Hall effect in topologically trivial isolated flat-band systems

TL;DR

This paper develops a quantum mechanical formula for the Hall effect in topologically trivial flat-band systems, revealing a universal contribution from quantum geometry even when traditional carriers are absent.

Contribution

It introduces a gauge-invariant quantum formula for Hall conductivity applicable to flat-band systems with zero Chern number, extending beyond semiclassical approximations.

Findings

Hall conductivity depends on energy differences and quantum geometric tensor in flat bands.

Universal Hall response exists even with zero Berry curvature.

Numerical validation on honeycomb and Kagome lattice models confirms theoretical predictions.

Abstract

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 $N=1$, $M=2$).