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

## TL;DR

This paper establishes a universal relation linking quantized Hall conductivity with charge and flux densities in 2D lattice insulators, revealing constraints on gapped states and their topological properties.

## Contribution

It derives a general, symmetry-based relation for Hall conductance in interacting lattice systems, extending Laughlin's argument and Lieb-Schultz-Mattis theorem.

## Key findings

- Gapped states at fractional filling must have nonzero Hall conductivity or anyons.
- The relation applies broadly to interacting many-body systems with magnetic translation symmetry.
- Constraints exclude trivial Mott insulators at fractional fillings.

## Abstract

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.