The Floquet-Boltzmann equation
Periodically driven quantum systems can be used to realize quantum pumps, ratchets, artificial gauge fields and novel topological states of matter. Starting from the Keldysh approach, we develop a formalism, the Floquet-Boltzmann equation, to describe the dynamics and the scattering of quasiparticles in such systems. The theory builds on a separation of time-scales. Rapid, periodic oscillations occurring on a time scale , are treated using the Floquet formalism and quasiparticles are defined as eigenstates of a non-interacting Floquet Hamiltonian. The dynamics on much longer time scales, however, is modelled by a Boltzmann equation which describes the semiclassical dynamics of the Floquet-quasiparticles and their scattering processes. As the energy is conserved only modulo , the interacting system heats up in the long-time limit. As a first application of this approach, we compute the heating rate for a cold-atom system, where a periodical shaking of the lattice was used to realize the Haldane model Jotzu et al. (2014).
Periodically modulated quantum systems can effectively be described by a static Hamiltonian. This theoretical concept has recently evolved into a major experimental tool used by many groups to generate new states of matter.
Early experiments Lignier et al. (2007); Struck et al. (2011) used, for example, that one can effectively change the strength as well as sign of the hopping of atoms in an optical lattice, allowing to realize new types of band structures. Periodic driving has also be used to realize directed transport in quantum ratchets Salger et al. (2009). More recently, the realization of emergent Gauge fields and topological band structures has been at the focus of many studies. Examples of such experiments include the generation of Gauge fields and superfluids with finite momentumStruck et al. (2012, 2013), the generation of topological quantum walks Kitagawa et al. (2012) and of effective electric fields in a discrete quantum simulator Genske et al. (2013), the realization of (Floquet-) topological insulators with photons Rechtsman et al. (2013), the generation of spin-orbit coupling Anderson, Spielman, and Juzeliūnas (2013), the direct measurement of Chern numbers and Berry phases in the Hofstadter Hamiltonian Aidelsburger et al. (2013, 2015) and the realization of quantum pumps Lohse et al. (2015). A recent experiment of the Esslinger group beautifully realized the Haldane model Jotzu et al. (2014), i.e., a model which demonstrates that a quantum-Hall state can exist without homogeneous external magnetic fields. In solid-state systems circularly polarized light has been used Wang et al. (2013) to manipulate the surface states of topological insulators.
Also from the theory side, many proposals have pointed out that periodically driven states can be used to realized a wide range of states of matter. Examples are photoinduced quantum Hall states Oka and Aoki (2009); Kitagawa et al. (2011) and various topological Floquet states Kitagawa et al. (2010); Lindner, Refael, and Galitski (2011); Gómez-León and Platero (2013); Rudner et al. (2013) including dissipative systems Dehghani, Oka, and Mitra (2014), quantum ratchets for Mott insulators Genske and Rosch (2014), Majorana Fermions in driven quantum wires Jiang et al. (2011), the generation of non-abelian gauge fields Hauke et al. (2012), Floquet fractional Chern insulators Grushin, Gómez-León, and Neupert (2014), Floquet-Anderson insulators and quantized charge pumps Titum et al. (2015).
In a periodically driven system, the Hamiltonian has only a discrete time-translational symmetry, . As a consequence, the total energy is not conserved but quantized changes of energy are possible, with and . For non-interacting systems the absence of energy conservation has mostly no effect. The situation is, however, different when interactions in a many-particle system are considered. For a generic closed system, one can expect that in the long-time limit, , the system approaches the state with the highest entropy consistent with the conservation laws. In the absence of some cooling mechanism, e.g., by an external bath Seetharam et al. (2015) or by emitting radiation, one can therefore expect that generic interacting Floquet systems heat up to infinite temperatures Eckstein and Werner (2011) (an exception are many-body localized systems Ponte et al. (2015)). This important (and well-known) aspect has received relatively little attention in previous studies. Eckstein and Werner used time-dependent dynamical mean-field theory to study heating effects. In Choudhury and Mueller (2014), the stability of BEC condensates in periodically driven systems discussed based on phase-space arguments for energy non-conserving scattering processes.
