Topological 'Luttinger' invariants for filling-enforced non-symmorphic semimetals

Luttinger's theorem is a fundamental result in the theory of interacting Fermi systems: it states that the volume inside the Fermi surface is left invariant by interactions, if the number of particles is held fixed. Although this is traditionally justified in terms of analytic properties of Green's functions, it can be viewed as arising from a momentum balance argument that examines the response of the ground state to the insertion of a single flux quantum (Oshikawa 2000 Phys. Rev. Lett. 84 3370). This reveals that the Fermi volume is a topologically protected quantity, whose change requires a phase transition. However, this sheds no light on the stability or lack thereof of interacting semimetals, that either lack a Fermi surface, or have perfectly compensated electron and hole pockets and hence vanishing net Fermi volume. Here, I show that semimetallic phases in non-symmorphic crystals possess additional topological 'Luttinger invariants' that can be nonzero even though the Fermi volume vanishes. The existence of these invariants is linked to the inability of non-symmorphic crystals to host band insulating ground states except at special fillings. I exemplify the use of these new invariants by showing that they distinguish various classes of two- and three-dimensional semimetals.