Topological micromotion of Floquet quantum systems

The Floquet Hamiltonian has often been used to describe a time-periodic system. Nevertheless, because the Floquet Hamiltonian depends on a micromotion parameter, the Floquet Hamiltonian with a fixed micromotion parameter cannot faithfully represent a time-periodic system, which manifests as the anomalous edge states. Here we show that an accurate description of a Floquet system requires a set of Hamiltonian spanning all values of the micromotion parameter, and this micromotion parameter can be viewed as an extra synthetic dimension of the system. Therefore we show that a $d$-dimensional Floquet system can be described by a $d+1$-dimensional static Hamiltonian, and the advantage of this representation is that the periodic boundary condition is automatically imposed along the extra dimension, which enables a straightforward definition of topological invariants. The topological invariant in the $d+1$-dimensional system can ensure a $d-1$-dimensional edge state of the $d$-dimensional Floquet system. We show two examples where the topological invariant is defined as the three-dimensional Hopf invariant.

Figure 1

(a) The Hamiltonian set $\{{\widehat{H}}_{\text{F}}(k_1,\cdots ,k_d,\alpha ),\alpha \subset [0,2\pi ]\}$ can be viewed as a Hamiltonian in $\widehat{H}(k_1,\cdots ,k_d,k_{d+1})$ in $(d+1)$ dimension. Here ${\mathbf{k}}_{\mathbf{R}}$ denotes $\{k_2,\cdots ,k_{d+1}\}$. (b) A real-space geometry for the $(d+1)$-dimensional system, with open boundary condition in one of the spatial dimension denoted by $R_1$ and periodic boundary condition in the rest dimensions denoted by $\mathbf{R}=\{R_2,\cdots ,R_{d+1}\}$. (c) Schematic of energy dispersion for systems shown in (b), as a function of good quantum numbers $k_2,k_3,\cdots ,k_{d+1}$. This dispersion is flat along $k_{d+1}$ direction.

Figure 2

Linking number [(a) and (c)], spectrum [(b) and (d)], and edge states [(e) and (f)] of model I. [(a) and (c)] The inverse images of the south (blue straight lines) and the north (red circle) poles of the three-dimensional Hamiltonian ${\widehat{H}}_{\text{F}}(k_1,k_2,k_3)$. [(b) and (d)] Spectrum for two-dimensional Floquet effective Hamiltonian ${\widehat{H}}_{\text{F}}(k_1,k_2,\alpha )$ with a fixed $\alpha$. Here we have fixed $\mu =-10$ in ${\widehat{H}}_1$ for all plots. We have chosen $\mu =-5$ for ${\widehat{H}}_2$ such that ${\widehat{H}}_2$ is a topologically trivial case in (a) and (b), and $\mu =-2$ such that ${\widehat{H}}_2$ is a topologically nontrivial case for (c) and (d). [(e) and (f)] The real-space distribution of the edge states correspond to the in-gap states shown in (d), with quasienergies located at zero-energy (e) and energy $\pi /T$ (f), respectively. Here, $t_0$ is chosen as 0.1.

Figure 3

Linking number [(a) and (c)], spectrum [(b) and (d)], and edge states [(e) and (f)] of model II. [(a) and (c)] The inverse images of the south (blue straight lines) and the north (red circle) poles of the three-dimensional Hamiltonian ${\widehat{H}}_{\text{F}}(k_1,k_2,k_3)$. [(b) and (d)] The spectrum for two-dimensional Floquet effective Hamiltonian ${\widehat{H}}_{\text{F}}(k_1,k_2,\alpha )$ with a fixed $\alpha$. Here we have chosen $\mu =-10$ and $\omega =12$ in (a) and (b), and $\mu =-2$ and $\omega =4$ for (c) and (d). [(e) and (f)] The real-space distribution of the edge states correspond to the in-gap states shown in (d), with quasienergies located at zero energy (e) and energy $\pi /T$ (f), respectively.