Floquet edge states in a harmonically driven integer quantum Hall system

Recent theoretical work on time-periodically kicked Hofstadter model found robust counterpropagating edge modes. It remains unclear how ubiquitously such anomalous modes can appear, and what dictates their robustness against disorder. Here we shed further light on the nature of these modes by analyzing a simple type of periodic driving where the hopping along one spatial direction is modulated sinusoidally with time while the hopping along the other spatial direction is kept constant. We obtain the phase diagram for the quasienergy spectrum at flux 1/3 as the driving frequency $\omega$ and the hopping anisotropy are varied. A series of topologically distinct phases with counterpropagating edge modes appear due to the harmonic driving, similar to the case of a periodically kicked system studied earlier. We analyze the time dependence of the pair of Floquet edge states localized at the same edge and compare their Fourier components in the frequency domain. In the limit of small modulation, one of the Floquet edge mode within the pair can be viewed as the edge mode originally living in the other energy gap shifted in quasienergy by $\hslash \omega$, i.e., by absorption or emission of a “photon” of frequency $\omega$. Our result suggests that counterpropagating Floquet edge modes are generic features of periodically driven integer quantum Hall systems, and not tied to any particular driving protocol. It also suggests that the Floquet edge modes would remain robust to any static perturbations that do not destroy the chiral edge modes of static quantum Hall states.

Figure 1

The bulk quasienergy spectrum $\epsilon$ versus the driving frequency $\omega$ for $j_y/j_x=1.5$. We focus on intervals of $\omega$ for which there are three well defined gaps in the quasienergy Brillouin zone $\epsilon \in [-\omega /2,\omega /2]$.

Figure 2

The phase diagram of harmonically driven Hofstadter model, Eqs. (1) and (2), in the $(j_y,\omega )$ plane for $\alpha =1/3$, and $\gamma =1$. The white regions are phases with three well defined quasienergy bands and three gaps. The dashed line depicts a cut through the phase diagram. The representative spectra of phases i to vi are shown in Fig. 3. Each phase has its characteristic Chern numbers and edge state spectrum. They are summarized in Fig. 4.

Figure 3

Representative quasienergy spectra of phase i to vi for the slab geometry. Each phase is labeled by a pair of numbers $(n_1,n_2)$, i.e., at each physical edge there are $n_1$ edge modes, regardless their chirality, in the gap near the QBZ boundary and $n_2$ edge modes in the other two gaps (see main text). From (i) to (vi), the parameters $(j_y,\omega )$ are (1.0, 3.3), (3.3, 6.7), (3.3, 5.0), (4.0, 5.3), (3.3, 3.3), (4.5, 3.3) measured in units of $j_x$.

Figure 4

Phases and phase transitions along a particular path in the phase diagram, the dashed line in Fig. 2. White regions are phases with three well defined gaps, while the gray “metallic” regions have at least one of the gaps closed. The table summarizes the Chern numbers, $(c_1,c_2,c_3)$, and the number of edge modes $(n_1,n_2)$ for all the phases i to vi.

Figure 5

The evolution of the spectrum for fixed driving frequency as the modulation strength $\gamma$ is increased. Upper panel: the spectrum of the static Hofstadter model, $\gamma =0$. Middle panel: the spectrum of a weakly driven system, $\gamma =0.01$. It nearly coincides with the static spectrum folded into the QBZ (the shaded region). Lower panel: the spectrum of a strongly driven system, $\gamma =1$. $\omega =2j_y=6.7j_x$, $\alpha =1/3$.

Figure 6

Wave function $\psi (x,t)$ of the two counterpropagating $\pi$ modes at the same quasienergy $\epsilon =-0.46\omega$ localized at the same edge $x=1$. They correspond to the star $*$ indicated in phase i of Fig. 3. The first mode (left panel) is at $k_y=-0.34\pi$, and the second mode (right panel) is at $0.59\pi$.

Figure 7

The corresponding frequency components of the edge state wave function shown in Fig. 6. Different Fourier components, ${\varphi }_m$ for $m=0,\pm 1,\pm 2...$, are shifted vertically for clarity.

Figure 8

The quasienergy eigenvalues of a harmonically driven Hofstadter model on a finite $30\times 30$ lattice for $j_y=j_x,\omega =3.3j_x$ (phase i), $\alpha =1/3$, and $\gamma =1$. The horizontal axis is the index of the eigenvalues. The leftmost is the spectrum without perturbation. The remaining spectra, a to d, are for different perturbations (see main text). They are shifted horizontally from each other for clarity. Static onsite or bond disorder does not open a gap in the spectrum, e.g., at the QBZ boundary (dashed line) $\epsilon =\pm \omega /2$.

Figure 9

The wave function $\psi (x,y,t=0)$ of the $\pi$ modes in the presence of disorder. The parameters used for panels a, b, c, d are shown as $\times$ in Fig. 8. They all correspond to quasienergy very close to $\epsilon =\omega /2$. These plots offer direct evidence for the robustness of the $\pi$ modes against static disorder.