  Recently, models with long-range interactions -- known as Hatsugai-Kohmoto
(HK) models -- have emerged as a promising tool to study the emergence of
superconductivity and topology in strongly correlated systems. Two obstacles,
however, have made it difficult to understand the applicability of these
models, especially to topological features: they have thermodynamically large
ground state degeneracies, and they tacitly assume spin conservation. We show
that neither are essential to HK models and that both can be avoided by
introducing interactions between tight-binding states in the orbital basis,
rather than between energy eigenstates. To solve these "orbital" models, we
introduce a general technique for solving HK models and show that previous
models appear as special cases. We illustrate our method by exactly solving
graphene and the Kane-Mele model with HK interactions. Both realize Mott
insulating phases with finite magnetic susceptibility; the graphene model has a
fourfold degenerate ground state while the Kane-Mele model has a nondegenerate
ground state in the presence of interactions. Our technique then allows us to
study the effect of strong interactions on symmetry-enforced degeneracy in
spin-orbit coupled double-Dirac semimetals. We show that adding HK interactions
to a double Dirac semi-metal leads to a Mott insulating, spin liquid phase. We
then use a Schrieffer-Wolff transformation to express the low-energy
Hamiltonian in terms of the spin degrees of freedom, making the spin-charge
separation explicit. Finally, we enumerate a broader class of
symmetry-preserving HK interactions and show how they can violate insulating
filling constraints derived from space group symmetries. This suggests that new
approaches are needed to study topological order in the presence of long-range
interactions of the HK type.
