High-frequency approximation for periodically driven quantum systems from a Floquet-space perspective

We derive a systematic high-frequency expansion for the effective Hamiltonian and the micromotion operator of periodically driven quantum systems. Our approach is based on the block diagonalization of the quasienergy operator in the extended Floquet Hilbert space by means of degenerate perturbation theory. The final results are equivalent to those obtained within a different approach [Phys.\ Rev.\ A {\bf 68}, 013820 (2003), Phys.\ Rev.\ X {\bf 4}, 031027 (2014)] and can also be related to the Floquet-Magnus expansion [J.\ Phys.\ A {\bf 34}, 3379 (2000)]. We discuss that the dependence on the driving phase, which plagues the latter, can lead to artifactual symmetry breaking. The high-frequency approach is illustrated using the example of a periodically driven Hubbard model. Moreover, we discuss the nature of the approximation and its limitations for systems of many interacting particles.

A prerequisite for Floquet engineering is a theoretical method to compute the effective Hamiltonian (as well as the micromotion operator), at least within a suitable approximation. In the high-frequency limit a rotatingwave-type approximation can be employed for this purpose. This approximation coincides with the leading order of a systematic high-frequency expansion that also provides higher-order corrections to the effective Hamiltonian and the micromotion operator [42][43][44][45]. In this paper we show that this high-frequency expansion can be obtained by employing degenerate perturbation theory in the extended Floquet Hilbert space. Our approach provides an intuitive picture of the nature of the approximation and the conditions under which it can be expected to provide a suitable description of a driven quantum system. We point out that the time scale on which the approximation is valid can be increased by increasing the order of the approximation for the effective Hamiltonian, while keeping a lower-order approximation for the time-periodic micromotion operator. We also address the relation between the high-frequency expansion derived here and the Floquet-Magnus expansion [46] (see also [47][48][49]). The origin of a spurious dependence of the quasienergy spectrum in Floquet-Magnus approximation on the driving phase is discussed (see also references [42,43,45]). Using the example of a circularly driven tight-binding lattice, this artifact is, moreover, shown to produce a non-physical breaking of the rotational symmetry in the approximate quasienergy band structure. Finally, we discuss the validity of the high-frequency approximation for systems of many interacting particles. This paper is organized as follows. Section 2 gives a brief introduction to the theory of periodically driven quantum systems (Floquet theory) and serves to define our notation. In section 3 we formulate the problem that is then attacked in section 4 by means of the degenerate perturbation theory developed in appendix C. The relation to the Floquet-Magnus expansion is discussed in section 5, and section 6 illustrates the approximation scheme using the example of a circularly driven hexagonal lattice [27,38,39]. Finally section 7 discusses effects of interactions within and beyond the high-frequency approximation, before we close with a brief summary in section 8.
2. Quantum Floquet theory and notation 2.1. Floquet states  . If the system is prepared in a coherent superposition of several Floquet states, the time evolution will no longer be periodic and will instead be determined by two contributions. The first contribution stems from the periodic time dependence of the Floquet modes ñ | ( ) u t n and is called micromotion. The second contribution, which leads to deviations from a periodic evolution, originates from the relative dephasing of the factors  e e t i n . Thus, beyond the periodic micromotion, the time evolution of a Floquet system is governed by the quasienergies e n of the Floquet states in much the same way as the time evolution of an autonomous system (with time-independent Hamiltonian) is governed by the energies of the stationary states.

Floquet Hamiltonian and micromotion operator
In order to study the dynamics over time spans that are long compared to a single driving period, one can ignore the micromotion by studying the time evolution in a stroboscopic fashion in steps of the driving period T. Such a stroboscopic time evolution is described by the time-independent Floquet HamiltonianĤ t F 0 . It is defined such that it generates the time evolution over one period, It is periodic in both arguments, , , F F F 2 1 2 1 2 1 . If the Floquet states and their quasienergies are known, e.g. from computing and diagonalizing the time evolution operator over one period, one can immediately write down the Floquet Hamiltonian and the micromotion operator by making use of equations (9) and (10). However, both the Floquet HamiltonianĤ t F 0 and the micromotion operator F might also be computed directly, without computing the Floquet states and the quasienergies beforehand. This will be the aim of the approximation scheme described in the main part of this paper. From the Floquet Hamiltonian and the micromotion operator one can then immediately write down the time evolution operator as e , , e . 12 The periodic time-dependence of the Floquet modes can subsequently be computed by employing the micromotion operator,