The goal of this paper is to derive and apply a Floquet-Boltzmann equation, i.e., a kinetic equation which can be used to describe the dynamics of weakly interacting Floquet systems. Such a kinetic equation is perhaps the simplest theoretical description which captures microscopically how interactions can equilibrate an interacting quantum system. Like other quasi-classical kinetic equations, our approach builds on a separation of time scales and describes situations, where the change of occupation functions is much slower than and the time-scales set by the bare parameters of the Hamiltonian. Motivated by the recent realization of the Haldane model by the Esslinger group Jotzu et al. (2014), we investigate the effects of local interactions in a fermionic system. Starting from the Keldysh-formalism, we obtain a quantum kinetic equation from the Dyson equation, which is then reduced to the Floquet-Boltzmann equation. As an example, we quantitatively investigate heating rates in a limit when energy-conserving processes dominate (realized in Ref. Jotzu et al. (2014)).
A similar semiclassical kinetic equation was also derived in a recent preprint of Seetharam et al. Seetharam et al. (2015). In contrast to our work, they considered electron-phonon coupling instead of fermion-fermion interactions. Technically, Ref. Seetharam et al. (2015) used an equation-of-motion approach. While this approach is, perhaps, less well suited to investigate limitations of semiclassics, we expect that in the semiclassical limit it gives results equivalent to our derivation.
I Keldysh approach and Quantum kinetic equation
Our aim is to develop a Floquet-Boltzmann approach for interacting many-particle systems. The derivation builds on two main elements: a separation of time scales and the limit of weak interactions. We consider a time-dependent many-particle Hamiltonian which has the property that on short, microscopic time scales it is approximately periodic,
with period . Furthermore, we also allow for a slow time-dependence on time-scales large compared to the and all microscopic time scales (inverse kinetic and potential energies and inverse scattering rates). The latter can, for example, be used to describe the influence of external forces or the slow (quasi-adiabatic) change of the Hamiltonian. The Hamiltonian can therefore be written in the form
with and with Fourier series coefficients which are time-dependent on time scales much larger than .
As the derivation of quantum kinetic equations is ultimately based on perturbation theory, we further require that this perturbation theory can indeed be applied. In our examples, this will be the case when interactions are weak. More generally, one can use the approach also in cases where interactions are strong, but scattering rates are nevertheless weak either due to phase space restriction (e.g., for the fermionic quasiparticles of a Fermi liquid close to the Fermi surface) or just because the density of excitations is low (e.g., a weakly excited bosonic Mott insulator Genske and Rosch (2014)).
To simplify the presentation, we will not discuss the most general setup but restrict ourselves to a simpler case. We consider weakly interacting Fermions in a lattice model that is described by the following Hamiltonian
with a static and local interaction
We assume in the following that the interaction strength is sufficiently small, i.e., , in order to allow a perturbative approach to solve the problem. Note that our approach can be generalized in a straightforward way to more complicated and time-dependent interactions. We will present a theory that will unite the Floquet theory describing the oscillatory character of our model with the Keldysh approach of quantum field theories capturing the non-equilibrium behaviour of the system due to interactions and adiabatic drifts.
In order to derive a quantum kinetic equation, we use the standard Keldysh approach Kamenev (2011); Rammer and Smith (1986). We will not give a review of this Keldysh approach here, but just give a few main definitions. More details can, e.g., be found in the book by KamenevKamenev (2011) on which the following discussion is based.
