experienced researchers who wish to gain new insight into familiar or possibly unfamiliar methods. This guide was developed out of supplementary material as a companion to our recent review, “Band structure engineering and non-equilibrium dynamics in Floquet topological insulators,” Nature Reviews Physics 2, 229 (2020). The primary focus is on analytical techniques relevant for Floquet-Bloch band engineering and related many-body dynamics. We will continue to update this document over time to include additional content, and welcome suggestions for further topics to consider. CONTENTS I. Introduction 1 II. Floquet theory basics and formalism 2 A. Frequency-space formulation 2 B. Truncation of frequency space 4 C. Floquet Fermi’s golden rule 5 III. Gap opening due to circularly polarized driving fields 9 IV. Vanishing of the winding number ν 1 for continuous time evolution 9 V. Definition of the time-averaged spectral function 9 A. Formal definitions 10 B. Properties of the Floquet retarded Green’s function 10 C. Evaluation of the Floquet retarded Green’s function for a non-interacting system 11 VI. Floquet-Kubo formula 12 A. Floquet-Kubo Formula for the electrical conductivity 13 VII. Acknowledgements 15 References 15 I. INTRODUCTION Time-periodic driving is a powerful tool for controlling quantum few- and many-body systems, with the poten- tial to enable “on-demand” dynamical control of mate- rial properties [1–5]. Periodic driving has been explored, for example, as a means to open and modify band gaps in graphene and topological insulator surfaces [2, 6–20], to create topologically non-trivial bands in topologically trivial systems [2–4, 16, 21–33], and to access entirely new types of non-equilibrium phases of matter with novel properties that may not exist in equilibrium [7, 24, 34– 51]. These applications and various aspects of the field have been summarized in a number of reviews [52–59]. The aim of this “Floquet engineer’s handbook” is to provide a pedagogical technical guide to many of the key theoretical tools and ideas that underlie work in this field. We hope that it will serve as a useful resource for new re- searchers aiming to enter the field, as well as experienced researchers who wish to gain new insight into familiar or possibly unfamiliar methods. As indicated above, the field is very broad and encompasses a wide variety of systems and phenomena. As such, a wide range of tech- niques are commonly employed. We primarily focus on analytical techniques relevant for Floquet-Bloch band en- gineering and related many-body dynamics. The topics covered below are organized as follows. In Sec. II we review the basics of Floquet theory, ap- plied to the time-evolution of periodically-driven quan- tum systems. We discuss the transformation to Fourier space, where the problem of time-evolution with a time- dependent Hamiltonian is recast as a problem of matrix diagonalization in an enlarged space. We further dis- cuss how this approach can be helpful for numerically- obtaining approximate Floquet states, and as a basis for analytical perturbative treatments. Then in Sec. III we present a simple explicit calculation that demonstrates how a circularly polarized driving field (analogous to circularly polarized light applied at normal incidence) induces a gap opening in a two-dimensional Dirac sys- tem such as graphene. In Sec. IV we briefly recount an important argument about a constraint on the winding numbers of Floquet-Bloch bands. In Sec. V we intro- duce and discuss an important object in Floquet analysis: the time-averaged spectral function. The time-averaged spectral function provides a useful way of “unfolding” the single-particle Floquet spectrum. This quantity al- lows a meaningful connection to be drawn between the quasienergies of Floquet states in a driven system and the energies of states in the system’s (non-driven) surround- ings. Finally, in Sec. VI we discuss the Floquet-Kubo formalism for describing the linear response properties arXiv:2003.08252v2 [cond-mat.mes-hall] 16 Jun 2020
Transcript
The Floquet Engineer’s Handbook
Mark S. Rudner1 and Netanel H. Lindner2
1Niels Bohr International Academy and the Center for Quantum Devices,Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark and
2Physics Department, Technion, 320003, Haifa, Israel
We provide a pedagogical technical guide to many of the key theoretical tools and ideas thatunderlie work in the field of Floquet engineering. We hope that this document will serve as auseful resource for new researchers aiming to enter the field, as well as experienced researchers whowish to gain new insight into familiar or possibly unfamiliar methods. This guide was developedout of supplementary material as a companion to our recent review, “Band structure engineeringand non-equilibrium dynamics in Floquet topological insulators,” Nature Reviews Physics 2, 229(2020). The primary focus is on analytical techniques relevant for Floquet-Bloch band engineeringand related many-body dynamics. We will continue to update this document over time to includeadditional content, and welcome suggestions for further topics to consider.
CONTENTS
I. Introduction 1
II. Floquet theory basics and formalism 2A. Frequency-space formulation 2B. Truncation of frequency space 4C. Floquet Fermi’s golden rule 5
III. Gap opening due to circularly polarized drivingfields 9
IV. Vanishing of the winding number ν1 forcontinuous time evolution 9
V. Definition of the time-averaged spectralfunction 9A. Formal definitions 10B. Properties of the Floquet retarded
Green’s function 10C. Evaluation of the Floquet retarded Green’s
function for a non-interacting system 11
VI. Floquet-Kubo formula 12A. Floquet-Kubo Formula for the electrical
conductivity 13
VII. Acknowledgements 15
References 15
I. INTRODUCTION
Time-periodic driving is a powerful tool for controllingquantum few- and many-body systems, with the poten-tial to enable “on-demand” dynamical control of mate-rial properties [1–5]. Periodic driving has been explored,for example, as a means to open and modify band gapsin graphene and topological insulator surfaces [2, 6–20],to create topologically non-trivial bands in topologicallytrivial systems [2–4, 16, 21–33], and to access entirely
new types of non-equilibrium phases of matter with novelproperties that may not exist in equilibrium [7, 24, 34–51]. These applications and various aspects of the fieldhave been summarized in a number of reviews [52–59].
The aim of this “Floquet engineer’s handbook” is toprovide a pedagogical technical guide to many of the keytheoretical tools and ideas that underlie work in this field.We hope that it will serve as a useful resource for new re-searchers aiming to enter the field, as well as experiencedresearchers who wish to gain new insight into familiaror possibly unfamiliar methods. As indicated above, thefield is very broad and encompasses a wide variety ofsystems and phenomena. As such, a wide range of tech-niques are commonly employed. We primarily focus onanalytical techniques relevant for Floquet-Bloch band en-gineering and related many-body dynamics.
The topics covered below are organized as follows.In Sec. II we review the basics of Floquet theory, ap-plied to the time-evolution of periodically-driven quan-tum systems. We discuss the transformation to Fourierspace, where the problem of time-evolution with a time-dependent Hamiltonian is recast as a problem of matrixdiagonalization in an enlarged space. We further dis-cuss how this approach can be helpful for numerically-obtaining approximate Floquet states, and as a basis foranalytical perturbative treatments. Then in Sec. III wepresent a simple explicit calculation that demonstrateshow a circularly polarized driving field (analogous tocircularly polarized light applied at normal incidence)induces a gap opening in a two-dimensional Dirac sys-tem such as graphene. In Sec. IV we briefly recount animportant argument about a constraint on the windingnumbers of Floquet-Bloch bands. In Sec. V we intro-duce and discuss an important object in Floquet analysis:the time-averaged spectral function. The time-averagedspectral function provides a useful way of “unfolding”the single-particle Floquet spectrum. This quantity al-lows a meaningful connection to be drawn between thequasienergies of Floquet states in a driven system and theenergies of states in the system’s (non-driven) surround-ings. Finally, in Sec. VI we discuss the Floquet-Kuboformalism for describing the linear response properties
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of periodically-driven systems, and the conditions un-der which a many-body system with Floquet-engineeredbands may be expected to exhibit response characteris-tics similar to those exhibited by a non-driven systemwith an analogous band structure.
II. FLOQUET THEORY BASICS ANDFORMALISM
Floquet’s theorem provides a powerful framework foranalyzing periodically-driven quantum systems. Fora system with evolution governed by a time-periodicHamiltonian H(t) with driving period T ,
ihd
dt|ψ(t)〉 = H(t)|ψ(t)〉, H(t+ T ) = H(t), (1)
Floquet states are stationary states of the stroboscopic(“Floquet”) evolution operator
U(T ) = T e−(i/h)∫ T0dt′H(t′). (2)
Here T denotes time ordering. The operator U(T ) prop-agates the system forward in time through one completeperiod of the drive. Once found, the Floquet states andtheir associated quasienergies, defined and discussed indetail below, may provide a useful basis for describingthe dynamics of the system. In this section we discussthe “extended space” formalism, which both provides anefficient scheme for numerically obtaining Floquet statesand a basis for further theoretical analysis including theapplication of perturbative techniques.
A. Frequency-space formulation
A conceptually natural approach to obtaining Floquetstates is to obtain (and then diagonalize) the stroboscopicFloquet evolution operator U(T ) in Eq. (2) through“brute force” direct integration of the time-dependentSchrodinger equation (1). Alternatively, as we now dis-cuss, we may use the structure of the Floquet states, asdescribed by Floquet’s theorem, to convert the problemof time-integration to a problem of diagonalization of asingle associated Hermitian operator defined on an en-larged space (compared to the Hilbert space of the orig-inal problem).