Quasienergy eigenvalue problem and extended Floquet Hilbert space
The phase factors  e e T i n and the Floquet states y ñ | ( ) t n , solving the eigenvalue problem of the time-evolution operator over one period, are uniquely defined (apart from the freedom to multiply each Floquet state by a time independent phase factor). In turn, the quasienergies e n , and with them also the Floquet modes  y ñ = ñ e | ( ) | ( ) u t t e n t n i n and the Floquet Hamiltonian (9), are not defined uniquely. Namely, adding an integer multiple of w to the quasienergy e n does not alter the phase factor  e e T i n . Fixing each quasienergy e n within this freedom fixes also the Floquet modes and the Floquet Hamiltonian. For example, one can choose all quasienergies to lie within the same interval of width w, often called a Brillouin zone. This term reflects a loose analogy to the theory of spatially periodic Hamiltonians, where the quasimomentum can be chosen to lie within a single reciprocal lattice cell such as the first Brillouin zone.
Starting from the known solution given by ñ | ( ) u t n and e n , one can label all possible choices for the quasienergy by introducing the integer index m, The corresponding Floquet mode reads When entering the right-hand side of equation (16) into the time-dependent Schrödinger equation (2), we arrive at This equation constitutes an eigenvalue problem in an extended Hilbert space    = Ä T [1,2]. This space is given by the product space of the state space  of a quantum system and the space of square-integrable Tperiodically time-dependent functions  T . Time is treated as a coordinate under periodic boundary conditions. In the extended Floquet Hilbert space  , the scalar product combines the scalar product of  with time averaging and is defined by We will use a double ket notation ññ |u for elements of  ; the corresponding state at time t in  will be denoted by including its full periodic time dependence, is denoted by ññ |v when considered as element of  . In the following sections, we will stick to this convention and conveniently switch between both representations. Likewise, an operator acting in  will be indicated by an overbar to distinguish it from operators acting in , which are marked by a hat. For example,Q denotes the  -space operator that in  is represented by The operatorQ is called quasienergy operator. It is hermitian in  and, as can be inferred from equation (17), its eigenstates and eigenvalues are the Floquet modes and their quasienergies, The complete set of solutions of the quasienergy eigenvalue problem (20) contains a lot of redundant information. In the extended space ññ |u nm and ññ ¢ |u nm constitute independent orthogonal solutions if ¢ ¹ m m. These solutions are, however, related to each other by equations (14) and (15), and give rise to the same Floquet state y ñ | ( ) t n . All Floquet states y ñ | ( ) t n of the system can, thus, be constructed, for example, from those Floquet modes whose quasienergies lie in a single Brillouin zone of the w-periodic quasienergy spectrum.
The quasienergy eigenvalue problem (20) provides a second approach for computing the Floquet states or the Floquet Hamiltonian, alternative to the computation and diagonalization of the time evolution operator over one driving period. It provides the Floquet modes not only at a time t 0 , but including their full periodic time dependence. Despite the drastically increased Hilbert space, treating the quasienergy eigenvalue problem (20) has also advantages. In order to diagonalize the hermitian quasienergy operatorQ, one can employ methods, concepts, and intuition from the physics of systems with time-independent Hamiltonians. When describing parameter variations, such as a smooth switching on of the driving amplitude, one can even derive a Schrödinger-type evolution equation acting in Floquet space and apply the adiabatic principle [50]. 4 A complete set of orthonormal basis states a ññ | m of  can be constructed by combining a complete set of orthonormal basis states añ | of  with the complete set of time-periodic functions w e im t labeled by the integer m,  acting in . The structure of the quasienergy operatorQ resembles that of the Hamiltonian describing a quantum system with Hilbert space  coupled to a photon-like mode in the classical limit of large photon numbers, where the spectrum becomes periodic in energy. In this picture m plays the role of a relative photon number. The quasienergy eigenvalue problem (20) is, thus, closely related to the dressed-atom picture [52,53] for a quantum system driven by coherent radiation [54]. Based on this analogy, one often uses the jargon to call m the 'photon' number. Moreover, the matrix elements ofĤ m are said to describe m-'photon' processes. This terminology suggests a very intuitive picture for the physics of time-periodically driven quantum systems and is also employed when the system is actually not driven by a photon mode.
In order to diagonalize or block diagonalize the quasienergy operator, it is natural and sufficient to consider unitary operatorsŪ that are translationally invariant with respect to the photon index m, a a a a áá ¢ ¢ ññ = á ¢ ñ acting in  (see also appendix B). From equation (19) we can infer that a unitary transformation with such an operatorŪ , is equivalent to a gauge transformation  y y y   The unitary operatorŪ D that diagonalizes the quasienergy operator with respect to a certain basis a ññ | m , is constructed such that it leads to a time-independent gauge-transformed Hamiltonian

Block diagonalization of the quasienergy operator and effective Hamiltonian
The quasienergy eigenvalue problem (20) which by construction is time independent. In fact, choosing an operatorŪ F that block diagonalizes the quasienergy operator is equivalent to choosingˆ( ) U t F such that the gauge transformation (33) leads to a timeindependent HamiltonianĤ F . This time-independent HamiltonianĤ F is called an effective Hamiltonian. Note that the unitary operatorŪ F is not determined uniquely. For example, multiplyingˆ( ) U t F with any timeindependent unitary operator from the right leads to a mixing of states within the diagonal blocks of¯¯ † U QU F F , but F one can then directly obtain the time evolution operator using equation (12). However, the time evolution operatorˆ( ) U t t , 2 1 can also be expressed directly in terms ofĤ F andˆ( . Namely [42], Compared to the representation (12) of the time-evolution operator in terms of the Floquet HamiltonianĤ t F 0 and the micromotion operatorˆ( ) U t t , F 2 1 , this expression has the disadvantage that it is a product of three operators and not just of two. However, using the representation (37) has also advantages. The micromotion has been expressed by the one-point micromotion operatorˆ( ) U t F , instead of by the two-point operator F , and the phase evolution is described by an effective HamiltonianĤ F without the parametric dependence on the switching time t 0 ofĤ t F 0 . The micromotion operatorˆ( ) U t F can also be expressed as in terms of an anti-hermitian operator = -ˆ † G G . The hermitian operator = ( )ˆ( ) K t G t i has recently been given the intuitive name kick operator [43].
The diagonalization of the effective HamiltonianĤ F , e ñ = ñ |˜|˜( ) H u u , 3 9 F n n n provides the Floquet modes and their quasienergies: , which describe the micromotion, are superpositions The strategy of computing the effective Hamiltonian directly, without computing the Floquet states beforehand, separates the Floquet problem into two distinct subproblems related to the short-time and the longtime dynamics, respectively. The first problem, computing the effective Hamiltonian (as well as the micromotion operator), concerns the short-time dynamics within one driving period only. The second problem consists of the integration of the time evolution generated by the effective Hamiltonian for a given initial state or even in the complete diagonalization of the effective Hamiltonian. This separation allows us to address the longtime dynamics over several driving periods in a very efficient way, without the need to follow the details of the dynamics within every driving period.
The advantage of splitting the Floquet problem into two parts becomes apparent especially when one of the two problems is more difficult than the other. A simple example for a case where computing the effective Hamiltonian is more difficult than diagonalizing it is a periodically driven two-level system corresponding to a spin-1/2 degree of freedom. While the block diagonalization of the quasienergy operator can generally not be accomplished analytically, the effective Hamiltonian describes (like every time-independent2 2 Hamiltonian) a spin-1/2 in a constant magnetic field leading to a simple precession dynamics on the Bloch sphere. Thus, once the effective Hamiltonian and the micromotion operator are computed, the time evolution is known. An example for the opposite case, where the effective Hamiltonian can be computed at least approximately while its diagonalization is much harder, is a time-periodically driven Hubbard-type model [19]. It describes interacting particles on a tight-binding lattice. This driven model allows for a quantitative description of experiments with ultracold atoms in optical lattices. In the limit of high-frequency forcing a suitable analytical approximation to the effective Hamiltonian can be well justified on the time scale of a typical optical lattice experiment. However, the effective Hamiltonian will constitute a many-body problem that is difficult to solve.
The possibility to compute the effective Hamiltonian for a many-body lattice system, at least within a suitable approximation, is also the basis for a novel and powerful type of quantum engineering. Here the properties of the effective HamiltonianĤ F are tailored by engineering the periodic time dependence of the Hamiltonianˆ( ) H t . This Floquet engineering has recently been successfully applied to ultracold atomic quantum gases (see references in the introduction). The fact that the effective Hamiltonian can possess properties that are hard to achieve otherwise, like the coupling of the kinetics of charge-neutral atoms to a vector potential describing an (artificial) magnetic field [27, 28, 30-32, 34-39, 55-60], makes Floquet engineering interesting for quantum simulation as well. Here, a quantum mechanical many-body model is realized accurately in the laboratory in order to investigate its properties by doing experiments. An essential prerequiste for Floquet engineering is an accurate approximation to the effective Hamiltonian. In the next section we will systematically derive a high-frequency approximation to both the effective Hamiltonian and the micromotion operator by block diagonalizing the quasienergy operator by means of degenerate perturbation theory.

