Exceptional Points in Hermitian Media, and Density of States Singularities in Anisotropic, Tellegen, Chiral, Moving-Medium, Omega and Other Isotropy-Broken Materials

Anisotropic, Tellegen, chiral, moving-medium-type, omega, gyrotropic, hyperbolic, and multi-hyperbolic materials form an important class of isotropy-broken photonic media in which wave propagation can no longer be characterized by the Fresnel wave surface alone. Here we show that Fresnel wave surfaces can be converted into propagation maps that organize positive- and negative-phase-velocity propagation together with attenuation and amplification. In Hermitian media, the boundary between forward and backward propagation forms the orthogonal-phase-velocity separatrix. This separatrix is also a continuous locus of orthogonal-phase-velocity exceptional points, where the index-of-refraction operator becomes defective even though the material medium remains Hermitian. In non-Hermitian media, the attenuation–amplification boundary forms the loss–gain separatrix. The associated loss–gain singularities occur where the handedness remains continuous while the gain–loss character changes sign. Their physical importance is revealed by the momentum-resolved density of states: at these points, the Lorentzian linewidth of the non-Hermitian momentum-resolved density of states (DOS) collapses, producing sharp DOS peaks whose sign reverses across the separatrix. Thus, loss–gain singularities are threshold-like singularities of the Fresnel wave-surface propagation map, generated by non-Hermitian linewidth collapse. The result is a compact geometric language for describing how handedness, degeneracy, loss, gain, and momentum-resolved DOS are organized in isotropy-broken electromagnetic materials.