Stability of zero-energy Dirac touchings in the honeycomb Hofstadter problem

We study the band structure of electrons hopping on a honeycomb lattice with $p/q$ ($p,q$ are coprime integers) flux quanta through each elementary hexagon. In the nearest-neighbor hopping model the two bands that eventually form the $n=0$ Landau level have $2q$ zero-energy Dirac touchings. In this work we study the conditions needed for these Dirac points and their stability to various perturbations. We prove that these touchings and their locations are guaranteed by a combination of an antiunitary particle-hole symmetry and the lattice symmetries of the honeycomb structure. We also study the stability of the Dirac touchings to one-body perturbations that explicitly lower the symmetry.