Orbital Analogue of the Quantum Anomalous Hall Effect in $p$-Band Systems

We investigate the topological insulating states of the $p$-band systems in optical lattices induced by the on site orbital angular momentum polarization, which exhibit gapless edge modes in the absence of Landau levels. This effect arises from the energy-level splitting between the on site $p_x+i{p}_y$ and $p_x-i{p}_y$ orbitals by rotating each optical lattice site around its own center. At large rotation angular velocities, this model naturally reduces to two copies of Haldane’s quantum Hall model. The distribution of the Berry curvature in momentum space and the quantized Chern numbers are calculated. The experimental realization of this state is feasible.

Figure 1

The band structure of Eq. (1) at ${\Omega }_z>0$ as shown in (a), (b), and (c). Only the lower two bands are presented, and the upper two are symmetric respect to the zero energy. (a) ${\Omega }_z/t_{\parallel }=0.3$; (b) ${\Omega }_z/t_{\parallel }=1.5$ where a single gapless Dirac cone appears; (c) ${\Omega }_z/t_{\parallel }=3$ where two massive Dirac cones appear at $K_{1,2}$ between the lower two bands and also between the upper two bands; (d) The pattern of the induced NNN hopping (complex valued) at ${\Omega }_z\gg t_{\parallel }$, which is generated by the virtual hopping between orbitals with opposite chirality.

Figure 2

The distribution of the Berry curvature $F_{n,xy}(\overrightarrow{k})$ in the Brillouin zone for the lower two bands at different ${\Omega }_z$. (a), (c), and (e) [(b), (d), and (f)] are $F_{n,xy}(\overrightarrow{k})$ of the first (second) band at ${\Omega }_z/t_{\parallel }=0.3$, 1.3 and 1.7, respectively. The Chern number of the first band is 1, and that of the second band changes from 0 to $-1$ at ${\Omega }_z/t_{\parallel }=\frac{3}{2}$.

Figure 3

The gapless edge excitations with the open boundary condition along the zigzag edge of the hexagon lattice. (a) ${\Omega }_z/t_{\parallel }=0.3$; (b) ${\Omega }_z/t_{\parallel }=1.7$. A topological phase transition occurs at ${\Omega }_z/t_{\parallel }=\frac{3}{2}$ above which the edge modes between the middle two bands disappear.

Figure 4

Edge and bulk states spectra of Eq. (3) with the open boundary condition along the zigzag edge. The flux per plaquette $\varphi /(2\pi )=1/20$. (a) ${\Omega }_z/t_{\parallel }=0.2$; (b) ${\Omega }_z/t_{\parallel }=1.5$.