Abstract

Hall effect is a hallmark of transport phenomena [1] that can bring us essential information on charge
carriers in materials, including their density, mass, the sign of charges, and scattering properties. At weak
magnetic fields, one can often adopt a semiclassical transport theory, in which the Hall conductivity is
proportional to the inverse of the effective mass of an electron. If one naively takes the large mass limit,
which corresponds to a flat-band limit, the Hall conductivity vanishes. 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 two-band model with one dispersive and one isolated flat band, and find that the total conductivity takes a universal form as an integral
of a product of the squared Berry curvature and energy difference between the two bands. In particular, the Hall coefficient can become nonzero in the flat-band systems with broken inversion symmetry.
We numerically confirm this Hall effect for an isolated flat-band lattice model on the honeycomb lattice [2].
[1] E. H. Hall, On a New Action of the Magnet on Electric Currents, Am. J. Math. 2, 287 (1879).
[2] R. Nagashima, M. Ogata, and N. Tsuji, Hall effect in isolated flat-band systems, arxiv.2506.09535