(a) Schematic structure of the ruby lattice with staggered fluxes. There are six atoms in the unit cell within the dashed hexagon, marked from 1 to 6. Here, we consider only the NN hopping process. Each NN bond of hexagons has the phase factor ±ϕ (i.e., the hopping terms teiϕ) because of the staggered fluxes threading the intracells, and other NN hopping potentials are t. The arrows denote the directions of fluxes, and along the arrows, ϕ is positive; otherwise, ϕ is negative. The total fluxes of each hexagon and square are +6ϕ and −2ϕ, and there is no flux in triangles. We can find that the total flux is zero [+6ϕ+6×(−ϕ)=0] in a unit cell. (b) First BZ of the ruby lattice. Some high-symmetry points are marked as Γ, K, and M. The band structures of the ruby lattice with torus geometry (c) without, ϕ=0, and (d) with, ϕ=0.2π, staggered fluxes. The staggered fluxes lead to the band gaps, for instance, the band gap Δ1 between the first- and second-lowest bands.