High-frequency expansion from degenerate perturbation theory
Degenerate perturbation theory is a standard approximation scheme for the systematic block diagonalization of a hermitian operator into two subspaces-a subspace of special interest on the one hand and the rest of state space on the other-that are divided by a large spectral gap. Here we adapt the method such that it allows for a systematic block diagonalization of the quasienergy operator with respect to the 'photon' index m (appendix C). Moreover, we will identify the system-independent 'photonic' part  -i d t of the quasienergy operator (19), with , as the unperturbed problem. As a consequence the system-specific Hamiltonianˆ( ) H t constitutes the perturbation. This will allow us to systematically derive simple and universal expansions for both the effective HamiltonianĤ F and the micromotion operatorˆ( ) U t F in the high-frequency limit, where w constitutes a large spectral gap between the unperturbed subspaces (see figure 1). We would like to point out that the application of degenerate perturbation theory in the extended Floquet Hilbert space is a well-established method. For example, it has recently been employed to estimate the matrix element for the resonant creation of collective excitations in a driven Bose-Hubbard model [51] and to treat a dissipative driven two-level system [61].
The basic strategy of our perturbative approach can be summarized as follows. The quasienergy operator is divided into an unperturbed partQ 0 and a perturbationV , The unperturbed operator can be diagonalized and separates the extended Floquet Hilbert space  into uncoupled subspaces  ( ) These subspaces shall be separated by unperturbed spectral gaps of the order of w, which are assumed to be large compared to the strength p of the perturbation coupling states of different subspaces. When smoothly switching on the perturbation, such that the spectral gaps do not close, the unperturbed subspaces  ( ) m 0 will be transformed adiabatically to the perturbed subspaces  m corresponding to a diagonal block of the perturbed problem. Since the perturbation is weak compared to the gap,  m will differ from  ( ) spanning the perturbed subspaces  m . In contrast, if the spectral gap separating different subspaces were to close, arbitrary weak coupling could hybridize degenerate states of different subspaces, contrary to the assumption of a weak perturbative admixture. The general formalism is developed in appendix C and will be applied to a specific choice of the unperturbed problem in the following. . The operator ¢ Q 0 is diagonal with respect to the basis states a ññ | m by construction and the corresponding perturbation ¢ = -¢¯V Q Q 0 consists of a block-diagonal part ¢ V D that couples states a ññ | m and a¢ ññ | m of the same 'photon' number m and a block-off-diagonal part ¢ V X that couples states a ññ | m and a¢ ¢ññ | m of different 'photon' numbers m′ and m. The problem to be solved by perturbation theory is visualized in figure 3(a). The unperturbed problem and the perturbation expansion depend on the choice of the basis states añ | . However, for the sake of simplicity we will not use equation (46). Instead we will simplify the unperturbed problem further, reducing it to the 'photonic' part of the quasienergy operator, . SoQ 0 is diagonal not only with respect to a specific set of basis states, but with respect to any set of basis states of the type a ññ | m . The perturbation is given by the Hamiltonian, It can be decomposed as Here the block-diagonal partV D comprises the ¢ = m m terms describing zero-'photon' processes determined by the time-averaged Hamiltonian, . The unperturbed problem is given by to be used here; all unperturbed states of identical 'photon' number m are degenerate.
The block-off-diagonal partV X describes Dm-'photon' processes determined by the Fourier components D H m of the Hamiltonian, The problem is visualized in figure 3(b). Its simple structure will allow us to write down universal analytical expressions for the leading terms of a perturbative high-frequency expansion of the effective Hamiltonian and the micromotion operator in powers of w p , with p symbolizing the perturbation strength. Before moving on, we note in passing that it can be useful to shift the 'photon' number of an unperturbed state a ññ | m by some integer Dm, before applying the high-frequency approximation. Such a procedure can be useful, if two states a ñ | 1 and a ñ . This redefinition is equivalent to a gauge transformation (25) 2 is employed to shift the time-averaged energy of a ñ | 2 by w -Dm . After this transformation the high-frequency approximation can be applied and used to describe the resonant coupling between both states a ñ | 1 and a ñ | 2 . Such a procedure can be employed, for example to describe resonant 'photon'-assisted (or AC-induced) tunneling against a strong potential gradient [12,45].

Micromotion
We wish to compute the unitary operatorŪ F that relates the unperturbed basis states a ññ | m to the perturbed basis states a ññ | m F that block diagonalize the quasienergy operator in a perturbative fashion. In the canonical van Vleck degenerate perturbation theory, it is written as In order to minimize the mixing of unperturbed states belonging to the same unperturbed subspace, it is, moreover, required thatḠ is block-off-diagonal. One can now systematically expandḠ as in powers of the perturbation. The general formalism for the perturbative expansion ofḠ in a situation where the state space is partitioned into more than just two subspaces is described in appendix C. Differences with respect to the standard procedure, where the state space is only bipartitioned, arise as a consequence of the fact that for multipartitioning it is generally no longer true that the product of two block-off-diagonal operators is block diagonal. The general form of the leading terms of the expansion (58) is given by equations (C. 40) and (C.41) of appendix C. Let us evaluate them for the particular choice of the unperturbed problem (47) for all diagonal matrix elements, following directly fromḠ being block- We can now also expand the unitary operatorŪ F in powers of the perturbation, where the second term of the last equation possesses matrix elements which are finite also for ¢ = m m. The corresponding operators in  can be constructed by employing the relation The leading terms of the perturbation expansion take the form  One can express these terms also as time integrals. For the leading order we obtain In the final result we have separated a factor of which is formula 1.441-1 of reference [62].
One can now approximateˆ( ) U t F up to a finite order ñ by simply truncating the perturbative expansion of ( ) However, this approximation has the disadvantage that it does not preserve unitarity at any finite order ñ . In turn, truncating the expansion ofˆ( ) G t leads to an approximation ⎛ The unitary two-point micromotion operator can be written as , , and comparing the epxansion of ¢ and so on. This gives the explicit expressions for the leading orders An approximation preserving the unitarity of the micromotion operator reads ⎛