To describe the time-evolution of the density matrix, one needs to keep track of two time-evolution operators and . Within the functional-integral version of the Keldysh approach one therefore introduces Grassmann fields on the forward branch, , and on the backward branch, . One then performs a rotation in Keldysh space using the following relations
After such a Keldysh rotation the action corresponding to the system excluding the interactions, i.e., , can be written in the form
where the integral should be understood as , and the fields as and . The non-interacting Green’s function, , is a 2x2 matrix with components
Generally, the Green’s functions of the full system are given by
with or within the functional integral by
Note that we choose throughout the paper. Here, and are the retarded and advanced Green’s functions, where , and is the Keldysh Green’s function (note that ). The latter can generally be parametrized in the following form
where is a Hermitian matrix and called distribution matrix. The “” represents matrix multiplication in all indices (space, time, spin). While carry information about the spectrum of the system, holds the information about the occupation as we will see below.
The Dyson equation
plays a central role in the derivation of the quantum kinetic equation. All objects are again matrices in Keldysh space, and the “” now also includes a matrix multiplication with respect to Keldysh indices. Here, one uses that the self-energy, , can be viewed as a functional of the full Green’s function, . The Dyson equation therefore is an integro-differental equation to determine in a time dependent system. The Keldysh component of the equation above can be identified with the quantum kinetic equation (see below). Upon further approximations, this equation can be simplified to obtain the semiclassical Boltzmann equation.
Using the fact that (in the fermionic case) the self-energy has the same structure as and , i.e.,
one can write down the Keldysh component of the Dyson equation, i.e. the quantum kinetic equation,
So far the expression is exact and no approximation has been made. For an interaction of the type as in Eq.(4), the interaction part of the action takes in Keldysh space the form
where () for (). Diagrammatically, yields four interaction vertices: two independent ones plus their complex conjugates (see Fig.1). To derive a quantum kinetic equation, we will consider (self-consistent) self-energy diagrams up to second order.
To linear order in , one obtains the familiar Hartree-Fock contributions: the energy of an electron is changed by . In the following, we will absorb all these completely into a redefinition of the non-interacting part . Note, however, that due to the time dependence of expectation values , the non-interacting part will obtain an extra time-dependence which has to be calculated self-consistently.
Due to the fact that , twelve independent diagrams are obtained in total for the second order expansion (see Fig.2) that contribute to the respective part of the self-energy. Exploiting the general identities for retarded and advanced Green’s functions and , where , one can write down the second order contributions to the individual parts of the self-energy as
where again () for () is the conjugate spin to and . The expression for can be straightforwardly obtained by using that .
The equations for the self-energies (15,16) as functions of the interacting Green’s function together with the Dyson equation (11) and a suitable initial condition define a quantum kinetic equation (QKE), which can be used to study the dynamics of a time-dependent interacting system.
A direct (numerical) solution of the QKE is very challenging, especially for a time-dependent Hamiltonian where the Green’s function depends on two time-variables separately. One can, however, make progress in situations where there is a clear separation of time scales.
The next step is to switch to a semi-classical representation via the Wigner transformation which we have to combine with the Floquet formalism to take into account rapid periodic oscillations.
Ii Floquet eigenstates and Floquet Green’s functions
The analysis of the dynamics of the Floquet system starts from the non-interacting, but time-dependent part of the Hamiltonian, , where varies only on time scales . More precisely, we will include in also all Hartree-Fock corrections arising from the interacting part of the Hamiltonian, i.e., terms like where will generically have components oscillating with frequencies plus an extra slow time-dependence arising, e.g., from heating. Using the assumed separation of time scales allows to define
as an approximation to for close to (on the time scale set by ) with . To solve such exact time-periodic Hamiltonians one uses the Floquet theorem (the analog of the Bloch theorem for periodic time- instead of space-dependencies) stating that solutions can be written in the form . Here, is a time-periodic Floquet state and is called the quasi energy. The eigenstates are obtained by diagonalizing the Floquet Hamiltonian
with . For practical calculations , the Floquet indices run from to , which is chosen such that is much larger than any other energy scale in the problem. Due to the ‘translation’ invariance (obtained for ), , one can obtain from each eigenstate with energy an eigenstate with energy by a simple translation of the Floquet indices, . Therefore, it is sufficient to consider only eigenstates with eigenenergies
Here, encodes the usual quantum number (spin, band-index, momentum).
