Landau quantization of C 3 high-fold fermions in honeycomb lattices

We present the theory of the Landau levels of C 3 high-fold fermions in honeycomb lattices. These fermions arise as a generalization of the tight-binding model in a honeycomb lattice with one orbital per site, known to describe graphene, to a family of models with several orbitals per site transforming according to representations of C 3, relevant for triangulene two-dimensional crystals. The models are defined by the number of orbitals n a and n b at the a and b sites of the unit cell. They feature energy bands that combine flat bands and Dirac cones, that are isomorphic to those of graphene, the Lieb, p xy honeycomb, and the Kagome tight-binding models. Combining tight-binding and k times p methods, here we compute the Landau levels of the (1,1), (1,2), (2,2) and (3,3) models, we compare them to those of the reference models, and we explore the similarities and differences with the Landau levels in graphene. Our findings reveal that the nature of C 3 high-fold fermions combine Dirac-type Landau quantization, originating from the linear crossings and associated relativistic band dispersion, with anomalous Landau quantization rooted in the quantum geometry of the singular flat bands.