  By a combined study with first-principles calculations and symmetry analysis,
we theoretically investigate the electronic properties of monolayer
MoSi$_2$N$_4$. While the spin-orbital coupling results in bands splitting, the
horizontal mirror symmetry locks the spin polarization along z-direction. In
addition, a three-band tight-binding model is constructed to describe the
low-energy quasi-particle states of monolayer MoSi$_2$N$_4$, which can be
generalized to strained MoSi$_2$N$_4$ and its derivatives. The calculations
using the tight-binding model show an undamped $\sqrt{q}$-dependent plasmon
mode that agrees well with the results of first-principles calculations. Our
model can be extended to be suitable for future theoretical and numerical
studies of low-energy properties in MoSi$_2$N$_4$ family materials.
Furthermore, the study of electronic properties of monolayer MoSi$_2$N$_4$
paves a way for its applications in spintronics and plasmonics.