Effective Hamiltonian
In order to obtain the effective Floquet Hamiltonian from equation (34), we need to compute the matrix elements (32) Expanding these matrix elements in powers of the perturbation, the leading terms a a n ¢ ( ) Q 0, are given by equations (C.50)-(C.53) of appendix C. Evaluating these expressions for the unperturbed problem (47), we obtain the perturbative expansion for the effective Hamiltonian . The leading terms are given by One can express these terms also in terms of time integrals. The leading order is given by the time-averaged Hamiltonian, The first correction takes the form where the sum over m has been evaluated using equation (75) and where we have separated a factor of ( ) T 1 2 2 representing the inverse integration area. In ñ th order the effective Hamiltonian is approximated by The results obtained here via degenerate perturbation theory in the extended Floquet Hilbert space are equivalent to the high-frequency expansion derived in references [42][43][44] by different means.

Role of the driving phase
An important property of the approximation (92) to the effective Hamiltonian is that it is independent of the driving phase. Namely, a shift in time 9 3 which leads to 9 4 m m m t m i does not alter the perturbation expansion ofĤ F , This is ensured by the structure of the perturbation theory, which restricts the products . Additionally, as an immediate consequence, the approximate quasienergy spectrum, obtained from the diagonalization ofĤ F , does not acquire a spurious dependence on the driving phase. In this respect, the high-frequency approximation obtained by truncating the high-frequency expansion ofĤ F at finite order, equations (76) and (92), is consistent with Floquet theory.
A time shift does, however, modify the terms of the unitary operatorˆ( ) U t F in the expected way, to the Floquet HamiltonianĤ t F 0 . The reason why it is generally useful to choose n¢ independent of ν is the following. In high-frequency approximation the time evolution from t 0 to t is described by captures the true micromotion of the system does not depend on the time span -( ) t t 0 of the integration, simply because this expression is time periodic. In turn, with increasing integration time - will deviate more and more from their actual value . Thus, the longer the time span t t 0 the better should be the approximation e e » n [ ] n n -that is, the larger ν should be. In contrast, the order n¢ can be chosen independently of -( ) t t 0 .

Relation to the Floquet-Magnus expansion
In this section we relate the high-frequency expansion ofĤ F andˆ( ) G t to the Floquet-Magnus expansion [46] (see also [47,48,63]). A discussion of this issue can also be found in references [42,43,45]. Recently, the Floquet-Magnus expansion has been employed frequently for the treatment of quantum Floquet systems. The starting point of the Floquet-Magnus expansion is the form (12) of the time evolution operator,    H F given by equation (92). This observation can be traced back further to the ansatz (103) for the time evolution operator. Bipartitioning the time-evolution operator into two exponentials like in equation (103) does not allow for disentangling the phase evolution from the micromotion. This is different for the tri-partitioning ansatz which underlies the perturbative approach presented in the previous section. In the tripartitioning ansatz (113), firstˆ( ) † U t F 0 transforms the state into a 'reference frame' where by construction no micromotion is present. Then the phase evolution is generated by the effective Hamiltonian before, at time t, the state is finally rotated back to the original frame byˆ( ) U t F . In contrast,Ĥ t F 0 , as it appears in the ansatz (103), also carries information about the micromotion. This fact is somewhat hidden, when the t 0 dependence of the Floquet Hamiltonian is not written out explicitly like in [47], where = t 0 0 is assumed. However, since we know that the effective HamiltonianĤ F and the Floquet HamiltonianĤ t F 0 possess the same spectrum, we also know that, when expanding bothĤ F andĤ t F 0 in powers of the inverse frequency, the spectra will also coincide up to this order. This means that the t 0 -dependent second term of equation (109) will not cause changes of the spectrum within the second order ( w µ -1 ). Instead, this second term can contribute to the third-order correction of the quasienergy spectrum, together with the terms ofˆ( . This argument generalizes to higher orders.
Let us illustrate our reasoning using a simple example. A spin-1/2 system shall be described by the timeperiodic Hamiltonian , 114 x y with spin operatorsŜ i and Fourier components According to equations (87) and (88), in second order the effective Hamiltonian is approximated by do not contain a second-order term. Third, we can see that, unlike the exact quasienergy spectrum, the approximate spectrum e  [ ] FM 2 depends on the time t 0 and, thus, also on the driving phase. However, this dependence on t 0 (or the driving phase) occurs only in terms w µ -2 that are not reproduced correctly within the second-order approximation. In a third-order approximation, the spectrum will be captured correctly and be independent of the driving phase up to the power w -2 and so on.
As a further example, we will discuss the circularly driven hexagonal lattice in the next section. There, we will see that the spurious driving-phase dependence of the Floquet-Magnus expansion will, additionally, also induce a spurious breaking of the rotational symmetry of the quasienergy dispersion relation (section 6.3). Thus, even a weak t 0 dependence can seemingly change the properties of the system in a fundamental way. Therefore, the Floquet-Magnus approximation should be used with care. The high-frequency approximation derived in the previous section (section 4) does not suffer from this problem.

