Tri- & Tetra-Hyperbolic Iso-frequncy Topologies Complete Classification of Bi-Anisotropic Materials

We describe novel topological phases of iso-frequency k-space surfaces in bi-anisotropic optical materials - tri- and tetra-hyperbolic materials, which are induced by introduction of chirality. This completes the classification of iso-frequency topologies for bi-anisotropic materials, since as we show all optical materials belong to one of the following topological classes: tetra-, tri-, bi-, mono- or non-hyperbolic. We show that phase transitions between these classes occur in the k-space directions with zero group velocity at high k-vectors. This classification is based on the sets of high-k polaritons (HKPs), supported by materials. We obtain the equation describing these sets and characterize the longitudinal polarization impedance of HKPs.