Isolated flat bands and spin-1 conical bands in two-dimensional lattices

Dispersionless bands, such as Landau levels, serve as a good starting point for obtaining interesting correlated states when interactions are added. With this motivation in mind, we study a variety of dispersionless (“flat”) band structures that arise in tight-binding Hamiltonians defined on hexagonal and kagome lattices with staggered fluxes. The flat bands and their neighboring dispersing bands have several notable features: (a) flat bands can be isolated from other bands by breaking time-reversal symmetry, allowing for an extensive degeneracy when these bands are partially filled; (b) an isolated flat band corresponds to a critical point between regimes where the band is electron-like or hole-like, with an anomalous Hall conductance that changes sign across the transition; (c) when the gap between a flat band and two neighboring bands closes, the system is described by a single spin-1 conical-like spectrum, extending to higher angular momentum the spin-1/2 Dirac-like spectra in topological insulators and graphene; (d) some configurations of parameters admit two isolated parallel flat bands, raising the possibility of exotic “heavy excitons”; and (e) we find that the Chern number of the flat bands, in all instances that we study here, is zero.

Figure 1

(Color online) Energy dispersions with staggered flux phases ${\varphi }_{+}$ and ${\varphi }_{-}$ on the up and down triangles of the kagome lattice. The dispersion on the left (type I) corresponds to ${\varphi }_{+}=2\pi$ and ${\varphi }_{-}=-\pi$ while the dispersion on the right (type II) corresponds to ${\varphi }_{+}={\varphi }_{-}=3\pi /2$.

Figure 2

The kagome lattice with fluxes ${\varphi }_{+}$ and ${\varphi }_{-}$ on alternating triangles, which correspond to a flux $-\left({\varphi }_{+}+{\varphi }_{-}\right)$ inside each hexagon. ${\mathbf{s}}_{1,2,3}$ are vectors pointing from the center of a down triangle to its three up neighbors.

Figure 3

(Color online) Left (right): middle band showing electron-like (hole-like) dispersion corresponding to $ϵ<0$ $\left(>0\right)$ in Eq. (10).

Figure 4

(Color online) Exact spectrum for $t=1$, $g=\sqrt{3}/2$, and ${\varphi }_{\pm }=3\pi /2$.