The Heisenberg operator
creates a fermion in such a Floquet eigenstate with (here includes all quantum numbers, e.g., lattice site and spin). Note that
the occupation of the Floquet states, is time independent for the non-interacting, periodic Floquet Hamiltonian (17).
Here, it is important to stress that a single function describes the occupation of the Floquet states, and it is not necessary to introduce separate occupations for each Floquet index (further Floquet eigenstates obtained by translations in Floquet space yield exactly the same Floquet-creation operator, ). Our main goal will be to find a semiclassical description of the time evolution of the Floquet-occupations .
In a state with one can calculate the Green’s functions (8) of the noninteracting system (17). They depend on two time variables but the dependence on is purely periodic and therefore it is useful to represent the Green’s function in Floquet space by introducing the Floquet-Fourier transformation to obtain
The distribution function defined in (10) is therefore given in Floquet representation by
Note that the Green’s functions defined above are solutions of the Dyson equation
Iii Floquet-Wigner formalism and Floquet-Moyal expansion
Our central goal is to separate the slow dynamics, treated within a semiclassical approach, from rapid, periodic oscillations which have to be treated fully quantum-mechanically. The latter aspect is treated within the Floquet formalism (see Sec. II) which uses that in a strictly periodic system with period , only Fourier modes of the form with occur. To derive the slow, semiclassical dynamics, the starting point is the use of a Wigner representation of the Green’s function , usually obtained by using a Fourier transformation of the relative time coordinate, , for fixed .
These two approaches can be combined in situations where there is a clear separation of time scales, as discussed above. We require that the oscillation period is much smaller than all time scales, , on which the occupation function changes or on which the oscillating Hamiltonian is modified. This allows us to introduce the following ‘Floquet-Wigner representation’ for functions (Green’s functions or self-energies) which depend on two time coordinates
is a version of the -function which is broadened on the time scale chosen much larger than the period , but much smaller than the time scale of slow modifications,
The use of the filter function guarantees that is not rapidly oscillating on the time scale . More precisely, all oscillating components at frequency are exponentially suppressed by the factor due to the convolution with the filter function. Later, will take over the role of introduced in the previous section.
The inverse transformation of the Floquet-Wigner representation is given by
Due to the finite width of the filter function , this back-transformation is not exact but it is valid (with exponential precision) in situations when Eq. (27) holds. To see just this, it is instructive to plug Eq. (28) into Eq. (25): The condition guarantees that the Floquet indices do not mix, while allows to use use as a true -function for all components which vary slowly in time.
By definition, the argument of the Floquet-Wigner representation is restricted to the interval (the analog of the reduced Brillouin zone for periodic systems in real space).
Within the quantum kinetic equation (13), one has to compute the product of matrices, , which takes the form
Into this equation we plug the inverse Floquet-Wigner transformation, Eq. (28), for and , and Taylor-expand the time and frequency arguments of and . It turns out, that the resulting expression can be written in the form
Here denotes the time- or frequency derivative of the functions or , respectively. By a Taylor-expansion of the exponential, one obtains the well-known Moyal expansion Kamenev (2011). The only difference in comparison to the standard Moyal expansion is that it is supplemented by a simple matrix multiplication of the Floquet indices. A short derivation of the Floquet-Moyal expansion can be found in appendix A.
For the computation of the self-energy, we also need the Floquet-Wigner transformation of a different type of product given by . In this case one obtains directly
Iv Floquet-Boltzmann Equation
Boltzmann equations are a powerful tool to describe how scattering processes affect the semiclassical dynamics. They do not aim at describing quantum-coherent processes at short times, but instead focus on the physics at time scales set by slow changes of external parameters and by the scattering time of particles. It therefore builds on a clear separation of the time scales for quantum-coherent processes (captured by us within the Floquet approach for periodically driven system) and for the semiclassical dynamics which changes occupation functions.
