Lattice generalization of the Dirac equation to general spin and the role of the flat band

We provide a novel setup for generalizing the two-dimensional pseudospin S=1/2 Dirac equation, arising in graphene's honeycomb lattice, to general pseudospin-S. We engineer these band structures as a nearest-neighbor hopping Hamiltonian involving stacked triangular lattices. We obtain multi-layered low energy excitations around half-filling described by a two-dimensional Dirac equation of the form H=v_F S\cdot p, where S represents an arbitrary spin-S (integer or half-integer). For integer-S, a flat band appears, whose presence modifies qualitatively the response of the system. Among physical observables, the density of states, the optical conductivity and the peculiarities of Klein tunneling are investigated. We also study Chern numbers as well as the zero-energy Landau level degeneracy. By changing the stacking pattern, the topological properties are altered significantly, with no obvious analogue in multilayer graphene stacks.