Example: circularly driven hexagonal lattice
In this section, we will discuss an instructive example of the physics of particles hopping on a hexagonal lattice (see figure 4(a)) subjected to a circular time-periodic force . 122 x y For a system of charged electrons such a force can be realized by applying circularly polarized light, whereas for a system of neutral particles (atoms in an optical lattice or photons in a wave guide) it can be achieved as an inertial force via circular lattice shaking [25,26,38,39]. The driven hexagonal lattice is particularly interesting, as it is the prototype of a Floquet topological insulator [27,[57][58][59][60]. It was pointed out by Oka and Aoki [27] that a nonvanishing forcing strength F opens a topological gap in the band structure of the effective Hamiltonian. As a consequence, the system possesses a quantized Hall conductivity, when the lowest band is filled completely with fermions. While the original proposal [27] is considering graphene irradiated by circularly polarized light (see also [64]), the topologically non-trivial band structure described by the effective Hamiltonian has been probed experimentally in other systems: with classical light in a hexagonal lattice of wave guides [39] and with ultracold fermionic atoms in a circularly shaken optical lattice [38].
We have decided to discuss the circularly driven hexagonal lattice here, even though its single-particle physics has already been described in detail elsewhere [27,38,58,65], for several reasons. First, it is a paradigmatic example of a system where the second-order high-frequency correction to the effective Hamiltonian gives rise to qualitatively new physics. Second, since both directions, x and y, are driven with a phase lag of p 2, the model is suitable to illustrate the difference between the high-frequency expansion advertised here and the Floquet-Magnus expansion. And third, it allows us to set the stage for the ensuing discussion (see section 7) of the role of interactions, which have been discussed controversially recently [66,67]. This issue includes two aspects: the impact of interactions on the validity of the high-frequency expansion as well as how interactions appear in the high-frequency expansion.
Let us consider the driven tight-binding Hamiltonian The first term describes the tunneling kinetics, with the sum running over all directed links á ¢ ñ ℓ ℓ connecting a site ℓ to its nearest neighbor ¢ ℓ on the hexagonal lattice depicted in figure 4(a). Hereˆℓ a is the annihilation operator for a particle (boson or fermion) at the lattice site ℓ located at ℓ r , and the tunneling parameter J is real and positive. The second sum runs over the lattice sites and describes the effect of the driving force in terms of the time-periodic on-site potential t and the number operator =ˆl ℓ ℓ † n a a . The direction of the vector pointing from site ℓ to a neighbor ¢ ℓ defines an angle j ¢ ℓ ℓ , . This angle determines the temporal driving phase of the relative potential modulation between both sites, . 125

Change of gauge
As will be seen shortly, we are interested in the regime of strong forcing, where the amplitude º K Fa of the relative potential modulation between two neighboring sites is comparable to or larger than w. Therefore, the Hamiltonianˆ( ) H t dr is not a suitable starting point for the high-frequency approximation. A remedy is provided by a gauge transformation with the time-periodic unitary operator [25] This gauge transformation induces a time-dependent shift in quasimomentum, and the second integral has been included to eliminate an overall quasimomentum drift. It provides a constant that subtracts the zero-frequency component of the first integral, thus making the time average of c ( ) ℓ t over one driving period vanish. One arrives at the translationally invariant time-periodic Hamiltonian Here the scalar potential ( ) ℓ v t is absent while the driving force is captured by the time-periodic Peierls phases . 129 Now we are in the position to apply the high-frequency approximation, even for w  K . The actual requirement is that w must be large compared to the tunneling matrix element J, which determines both the spectral width ofĤ 0 and the strength of the coupling termsĤ m with ¹ m 0.

Effective Hamiltonian
The leading term in the expansion of the effective Hamiltonian is, according to equation (87), given by the timeaverage of the driven Hamiltonian  This Bessel-function-type renormalization of the tunnel matrix element (see figure 5 for a plot) allows us to effectively reduce or even completely 'switch off' the nearest-neighbor tunneling matrix element. This effect is known as dynamic localization [3], coherent destruction of tunneling [4,6], or band collapse [5]. It has been observed in the coherent expansion of a localized Bose condensate in a shaken optical lattice [7]. The effect has also been used to induce the transition between a bosonic superfluid to a Mott insulator (and back) by shaking an optical lattice [19,20]. The possibility to make the tunneling matrix element negative has moreover been exploited to achieve kinetic frustration in a circularly forced triangular lattice and to mimic antiferromagnetism with spinless bosons [25,26]. The second-order contribution to the effective Hamiltonian is given by equation (88) and can be written as where the sum runs over next-nearest neighbors ¢ ℓ and ℓ. The effective tunneling matrix element is given by  (dotted red line). 5 In other lattice geometries, several two-step paths between ¢ ℓ and ℓ can exist. In this case one must sum over all of them.
forming the pattern of effective tunneling matrix elements depicted in figure 4(b). Here the sum is to good approximation exhausted by its first term, as is demonstrated also in figure 5.
The as it is depicted in figure 4(b), directly corresponds to the famous Haldane model [68] (see also reference [69]), being the prototype of a topological Chern insulator [70,71]. The next-nearest neighbor tunneling matrix elements open a gap between the two low-energy Bloch bands of the hexagonal lattice, such that the bands acquire topologically non-trivial properties of a Landau level characterized by a non-zero integer Chern number [72]. As a consequence, the system features chiral edge states, which have been observed experimentally with optical wave guides [39], and a finite Hall conductivity, as has been observed with ultracold fermionic atoms [38], which is quantized for a completely filled lower band. The fact that circular forcing can induce such non-trivial properties to a hexagonal lattice has been pointed out in [27]. This is the first proposal for a Floquet-topological insulator [60] (later proposals include [57,59]). These systems can be defined as driven lattice systems with the effective Hamiltonian featuring new matrix elements that open topologically non-trivial gaps in the Floquet-Bloch band structure. Such new matrix elements appear in the second (or higher) order of the high-frequency approximation that capture processes where a particle tunnels twice (or several times) during one driving period and that are of the order of wJ 2 . Therefore, Floquet topological insulators require the driving frequency to be at most moderately larger than the tunneling matrix element J. This is different for another class of schemes for the creation of artificial gauge fields and topological insulators recently pushed forward mainly in the context of ultracold quantum gases [28][29][30][31][32][34][35][36][37]. In these schemes non-trivial effects enter already in the leading first-order of the high-frequency expansion, so that they also work in the high-frequency limit w  J .
The non-interacting driven hexagonal lattice considered in this section can easily be solved numerically, without further approximation. The driven Hamiltonian (128) obeys the discrete translational symmetry of the lattice, with two sublattice states per lattice cell. As a consequence, the single-particle state space is divided into uncoupled sectors of sharp quasimomentum, each containing two states. The problem is reduced to that of a family of driven two-level systems labeled by the quasimomentum wave vector k. The high-frequency approximation (140) is nevertheless useful. First, it gives rise to an analytical approximation to the Floquet Hamiltonian, directly corresponding to the paradigmatic Haldane model [38,58]. Second, as we will argue in the next paragraph, the second-order approximation captures already the essential physics of the non-interacting system in the regime of large frequencies. And third, it also allows us to take into account some of the effects related to interactions (see section 7.1).
Let us briefly argue why the approximate effective Hamiltonian (140) captures the essential properties of the full one for the translationally invariant non-interacting system in the limit of large driving frequencies. The two-level Hamiltonian acting in the single-particle space of states with quasimomentum k is represented by á 2 2matrix of the general form , , 0 , with s denoting the vector of Pauli matrices and vector = ( are small compared to w, the perturbation expansion can be expected to converge. It is then left to argue that corrections beyond the second-order approximation (140) do not lead to qualitatively new behavior. This can be done by employing the arguments by Haldane [68] (see also reference [69]). In the subspace of quasimomentum k, the effective HamiltonianĤ F is represented by a time-independent matrix Its components determine the single-particle quasienergy dispersion relation is non-zero, since next-nearest-neighbor tunneling is sublattice preserving. Moreover, the time-reversal symmetry breaking associated with the phase ϑ makes this z-component non-zero at the Dirac points, such that = -¹ + - . In this way the second-order correction removes the band touching and opens a gap between both bands. The opposite sign of ). Even higher-order corrections, which in real space describe tunneling at even longer distances associated with a particle tunneling three or more times during one driving period, will be of the order of w ( ) J 3 2 . They will no longer be able to close this gap and change the topological properties of the bands. Thus, the essential physics of the driven system is captured by the approximation (140).
Qualitatively new behavior beyond the high-frequency approximation occurs, however, when w becomes small enough that

