Fig. 1

Bulk and surface states in an effective-medium model of chiral hyperbolic metamaterials. Schematic view of energy dispersion near a type-II Weyl point for the chiral hyperbolic metamaterials (CHM) with respect to a x–y and b y–z momentum space. Bottom planes show momentum resolved density of states (DOS) at the ‘Fermi energy’ E _F indicated by the purple plane. c Effectively modeled band structures of bulk (7th and 8th bands) and surface states for the CHM. Red dots (W1) indicate one pair of type-II Weyl points. Blue dots (W2) (one is hidden) are their chiral partners. One surface state between two topological partners is indicated by the red surface, on which the cyan lines highlight surface-state arcs. To show the gap, the other surface state is not plotted; it can be obtained after time-reversal operation of the present one. d Bulk bands along Y(0,−3,0)–Γ(0,0,0)–X(3,0,0). e Surface band structure on a varying-frequency k _x –k _y map. The arc (k _y  > 0) is constructed from intersection between the surface state (red surface in c) and a plane with constraints: ω is proportional to k _y and k _x is arbitrary. f–h, show three equi-frequency contours (EFCs) containing both bulk and surface states on the k _x –k _y plane. Their corresponding frequencies are indicated in c with cyan lines and in d with dashed lines, respectively. In f, the dashed line indicates the flat band (Γ–X) in d. Here, ε ₀ = μ ₀ = c = 1, ω, k _x, and k _y are normalized with respect of ω _p (at k _y  = 0), which is the plasma frequency