In the context of time evolution of quantum systemswith time-periodic Hamiltonians, Floquet’s theorem [60]can be stated as follows:
Floquet’s Theorem – Consider a quantum systemwith dynamics governed by a time-periodic Hamiltonian
H(t + T ) = H(t), with driving period T . The system’sevolution can be expanded in a complete basis of or-thonormal quasi-stationary “Floquet states” |ψn(t)〉,which under the time-evolution generated by H(t)satisfy |ψn(t+ T )〉 = e−iεnT/h|ψn(t)〉. The parameter εnis the “quasienergy” of Floquet state |ψn(t)〉.
Bloch’s theorem, which is widely familiar in solid statephysics, is another special case of Floquet’s theorem, ap-plied to the eigenstates of an electron in a crystal witha potential that has a periodic structure in space. Anal-ogous to how a Bloch state can be decomposed into aproduct of a plane wave and a periodic function (withthe same periodicity as the underlying lattice), each Flo-quet state can be decomposed as:
|ψn(t)〉 = e−iεnt/h|Φn(t)〉, |Φn(t+ T )〉 = |Φn(t)〉. (3)
Importantly, due to the fact that |Φn(t)〉 exhibits thesame time-periodicity as the drive, it can be representedin terms of a discrete Fourier series in terms of harmonicsof the drive frequency, ω = 2π/T :
|Φn(t)〉 =∑m
e−imωt|φ(m)n 〉, (4)
where |φ(m)n 〉 is the m-th Fourier coefficient of |Φn(t)〉.
Note that i) the Fourier coefficients |φ(m)n 〉 are not nor-
malized, i.e., generically 〈φ(m)n |φ(m)
n 〉 < 1, and ii) thereis no simple orthogonality relation between different
Fourier coefficients |φ(m)n 〉 and |φ(m′)
n 〉, apart from the(nontrivial) fact that they must add up to produce or-thogonal Floquet states |ψn(t)〉 at every time, t.
Our goal is now to find an algebraic equation that
yields the Fourier coefficients |φ(m)n 〉 for the Floquet
state solutions to the Schrodinger equation (1). Substi-tuting Eq. (3) on both sides of Eq. (1) and cancelingfactors of e−iεnt/h yields[
εn + ihd
dt
]|Φn(t)〉 = H(t)|Φn(t)〉. (5)
Using the Fourier decomposition of |Φn(t)〉 in Eq. (4) andH(t) =
∑m e−imωtH(m), which follows from the period-
icity of H(t), Eq. (5) yields:
(εn +mhω) |φ(m)n 〉 =
∑m′
H(m−m′)|φ(m′)n 〉. (6)
As a final step, we express Eq. (6) as an eigenvalueequation in Fourier harmonic space. Specifically, we cre-ate a vector ϕn by “stacking up” the Fourier coefficients
|φ(m)n 〉, and rearrange the coefficients in Eq. (6) into a
matrix H acting on vectors in this space:
3
Hϕn = εnϕn, H =
. . . H(−1) H(−2)
H(1) H0 −mhω H(−1) H(−2)
H(2) H(1) H0 − (m+ 1)hω H(−1)
H(2) H(1) . . .
, ϕn =
...
|φ(m)n 〉
|φ(m+1)n 〉
...
. (7)
Here, H0 ≡ H(0) = 1T
∫ T0dtH(t) is the dc (time-
averaged) part of the Hamiltonian. Note that the matrixH in Eq. (7) has a block structure: each block is of sized × d, where d is the dimension of the Hilbert space ofthe system [i.e., the dimension of H(t)]. The number ofblocks, which are labeled by Fourier harmonic indices, isformally infinite, since the sum over m in Eq. (3) rangesover all integers.
The time-periodic part of the Floquet state wavefunction, |Φn(t)〉, is obtained from the vector ofFourier coefficients ϕn through multiplication witha rectangular matrix of oscillatory phase factors,P(ωt) = (· · · e−imωt1d×d e−i(m+1)ωt1d×d · · · ). Here1d×d stands for the identity operator in the physicalHilbert space, taken to have dimension d (see above).Explicitly, this gives
|Φn(t)〉 = P(ωt)ϕn
=∑m
e−imωt|ϕ(m)n 〉. (8)
The projector P(ωt) thus plays a central role in themapping between extended space and the physicalHilbert space.
Overcompleteness of solutions to Eq. (7).— As mentionedabove, the dimension of the matrix H in Eq. (7) is largerthan that of the Hilbert space of the original problem inEq. (1). Thus the solutions to Eq. (7) must correspond tolinearly-dependent quantum states. In fact, the Fourierspace formulation of the Floquet problem encodes a re-dundancy, such that the same physical solutions of theform in Eq. (3) are encoded infinitely-many times in theeigenvectors of H.
To see where the redundancy of solutions to Eq. (7)comes from, we write a general Floquet state in the form
|ψn(t)〉 = e−iεnt/h∑m e−imωt|φ(m)
n 〉 [see Eqs. (3) and(4)]. Note that, without changing the state |ψn(t)〉, itsquasienergy can be shifted by any integer multiple of hωby adding and subtracting any integer multiple of iωt inthe pair of exponentials in this expression:
|ψn(t)〉 = e−i(εn+m′hω)t/h∑m
e−i(m−m′)ωt|φ(m)
n 〉
≡ e−iεnt/h∑m
e−imωt|φ(m)n 〉, (9)
where εn = εn +m′hω and |φ(m)n 〉 = |φ(m+m′)
n 〉. Thus alldistinct Floquet state solutions to the Schrodinger equa-tion can be indexed with quasienergies that fall within a
FIG. S1. Floquet spectrum and time-averaged spectral func-tion. a) The Floquet (quasienergy) spectrum, obtained bydiagonalizing the “extended-zone” Floquet Hamiltonian H inEq. (7), is periodic with a “Floquet Brillouin zone” of widthequal to the driving field photon energy, hω. For illustra-tion, here we show the spectrum of graphene subjected tocircularly polarized light. Colors indicate overlaps of the Flo-quet states with corresponding valence (blue) and conduc-tion (pink) band states of the system in the absence of thedrive. b) The time-averaged spectral function provides meansto “unfold” the quasienergy spectrum, and to visualize howthe Floquet states of the driven system may couple to exter-nal (non-driven) degrees of freedom at specific energies. Asshown here, for the same system as displayed in panel a), thespectral weights of the sidebands (arising from the originalconduction and valence bands being shifted up and down byan integer multiple of the driving field photon energy hω) de-cay rapidly with the number of photons absorbed/emitted (in-dicated by the intensities of the points). The rates of photon-assisted tunneling between an external lead and the Floquetsystem, and for example, of photon-assisted electron-phonon“Floquet-Umklapp” scattering, are controlled by these side-band intensities.
single “Floquet-Brillouin zone” of width hω, εmin ≤ ε <εmin + hω. Any solution to Eq. (7) with quasienergy out-side of the Floquet-Brillouin zone encodes [via Eq. (9)] anidentical physical state to one corresponding to a solutionof Eq. (7) with quasienergy within the Floquet-Brillouinzone. Consequently, as illustrated in Fig. S1a, the spec-trum of H in Eq. (7) consists of an infinite number ofcopies of the system’s Floquet spectrum, rigidly shiftedup and down by integer multiples of hω.
4
At a fundamental level, the redundancy of the Floquetspectrum arises due to the discrete time-translationinvariance of the system. The Floquet states areeigenstates of the stroboscopic evolution (discretetime-translation) operator, U(T ), in Eq. (2). Due to theunitarity of U(T ), its eigenvalues are all unit-moduluscomplex numbers: U(T )|ψn(t)〉 = e−iθn |ψn(t)〉, whereθn is real. By parametrizing the eigenvalues of U(T ) interms of quasienergies, θn = εnT/h, we introduce anartificial multi-valuedness: quasienergy values separatedby an integer multiple of hω correspond to the sameeigenvalue e−iεnT/h of U(T ). In physical terms, theperiodicity of quasienergy expresses the fact that, inthe presence of a periodic drive, the system’s energy isconserved up to the absorption or emission of integermultiples of the driving field photon energy, hω.
B. Truncation of frequency space
In the previous subsection, we described how theproblem of finding Floquet state solutions of the time-dependent Schrodinger equation for a periodically-drivensystem, Eq. (1), can be recast from one of computinga time-ordered exponential [Eq. (2)], to one of findingeigenvalues of an enlarged “extended-zone” or “extended-space” Floquet Hamiltonian, H [Eq. (7)]. Given that His an infinite-dimensional matrix, even for a system witha finite-dimensional Hilbert space, it may not be clearwhat we have gained through this transformation. For-tunately, as we now discuss, H has a particular structurethat can be exploited to enable approximate solutionsto Eq. (7) to be found efficiently. This structure more-over has a clear physical interpretation, which may helpprovide further insight into the dynamics of the system.
