In the last years, kagome materials have received massive attention by virtue of being candidate hosts for a large variety of quantum phases: spin liquids, unconventional superconductivity, and topological phases of matter, to name the more exotic. One of the most interesting features is tunability: changing the filling, the noninteracting band structure can be tuned from flat bands to conventional metallic phases as well as to semimetals. In this paper, we concentrate on the latter. At specific lattice filling, the electronic bands have a semimetallic structure, hosting Dirac, massless quasiparticles, such as in graphene or other layered two dimensional materials. Specifically, we determine what terms can be added to the nearest-neighbor hopping that opens at the gap at the said Dirac point. These terms can, in principle, arise through external perturbations, interactions, or collective instabilities. We classify the 16 possible gap-opening terms according to the broken symmetries. Furthermore, we identify concrete microscopic realizations, allowing for an interpretation of these phases.
Ciceri et al. (Mon,) studied this question.