Floquet-Boltzmann equation for periodically driven Fermi systems

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 $T_0=2\pi /\mathrm{\Omega }$ are treated using the Floquet formalism and quasiparticles are defined as eigenstates of a noninteracting Floquet Hamiltonian. The dynamics on much longer time scales, however, is modeled by a Boltzmann equation which describes the semiclassical dynamics of the Floquet quasiparticles and their scattering processes. As the energy is conserved only modulo $\hslash \mathrm{\Omega }$, 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 [G. Jotzu et al., Nature (London) 515, 237 (2014)].

Figure 1

Two of four interaction vertices generated by interaction of type Eq. (5). The remaining two diagrams are obtained by complex conjugation of (a) and (b), which simply corresponds to inverting the direction of the arrows.

Figure 2

Second-order diagrams due to onsite interactions. (a)–(e) contribute to ${\mathrm{\Sigma }}^R$ and (f)–(l) contribute to ${\mathrm{\Sigma }}^K$. ${\mathrm{\Sigma }}^A$ is obtained by realizing that ${\mathrm{\Sigma }}^A={\mathrm{\Sigma }}^{R†}$. Note that the assignment of the spin indices as in (a) is the same for all diagrams.

Figure 3

Scheme of the hexagonal lattice used to realize the Haldane model [13]. For every tunneling amplitude $t_j\left(t_j^A\right)$ there is an associated lattice vector ${\mathbf{v}}_j\left({\mathbf{u}}_j\right)$. Note that $t_j^A=t_j^B$ and that the phase of complex hopping strengths is defined along the direction of the respective vector in this figure.

Figure 4

Two-particle collision processes for (a) the energy-conserving case $(n=0)$ and (b) the energy-violating case $(n=1)$. The energy bands of the system described in the main text ($\mathrm{\Omega }=6.3\left|J\right|=2\pi \times 1080$ Hz) are shown along the diagonal of the quadratic Brillouin zone ($k_x=0$ within our conventions). Scattering is indicated by arrows from initial states (black) into final states (blue or red, respectively).

Figure 5

Heating rate per lattice site (in units of $U^2$) plotted against dimensionless temperature $T/J$, where $J$ is the hopping amplitude of the isotropic Hubbard model at hand. Different curves describe different driving frequencies $\mathrm{\Omega }$, ranging from $5\left|J\right|=2\pi \times 864$ Hz to $8.8\left|J\right|=2\pi \times 1512$ Hz (see legend). The inset shows a double-logarithmic plot.

Figure 6

Specific heat per lattice site $c\left(T\right)$ plotted as function of dimensionless temperature $T/J$ for a driving frequency $\mathrm{\Omega }=6.3\left|J\right|=2\pi \times 1080\mathrm{Hz}$. The inset shows the corresponding entropy per lattice site $(k_B=1)$ as function of $T/J$. $c\left(T\right)$ depends only very weakly on $\mathrm{\Omega }$ in the considered parameter regime.

Figure 7

Dimensionless temperature $T/J$ of the interacting system as a function of time for different driving frequencies $\mathrm{\Omega }$ ($\mathrm{\Omega }=2\pi \times 864,\cdots ,1512$ Hz) (see legend of Fig. 5). The inset clarifies that the curves undergo two distinct regimes before rising with exponential speed.

Figure 8

Entropy per lattice site (per particle) as a function of time for different driving frequencies $\mathrm{\Omega }$ ($\mathrm{\Omega }=2\pi \times 864,\cdots ,1512$ Hz) (see legend of Fig. 5). The solid curve ($\mathrm{\Omega }=6.3\left|J\right|=2\pi \times 1080$ Hz) is the lowest frequency used in the experiment [13].

Figure 9

The ratio of scattering times [Eq. (65)] for energy-nonconserving $\left(W^{\pm 1}\right)$ and energy-conserving processes $\left(W^0\right)$ plotted as a function of the driving frequency ($\mathrm{\Omega }/2\pi$ is given in units of Hz). For large frequencies, $\mathrm{\Omega }>6\left|J\right|\approx 2\pi \times 1000$ Hz energy-conserving scattering dominates by several orders of magnitude. Inset: double-logarithmic plot.

Figure 10

Heating rate per lattice site (in units of $U^2$) plotted against dimensionless temperature $T/J_{\max }$, where $J_{\max }$ is the maximal hopping amplitude of the anisotropic Hubbard model at hand. Different curves describe different driving frequencies $\mathrm{\Omega }$ ranging from $4|J_{\max }|\approx 2\pi \times 3000$ Hz to $8|J_{\max }|\approx 2\pi \times 6000$ Hz (see legend). The inset shows a double-logarithmic plot. The solid curve ($\mathrm{\Omega }=2\pi \times 1080\mathrm{Hz}\approx 5.4|{\mathrm{J}}_{\max }|$) is the frequency used in the main text of [13].

Figure 11

Specific heat per lattice site $c\left(T\right)$, plotted as function of dimensionless temperature $T/J_{\max }$ for a driving frequency $\mathrm{\Omega }=2\pi \times 4000\mathrm{Hz}\approx 5.4|{\mathrm{J}}_{\max }|$. The inset shows the corresponding entropy per lattice site $(k_B=1)$ as function of $T/J_{\max }$.

Figure 12

Dimensionless temperature $T/J_{\max }$ of the interacting system as a function of time for different driving frequencies $\mathrm{\Omega }$ ($\mathrm{\Omega }=2\pi \times 3000,\cdots ,6000$ Hz) (see legend of Fig. 10). The inset clarifies that the curves undergo two distinct regimes before rising with exponential speed.

Figure 13

Entropy per lattice site (per particle) as a function of time for different driving frequencies $\mathrm{\Omega }$ ($\mathrm{\Omega }=2\pi \times 3000,\cdots ,6000$ Hz) (see legend of Fig. 10).