Quadratic band touching points and flat bands in two-dimensional topological Floquet systems

In this paper we theoretically study, using Floquet-Bloch theory, the influence of circularly and linearly polarized light on two-dimensional band structures with Dirac and quadratic band touching points, and flat bands, taking the nearest neighbor hopping model on the kagome lattice as an example. We find circularly polarized light can invert the ordering of this three-band model, while leaving the flat band dispersionless. We find a small gap is also opened at the quadratic band touching point by two-photon and higher order processes. By contrast, linearly polarized light splits the quadratic band touching point (into two Dirac points) by an amount that depends only on the amplitude and polarization direction of the light, independent of the frequency, and generally renders dispersion to the flat band. The splitting is perpendicular to the direction of the polarization of the light. We derive an effective low-energy theory that captures these key results. Finally, we compute the frequency dependence of the optical conductivity for this three-band model and analyze the various interband contributions of the Floquet modes. Our results suggest strategies for optically controlling band structure and interaction strength in real systems.

Figure 1

(a) Kagome lattice with nearest neighbor vectors ${\mathbf{a}}_1,{\mathbf{a}}_2$ labeled, and the three sites in the unit cell labeled. (b) First Brillouin zone of the underlying triangular Bravais lattice. In the nearest-neighbor hopping model, Eq. (1), Dirac points occur at the $\mathbf{K}$ and ${\mathbf{K}}^{\prime }$ points, and a quadratic band touching point at the $\mathrm{\Gamma }$ point. All red (gray) circles are equivalent to $\mathbf{K}$ and all open circles are equivalent to ${\mathbf{K}}^{\prime }$.

Figure 2

The Floquet-Bloch band structure of the kagome lattice embedded in a normally incident polarized light. The frequency of the pump light is $\mathrm{\Omega }=6t_{\mathrm{h}}$. (a) The band structure in equilibrium, without any incident light, given by Eq. (1); (b) the Floquet-Bloch band structure with circularly polarized light $A_0=1.0$; (c) the Floquet-Bloch band structure with circularly polarized light $A_0=3.8$; (d) the Floquet-Bloch band structure with linearly polarized light $A_0=1.0$.

Figure 3

The Floquet-Bloch band structure of the kagome lattice embedded in a normally incident polarized light. The frequency of pump light is $\mathrm{\Omega }=6t_{\mathrm{h}}$. The color of the lines are the same as in Fig. 2. (a) The band structure in equilibrium at $k_y=0.0$. (b) The Floquet-Bloch band structure with circularly polarized light $A_0=1.0$ at $k_y=0.0$. (c) The Floquet-Bloch band structure with circularly polarized light $A_0=3.8$ at $k_y=0.0$. (d) The Floquet-Bloch band structure with linearly polarized light $A_0=1.0$ at $k_y=0.0$.

Figure 4

The optical Hall conductivity $\mathbf{Re}\left[{\sigma }_{xy}\left(\omega \right)\right]$ as a function of the frequency of the probe light, in units of $e^2/h$. The driving laser frequency, laser amplitude, and the Chern number (zero frequency limit of $\frac{{\sigma }_{xy}\left(0\right)}{e^2/h})$ are (a) $\mathrm{\Omega }=5,A_0=0.5,C=-2$, (b) $\mathrm{\Omega }=5,A_0=1.5,C=-1$, (c) $\mathrm{\Omega }=10,A_0=0.5,C=-1$, (d) $\mathrm{\Omega }=10,A_0=1.5,C=-1$. A disorder broadening in Eq. (21) is set to be $\delta =\mathrm{\Omega }/100$. For a description of ideal and quench cases, see text.

Figure 5

$F_k^{m,\alpha \beta }$ as a function of $({\epsilon }_{k\beta }-{\epsilon }_{k\alpha }-m\mathrm{\Omega })/\mathrm{\Omega }$ where band index $(\alpha ,\beta )=(1,2),(1,3)$ at (a) $\mathrm{\Omega }=5,A_0=0.5,C=-2$, (b) $\mathrm{\Omega }=5,A_0=1.5,C=-1$, (c) $\mathrm{\Omega }=10,A_0=0.5,C=-1$, (d) $\mathrm{\Omega }=10,A_0=1.5,C=-1$. Here all the “k” points in the first Brillouin zone are plotted.