Filling constraints for spin-orbit coupled insulators in symmorphic and non-symmorphic crystals

Understand

We determine conditions on the filling of electrons in a crystalline lattice to obtain the equivalent of a band insulator -- a gapped insulator with neither symmetry breaking nor fractionalized excitations.

• We allow for strong interactions, which precludes a free particle description.
• Previous approaches that extend the Lieb-Schultz-Mattis argument invoked spin conservation in an essential way, and cannot be applied to the physically interesting case of spin-orbit coupled systems.
• Here we introduce two approaches, the first an entanglement based scheme, while the second studies the system on an appropriate flat `Bieberbach' manifold to obtain the filling conditions for all 230 space groups.