Hall effect in topologically trivial isolated flat-band systems

We study the Hall effect in topologically trivial isolated flat-band systems (i.e., flat bands are separated from other bands and have zero Chern number) for a weak magnetic field. In a naive semiclassical picture, the Hall conductivity vanishes when dispersive bands are unoccupied, since there are no mobile carriers. To go beyond the semiclassical picture, we establish a fully quantum mechanical gauge-invariant formula for the Hall conductivity that can be applied to any lattice models. We apply the formula to a general $N+M$-band model with $N$ dispersive bands and $M$-fold degenerate isolated flat bands, and find that when the dispersive bands are unoccupied, the total conductivity takes a universal form consisting of the energy difference between the dispersive and flat bands, and the non-Abelian quantum geometric tensor of the flat bands, which can be nonzero in systems with vanishing Berry curvature. We numerically confirm the Hall effect for isolated flat-band lattice models on the honeycomb lattice ($N=M=1$) and two different Kagome lattices ($N=2$, $M=1$ and $N=1$, $M=2$).