We overview results on the dynamics of waves (quantum, E&M,…) in continuum honeycomb structures and their deformations. We study phenomena which arise from the presence of Dirac points in the bulk band structure. These include the existence of robust edge states, which localize along zigzag sharp terminations, and along general “rational” domain walls. Finally we discuss the emergence of pseudo-magnetic fields
in non-uniformly deformed honeycomb structures. Our asymptotic theory applies in settings where there is no tight-binding regime. We apply these results to a predict Landau levels in photonic crystals, and present numerical confirmation.
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