Floquet topological phase transition in the $\alpha \text{−}{\mathcal{T}}_3$ lattice

We investigate topological characteristics of the photon-dressed band structure of the $\alpha \text{−}{\mathcal{T}}_3$ lattice on being driven by off-resonant circularly polarized radiation. We obtain exact analytical expressions of the quasienergy bands over the first Brillouin zone. The broken time-reversal symmetry caused by the circularly polarized light lifts the triple point degeneracy completely at both the Dirac points. The gaps become unequal at $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ (except at $\alpha =0$ and 1), which reveals the absence of inversion symmetry in the system. At $\alpha =1/\sqrt{2}$, the gap between the flat and valence bands closes at $\mathbf{K}$, while that between the conduction and flat bands closes at ${\mathbf{K}}^{\prime }$, thereby restoring a semimetalic phase. At the gap closing point ($\alpha =1/\sqrt{2}$) which is independent of the radiation amplitude, there is a reappearance of low-energy Dirac cones around $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ points. Under the influence of the circularly polarized radiation, the $\alpha \text{−}{\mathcal{T}}_3$ lattice is transformed from semimetal to a Haldane-like Chern insulator characterized by nonzero Chern number. The system undergoes a topological phase transition from $\mathcal{C}=1(-1)$ to $\mathcal{C}=2(-2)$ at $\alpha =1/\sqrt{2}$, where $\mathcal{C}$ is the Chern number of the valence (conduction) band. This sets an example of a multiband system having larger Chern number. These results are supported by the appearance of chiral edge states in an irradiated $\alpha \text{−}{\mathcal{T}}_3$ nanoribbon.

Figure 1

Schematic diagram of the $\alpha \text{−}{\mathcal{T}}_3$ lattice illuminated by off-resonant circularly polarized light.

Figure 2

Plots of low-energy Floquet bands for various values of $\alpha$: (a) $\alpha =0$, (b) $0.6$, (c) $1/\sqrt{2}$, (d) 1, (e) $1.2$, and (f) $\sqrt{2}$. The bands are plotted along the line joining the high-symmetry $\mathbf{K},\mathbf{M}$, and ${\mathbf{K}}^{\prime }$ points. Bands are no longer symmetric under exchange of valleys except at $\alpha =0$ and 1. Here we have taken $J_1\left(\eta \right)=0.57$.

Figure 3

Plots of the photoinduced gaps at the Dirac point $\mathbf{K}$ as a function of $\alpha$.

Figure 4

Density-contour plots of the Berry curvature ($\mathrm{\Omega }\left(\mathbf{k}\right)$) around the two Dirac points $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ for different values of $\alpha$: (top) 0, (middle) $0.5$, and (bottom) 1. Here, $k_x$ and $k_y$ are in units of $a^{-1}$.

Figure 5

(Top) Plots of $s_2\left(\mathbf{K}\right)$ and $s_2\left({\mathbf{K}}^{\prime }\right)$ vs $\alpha$. (Bottom) Plots of the Chern number ${\mathcal{C}}_2$ vs $\alpha$.

Figure 6

Sketch of the FBZ with the locations of the singular points.

Figure 7

Radiation-dressed band structure of the $\alpha \text{−}{\mathcal{T}}_3$ nanoribbon with armchair edges for (a) $\alpha =0$, (b) $0.5$, (c) $0.8$, and (d) $1.0$. The red/blue curves represent edge states propagating on top/bottom edges, while the black curves represent the bulk bands.