Decoupling the energy and momentum of photons in the quasistatic limit

We theoretically show that the frequency and momentum of a photon are not necessarily proportional to one another at low frequencies in photonic crystals comprising materials with positive- and negative-valued material properties. We rigorously determine closed-form conditions for the light cone to emanate from points other than the origin of $k$ space, ultimately decoupling the first band from the origin and demonstrating light propagation at zero energy with nonzero crystal momentum. We also numerically show that first bands can originate from an arbitrary Bloch coordinate as well as from multiple coordinates simultaneously.