Interplay between boundary conditions and Wilson's mass in Dirac-like Hamiltonians

Dirac-like Hamiltonians, linear in momentum k, describe the low-energy physics of a large set of novel materials, including graphene, topological insulators, and Weyl fermions. We show here that the inclusion of a minimal k2 Wilson's mass correction improves the models and allows for systematic derivations of appropriate boundary conditions for the envelope functions on finite systems. Considering only Wilson's masses allowed by symmetry, we show that the k2 corrections are equivalent to Berry-Mondragon's discontinuous boundary conditions. This allows for simple numerical implementations of regularized Dirac models on a lattice, while properly accounting for the desired boundary condition. We apply our results on graphene nanoribbons (zigzag and armchair), and on a PbSe monolayer (topological crystalline insulator). For graphene, we find generalized Brey-Fertig boundary conditions, which correctly describe the small gap seen on ab initio data for the metallic armchair nanoribbon. On PbSe, we show how our approach can be used to find spin-orbital-coupled boundary conditions. Overall, our discussions are set on a generic model that can be easily generalized for any Dirac-like Hamiltonian.