Cosine edge modes in a periodically driven quantum system

Time-periodic (Floquet) topological phases of matter exhibit bulk-edge relationships that are more complex than static topological insulators and superconductors. Finding the edge modes unique to driven systems usually requires numerics. Here we present a minimal two-band model of Floquet topological insulators and semimetals in two dimensions where all the bulk and edge properties can be obtained analytically. It is based on the extended Harper model of quantum Hall effect at flux one-half. We show that periodical driving gives rise to a series of phases characterized by a pair of integers. The model has a most striking feature: the spectrum of the edge modes is always given by a single cosine function, $\omega \left(k_y\right)\propto cos{k}_y$ where $k_y$ is the wave number along the edge, as if it is freely dispersing and completely decoupled from the bulk. The cosine mode is robust against the change in driving parameters. It also persists in the semimetallic phases with Dirac points.

Figure 1

Phase diagrams of the periodically kicked model given by Eq. (6). (a) Schematic of the quasienergy bands (shaded region) and gaps (empty region) as $\overline{J_y}$ is increased for fixed $\overline{J_x}<1/2$ and $\alpha <1/2$. A series of insulating (I) phases ${\mathrm{I}}_{w_0,w_{\pi }}$ are identified by examining the band Chern numbers $C_{+},C_{-}$ and the winding numbers $w_0,w_{\pi }$. (b) Topological phases with fixed $\alpha =1/2$ and $\overline{J_x}=0.75$. ${\mathrm{S}}_{1,2}$ and ${\tilde{\mathrm{S}}}_{3,2}$ are semimetallic (S) phases described in the main text. Their spectra are shown in Figs. 2 and 2. (c) Topological phases along the line $2\alpha =\overline{J_x}=\overline{J_y}$. SS denotes (double) semimetallic phases with Dirac points at both quasienergy 0 and $\pi /T$.

Figure 2

Quasienergy spectrum of a slab of finite width $L_x=61$ in the $x$ direction, but periodic in $y$ direction. (a)–(d) are for $\overline{J_y}=0.4$, 0.6, 0.8, and 1.2, respectively, corresponding to phases ${\mathrm{I}}_{1,0},{\mathrm{S}}_{1,2},{\mathrm{I}}_{1,2}$, and ${\tilde{\mathrm{S}}}_{3,2}$. Here, $\alpha =1/2,{\overline{J}}_x={\overline{J}}_y$, except for (d), where $\overline{J_x}=0.75$.