  We calculate the momentum dependence of the $\rho^0-\omega$ mixing amplitude
in vacuum with vector nucleon-nucleon interaction in presence of a constant
homogeneous weak magnetic field background. The mixing amplitude is generated
by the nucleon-nucleon ($NN$) interaction and thus driven by the neutron-proton
mass difference along with a constant magnetic field. We find a significant
effect of magnetic field on the mixing amplitude. We also calculate the Charge
symmetry violating (CSV) $NN$ potential induced by the magnetic field dependent
mixing amplitude. The presence of the magnetic field influences the $NN$
potential substantially which can have important consequences in highly
magnetized astrophysical compact objects, such as magnetars. The most important
observation of this work is that the mixing amplitude is non-zero, leading to
positive contribute to the CSV potential if the proton and neutron masses are
taken to be equal.
