The low-energy effective theories for gapped insulators are classified by three parameters: permittivity ϵ, permeability μ, and theta angle θ. Crystals with periodic ϵ are known as photonic crystals. We here study the band structure of photons in a new type of crystals with periodic θ (modulo 2π) in space, which we call the axion crystals. We find that the axion crystals have a number of new properties that the usual photonic crystals do not possess, such as the helicity-dependent mass gap and nonrelativistic gapless dispersion relation at small momentum. We briefly discuss possible realizations of axion crystals in condensed matter systems and high-energy physics.
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