Nearly flat band with Chern number $C=2$ on the dice lattice

We point out the possibility of a nearly flat band with Chern number $C=2$ on the dice lattice in a simple nearest-neighbor tight-binding model. This lattice can be naturally formed by three adjacent $\left(111\right)$ layers of cubic lattice, which may be realized in certain thin films or artificial heterostructures, such as the SrTiO${}_3$/SrIrO${}_3$/SrTiO${}_3$ trilayer heterostructure grown along the $\left(111\right)$ direction. The flatness of two bands is protected by the bipartite nature of the lattice. Including the Rashba spin-orbit coupling on nearest-neighbor bonds causes the flat bands to separate from the others but maintain their flatness. Repulsive interaction will drive spontaneous ferromagnetism on the Kramer pair of the flat bands and split them into two nearly flat bands with Chern number $C=\pm 2$. We thus propose that this may be a route to the quantum anomalous Hall effect and further conjecture that the partial filling of the $C=2$ band may realize exotic fractional quantum Hall effects.

Figure 1

Top: The dice lattice. Small upward triangles (bottom layer), downward triangles (top layer), and hexagons (middle layer) indicate the three sublattices. Numbers 1, 2, and 3 label the three basis sites in the unit cell at origin. Coordination-number-3 sites (1 and 2) and coordination-number-6 sites (3) are the two subsets of this bipartite lattice. Blue dashed arrows labeled by ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ indicate the two translations of the dice lattice. Thick green arrows labeled as $D_{ij}$ indicate the Rashba SOC directions on those bonds $ij$, with coordination-number-6 site $j$. Red dotted arrows with the labels $+x$, $+y$, and $+z$ indicate of the projection of the cubic lattice axis. Capital letters $X$, $Y$, and $Z$ are axes for spin space in Rashba SOC. $Z$ is the original $\left(111\right)$ direction. Bottom: Perspective view of three adjacent $\left(111\right)$ layers of the cubic lattice. The middle layer has a different color for easy recognition. The top view of this trilayer is the dice lattice.

Figure 2

Dispersions of NN tight-binding models on dice lattice along high-symmetry directions. Parameters used are $\epsilon =0.6t$, $\lambda =0.3t$, and $B=0.2t$. The bands marked as red in the middle are the (nearly) flat bands. Top left corner is the Brillouin zone with the high-symmetry lines indicated by dashed blue lines. (a) Spin-independent hoppings only. (b) Spin-independent hoppings plus Rashba SOC $\lambda$. (c) Spin-independent hoppings plus Rashba SOC $\lambda$ and magnetic field $B$ along the $Z$ (111) direction. The Chern number of each band is indicated.

Figure 3

Local Wannier function of one of the flat bands of the NN model on a dice lattice (black dotted lines) with Rashba SOC $\lambda$ ($t=1$ for simplicity). The two-component vector on each coordination-number-3 site indicates spin-up and spin-down amplitudes on that site. Amplitudes on coordination-number-6 sites vanish, and therefore the parameter $\epsilon$ has no effect. The small hexagon is the “guiding center” of this Wannier function.

Figure 4

Left: The two nearly flat bands (red) with Chern numbers $C=\pm 2$. Parameters are $\epsilon =0.6t$, $\lambda =0.3t$, $B_1=0.2440t$, and $B_3=-0.0162t$. Right: Dispersion of a cylinder with 32-unit-cell open boundary condition along ${\mathbf{e}}_2$ and periodic boundary condition along ${\mathbf{e}}_1$, showing the edge states between the nearly flat bands.