Topological Order and Absence of Band Insulators at Integer Filling in Non-Symmorphic Crystals

TL;DR

This paper demonstrates that in non-symmorphic crystals, certain integer fillings cannot host band insulators due to symmetry constraints, implying the necessity of topological order in such systems.

Contribution

It reveals that non-symmorphic symmetries can prevent band insulators at specific integer fillings, indicating topological order is required for a gapped symmetric ground state.

Findings

Non-symmorphic symmetries can forbid band insulators at certain integer fillings.

Presence of topological order is inferred in these cases.

Flux threading arguments apply to various quantum systems.

Abstract

Band insulators appear in a crystalline system only when the filling -- the number of electrons per unit cell and spin projection -- is an integer. At fractional filling, an insulating phase that preserves all symmetries is a Mott insulator, i.e. it is either gapless or, if gapped, displays fractionalized excitations and topological order. We raise the inverse question -- at an integer filling is a band insulator always possible? Here we show that lattice symmetries may forbid a band insulator even at certain integer fillings, if the crystal is non-symmorphic -- a property shared by a majority of three-dimensional crystal structures. In these cases, one may infer the existence of topological order if the ground state is gapped and fully symmetric. This is demonstrated using a non-perturbative flux threading argument, which has immediate applications to quantum spin systems and bosonic insulators in addition to electronic band structures in the absence of spin-orbit interactions.