Floquet topological systems with flat bands: Edge modes, Berry curvature, and orbital magnetization

Results are presented for Floquet systems in two spatial dimensions where the Floquet driving breaks an effective time-reversal symmetry. The driving protocol also induces flat bands that correspond to anomalous Floquet phases where the Chern number is zero and yet chiral edge modes exist. Analytic expressions for the edge modes, Berry curvature, and orbital magnetization are derived for the flat bands. Results are also presented for the static Haldane model for parameters when the bands are flat. Floquet driving of the same model is shown to give rise to Chern insulators as well as anomalous Floquet phases. The orbital magnetization for these different topological phases are presented and are found to be enhanced at half filling by the broken particle-hole symmetry of the Haldane model.

Figure 1

(a) Schematic of static graphene with a zigzag boundary. The dotted and solid black lines indicate different coupling strengths on the cylinder, resulting in an effective Su-Schrieffer-Heeger model [47, 48]. (b) Schematic of Floquet-driven graphene with a zigzag boundary. The first, second, and third steps of the Floquet protocol are represented by green solid, purple dashed, and orange dotted lines respectively.

Figure 2

(a) Static Haldane model on a cylinder with a zigzag boundary. The black and purple dotted lines connect odd (green) and even (orange) sites, respectively, whose coupling strengths differ on the boundary due to broken TRS [second line of Eq. (5)]. Horizontal lines denote NNN couplings, and on the boundary these again contribute differently for even and odd sites [third line of Eq. (5)]. The TRS breaking phase $\varphi$ is either positive or negative depending on the direction of the arrows. (b) Floquet-driven Haldane model on a cylinder where the first, second, and third steps of the protocol are represented by green solid, purple dashed, and orange dotted lines respectively. The NNN couplings are pictured as gray and indicate that they are not driven but remain fixed at a nonzero value.

Figure 3

(a) Band structure of driven graphene with zigzag boundaries for small detuning parameter $JT/3=0.01\times \pi /2$ from the flat-band limit while keeping $\lambda JT/3=\pi /2$ fixed. The bands are not perfectly flat because $JT\ne 0$. (b). Berry curvature for the model on a torus for the same parameters. The integral of the Berry curvature over the BZ vanishes, i.e., $C=0$. (c). Orbital magnetization per unit area on the torus, in units of $-e/\hslash T$, for the same parameters with a temperature ${\beta }^{-1}=0.05\times JT/3$ (blue circles), and a temperature ${\beta }^{-1}=0.05\times \lambda JT/3$ (yellow diamonds). Vertical red lines denote the band edges.

Figure 4

(a) $\overline{M}/A$ in units of $-e/\hslash T$ versus $\beta$ the inverse temperature for $\mu =\pi /2$, $JT/3={10}^{-4}\times (\pi /2+\xi )$ and $\lambda JT/3=\pi /2+\xi$. Here we plot $\overline{M}/A$ for different $\xi =0.16,0.016,0.0016$ (diamonds, filled squares and circles, respectively). Solid-red line is $\propto {\beta }^3$. (b) $\overline{M}/A$ in units of $-e/\hslash T$ versus the detuning $\xi$, at a low temperature of $\beta \approx 1/1.6\times {10}^6$ and $\mu =\pi /2$. We choose $\lambda JT/3=\pi /2+\xi$ and $JT/3=(\pi /2+\xi )\times [0,{10}^{-2},{10}^{-4}]$ corresponding, respectively, to filled squares, diamonds, and circles. As $\beta \to \infty$ and $JT/3\to 0$, $\overline{M}/A\propto \xi$ for all $\xi$. On detuning with small but nonzero $JT/3\ll 1$, the linear trend saturates as $\xi \to 0$.

Figure 5

Results for the static Haldane model with $t_2=J/\left(12\sqrt{3/43}\right)$ and $\varphi =arccos\left(3\sqrt{3/43}\right)$. (a) Spectrum on the cylinder with edge modes. (b) Berry curvature for the torus geometry. (c) Orbital magnetization per unit area on the torus, in units of $-e/\hslash T$ for temperature ${\beta }^{-1}=0.05\times JT/3$. Vertical red lines denote the band edges.

Figure 6

(a)–(c) Spectra on a cylinder for $N=100$. (d)–(f) Orbital magnetization per unit area in units of $-e/\hslash T$ for the model on the torus for temperature ${\beta }^{-1}=0.05\times JT/3$. All figures are for $JT=\pi /8$, $t_2=J/\left(12\sqrt{3/43}\right)$ and $\varphi =arccos\left(3\sqrt{3/43}\right)$ but different $\lambda$. (a), (d) $\lambda =5$ with $C=1$ where the edge modes are inherited from the static model. (b), (e) $\lambda =7$ with $C=0$. This is an anomalous phase that shows two pairs of chiral edge modes. (c), (f) $\lambda =15$ with $C=1$. Note the shifts in the quasienergy in (d), (e) by $-0.5$ and $-0.44$, respectively. This is needed for the proper calculation of the Berry curvature. See text for the details.