The derivation of the Floquet-Boltzmann equation can be divided into four steps. (i) Starting point is the quantum kinetic equation (13) together with the calculation of self-energy diagrams, which are functionals of the Green’s function, see Eq. (15,(16). The next goal is to use the separation of time scales. Therefore, (ii) one uses the Floquet-Wigner representation, introduced in Sec. III, for all Green’s functions and self energies. Convolutions, ’’, can be written in terms of a Floquet-Moyal product, (30). Using the separation of time scales which implies that terms proportional to give small contributions, we can (iii) Taylor-expand to leading order, i.e., to linear order on the left-hand and to zeroth order on the right-hand side of the quantum kinetic equation (13). For problems which are not spatially homogeneous a similar Moyal expansion is also used for the spatial coordinates. Finally, (iv) the resulting equation is projected onto on-shell processes, e.g., by an integration over frequencies.
The final result of these steps is an equation for the occupation functions of the Floquet eigenstates at time . Here is the momentum and includes band- and spin indices. The Floquet states at time are the eigenstates, , of the Floquet Hamiltonian (18) with and we have to set . In complete analogy to the treatment of the variable , which we used in Sec. II to deal with the slow time dependence, we also allow that the Hamiltonian depends smoothly on the spatial parameter . For a lattice model with sites per unit cell, the eigenfunctions are calculated in momentum space by diagonalizing a dimensional matrix. In the following we will denote the corresponding eigenfunctions in momentum space by where is the Floquet index and describes the structure of the Bloch-Floquet wave function within the unit cell. Sometimes, we will omit the and index to simplify notations and just write .
For the following discussion, we will not discuss the (main) part of the derivation which is identical for Floquet systems and conventional cases, as these are well described in the literature Rammer and Smith (1986); Wickles and Belzig (2013) and textbooks Kamenev (2011). Instead, we will only describe those aspects which are different in the Floquet case.
iv.1 Semiclassical dynamics and left-hand side of the Floquet-Boltzmann equation
The Floquet-Moyal expansion, Eq. (30), differs from the standard Moyal expansion only by the presence of the extra Floquet indices. This gives rise to a simple matrix multiplication. To be able to describe also situations where the occupation functions are not spatially translational invariant, but vary smoothly (on length scales large compared to the lattice spacing), one uses a Wigner representation and Moyal expansion for space and momentum degrees of freedom, similar to the one described above for time and frequency variables Kamenev (2011); Rammer and Smith (1986); Wickles and Belzig (2013). A major difference between the momentum and the frequency dependence is, however, that the semiclassical occupation functions depend on the quantum numbers momentum and band index. but not on frequency and Floquet indices.
A derivation of the conventional collisionless Boltzmann equation based on the quantum kinetic approach, which (in contrast to previous derivations) includes all Berry-phase correction to leading-order, has recently been given by Wickels and Belzig Wickles and Belzig (2013). One can check (see Appendix B) that the only difference arising in the Floquet case is that all Berry curvatures have to be computed from the Floquet-eigenfunctions introduced in Sec. II,
where the scalar product involves also a summation over the Floquet index . Here and , stands for derivatives in time, space and momentum variables Freimuth et al. (2013). These Berry curvatures modify the semiclassical equations of motion in phase space and therefore also the left-hand side of the Boltzmann equation which takes the form
While , and are matrices with , and are vectors . Note also that , can be related to effective magnetic fields, and , are referred to effective electric fields.
Since many modern applications of Floquet Hamiltonians Hauke et al. (2012); Struck et al. (2013); Aidelsburger et al. (2013); Jotzu et al. (2014); Aidelsburger et al. (2015); Lohse et al. (2015) have as a goal to realize systems with non-trivial Berry phases, it is important to keep track of these effects on the left-hand side of the Boltzmann equation. Consider, for example, an interacting Floquet system which heats up as function of time (see Sec. V). The (slow) change of occupation functions can trigger a change of the momentum-space Berry curvature , a momentum-space ‘magnetic’ field. This implies that also corresponding momentum-space ‘electric’ fields and are generated. They can, e.g., induce a macroscopic rotation of the cold-atom system.
