Four-dimensional Floquet topological insulator with an emergent second Chern number

Floquet topological insulators have been widely investigated in lower-dimensional systems. However, Floquet topological insulators induced by time-periodic driving in higher-dimensional systems remain unexplored. In this work, we study the effects of time-periodic driving in a four-dimensional (4D) normal insulator, focusing on topological phase transitions at the resonant quasienergy gap. We consider two types of time-periodic driving, including a time-periodic onsite potential and a time-periodic vector potential. We reveal that both types of time-periodic driving can transform the 4D normal insulator into a 4D Floquet topological insulator characterized by an emergent second Chern number. Moreover, it is found that the topological phase of the 4D system can be modulated by tuning the strength and frequency of the time-periodic driving. Our work will be helpful for the future investigation of Floquet topological insulators in higher dimensions.

Figure 1

(a) The bulk quasienergy spectra for the Floquet Hamiltonian $H_F$ when $k_y=\pi ,k_z=0,k_w=0$. (b) shows the quasienergy spectra of the cyan region in (a). The gray dashed line (red solid line) represents the amplitude of the time-periodic onsite potential $V=0$ ($V=1$). (c) The bulk gap ${\varepsilon }_g$ and (d) the second Chern number $C_2$ as a function of the amplitude of the time-periodic onsite potential $V$. Those red solid dots in (c) represent the bulk gap of the effective Hamiltonian $H_{\mathrm{RWA}}$ obtained by the rotating wave approximation. Here, we choose $\omega =11$ and $m=-6$.

Figure 2

(a) Band structure in the plane $(\varepsilon ,\mathbf{k})$ when the open boundary condition is along the $x$ direction. The indicators of the horizontal axis are the high symmetry $\mathbf{k}$ points in the first Brillouin zone of the quasi-3D system, $\mathrm{\Gamma }=(k_y=0,k_z=0,k_w=0)$, $\mathrm{Y}=(\pi ,0,0)$, $\mathrm{Z}=(0,\pi ,0)$, $\mathrm{W}=(0,0,\pi )$, ${\mathrm{M}}_{\mathrm{yz}}=(\pi ,\pi ,0)$, ${\mathrm{M}}_{\mathrm{yw}}=(\pi ,0,\pi )$, ${\mathrm{M}}_{\mathrm{zw}}=(0,\pi ,\pi )$, $\mathrm{R}=(\pi ,\pi ,\pi )$. (b) Band structure in the quasienergy interval $\varepsilon \in (4,7)$ in which there are gapless boundary states. Here, we choose $\omega =11$, $V=1$, and $m=-6$.

Figure 3

(a) The bulk gap ${\varepsilon }_g$ and the second Chern number $C_2$ as a function of the frequency $\omega$ when $V=1$. (b) The second Chern number $C_2$ as a function of $V$. The blue, green, and red dotted lines correspond to $\omega =11$, $\omega =13$, and $\omega =17$, respectively. Band structure of the resonant quasienergy region in the plane $(\varepsilon ,\mathbf{k})$ under the open boundary condition along the $x$ direction for (c) $\omega =13$ and (d) $\omega =17$. The indicators of the horizontal axis are the high symmetry $\mathbf{k}$ points in the first Brillouin zone of the quasi-3D system, $\mathrm{\Gamma }=(k_y=0,k_z=0,k_w=0)$, $\mathrm{Y}=(\pi ,0,0)$, $\mathrm{Z}=(0,\pi ,0)$, $\mathrm{W}=(0,0,\pi )$, ${\mathrm{M}}_{\mathrm{yz}}=(\pi ,\pi ,0)$, ${\mathrm{M}}_{\mathrm{yw}}=(\pi ,0,\pi )$, ${\mathrm{M}}_{\mathrm{zw}}=(0,\pi ,\pi )$, $\mathrm{R}=(\pi ,\pi ,\pi )$. In (c) and (d), $V=1$. And in all plots, we choose $m=-6$.

Figure 4

(a) The bulk quasienergy spectra for the Floquet Hamiltonian ${\mathcal{H}}_F$ when $k=k_n$ ($n=x,y,z,w$). (b) shows the quasienergy spectra of the region $\varepsilon \in (0,\omega )$ in (a). The gray dashed line (red solid line) represents the amplitude of the time-periodic vector potential $A=0$ ($A=1.5$). (c) Band structure in the plane $(\varepsilon ,k=k_y=k_z=k_w)$ when the open boundary condition is along the $x$ direction. (d) Band structure in the quasienergy interval $\varepsilon \in (4.5,6.5)$ in which there are gapless boundary states. In (c) and (d), the amplitude is $A=1.5$. Here, we choose $\omega =11$ and $m=-6$.

Figure 5

(a) The bulk gap ${\varepsilon }_g$ for the Floquet Hamiltonian ${\mathcal{H}}_F$ as a function of $A$. (b) The second Chern number $C_2$ as a function of $A$. The color of the solid line represents the number of $\mathbf{k}$ points in the first Brillouin zone for the calculation of the second Chern number, which is $N^4$. (c) $\ln |{\mathrm{C}}_2-2|$ as a function of $N$ for different amplitudes $A$. The red, green, and cyan dotted lines correspond to $A=0.5$, $A=1$, and $A=1.5$, respectively. The dashed lines of the linear distribution are obtained by fitting the dotted lines. (d) Band structure in the plane $(\varepsilon ,k=k_y=k_z=k_w)$ when the open boundary condition is along the $x$ direction. We choose the amplitude of the time-periodic vector potential $A=0.5$ in (d). Here, the parameters are $\omega =11$ and $m=-6$.