Saddle point singularity and optical phase transition in bilayer hyperbolic metamaterials

We study theoretically and numerically high density of states for hyperbolic bilayered metamaterials (HMM). It reveals that density response of HMM is reminiscent of Fermi electronic band structure of metal or semiconductors. By the method of Green function a van Hove type singularity is found in photonic density spectra of HMM with saddle point localization on photonic Fermi surface (FS) of metamaterial. Similar to the electronic systems, the photonic FS experiences instabilities induced by the changes in volume fractions of its constituents that leads to the Lifshitz type zero-temperature phase transition between FS of types I and II hyperbolic states at the topology protected critical point.