Polarization of vacuum by an external magnetic field in the $(2+1)$-dimensional quantum electrodynamics with a nonzero fermion density

The Green's function of the Dirac equation with an external stationary homogeneous magnetic field in the $(2+1)$-dimensional quantum electrodynamics ($\mathrm{QED}_{2+1}$) with a nonzero fermion density is constructed. An expression for the polarization operator in an external stationary homogeneous magnetic field with a nonzero chemical potential is derived in the one-loop $\mathrm{QED}_{2+1}$ approximation. The contribution of the induced Chern–Simons term to the polarization operator and the effective Lagrangian for the fermion density corresponding to the occupation of $n$ relativistic Landau levels in an external magnetic field are calculated. An expression of the induced Chern–Simons term in a magnetic field for the case of a finite temperature and a nonzero chemical potential is obtained.