To most clearly exhibit the key features of theextended-zone Floquet Hamiltonian, we will focus fornow on the case of a single harmonic drive:
H(t) = H0 + V (t), V (t) = V eiωt + V †e−iωt. (10)
The corresponding extended-zone Floquet HamiltonianH then takes the form in Eq. (7) with H(−1) = V , H(1) =V †, and H(∆m) = 0 for |∆m| > 1. In this case, H has ablock-tridiagonal form, with the non-driven HamiltonianH0 (shifted by multiples of hω) copied over and over inthe blocks along the diagonal, and the drive coupling V(V †) placed in the blocks above (below) the diagonal:
H =
. . . V 0V † H0 −mhω V
0 V †. . .
. (11)
Importantly, the (block) tridiagonal structure of H isclosely analogous to that of a tight-binding lattice withnearest-neighbor hopping and a linear potential (i.e., tothe problem of an electron in a one-dimensional lattice
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FIG. S2. The extended-zone Floquet Hamiltonian, Eq. (7),can be interpreted as describing the one-dimensional tight-binding dynamics of an effective particle hopping in Fourierharmonic space. The site energies and couplings within eachunit cell, m, are described by the time-averaged Hamilto-nian, H0. For a drive with a single harmonic, as describedby Eqs. (10) and (11), the drive V induces hopping betweenneighboring unit cells, in which the Fourier harmonic shiftsby one. The on-site energies vary linearly with m due to theshifts −mhω appearing in the diagonal blocks of Eqs. (7) and(11). This linear potential acts as a uniform electric field onthe effective particle. In the absence of the linear potential,the eigenstates would be Bloch waves in m-space, with cor-responding total bandwidth W ∼ max||V ||, ||H0||. From asemiclassical point of view, in the presence of the linear poten-tial the effective particle may explore a range of harmonics ofwidth `m ∼ W/(hω), where its kinetic and potential energiesmay be balanced; the Floquet state wave functions are there-fore localized in harmonic space and the infinite-dimensionalmatrices in Eqs. (7) and (11) can be safely truncated.
with a uniform electric field), see Fig. S2. By mak-ing analogies to Bloch oscillations [61–63] and to the“Wannier-Stark ladder” [64], we may gain important in-sight into the nature of the Floquet states encoded inthe solutions to Eq. (7). These insights will furthermoreshed light on the utility of this approach as a numericalmethod for obtaining Floquet states.
Consider the situation where the system described byH0 has a finite dimensional Hilbert space (e.g., for a two-band Floquet-Bloch system, H0 could be the 2×2 BlochHamiltonian describing the dynamics for a given crystalmomentum, k). Neglecting the “linear potential” −mhωin Eq. (11), we interpret the matrixH as the Hamiltonianof an effective particle hopping between sites/unit cells ofa one-dimensional lattice in Fourier harmonic space. Theon-site energies and intra-unit-cell hoppings are describedby H0, while the inter-unit-cell hopping is described byV , see Fig. S2. In particular, for a system where H0 hasdimension one, the kinetic energy of the effective particlehas a bandwidth W = 2|V |. Crucially, even when thedimension of H0 is larger than one (but finite), the effec-tive particle system (in the absence of the mhω terms)still has a finite bandwidth, W, approximately given bythe maximum between the norms ||V || and ||H0||.
In the absence of the linear potential −mhω, the eigen-states of the effective particle are extended Bloch wavesin the Fourier harmonic space. These eigenstates becomelocalized in Fourier harmonic space when the −mhω “po-
5
tential energy” terms are introduced. From a semiclas-sical point of view, this localization arises from the factthat, due to the finite bandwidth of the effective parti-cle’s kinetic energy, the total energy given by the sum ofpotential and kinetic energies can only take a constantvalue over a finite range of m-values, `m, see Fig. S2.The range `m of Fourier coefficients accessible at a givenquasienergy ε can be estimated as `m ∼ W/(hω), withthe Fourier-space wave function falling off (faster than)exponentially in m outside of this “classically-allowed”region [65].
Importantly, due to the fact that the each eigenvectorof the infinite-dimensional matrix H in Eq. (11) hassupport only over a limited range of Fourier harmonics,m, a complete set of Floquet states with quasienergiesin one Floquet-Brillouin zone can be found to arbi-trary accuracy by using a truncated, finite-dimensionalsub-matrix of H, which we call Htrunc. In particular,by restricting the set of Fourier coefficients to a rangeof (2M + 1) values of width much larger than thelocalization radius of the Fourier-space wave functions,(2M + 1) `m, the difference between the true Floquetstate wave functions and the approximate wave functionsobtained as eigenvectors of the truncated extended-zoneFloquet Hamiltonian can be made exponentially small.Importantly, the truncation of H breaks the perfectrepetition/redundancy of the Floquet spectrum betweendifferent Floquet-Brillouin zones. For Floquet-zonescentered at m-values near the truncation boundaries,the spectrum and eigenstates may be severely distorted.However, as long as the truncation range is taken tobe large enough, the eigenvectors and eigenvalues nearthe middle of the spectrum of Htrunc can be used tonumerically obtain approximate Floquet states with anerror that decays exponentially with M .
Relation to system coupled to quantized harmonicmode.— Despite the fact that Eqs. (1), (7) and (11) de-scribe a system subjected to a classical time-dependentdrive, the structure of the extended-zone Floquet Hamil-tonian H is similar to that of the same system with thedrive replaced by a quantized harmonic oscillator mode.In this analogy, the Fourier harmonic indices label Fockstates of the oscillator mode. The diagonal blocks, whichtake the form H0 − mhω, describe the system togetherwith the oscillator, where m photons have been removedfrom the oscillator state relative to an arbitrary refer-ence Fock state (which must be assumed to contain alarge number of photons). The off-diagonal blocksH(∆m)
describe couplings in which ∆m photons are absorbed(∆m > 0) or emitted (∆m < 0) by the system.
While the similarities mentioned above may be use-ful for gaining conceptual insight into the Floquet prob-lem, there are several key differences between H and atrue system-oscillator coupling Hamiltonian. First, recallthat, when applied to a Fock state, the creation and anni-hilation operators a† and a of a harmonic mode generatefactors proportional to the square root of the number of
photons present: a|n〉ω =√n|n− 1〉ω, where |n〉ω labels
a Fock state with n photons in the mode with frequencyω. Therefore, in a true system-oscillator Hamiltonian,the off-diagonal blocks H(∆m) would contain an explicitdependence on the photon number index; in contrast,the off-diagonal blocks in H are translation-invariant, de-pending only on the net number of photons absorbed,∆m. Second, while a true oscillator mode has a vac-uum state |n = 0〉ω, such that the photon-number spaceis semi-infinite (with the photon number index rangingonly over non-negative integers), the Fourier harmonicindex labeling the blocks of H runs over all integers. Thesystem dynamics induced by the classical drive, as de-scribed in Eq. (1), can be effectively recovered by placingthe oscillator in a large-amplitude coherent state, wherethe square-root dependencies of the matrix elements maybe ignored and where the support of the oscillator wavefunction is localized far from the vacuum state in photon-number space.
C. Floquet Fermi’s golden rule
In this subsection we derive a Floquet-version ofFermi’s golden rule, and show how it can be evaluatedwithin the extended space formalism. Our aim is to cal-culate the rate of transitions between (unperturbed) Flo-quet states, induced by a weak perturbation. Of particu-lar relevance for FTIs, as an application we will use Flo-quet Fermi’s golden rule in the extended space to gaininsight into the nature of Floquet-Umklapp scatteringprocesses (as discussed in Sec. V of the Ref. [59] and,e.g., Refs. [11, 66–71]).
Consider a periodically-driven system, with Hamilto-nian H(t) = H(t+ T ), subjected to a weak perturbationHint. For simplicity, here we consider the case whereHint is a static (time-independent) perturbation. Thetime-evolution of the system is governed by ih d
dt |ψ(t)〉 =[H(t) + Hint]|ψ(t)〉. We emphasize that the drive itselfmay be strong: only the additional perturbation Hint istaken to be weak.
As a starting point, we assume that a complete setof Floquet states of the unperturbed problem (withHint = 0) are known. We denote these unperturbed Flo-quet states as |ψn(t)〉, with corresponding quasiener-gies εn. As discussed above, a complete set of Flo-quet states can be indexed within a single Floquet-Brillouin zone. Therefore, we take all εn to satisfyεmin ≤ εn < εmin + hω.
Suppose that the system is in the unperturbed Floquetstate |ψi(0)〉 at time t = 0. We calculate the probabilitythat, due to the perturbation Hint, the system has madea transition to unperturbed Floquet state |ψf (NT )〉 afterN periods of the drive, where N is an integer. After timeNT , the transition probability Pfi(NT ) from |ψi(0)〉 to(unperturbed) Floquet state |ψf (NT )〉 is given by
Pfi(NT ) = |〈ψf (NT )|ψ(NT )〉|2, (12)
6
where |ψ(NT )〉 is the state of the system evolved in thepresence of the perturbation. Below we will see that,by focusing on these stroboscopic probabilities, we willobtain a simple expression for the transition rate in termsof the Fourier component vectors ϕi and ϕf in extendedspace. We will further show that these results directlyextend to times that are not necessarily integer multiplesof the driving period.
