  We study superfluid properties of alkali-earth-like Fermi atomic systems in
the presence of orbital Feshbach resonance. Using a two-band description of the
ground state and excited state and a mean-field approximation of the intra-band
atomic pairing, we investigate the phase transitions and crossover between
superfluid/normal phases. Defining an effective scattering length by combining
both inter-band and intra-band interactions, we derive closed form gap and
number density equations for both ground state and excited state atomic bands.
We find that our zero-temperature analytical results and finite-temperature
numerical results indicate that the system can show smooth crossover between
Bardeen, Cooper, and Schreifer (BCS) and Bose-Einstein Condensate (BEC)
superfluidity for atoms in each band. In addition, we find that inter-band and
intra-band interactions can induce quantum phase transitions between BCS/BEC
superfluid states of atoms in one band to that of the other. We anticipate that
our closed form analytical results can be used as a bench mark for future
experimental and theoretical investigations and will have an impact on the
current understanding of two-band superconductors such as MgB$_2$.