Comparison with Floquet-Magnus expansion
The circularly driven hexagonal lattice is also an instructive example that illustrates the difference between the high-frequency expansion of the effective HamiltonianĤ F on the one hand and of the Floquet HamiltonianĤ t , with k defined as the intermediate site between ℓ and ¢ ℓ . It is easy to check that the tunneling matrix element (147) depends on the direction of tunneling and not only on whether a particle tunnels in clockwise or anticlockwise direction around a hexagonal plaquette. This directional dependence is determined by t 0 and concerns not only the phase but also the amplitude of the next-nearest-neighbor tunneling matrix element (147). The consequence is a spurious t 0 -dependent breaking of the discrete rotational symmetry of the band structure of the approximate Floquet Hamiltonian +( . This can be seen as follows. The driven Hamiltonianˆ( ) H t obeys the discrete translational symmetry of the hexagonal lattice so that quasimomentum is a conserved quantity. Therefore,Ĥ t F 0 andĤ F possess not only the same spectrum but also the same single-particle dispersion relation e  ( ) k . The symmetry of the hexagonal lattice with respect to discrete spatial rotations by p 2 3is broken by the periodic force. However, the force leaves the Hamiltonianˆ( ) H t unaltered with respect to the joint operation of a rotation by p 2 3combined with a time shift by -T 3. This spatio-temporal symmetry ensures that the effective HamiltonianĤ F again possesses the full discrete rotational symmetry of the hexagonal lattice. This is reflected in the leading terms (130) and (135) of the high-frequency expansion. As a consequence, the quasienergy band structure, given by the single-particle dispersion relation e  ( ) k , is also symmetric with respect to a spatial rotation by p 2 3. The Floquet HamiltonianĤ t F 0 , whose parametric dependence on the time t 0 indicates that it also depends on the micromotion, does not obey the discrete rotational symmetry of the hexagonal lattice when evaluated for a fixed time t 0 . However, the singleparticle spectrum ofĤ t F 0 is still given by e  ( ) k and, thus, is rotationally symmetric. This latter property of the exact Floquet HamiltonianĤ t F 0 is not preserved by the Floquet-Magnus approximation . Here the second-order term not only leads to a spurious t 0 dependence of the spectrum, but also breaks the rotational symmetry of the quasienergy band structure. This is illustrated in figure 6, where we compare the approximate dispersion relation e + ( ) [  is inconsistent with Floquet theory. The origin of this inconsistency is that the dispersion relation in the Floquet-Magnus approximation depends on the driving phase, which, in turn, depends on the direction. Even though the spurious symmetry breaking should be small and of the order of w fixed], corresponding to the neglected third order, it still changes the property of the system in a fundamental way. Therefore, the Floquet-Magnus expansion must be employed with care.

Micromotion
The micromotion operator = and, employing equation (69), we find is anti-hermitian as required. With respect to the original frame of reference, where the system is described by the driven Hamiltonianˆ( ) H t dr , the micromotion operator is given by ⎛ The dynamics that is described byˆ( ) U t does not happen in real space, but corresponds to a global timeperiodic oscillation in quasimomentum by y . This momentum oscillation is significant when w K and it is taken into account via the initial gauge transformation in a non-perturbative fashion, as can be seen from the Bessel-function-type dependence of the effective tunneling matrix elements on w K . In turn,ˆ( ) U t F conserves quasimomentum and describes a micromotion in real space. This real-space micromotion becomes significant when the tunneling time  p J 2 is not too large compared to the driving period T, i.e. for w J not too small. A significant second-order correctionˆ( The second-order correction (88) vanishes, because all Fourier componentsĤ m int with > | | m 0 vanish. Therefore, the leading correction involving the interactions appears in third order (equation (89)) and reads ⎛  int , as they result from real-space micromotion, was overlooked in a recent work investigating the possibility of stabilizing a fractional-Chern-insulator-type many-body Floquet state with interacting fermions in the circularly driven hexagonal lattice [66]. A recent study, which is based on the results presented here, investigates the impact of the correction (157) on the stability of both fermionic and bosonic Floquet fractional Chern insulators [74]. It is found that the correction tends to destabilize such a topologically ordered phase.
Note that the high-frequency expansion of the Floquet HamiltonianĤ t F 0 , as it appears in the Floquet-Magnus expansion, also produces a second-order correction (109) that involves the interactions. It is given by [48,49,67]  This model system provides a quantitative description of ultracold bosonic atoms in an optical lattice and is interesting also because (in addition to the fermionic system [66]) it might be a possible candidate for a system stabilizing a Floquet fractional Chern insulator state. Using equation (157), we find those third-order correction terms that involve the interactions to be given by å å å = -- in the extended Hubbard model and a pair-tunneling term. And the third sum, finally describes density-assisted tunneling between next-nearest neighbors, as well as the joint tunneling of two particles into or away from a given site.