To obtain the rate of transitions, i.e., of linear growth ofthe Pfi(NT ) with NT , we evaluate the transition prob-
ability P(1)fi (NT ) using the evolution expanded to first
order in Hint. As a first step, we move to the inter-action picture |ψ(t)〉I = U†(t, 0)|ψ(t)〉, where U(t, 0) =
T e−(i/h)∫ t0dt′H(t′) is the evolution operator of the unper-
turbed system. In this interaction picture, the Floquetstates |ψn(t)〉 are stationary: |ψn(t)〉I = |ψn(0)〉.
Writing the overlap in Eq. (12) using the interactionpicture states and evolution operator, and expanding tofirst order in Hint,I(t) = U†(t, 0)Hint U(t, 0), we obtain
P(1)fi (NT ) =
∣∣∣∣∣−ih∫ NT
0
dt 〈ψf (0)|Hint,I(t)|ψi(0)〉∣∣∣∣∣2
. (13)
Here we used |ψf (NT )〉I = |ψf (0)〉. The correspondingtransition rate, Wfi, is then given by
Wfi = limN→∞
1
NTP
(1)fi (NT ). (14)
Below we derive an exact expression for P(1)fi (NT ), and
then obtain Wfi by taking the limit in Eq. (14).Returning to Eq. (13), we apply the evolution opera-
tors U(t, 0) and U†(t, 0) in Hint,I(t) to the right and left,respectively. Writing the Floquet states as |ψi,f (t)〉 =
e−iεi,f t/h|Φi,f (t)〉, with |Φi,f (t)〉 = |Φi,f (t+ T )〉, gives
P(1)fi (NT ) =
∣∣∣∣∣−ih∫ NT
0
dt ei∆εt/h〈Φf (t)|Hint|Φi(t)〉∣∣∣∣∣2
,
(15)where we have defined ∆ε ≡ εf − εi for brevity.
The integral in Eq. (15) can be split into a sum of inte-grals over each complete period, nT ≤ t < (n+ 1)T , forn = (0, 1, . . . , N−1). Due to the periodicity of |Φi(t)〉 and|Φf (t)〉, for each n the corresponding integral is given by
ei∆ε(nT )/h∫ T
0dt ei∆εt/h〈Φf (t)|Hint|Φi(t)〉. We thus ex-
press P(1)fi (NT ) in terms of the time-averaged matrix el-
ement Mfi ≡ 1T
∫ T0dt ei∆εt/h〈Φf (t)|Hint|Φi(t)〉 as
P(1)fi (NT ) =
T 2
h2 |Mfi|2N−1∑n,n′=0
ei∆ε(n−n′)T/h. (16)
The factor of T 2 in Eq. (16) was obtained by multiplying
and dividing by T 2 in order to express P(1)fi (NT ) in terms
of the time-averaged matrix element Mfi.The sums over n and n′ in Eq. (16) can be evaluated
by changing to sum and difference indices n = n + n′
and ∆n = n − n′. Heuristically, if the sum over ∆nis taken from −∞ to ∞, we obtain a periodic array ofdelta functions (a “Dirac comb”):
∑∆n e
i(εf−εi)∆nT/h ∼2πhT
∑∞k=−∞ δ(εf − εi − khω). The sum over n gives a
contribution proportional to N . All together we thus
obtain a transition probability P(1)fi (NT ) that grows lin-
early with NT , which gives a finite rate in Eq. (14).More rigorously, the sums appearing in Eq. (16) can
be evaluated exactly, analytically. For each value of n,the sum over ∆n can be expressed in the form of a finitegeometric series with limits that depend on n. Definingx = e2i∆θ, with ∆θ = ∆εT/h, the sum over ∆n can beperformed using
k∑n=0
xn =1− xk+1
1− x , (17)
where k depends on n. Once the sum over ∆n is evalu-ated as a function of n, the sum over n can be performedpiecewise, for i) 0 ≤ n ≤ N − 1 and ii) N ≤ n ≤ 2N − 2,using the summation formula (see, e.g., Wikipedia)
M∑n=0
sin(∆θ+n∆θ) =sin (M+1)∆θ
2 sin(∆θ + M∆θ2 )
sin ∆θ2
. (18)
Evaluating the two sums using upper limits M = N − 1and M = N − 2, for cases i) and ii), respectively, weobtain:
N−1∑n,n′=0
ei∆θ∆n =sin N∆θ
2
(sin (N−1)∆θ
2 + sin (N+1)∆θ2
)sin ∆θ sin ∆θ
2
.
(19)As a function of ∆θ, the expression on the right hand
side of Eq. (19) consists of a periodic array of sharppeaks, centered where ∆θ is equal to an integer multi-ple of 2π. The peaks become narrower and narrower asN becomes large. Importantly, because we parametrizethe initial state and the complete set of possible finalstates by quasienergies εi and εf within a single Flo-quet Brillouin zone, the quasienergy difference εf − εimust always be smaller than hω. Correspondingly, wehave that |∆θ| < 2π. As a result, only the peak around∆θ = 0 is relevant.
Due to the fact that∑n,n′ ei∆θ(n−n
′) is only significant
for ∆θ 1, we expand Eq. (19) for small ∆θ. This gives,for large N and ∆θ 1,
limN→∞
1
N
∑n,n′
ei∆θ(n−n′) = lim
N→∞1
N
sin2 N∆θ2
(∆θ/2)2. (20)
Substituting this result into Eq. (16), using Eq. (14), andrestoring ∆θ = ∆εT/h, gives
Wfi = limN→∞
1
NT
sin2 ∆εNT2h(
∆ε2
)2 |Mfi|2. (21)
7
Recognizing precisely the same squared sinc function thatappears in typical derivations of Fermi’s golden rule (see,e.g., Ref. [72]), we finally obtain the transition rate
Wfi =2π
h|Mfi|2δ(εf − εi). (22)
Note that in obtaining Eq. (22) we have assumed that theperturbation Hint is sufficiently weak such that Wfi ω.This gives a sufficient time window for the peaks of thefunction in Eq. (19) to become sharp on the scale of ωbefore the transition probability saturates.
To evaluate the time-averaged matrix element Mfi [seedefinition above Eq. (16)], we note that the delta func-tion sets ∆ε = 0. Thus, expanding |Φi,f (t)〉 in terms of
Fourier harmonics |φ(m)i,f 〉, we obtain
Mfi =∑mm′
1
T
∫ T
0
dt e−i(m−m′)ωt〈φ(m′)
f |Hint|φ(m)i 〉
=∑m
〈φ(m)f |Hint|φ(m)
i 〉. (23)
Remarkably, the sum over Fourier harmonic components
in Eq. (23) is precisely given by Mfi = ϕ†fHintϕi,where ϕi and ϕf are the extended space Fourier com-ponent vectors corresponding to Floquet states |ψi(t)〉and |ψf (t)〉 with quasienergies taken in the specified zoneεmin ≤ εi,f < εmin + hω, and Hint is the perturbationHamiltonian in extended space. Importantly, due to thefact that Hint is independent of time, Hint is diagonal inits Fourier component indices (i.e., all off-diagonal blocksare zero). All together, this leads to the final expressionfor Floquet Fermi’s golden rule, in terms of the extendedspace perturbation Hamiltonian and eigenvectors:
Wfi =2π
h|ϕ†f Hintϕi|2δ(εf − εi). (24)
With this expression, transition rates between Floquetstates can be calculated straightforwardly from their ex-tended space representations.
Interestingly, we note that Eq. (24) is precisely whatone would obtain for Fermi’s golden rule based on naivelyevolving in time in extended space. Specifically, considerevolution in extended space with respect to an auxiliarytime variable, τ :
ihd
dτΨ(τ) = [H+Hint]Ψ(τ), (25)
where Ψ(τ) is a vector in the extended (Fourier coeffi-cient) space, H is the extended space Hamiltonian (ofthe unperturbed system), defined as in Eq. (7), and Hint
is the extended space representation of the perturba-tion Hint. Importantly, as noted above, Hint is diago-
nal in Fourier harmonics, and takes the form H(m,m′)int =
Hintδmm′ . Here the superscript (m,m′) indicates theblock coupling Fourier component sectors m and m′.
