  Discovering the physical requirements for meeting the indefinite permittivity
in natural material as well as proposing a new natural hyperbolic media offer a
possible route to significantly improve our knowledge and ability to confine
and controlling light in optoelectronic devices. We demonstrate the
hyperbolicity in a class of materials with hexagonal P6/mmm and P6$_{3}$/mmc
layered crystal structures and its physical origin is thoroughly investigated.
By utilizing density functional theory and solving the Bethe-Salpeter equation
(BSE), we find that the layered crystal structure and symmetry imposed
constraints in Li$_{3}$N gives rise to an exceedingly strong anisotropy in
optical responses along in- and out-of-plane directions of the crystals making
it a natural hyperbolic in a broad spectral range from the visible spectrum to
the ultraviolet. More excitingly, the hyperbolicity relation to anisotropic
interband absorption in addition to the impressive dependency of the conduction
band to the lattice constant along the out-of-plane direction provide the
hyperbolicity tunability in these hexagonal structures under strain, doping,
and alloying. Our findings not only suggest a large family of real hexagonal
compounds as a unique class of materials for realization of the highly tunable
broad band hyperbolicity but also offers an approach to search for new
hyperbolic