On the validity of the high-frequency approximation for many-body systems
The high-frequency expansion of the effective HamiltonianĤ F and the micromotion operatorˆ( ) U t F can generally not be expected to converge for a system of many interacting particles. Namely, the time averageĤ 0 of the full many-body Hamiltonianˆ( ) H t , which determines the spectrum of the diagonal blocks of the quasienergy operatorQ depicted in figure 1, will possess collective excitations also at very large energies. Therefore the energy gaps of w, which separate the subspaces of different 'photon' numbers m in the unperturbed problemQ 0 , will close when the perturbation is switched on (unless w was a macroscopic energy).
Nevertheless, the fact that the unperturbed problemQ 0 is given by the 'photonic' part of the quasienergy operator, with exactly degenerate eigenvalues w m in each subspace of photon number m, allows one to formally write down the perturbation expansion. The energy denominators will not diverge. This is illustrated in figure 7. And, even if the perturbation expansion can generally not be expected to converge, these terms can still provide an approximate description of the driven many-body system on a finite time scale. This can be the case if the νth-order approximate effective Hamiltonian H F is governed by energy scales that are small compared to w. Then the creation of a collective excitation of energy w corresponds to a significant change in the structure of the many-body wave function and is a process associated with a very small matrix element only. As a consequence, on time scales that are small compared to the inverse of such residual matrix elements, the approximate effective Hamiltonian can be employed to compute the dynamics and approximate Floquet states of the system. This is the basis for Floquet engineering in interacting systems.
Let us illustrate this rather abstract reasoning by a concrete example. Consider again spinless bosons in the circularly forced hexagonal lattice, described by the driven Bose-Hubbard Hamiltonian (153). The first-order approximation to the effective Hamiltonian is given by In the thermodynamic limit (where the number of lattice sites M is taken to infinity, while keeping the filling of n particles per lattice site fixed), the system possesses excited states at arbitrarily large energies. Therefore, the width of the spectrum is obviously larger than w. As a consequence, the first-order quasienergy spectrum that describes the subspace of 'photon' number m is much wider than w. It is no longer separated by a spectral gap from subspaces of different 'photon number', but overlaps with quasienergies from these spectra. Therefore, the perturbation expansion, which is based on the assumption that the unperturbed spectral gaps do not close in response to the perturbation, can generally not be expected to converge 6 . Namely, if we would consider the full diagonal blocks of the quasienergy matrix (figure 1) as an unperturbed problem (that is if we would add the block-diagonal part of the supposedly small perturbation to the unperturbed quasienergy operator considered before), the unperturbed quasienergy spectra of different subspaces m would already overlap. As a consequence, the energy denominators appearing in the perturbation expansion would diverge, whenever a degeneracy between unperturbed states of different 'photon' number m occurs.
We can now identify processes that spoil the high-frequency expansion for the driven Bose-Hubbard model (153) and estimate the time scale on which they occur. For simplicity, we focus on the limit of strong interactions  U J and assume a filling of n = 1. In the spirit of the argument presented at the end of the preceding paragraph, we can now add part of the perturbation to the unperturbed problem. We include the interactions in the definition of the unperturbed quasienergy operator,  6 Note that there are exceptions to this expectation. For example, the non-interacting system with U = 0 is described by a quadratic Hamiltonian. In this case the problem can be reduced to a single-particle problem. As a consequence, convergence is already expected for w that is sufficiently large compared to the width of the single-particle spectrum~J . This is true even if w is small compared to the width of the N-particle spectrum~NJ of the Hamiltonian (165) with U = 0. Namely, the matrix elements for the joint creation of several single-particle excitations at once vanish. However, if the particles are coupled by a finite interaction strength U, the single-particle picture no longer applies.
. 168 m t i They diagonalize the unperturbed problem, The high-frequency approximation is spoiled whenever two unperturbed states ññ |{ } ℓ n m and ¢ ¢ññ |{ } ℓ n m with different photon numbers ¹ ¢ m m are nearly degenerate and coupled to each other (either directly or in a higher-order process). In such a situation a strong hybridization between these states of different 'photon' number occurs, rather than a perturbative admixture of one state to the other as required by the high-frequency approximation.
Let us investigate how far the unperturbed 'ground state' of the m = 0 manifold, y ññ | 0 , suffers from such a detrimental coupling. This state corresponds to a Mott-insulator with one particle per site, It is the only unperturbed m = 0 state without any doubly occupied site, so that the interaction energy vanishes. Its quasienergy reads e¢ = ( ) 0 0 0 . It is the approximate ground state of the approximate effective Hamiltonian (165) in the regime  U J [75]. We can now systematically study the most relevant processes that couple y ññ | 0 to states of 'photon' number ¹ m 0. For that purpose, we will follow the procedure applied to a one-dimensional chain in reference [51].
The state y ññ | 0 is coupled directly (in first-order 7 ) to unperturbed states . The corresponding coupling matrix elements is of the order of J and reads  7 This order does not refer to the high-frequency approximation discussed so far, but to a perturbative approach treating y ññ | 0 and collective excitations with < m 0 photons as nearly degenerate. The time scale for the resonant creation of particle-hole excitations in the effective ground state can thus be estimated to be given by  p ( ) C 2 m 1 . If w is larger than U, the direct excitation of particle-hole pairs is not resonant, so that this type of heating process cannot spoil the high-frequency approximation.
The next-strongest type of process is the creation of a collective excitation consiting of two coupled (i.e. overlapping) particle-hole pairs, with two extra particles on a site ℓ 3 and no particles on two neighboring sites ℓ 1 and ℓ 2 8 . Such a state is given by 1 175  It is increased by a factor of the order of w J with respect to the creation of a single particle-hole pair, since it appears one order higher in perturbation theory. In an experiment one can avoid heating processes associated with the creation of two coupled particle-hole excitations by choosing w well above U 3 . In this case, the most detrimental process will be the creation of a collective excitation given by three overlapping particle-hole pairs in a third-order process, with matrix element w ( ) 2 . These third-order heating processes can, in turn, be avoided for w well above U 12 , and so on. Thus, by increasing the driving frequency, the time scale for detrimental heating due to the creation of collective excitations increases.
The order of perturbation theory in which detrimental heating processes beyond the high-frequency approximation occur increases like a power law with the frequency w, at least for the model and the parameters considered. This suggests that the time scale for heating due to the creation of collective excitations increases exponentially with the driving frequency (i.e. the higher the energy of a collective excitation the more complex it is, and the smaller will be the matrix element for its creation). Such a scaling is favorable for quantum engineering based on the high-frequency approximation. However, one must note that the driving frequency cannot simply be increased to arbitrarily large values, because with increasing driving frequency another type of heating process will become more relevant. Namely, the resonant creation of high-energy single-particle states neglected in a low-energy model will become more and more relevant. In the driven lattice system these states belong to excited Bloch bands above a large energy gap Δ that are not taken into account in the tight binding description. If the resonance condition w D » m is fulfilled, these states can be populated in m-photon processes [76][77][78], the time scale of which tends to increase with m. Thus, Floquet engineering requires a window of suitable driving frequencies that are both large enough to suppress heating due to the creation of collective excitations and small enough to suppress heating due to interband transitions. In optical lattice systems, the existence of a window of suitable driving frequencies has been demonstrated in a number of experiments (see first paragraph of section 1).
In the preceding paragraph the requirement 'to suppress heating' should be read as 'to suppress heating on the time scale of the experimental protocol'. Thus, the window of suitable frequencies depends also on the duration of the experiments, which in turn is determined by the effects to be studied. This dependence on the duration of the experiment is also reflected, in the response of a system to parameter variations. An important experimental protocol in the context of Floquet engineering is, for example, the smooth switching-on of the driving amplitude. Here the aim is to start from the ground state of the undriven model and to adiabatically prepare the ground state of the effective Hamiltonian obtained using the high-frequency approximation. In order to understand such parameter variations, one can employ the adiabatic principle for Floquet states [50]. In order to achieve the desired dynamics in a driven many-body system, two things are required: first, that the parameter variation is slow enough to ensure adiabatic following with respect to the high-frequency approximate effective Hamiltonian; and second, that the parameter variation is sufficiently fast. The second requirement can be understood as follows. The matrix elements for the resonant creation of collective excitations discussed above will lead to tiny avoided crossings in the quasienergy spectrum between states of different 'photon' number. These avoided crossings are not captured by the high-frequency approximation and, in order to avoid heating, must be passed diabatically [51].
The window of suitable driving frequencies is not necessarily determined by heating processes alone. Another limitation comes in, if the second-order term of the high-frequency approximation,ˆ( ) H F 2 , is relevant for the model system to be realized via Floquet engineering. The opening of a topological band gap in the circularly driven hexagonal lattice, discussed in section 6.2, is a prime example. This band gap appears in second order and depends inversely on the driving frequency Thus, if the band gap is required to be larger than some minimum value (determined, for example, by interactions, temperature, or the rate of a parameter variation), this sets an upper limit for the driving frequency. The existence (and the identification) of system parameters for which a window of suitable driving frequencies exists is an important issue of Floquet engineering.
The reasoning of this subsection suggests that, when dealing with a large system of interacting particles, on long time scales the high-frequency approximation can be expected to break down due to heating. In fact, the expected generic behavior of a driven many-body system in the thermodynamic limit is that it eventually approaches an infinite-temperature-like state in the long-time limit [79,80]. 9 In the sense of eigenstate thermalization [88,89], the conjectured infinite-temperature behavior is attributed to the resonant (m-'photon'-type) coupling, and the resulting hybridization, of states with very different mean energies. However, irrespective of the long-time behavior, the high-frequency approximation might still provide an accurate description of a driven many-body system on a finite time scale.