To see the physical significance of the auxiliary timeevolution in extended space, consider the system in
the absence of the perturbation, Hint = 0, and letΨn(τ = 0) = ϕn, corresponding to a Floquet statewith quasienergy εn as in Eq. (7): Hϕn = εnϕn. Un-der the auxiliary time evolution in Eq. (25), the vectorΨn evolves to Ψn(τ) = e−iεnτ/hϕn. Using the projectorP(ωτ) defined above Eq. (8) to carry out the mappingfrom extended space back to the physical Hilbert spaceas in Eq. (8), we obtain
P(ωτ)Ψn(τ) = e−iεnτ/h∑m
e−imωτ |φ(m)n 〉
= |ψn(τ)〉. (26)
Thus the physical time evolution of each Floquet state isgenerated by auxiliary time evolution in extended space(with constant extended space Hamiltonian H) followedby projection to the physical Hilbert space via P(ωτ),and finally setting τ = t to obtain the state at time t. Bylinearity of Eq. (25), the same arguments can be appliedto the evolution of any superposition of Floquet states.
Noting that Eq. (25) defines a Schrodinger-like prob-lem with a constant perturbation, a textbook deriva-tion of Fermi’s golden rule applied directly to the ex-tended space evolution of Ψ(τ) would directly yieldEq. (24). This association is nontrivial, as the over-
lap Ψ†1(t)Ψ2(t) between two Fourier coefficient vectorsΨ1(τ = t) and Ψ2(τ = t) is generally not equal to theoverlap between the corresponding physical states |ψ1(t)〉and |ψ2(t)〉. Rather, the correct relation between the
overlaps is 〈ψ1(t)|ψ2(t)〉 = Ψ†1(t)P†(ωt)P(ωt)Ψ2(t).To characterize the action P(ωτ), it is helpful to ex-
pand the auxiliary-time-evolved Fourier component vec-tor Ψ(τ) in the basis of eigenvectors of H, ϕn:
Ψ(τ) =∑n
cn(τ)e−iεnτϕn. (27)
Generically, under the projection P(ωτ) from extendedspace to the physical Hilbert space, multiple componentsof the expansion of Ψ(τ) in Eq. (27) may map to thesame Floquet state due to the redundancy described inEq. (9) and the surrounding text. Due to this possibil-ity, the coefficients cn(τ) may not be simply related tothe amplitudes of the physical state |ψ(t)〉 expanded inthe basis of unperturbed Floquet eigenstates. Crucially,however, the conservation of (quasi)energy that emergesat long times ensures that Ψ(τ) can be decomposed asa superposition of Fourier coefficient vectors with nearlyequal quasienergies: limτ→∞ cn(τ) = 0 for εn 6= εi, wherewe recall that εi is the quasienergy of the initial state,evaluated for the unperturbed (extended space) Hamil-tonian H. This in particular implies that all componentsof limτ→∞Ψ(τ) map to physically-distinct (unperturbed)Floquet eigenstates P(ωτ)e−iεnτϕn = |ψn(t = τ)〉 underthe action of P(ωτ). Therefore, in the long-time limitwhere the transition rate is evaluated, the squares of thecoefficients in Eq. (27), |cn(τ)|2, directly give the prob-abilities of finding the physical state of the system inFloquet state |ψn(t)〉. Similar considerations apply for
8
Ψ(t) that would be obtained using perturbation theoryin Hint, as used in the derivation of Fermi’s Golden Rulein extended space, Eq. (24).
Note that we derived the transition rate Wfi by
considering the transition probability P(1)fi (NT ) at
integer multiples of the driving period [see Eq. (13)].However, the argument above shows that, at late times,the transition probability can be computed from theextended space evolution, even for non-integer multiplesof the driving period [provided that Hint is weak enoughthat Wfi ω, see comment below Eq. (22)]. Thusthe specialization to stroboscopic times was performedwithout loss of generality.
Example: Scattering rate between Floquet-Bloch statesdue to spontaneous phonon emission.— To illustrate theutility of Eq. (24), we now discuss how to use the ex-tended space formalism to analyze the scattering rate foran electron in a Floquet-Bloch band due to its couplingwith a bosonic bath of phonons. The noninteracting partof the Hamiltonian is quadratic in both the electron andphonon creation and annihilation operators,
H(t) = Hel(t) +∑q
hωqb†qbq. (28)
For simplicity, we consider a quadratic Hamiltonian forspinless electrons in a single band,
Hel(t) =∑k
c†kHk(t)ck. (29)
The discussion below can be easily generalized to elec-tronic dispersions containing multiple bands. We con-sider an electron-phonon interaction of the typical form
Hint =∑k,q
U(q)c†kck+q(b†q + b−q) + h.c., (30)
where the matrix element U(q) characterizes the strengthand form of the interaction.
For illustration, we consider a single electron in anotherwise empty Floquet band, and a zero-temperaturephonon bath. This situation will allow us to illustratethe intrinsic rates for spontaneous transitions betweenFloquet-Bloch states. Specifically, we use Eq. (24) tocalculate the rate for an electron initially in a Floquetstate |ψi(t)〉 with momentum k to scatter to a Floquetstate |ψf (t)〉 with final momentum k′, while emitting aphonon with momentum q = k− k′.
As described above, we translate the problem to ex-tended space where the Hamiltonian Htot(t) = H(t) +Hel−ph [see Eqs. (28)-(30)] is represented by the Fourier-space matrix Htot = H +Hint. We associate a completebasis of Floquet states of the noninteracting Hamilto-nian, Eq. (28), with eigenvectors of H in the extendedspace, with quasienergies within a single (the “first”)Floquet-Brillouin zone, εmin < ε < εmin + hω. Impor-tantly, the unperturbed Hamiltonian H(t) (and hence H
in extended space) acts on states in the combined Hilbertspace of electrons and phonons. Therefore each unper-
turbed quasienergy εk,nq = E(el)k + E(ph)
nq consists of a
sum of the electronic quasienergy E(el)k , which is an eigen-
value ofHel [the extended space representation of Hel(t)],
and the phonon (quasi)energy E(ph)nq =
∑q hωqnq, where
nq is the occupation number of mode q. Crucially, un-der this convention, in order for all total quasienergiesto fall within the first Floquet-Brillouin zone, the sepa-
rate electron and phonon contributions E(el)k and E(ph)
nqgenerically do not fall within the first Floquet-Brillouinzone (it is only their sum that does). The value of
the total phonon (quasi)energy E(ph)nq thereby determines
which extended space representation of the electronicstate |ψi(t)〉 is needed, using the freedom expressed inEq. (9), in order to yield a total quasienergy within theselected zone.
For the case of a zero temperature phonon bath thatwe consider, the quasienergy εi of the initial state inEq. (24) consists solely of that of the electron: εi =
E(el)i (recall that E(el)
i is an eigenvalue of Hel), with
εmin < E(el)i < εmin + hω. The quasienergy of the fi-
nal state is a sum of the electron and (emitted) phonon
contributions, εf = E(el)f + hωq. Importantly, the re-
quirement that the quasienergy is conserved in Eq. (24),
i.e., εf = E(el)f + hωq, implies that E(el)
f may not nec-essarily remain within the first Floquet-Brillouin zone.
Specifically, we may find that E(el)f lies in the interval
ger m. Such transitions, in which an electron is scatteredbetween different Floquet-Brillouin zones, are referred toas “Floquet-Umklapp” processes.
To further illustrate the nature of Floquet-Umklappprocesses, consider the case m = −1. Here the elec-tron spontaneously “falls” out the bottom of the Floquet-Brillouin zone from which it started, landing in a finalstate in the zone below. In this sense, the many redun-dant copies of the Floquet spectrum in extended space(e.g., as depicted in Fig. S1a) may act like additionalbands in the spectrum. Importantly, if the quasienergy ofthe final electronic state |ψf (t)〉 is “folded back” into thefirst Floquet-Brillouin zone [via Eq. (9)], it may appear asif the electronic quasienergy has spontaneously increasedwithin the zone. For this reason, Floquet-Umklapp pro-cesses are an important source of “quantum heating” inFloquet systems [73].
To assess the rates of Floquet-Umklapp processes andthe significance of the heating they may produce, itis necessary to evaluate the transition matrix element
|ϕ†f Hintϕi|2, where ϕi and ϕf are the extended space
Fourier component vectors describing the (tensor prod-uct) initial and final states of the electrons and phonons,together. When an electron changes Floquet-Brillouinzones, this corresponds to the transformation in Eq. (9):a shift of the electronic quasienergy by m zones shifts
9
all Fourier harmonics in the corresponding vector ϕ by
m. The matrix element |ϕ†f Hintϕi|2 is only signifi-cant if both ϕi and ϕf have their support in the sameregion of Fourier harmonic space (see Sec. II B). Thisleads to a suppression of the rates in some (but not all)Floquet-Umklapp scattering processes. Similar consider-ations apply to scattering processes arising from interac-tions of electrons with the electromagnetic environment,electron-electron interactions, and coupling to externalleads. For a detailed discussion and calculation of scatter-ing rates for Floquet-Bloch states, see, e.g., Refs. 11, 66–71.
