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

Significance Quantum spin liquids are exotic insulators in which the electron’s spin, charge, and fermionic statistics are carried by separate excitations. They have received attention for their potential as a quantum memory and as a parent state for high-temperature superconductivity. A key principle guiding the search for experimental candidate materials is the absence of conventional order in insulators at “fractional” electron filling. Previous results established what constitutes fractional filling, but assumed symmetries that are typically not respected in real materials or considered only simple lattices. Here, assuming only physical symmetries we establish filling conditions for all 230 space groups. This should aid in the search for quantum spin liquids and topological semimetals. 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 one is an entanglement-based scheme, and the second one studies the system on an appropriate flat “Bieberbach” manifold to obtain the filling conditions for all 230 space groups. These approaches assume only 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.