iv.2 Scattering and the right-hand side of the Floquet-Boltzmann equation
To calculate the right-hand side of the Floquet-Boltzmann equation, we start from the self-energies Eq. (15) and (16). First, we need an expression for the Green’s function. Due to the assumed separation of time scales, it is sufficient to evaluate the Green’s function using a zeroth-order Floquet-Moyal expansion of the Dyson equation (11). Furthermore, within our perturbative approach, we do not have to include any self-energy corrections (as the self-energy is already ). Using the Floquet-Wigner representation of both the Hamiltonian and the Green’s function, the Dyson equation takes with these approximations exactly the form of Eq. (24). This implies that we are allowed to use directly the Green’s functions of Eq. (II) with and .
To evaluate the Floquet-Wigner representation of the self-energies Eq. (15),(16) and the right-hand side of quantum-kinetic equation (13), we use the convolution formula Eq. (31) twice. For the first line of the formula in Eq. (15), for example, we obtain after a few steps of simplification a contribution of the form
where () for (), the superscript ’’ refers to the first line of Eq. (15), and
where are Floquet indices, are band indices and denote sites within the unit cell. Here, is the volume of the Brillouin zone. We have omitted extra and labels which each function obtains to reflect the smooth time and spatial dependencies of the system.
The last remaining step is to evaluate the resulting formula on-shell: we multiply the right-hand side of the quantum kinetic equation (13) by the Floquet-spectral function of the state with quantum numbers and , integrate over frequencies and trace over Floquet- and space indices. Note that considering only diagonal, on-shell contributions implies that Boltzmann-type equations cannot describe coherent quantum-oscillations. The resulting equations are therefore only valid on time scales longer than the decay time of such oscillations. This is consistent with our assumptions on the separation of time scales underlying our analysis. Furthermore, a quasiparticle has to be well defined, implying that the broadening of the spectral function by scattering is small compared to the energy of the quasiparticles (and therefore also small compared to ).
After this last transformation Eq. (35) takes, for example, the form
with , and the transformed matrix element
It is convenient to introduce for each occupation function a separate momentum variable and a -function which guarantees momentum conservation (modulo reciprocal lattice vectors ). Performing the entire procedure for all terms of Eq. (15) and likewise for all contributions associated with the second term on the right-hand side of the quantum kinetic equation (13), one eventually finds an expression for the collision integral
where we have introduced the integers to account for Umklapp scattering in momentum- and frequency space, respectively, and is the scattering rate for a process involving an energy transfer to the system of , . We obtain
with the amplitude
Note that the Floquet- and momentum indices enter the matrix elements, and therefore the collision integral, in a completely different way: occupation functions depend on momentum and band indices, but do not depend on the Floquet indices. Correspondingly, we sum over Floquet indices in Eq. (41), but not over momentum or band indices. We will discuss this important difference again in the concluding section.
The collision integral in Eq. (IV.2) forms the right-hand side of the Floquet-Boltzmann equation
where, in general, also the effective forces and velocities depend smoothly on time and space, and . This dependence can either arise from an explicit and dependence of the Hamiltonian or arise from Hartree-Fock corrections to the Hamiltonian, which have to be computed using .
V Haldane model
In the following we want to apply the Floquet-Boltzmann equation to a concrete example. In a recent experiment with ultracold atoms in an optical lattice, the Haldane model was realized by means of periodic shaking of the lattice Jotzu et al. (2014). The Haldane model is the prototypical example of a topological insulator: Haldane showed that an integer quantum Hall state can be realized without any external magnetic field on average, but just by arranging complex hopping parameters on a hexagonal lattice Haldane (1988).
The experiment can be described (see supplementary information of Ref. Jotzu et al. (2014)) by a (distorted) honeycomb lattices, see Fig. 3, with two sites per unit cell, which form two chequerboard sublattices and . The static Hamiltonian can be described by real nearest-neighbour and next-nearest-neighbour hopping amplitudes