III. GAP OPENING DUE TO CIRCULARLYPOLARIZED DRIVING FIELDS
In this section we provide an explicit calculation todemonstrate how a circularly polarized driving field canopen a gap at the Dirac point in the massless Dirac modelof Eq. (1) in Ref. [59]; see also Refs. [2, 3]. Rather thanconsidering a continuously rotating circularly polarizedharmonic driving field, we consider a four-step, piecewise-constant approximation to the circularly polarized driveas depicted in Fig. S3. In each step n, the vector potentialis held constant for a time T/4 with magnitude A0 alongthe direction en corresponding to point n in the figure.For each k, the corresponding Bloch Hamiltonian for stepn is given by Hn(k) = v[hk− eA0en] · σ.
The four-step drive maintains the chirality of the con-tinuously rotating field, while allowing for a simple exactsolution in the two-band model of Eq. (1) of Ref. [59]. Toobtain the effective Hamiltonian, we calculate the Flo-quet operator U(k, T ) = U4(k)U3(k)U2(k)U1(k), whereUn(k) = e−iHn(k)T/(4h). For each n and k, Un(k) can beobtained simply by exponentiating Pauli matrices.
To illustrate gap opening we focus on the Dirac pointand evaluate U(k = 0, T ) = e−iφσye−iφσxeiφσyeiφσx ,with φ = evA0T/(4h). In the high frequency (short pe-
1
2
3
4
Ax Ax
Ay Ay
FIG. S3. To demonstrate Floquet gap opening at a Diracpoint [Eq. (1) of Ref. [59]t], we consider a piecewise-constantdriving protocol that mimics the effect of a circularly polar-ized field. Rather than continuously rotating in the xy-plane,the vector potential with strength A0 is taken to point in se-quence along x, y, −x, and −y. In each step, the field isapplied for a duration T/4, such that the total driving periodis T ≡ 2π/ω. The Floquet evolution operator U(T ) can befound exactly within this model.
riod, T ) limit φ 1, the exponentials can be expandedto obtain
U(k = 0, T ) = 1 + φ2[σx, σy] +O(φ3) (31)
= 1 + 2iφ2σz ≈ e2iφ2σz . (32)
Using U(k = 0, T ) = e−iHeff (k=0)T/h, and T = 2π/ω, we
find Heff(k = 0) = ∆σz with ∆ = −π(evA0)2/(4hω).As this explicit calculation shows, the chiral driving
field induces a gap at the Dirac point [cf. Eq. (1) ofRef. [59]], with a magnitude that grows quadraticallywith the amplitude (linearly with the intensity) of thedrive. A very similar result is obtained for the case ofthe continuously rotating field, with a different prefac-tor. The σz term in the effective Hamiltonian arises dueto the fact that H(k, t) does not commute with itself atdifferent times. Reversing the handedness of the drivewould reverse the sign of the commutator in Eq. (31),and hence also reverse the sign of the induced gap.
IV. VANISHING OF THE WINDING NUMBERν1 FOR CONTINUOUS TIME EVOLUTION
As stated in Sec. II of Ref. [59], the winding num-ber ν1 that counts the net winding of all Floquet bandsaround the quasienergy zone must vanish for any con-tinuous evolution generated by a local, bounded Hamil-tonian. To see this, it is helpful to consider how thespectrum (band structure) of the evolution operator
U1D(k, t) = T e−(i/h)∫ t0dt′H1D(k,t′) builds up as a func-
tion of time, t. At t = 0, the evolution operator is theidentity, U1D(k, t = 0) = 1. At a small time δt > 0,the spectrum of U1D(k, δt) reflects the band structure ofthe instantaneous Hamiltonian, H1D(k, t = 0); crucially,the eigenvalues of U1D(k, δt) are periodic in k and donot exhibit any nontrivial winding when k traverses theBrillouin zone. As time advances, the spectrum remainsperiodic in k for all times, and by continuity cannot sud-denly develop a net winding. Therefore the spectrum ofthe Floquet operator U1D(k, T ) cannot host a net wind-ing of all of its bands; thus ν1 = 0.
V. DEFINITION OF THE TIME-AVERAGEDSPECTRAL FUNCTION
In Sec. IV.A of Ref. [59] (see in particular Fig. 3 andFig. S1b herein) we introduced the “time-averaged spec-tral function” as a helpful tool for visualizing how meso-scopic transport occurs in Floquet systems. In this sec-tion we give a formal definition of this quantity in termsof the system’s retarded single-particle Floquet Green’sfunction. For completeness we review some basic fea-tures of Green’s functions in Floquet systems, and discusskey similarities and differences between driven and non-driven systems (focusing on the non-interacting case).Throughout this section we set h = 1.
10
A. Formal definitions
We consider a Floquet system governed by a time-periodic Hamiltonian H(t) satisfying H(t + T ) = H(t).For a time-dependent Hamiltonian, we recall that thetransformation to the Heisenberg picture is defined by
the time evolution operator U(t, t0) = T e−i∫ tt0dt′H(t′)
,which propagates the system forward in time from timet0 to time t. (Here t0 is an arbitrary reference time,with respect to which the transformation is defined.) The(time-independent) state of the system in the Heisenbergpicture, |ψ〉H , is obtained from the Schrodinger picturestate |ψ(t)〉 via |ψ〉H = U†(t, t0)|ψ(t)〉. Importantly, ageneric time-dependent operator A(t) in the Schrodingerpicture transforms as
AH(t) = U†(t, t0)A(t)U(t, t0). (33)
Below we will work within the formalism of secondquantization. For brevity we will neglect spin through-out; the extension to include spin is straightforward. Wedefine the fermionic (“field”) creation operator Ψ†(r),that creates a particle at position r, and the annihilationoperator Ψ(r), that removes a particle from position r.These operators obey usual fermionic anticommutationrelations, Ψ(r),Ψ†(r′) = δ(r− r′).
In the position representation, the single-particle re-tarded Green’s function [74] is defined as
where the average is taken with respect to the many-bodystate of the system. Here, the Heisenberg picture cre-
ation and annihilation operators ΨH(r, t) and Ψ†H(r′, t′)are obtained from Ψ(r) and Ψ†(r′), respectively, usingEq. (33). The retarded Green’s function contains impor-tant information about the modes in which particles maypropagate in the system, and will be the central objectin the analysis below.
Similar to the case in equilibrium, for non- or weakly-interacting Floquet systems it is convenient to evalu-ate the Green’s function in Eq. (34) in the basis of thesingle-particle Floquet states (“modes”) of the systemin the absence of interactions, |ψν(t)〉. The Floquetmodes are defined in the Schrodinger picture as |ψν(t)〉 =e−iεν(t−t0)|φν(t)〉, with |φν(t + T )〉 = |φν(t)〉. Here εν isthe single-particle quasienergy associated with mode ν.In the position representation, mode ν is described bythe time-periodic wave function φν(r, t) = 〈r|φν(t)〉.
In the Schrodinger picture, we define the Floquet statecreation operator at time t as
c†ν(t) =
∫drφν(r, t)Ψ†(r). (35)
Thus, c†ν(t) creates a particle in Floquet state ν at timet. [Note that we omit the quasienergy phase factor fromthe definition of c†ν(t).] The annihilation operator cν(t)is defined analogously through Hermitian conjugation.
Crucially, for a non-interacting system, the Heisenberg
operators c†ν,H(t) and cν,H(t), defined via Eq. (33), takeon a simple form in terms of the corresponding operatorsat the reference time t0:
c†ν,H(t) = eiεν(t−t0)c†ν(t0)
cν,H(t) = e−iεν(t−t0)cν(t0). (36)
To see why Eq. (36) is true, note that the transforma-tion to the Heisenberg picture corresponds to evolvingbackwards in time from t to t0: if a particle is cre-ated in Floquet mode ν at time t, then by its defini-tion as a Floquet state it will evolve back to the cor-responding mode ν at time t0 under this transforma-tion. More formally, this property is reflected in thefact that in first quantized representation the evolutionoperator U(t, t0) for a single particle is diagonal withrespect to the Floquet states, and can be written asU(t, t0) =
∑ν e−iεν(t−t0)|φν(t)〉〈φν(t0)|.
Using Eq. (36), we obtain a crucial relation that wewill use below for the evaluation of the retarded Green’sfunction for non-interacting Floquet systems:
cν,H(t), c†ν′,H(t′) = e−iεν(t−t′)δνν′ . (37)
To obtain this result, we used the orthogonalityof the single-particle Floquet states at equal times:〈φν(t0)|φν′(t0)〉 = δνν′ . Note the similarities betweenthe relations in Eqs. (36) and (37) and the correspondingrelations for the Heisenberg picture creation and annihi-lation operators in a non-driven, non-interacting system.
B. Properties of the Floquet retardedGreen’s function
In equilibrium, with or without interactions, the single-particle retarded Green’s function [Eq. (34)] has manyuseful properties. For example GR(t, t′) (with positionor orbital indices suppressed) depends on t and t′ onlythrough the time difference t − t′. After transform-ing to frequency space, the spectral function A(ω) =− 1π Im[GR(ω)] carries information about the energies of
the characteristic single-particle modes of the system.The trace (i.e., sum over all states) of the spectral func-tion yields the density of states of the system. Impor-tantly, A(ω) is a positive semi-definite matrix (in thespace of single particle orbitals), and for non-interactingsystems does not depend on the state of the system.
