  The discovery of graphene makes it highly desirable to seek new
two-dimensional materials. Through first-principles investigation, we predict
two-dimensional materials of ReN$_{2}$: honeycomb and tetragonal structures.
The phonon spectra establish the dynamical stability for both of the two
structures, and the calculated in-plane stiffness constants proves their
mechanical stability. The energy bands near the Fermi level consist of N-p and
Re-d orbitals for the honeycomb structure, and are mainly from Re d orbitals
for the tetragonal structure. While the tetragonal structure is non-magnetic,
the honeycomb structure has N-based ferromagnetism, which will transit to
anti-ferromagnetism under 14$\%$ biaxial strain. The calculated electron
localization function and spin density indicate that direct N-N bond can occur
only in the honeycomb structure. The ferromagnetism allows us to distinguish
the two 2D phases easily. The tetragonal phase has lower energy than the
honeycomb one, which means that the tetragonal phase is more stable, but the
hexagonal phase has much larger bulk, shear, and Young's muduli than the
tetragonal phase. The tetragonal phase is a three-bands metal, and the
hexagonal phase is a ferromagnetic semi-metal. The special structural,
electronic, magnetic, and optical properties in the honeycomb and tetragonal
structures make them promising for novel applications.
