Topological defects in Floquet systems: Anomalous chiral modes and topological invariant

Backscattering-immune chiral modes arise along certain line defects in three-dimensional materials. Here, we study the Floquet chiral modes along Floquet topological defects, namely, the defects come entirely from spatial modulations of periodic driving. We define a precise topological invariant that counts the number of Floquet chiral modes, which is expressed as an integral on a five-dimensional torus parametrized by $(k_x,k_y,k_z,\theta ,t)$. This work demonstrates the possibility of creating chiral modes in three-dimensional bulk materials by modulated driving.

Figure 1

Sketch. (a) The time-periodic driving is spatially modulated as a function of the polar angle $\theta$, creating a Floquet line defect along $r=0[r\equiv \sqrt{{(x-x_0)}^2+{(y-y_0)}^2},tan\theta \equiv (y-y_0)/(x-x_0)$; $(x_0,y_0)$ is the location of the defect]. The blue arrow stands for the Floquet chiral modes inside the bulk energy gap. (b) Sketch of the quasienergy bands, with the shadowed region representing the bulk bands, connected by the Floquet chiral modes (blue lines).

Figure 2

(a) The resonance surface $d\left(\mathbf{k}\right)=\omega /2$, where the $m=0$ and $m=1$ Floquet bands cross each other (when $D=0$). (b) Floquet bands along the 111 direction ($k_x=k_y=k_z=k$) for $D=0$ (dashed curves, with the Floquet index $m$ marked), and $D=0.6$ (solid curves). All bands are doubly degenerate.

Figure 3

(a) Quasienergy bands $\varepsilon \left(k_z\right)$ of an open-boundary sample with a Floquet line defect (with $n=1$). The system size is $L_x\times L_y\times L_z=20\times 20\times \infty$. The thick blue curve represents the chiral modes localized near $r=0$, while the thin green curve represents the back-propagating modes at the system boundary. Each band is doubly degenerate. (b) The wave-function profiles of the two energetically degenerate chiral modes at $k_z=0.1\pi$ [indicated by an open circle in (a)]. (c) shows bands for the $n=2$ defect. The chiral-mode profiles at $k_z=0.3\pi$ and $-0.3\pi$ are shown in (d) and (e), respectively.

Figure 4

The bulk quasienergy gap $\mathrm{\Delta }\varepsilon$ around $\omega /2$ and the topological invariant $W(\omega /2)$ for $n=1$ (in the gapped regime).