In contrast, for a driven system, GR(t, t′) is a functionof both t and t′, not only the time difference. Nonethe-less, analogous spectral information to that familiar fromequilibrium systems can be obtained for Floquet systems(see Refs. [9, 10, 75–80]), provided that the system isnon-interacting and/or that it is in a time-periodic steadystate. Under these conditions, GR(t, t′) is periodic in theaverage (or “center of mass”) time t = 1
2 (t+ t′) with pe-riod T : shifting both t and t′ by T leaves the state ofthe system invariant. As shown rigorously in Ref. [80], a
11
positive spectral density analogous to that in equilibriumis then obtained by averaging GR(t + 1
2τ, t − 12τ), with
τ = t − t′, over one full period in the average time t,and then Fourier transforming with respect to the time-difference, τ .
C. Evaluation of the Floquet retarded Green’s function for a non-interacting system
In the remainder of this section we illustrate the properties above with an explicit calculation of the retarded Green’s
function for a non-interacting Floquet system. Returning to Eq. (34), we express the field operators Ψ†H(r′, t′)and ΨH(r, t) in terms of Floquet state creation and annihilation operators. There is some freedom in how to do
this; the most fruitful way is to write Ψ†H(r′, t′) = U†(t′, t0)Ψ†(r′)U(t′, t0) and Ψ(r, t) = U†(t, t0)Ψ(r)U(t, t0), and
then to express Ψ(r) and Ψ†(r′) in terms of the (Schrodinger picture) operators cν(t) and c†ν′(t′). Using the
inverse transformation of Eq. (35), this yields ΨH(r, t) =∑ν φν(r, t)cν,H(t) and Ψ†H(r′, t′) =
∑ν φ∗ν(r′, t′)c†ν′,H(t).
Substituting into Eq. (34) and using Eq. (37), we obtain
GR0 (rt; r′t′) = −iθ(t− t′)∑ν
φν(r, t)φ∗ν(r′, t′)e−iεν(t−t′). (38)
Here we use the subscript 0 to emphasize that the result holds only for non-interacting systems.Due to the periodicity of the wave functions φν(r, t), we express φν(r, t)φ∗ν(r′, t′) as a Fourier series in terms of har-
monics φ(m)ν (r) of the drive frequency, ω: φν(r, t)φ∗ν(r′, t′) =
∑mm′ e−imω(t+τ/2)eim
′ω(t−τ/2)φ(m)ν (r)
[φ
(m′)ν (r′)
]∗.
Inserting this expression into Eq. (38) and averaging with respect to t over one full period (for fixed τ), we obtain thetime-averaged single-particle retarded Green’s function, GR0 (r, r′; τ):
GR0 (r, r′; τ) ≡ 1
T
∫ t0+T
t0
dtGR0 (r,t+ τ/2; r′, t− τ/2) (39)
= −iθ(τ)∑ν,m
e−i(εν+mω)τφ(m)ν (r)
[φ(m)ν (r′)
]∗.
Taking the Fourier transform GR0 (r, r′; Ω) = limη→0+
∫∞−∞ dτ ei(Ω+iη)τ GR0 (r, r′; τ), we obtain
GR0 (r, r′; Ω) = limη→0+
∑ν,m
φ(m)ν (r)
[φ
(m)ν (r′)
]∗εν +mω − Ω + iη
. (40)
Analogous to equilibrium, we define the “time-averaged density of states” via ρ0(Ω) = − 1πTr Im[GR0 (Ω)]:
ρ0(Ω) =∑ν,m
A(m)ν δ(εν +mω − Ω), A(m)
ν = 〈φ(m)ν |φ(m)
ν 〉, (41)
where A(m)ν captures the spectral weight of the m-th harmonic/sideband component |φ(m)
ν 〉 in the Floquet state |ψν(t)〉.(Here |φ(m)
ν 〉 is the ket corresponding to the component φ(m)ν (r) of the position-space Floquet state wave function
defined above.)As seen in Eq. (41), the time-averaged density of states is comprised of delta-function peaks at frequencies corre-
sponding to the quasienergies of the single-particle Floquet states of the system, plus or minus all integer multiples
of the driving field photon energy, hω. The weight of each peak is governed by the spectral weight A(m)ν of the
corresponding sideband component of the Floquet state.To obtain the time-averaged spectral function, we single out the contribution from a given Floquet eigenstate ν:
Aν(Ω) =∑mA
(m)ν δ(εν + mω − Ω). The time-averaged spectral function captures the fact that the spectral weight
of each Floquet state is spread in frequency (energy) over several discrete harmonics. As displayed in Fig. S1b, thespectral function for graphene subjected to circularly polarized light shows, in highest weight, the original graphene(projected) band structure, interrupted by dynamically-induced band gaps at zero energy and at resonances wherestates in the original conduction and valence bands are separated by integer multiples of hω. Sidebands (of lowspectral weight) appear as faint copies of these bands, shifted up and down by integer multiples of hω. As discussedin Ref. [59], the time-averaged spectral function thus provides a helpful way to resolve how states at specific energiesin the leads couple to Floquet states with given quasienergies in the system [10, 81–83].
12
VI. FLOQUET-KUBO FORMULA
In this section we discuss the linear response of a Floquet system to the application of an additional weak perturbingfield, oscillating at a frequency Ω that we assume to be much smaller than the driving frequency ω. (For an extendeddiscussion of linear response theory for Floquet systems, see Ref. 84.) The Hamiltonian, including the perturbation, is
given by H(t) = H0(t) +F (t)B(t), where F (t) is a general function of time, and B(t) is an operator that may depend
on time explicitly in the Schrodinger picture, albeit in a periodic manner with the same period as H0: B(t) = B(t+T ).The derivation given in this section closely follows the derivation of the Kubo formula for equilibrium systems [74].For a derivation of the Floquet-Kubo formula based on the tt′ formalism, see Ref. 85. Throughout this section wewill denote operators using hats to avoid possible ambiguity about which objects are operators and which are simplyreal or complex numbers.
As a starting point, we assume that in the absence of the perturbing field the system has reached a time-periodicstationary state, described by a density matrix ρ0(t), with ρ0(t+ T ) = ρ0(t). The change in the expectation value of
a constant or time-periodic operator A(t) in response to the perturbing field F (t) is given by
δ〈A(t)〉 =
∫ ∞−∞
dt′ χ(t, t′)F (t′), (42)
where δ〈A(t)〉 = Tr[ρ(t)A(t)
]− Tr
[ρ0(t)A(t)
]. Here, ρ(t) is the state of the system in the presence of the pertur-
bation. Analogous to the case in equilibrium, the retarded response function χ(t, t′) is defined as
χ(t, t′) = −iθ(t− t′) Trρ0(t0)
[AI(t), BI(t
′)]
, (43)
where the interaction picture operator AI(t) is defined by the transformation
AI(t) = U†0 (t, t0)A(t)U0(t, t0), (44)
with U0(t, t0) = T e−i∫ tt0H0(t′)dt′
, where T denotes time ordering. The operator BI(t′) is defined in a similar manner.
Note that although the arbitrary reference time t0 appears explicitly in Eq. (43), the response function χ(t, t′) is in
fact independent of t0. This can be checked by using the definitions of ρ0(t0) and of the operators AI(t) and BI(t′).
Since the system is assumed to be in a steady state, the response function χ(t, t′) is invariant under shifting both tand t′ by T , the driving period. To see why, we note the steady state condition implies that ρ0(t0) = ρ0(t0− T ). The
time-periodicity of the Hamiltonian implies U0(t+ T, t0) = U0(t, t0 − T ), and therefore
Trρ0(t0)
[AI(t+ T ), BI(t
′ + T )]
= Trρ0(t0 − T )U†0 (t0, t0 − T )
[AI(t), BI(t
′)]U0(t0, t0 − T )
= Tr
ρ0(t0)
[AI(t), BI(t
′)]
.(45)
Transforming variables using τ ≡ t− t′ we see that the function χ(τ, t′) ≡ χ(t′ + τ, t′), given by
χ(τ, t′) = −iθ(τ) Trρ0(t0)
[AI(t
′ + τ), BI(t′)]
, (46)
is periodic in t′ with period T (for fixed τ). Thus χ(τ, t′) has a mixed Fourier representation in terms of one continuousfrequency variable, Ω, conjugate to τ , and a discrete set of harmonics, m, which capture its periodicity in t′:
χ(t− t′, t′) =∑m
∫ ∞−∞
dΩ e−iΩ(t−t′)e−imωt′χ(m)(Ω). (47)
Inserting Eq. (47) into Eq. (42) and taking the Fourier transform of both sides with respect to t, we obtain
δ〈A〉(Ω) =∑m
χ(m)(Ω)F (Ω−mω). (48)
Suppose that i) we probe the observable A(t) at frequencies Ω that are much smaller than the drive frequency, ω,and ii) the Fourier transform of the perturbing field, F (Ω′), only has support at frequencies Ω′ ω. (The lattercaptures our original assumption that the system is perturbed at a frequency much less than that of the drive.) Underthese conditions, the only significant term in the sum in Eq. (48) corresponds to m = 0, giving
δ〈A〉(Ω) = χ(0)(Ω)F (Ω). (49)
Note that χ(0)(Ω) is just the Fourier transform of the “time-averaged response function,” χ(τ) = 1T
∫ T0dt′χ(τ, t′),
with respect to the variable τ .
