A symmetry proof shows that 2q zero-energy Dirac touchings in the honeycomb Hofstadter model are protected by an anti-unitary particle-hole symmetry together with lattice symmetries, for any bipartite hopping range.
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Stability of zero energy Dirac touchings in the honeycomb Hofstadter problem
A symmetry proof shows that 2q zero-energy Dirac touchings in the honeycomb Hofstadter model are protected by an anti-unitary particle-hole symmetry together with lattice symmetries, for any bipartite hopping range.