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

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. A variety of complementary arguments, based on a cut-and-glue procedure, the many-body electric polarization, and a fractionalization algebra of magnetic translation symmetry, are given. Our universal relation is applied to several examples to show nontrivial constraints. In particular, a gapped ground state at a fractional charge filling per physical unit cell must have either a nonvanishing Hall conductivity or anyon excitations, excluding a trivial Mott insulator.