13
A. Floquet-Kubo Formula for the electrical conductivity
We now focus on the response of an electromagnetically-driven electronic Floquet system to a uniform probing ACelectric field oscillating at a frequency Ω, where as before Ω ω [2, 86]. The extension to finite wave vector probefields can be performed using a similar approach. The electromagnetic gauge potential is then a sum of two terms,A(t) = A0(t) + δA(t), where A0(t) = A0(t + T ) describes the time-periodic driving field (which we aim to includeexactly), and δA(t) describes the weak probe field. To second order in δA(t), the Hamiltonian is given by
H(t) = H(t)∣∣∣δA(t)=0
+∂H(t)
∂A(t)
∣∣∣∣∣δA(t)=0
· δA(t) +∂2H(t)
∂Aα(t)∂Aβ(t)
∣∣∣∣∣δA(t)=0
δAα(t)δAβ(t), (50)
where α, β = x, y, z. The current operator J(t) =(Jx(t), Jy(t), Jz(t)
)is given by
J(t) = −∂H(t)
∂A(t). (51)
Importantly, due to the time-dependence of A0(t) in H(t), the current operator J(t) is time-dependent (in theSchrodinger picture). In analogy to common practice in equilibrium systems, we now define a “paramagnetic current
operator” J(p)(t) and the “kinetic” operator Kαβ(t) as [87, 88]:
J(p)(t) = −∂H(t)
∂A(t)
∣∣∣∣∣δA(t)=0
, Kαβ(t) = − ∂2H(t)
∂Aα(t)∂Aβ(t)
∣∣∣∣∣δA(t)=0
. (52)
Using Eqs. (50)–(52), the expectation value of the current, up to first order in δA(t), can be written as a sum of“paramagnetic” and “diamagnetic” contributions,
〈Jα(t)〉 = 〈J (p)α (t)〉+
∑β
Tr[ρ0(t)Kαβ(t)
]δAβ(t). (53)
For clarity we note that the expectation value 〈Jα(t)〉 ≡ Tr [ρ(t)Jα(t)] of the time-dependent operator Jα(t) is takenwith respect to the (full) state of the system ρ(t) at the same time, t. To obtain an equation valid within the regime
of linear response, in Eq. (53) we expand the expectation value 〈J (p)α (t)〉 up to linear order in the probe field δA(t),
while we take the expectation value in the second term on the RHS with respect to the unperturbed state ρ0(t).The change in the paramagnetic current due to the probe field δA(t) (relative to any time periodic current that
may flow in the steady state) is given by
δ〈J (p)α (t)〉 =
∫ ∞−∞
dt′χαβ(t− t′, t′)δAβ(t′), (54)
where
χαβ(τ, t′) = −iθ(τ) Trρ0(t0)
[J
(p)α,I(t
′ + τ), J(p)β,I(t
′)]
. (55)
Taking the Fourier transform of Eq. (54) with respect to t, and using the periodicity of χαβ(t− t′, t′) with respect tot′, we obtain
δ〈J (p)α 〉(Ω) =
∫ ∞−∞
dt′∑m
ei(Ω−mω)t′χ(m)αβ (Ω)δAβ(t′). (56)
For the electromagnetic gauge potential, we choose a gauge for which E(t) = − ddtA(t). Using integration by parts on
Eq. (56) to obtain a time derivative on δAβ(t) yields
δ〈J (p)α 〉(Ω) =
∑m
χ(m)αβ (Ω)
i(Ω−mω)δEβ(Ω−mω). (57)
For electric fields that contain only low frequencies, Ω ω, only the m = 0 component contributes to Eq. (57).
14
In the following we will assume that the gauge (probe) field δA(t) appears only in the quadratic part of the
Hamiltonian, H(2)(t) =∑ij hij(t)d
†i dj , where d†i is an electronic creation operator with respect to a fixed, time-
independent basis. In terms of the basis of Floquet states, the quadratic part of the Hamiltonian can be written as
H(2)(t) =∑
k hν1ν2(k, t)c†kν1(t)ckν2(t), where c†kν(t) are creation operators of electrons in single particle Floquet
states, see Eq. (35), and ν is a Floquet band index. We then express the paramagnetic part of the current operator as
J (p)α (t) =
∑k
jα,ν1ν2(k, t)c†kν1(t)ckν2(t), (58)
where the matrix elements jα,ν1ν2(k, t) are given by
jα,ν1ν2(k, t) = −∂hν1ν2(k, t)
∂Aα(t)
∣∣∣∣∣δA(t)=0
= −⟨φk,ν1(t)
∣∣ ∂h(t)
∂Aα(t)
∣∣φk,ν2(t)⟩∣∣∣δA(t)=0
, (59)
where h(t) is the single-particle (first quantized) operator corresponding to the matrix hij(t).We focus on the case of a diagonal steady state of the form
ρ0(t) =∏kν
[fkν c†kν(t)ckν(t) + (1− fkν)ckν(t)c†kν(t)], (60)
where fkν is the population of the Floquet state created by c†kν(t). As a useful preliminary for the evaluation of theresponse function, we note that for a steady state of the form in Eq. (60), the analogue of Eq. (36) for the interactionpicture yields
Tr[ρ0(t0)c†I,kν(t1)cI,k′ν′(t2)
]= fkνe
iεkν(t1−t2)δkk′δνν′ . (61)
We now use the results and definitions above to evaluate the linear response conductivity of the Floquet system.Using Eq. (58)–Eq. (61), the response function χαβ(τ, t′) is evaluated to be
Using the time-periodicity of jα,ν1ν2(k, t) [see definition in Eq. (59)], we expand jα,ν1ν2(k, t) =∑m e−imωtj(m)
α,ν1ν2(k).Averaging the time variable t′ over one period and taking the Fourier transform with respect to τ , we obtain thetime-averaged response function
χ(0)αβ(Ω) = lim
η→0+
∑k
∑ν1ν2
∑m
(fkν1 − fkν2) j(m)α,ν1ν2(k)j
(−m)β,ν2ν1
(k)
Ω−mω + (εkν1 − εkν2) + iη. (63)
Using the gauge E(t) = − ddtA(t) as before, we obtain a contribution from the diamagnetic (second) term in Eq. (53)
to the total current 〈Jα〉(Ω) ≡∫∞−∞ dt eiΩt〈Jα(t)〉 that is given by:∫ ∞
−∞dt eiΩt Tr
[ρ0(t)Kαβ(t)
]δAα(t) =
∑m
K(m)αβ
i(Ω−mω)δEα(Ω +mω). (64)
In writing Eq. (64) we have used the time-periodicity of Kαβ(t) = Tr[ρ0(t)Kαβ(t)
]and expanded Kαβ(t) =∑
mK(m)αβ e
−imωt. Using this expansion, Eq. (64) is obtained by integration by parts. Note that for electric fields
that contain only low frequency components Ω ω, only the term m = 0 in Eq. (64) gives a contribution to thecurrent. In this case, this second contribution to the current is given by the time-average of Kαβ(t),
K(0)αβ =
1
T
∫ T
0
dtKαβ(t). (65)
Finally we obtain Ohm’s law for the response to the probe field, δ〈Jα〉(Ω) = σαβ(Ω)δEβ(Ω):
σαβ(Ω) =χ
(0)αβ(Ω) +K(0)
αβ
iΩ, (for Ω ω). (66)
15
Note the similarity between this result and the analogous expression for the conductivity in equilibrium systems, as
well the form of the response function χ(0)αβ(Ω) in Eq. (63).
VII. ACKNOWLEDGEMENTS
We thank all of our collaborators on FTI-related work,with whom we have had many stimulating interactions.In particular, we acknowledge Erez Berg, Eugene Dem-ler, Victor Galitski, Takuya Kitagawa, Michael Levin andGil Refael, with whom we began our journey in this field.N. L. acknowledges support from the European ResearchCouncil (ERC) under the European Union Horizon 2020
Research and Innovation Programme (Grant AgreementNo. 639172), and from the Israeli Center of ResearchExcellence (I-CORE) “Circle of Light”. M. R. grate-fully acknowledges the support of the European ResearchCouncil (ERC) under the European Union Horizon 2020Research and Innovation Programme (Grant AgreementNo.678862), the Villum Foundation, and CRC 183 of theDeutsche Forschungsgemeinschaft.
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