Summary
We have used degenerate perturbation theory in the exended Floquet space to derive a high-frequency expansion of the effective Hamiltonian and the micromotion operator of periodically driven quantum systems. This approach provides an intuitive picture of the nature of the high-frequency approximation and its limitations. We have, moreover, related our approach to the Floquet-Magnus expansion. We have discussed the fact that the latter is not only plagued by a spurious t 0 dependence of the quasienergy spectrum, but that it can also violate further symmetries of the exact result, like the rotational symmetry of a Floquet band structure. Finally, we have addressed the possibly detrimental impact of interactions on the validity of the high-frequency approximation beyond a certain time scale. Since , on both sides of the eigenvalue equation (A.1) and multiplies the equation by ¢ ( ) U t t , from the left. Furthermore, the periodic timedependence of the Hamiltonian must be employed by using . One arrives at . (iii) A multiplication of two TPTL operators in  directly corresponds to a multiplication in  , The proof is straightforward. This implies also that where the function f is defined via its Taylor expansion.
(v) A TPTL operatorˆ( ) U t that for all times t is unitary in  corresponds to an operatorŪ in  that is translationally invariant with respect to the 'photon' number m and unitary, and vice versa, This is a direct consequence of (iii) and (iv).

Appendix C. Degenerate perturbation theory in the extended Floquet Hilbert space
Degenerate perturbation theory is an approximation scheme that allows for the systematic block diagonalization of a hermitian operator into two subspaces separated by a spectral gap. Here we apply the canonical van Vleck degenerate perturbation theory [90] to the quasienergy operatorQ in the extended Floquet Hilbert space  and generalize it into an approximation scheme for the systematic block diagonalization ofQ into multiple (more than two) subspaces separated by spectral gaps. The generalized formalism is found to contain additional terms that do not appear in the standard scheme for bipartitioning. The procedure is closely related to the dressedatom approach described in reference [53] and has previously also been developed for the concrete example of a driven two-level system [61]. The index m separates the eigenstates into multiple subsets. The states within each subset m are labeled by the index α and span the unperturbed subspace  ( ) m 0 related to m. The quasienergies of two subsets m and m′ shall be separated by a quasienergy gap that is large compared to the matrix elements of the perturbationV . When the perturbation is switched on smoothly, without closing the spectral gaps, the unperturbed subspaces  ( ) m 0 will be transformed adiabatically to the perturbed subspaces  m , corresponding to a diagonal block of the perturbed problem. These subspaces  m will be spanned by new basis states a ññ | m B that deviate from the unperturbed states a ññ | m by small perturbative admixtures of states from other unperturbed subspaces.  The fact that the second term on the right-hand side is finite will give rise to additional terms in the perturbation expansion that do not appear in the standard formalism. We wish to block diagonalize the full unperturbed quasienergy operatorQ by means of a unitary operatorÛ , such that   we find the leading orders to be given by