Filling-enforced constraint on the quantized Hall conductivity on a periodic lattice

Understand

We discuss quantum Hall effects in a gapped insulator on a periodic two-dimensional lattice.

• We derive a universal relation among the the quantized Hall conductivity, and charge and flux densities per physical unit cell.
• This follows from the magnetic translation symmetry and the large gauge invariance, and holds for a very general class of interacting many-body systems.
• It can be understood as a combination of Laughlin's gauge invariance argument and Lieb-Schultz-Mattis-type theorem.