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

TL;DR

This paper establishes filling constraints for spin-orbit coupled insulators in all 230 space groups, using entanglement and manifold methods, without relying on spin conservation, to identify conditions for gapped, non-fractionalized phases.

Contribution

It introduces two novel approaches to determine filling constraints in spin-orbit coupled systems, applicable to all space groups and only assuming time-reversal symmetry.

Findings

Filling constraints depend on whether the crystal is symmorphic or non-symmorphic.

Conditions for gapped insulators are clarified without spin conservation.

Applications to real materials and potential for exploring exotic ground states.

Abstract

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. These approaches only assume time reversal rather than spin rotation invariance. The results depend crucially on whether the crystal symmetry is symmorphic. Our results clarify when one may infer the existence of an exotic ground state based on the absence of order, and we point out applications to experimentally realized materials. Extensions to new situations involving purely spin models are also mentioned.