Perturbative/nonperturbative aspects of Bloch electrons in a honeycomb lattice

We revisit the spectral problem for Bloch electrons in a two-dimensional bipartite honeycomb lattice under a uniform magnetic field. It is well-known that such a honeycomb structure is realized in graphene. We present a systematic framework to compute the perturbative magnetic flux expansions near two distinct band edges. We then analyze the nonperturbative bandwidth of the spectrum. It turns out that there is a novel similarity between the spectrum near the Dirac point in the honeycomb lattice and the spectrum in the supersymmetric sine-Gordon quantum mechanics. We finally confirm a nontrivial vacuum-instanton-bion threesome relationship. Our analysis heavily relies on numerical experiments.