  We reexamine effects of the $\rho-\omega$ meson mixing mediated by nucleon
polarizations on the symmetry energy in isospin-asymmetric nuclear matter.
Taking into account the rearrangement term neglected in previous studies by
others, we evaluate the $\rho-\omega$ mixing angle in a novel way within the
Relativistic Mean-Field Models with and without chiral limits. It is found that
the symmetry energy is significantly softened at high densities contrary to the
finding in earlier studies. As the first step of going beyond the lowest-order
calculations, we also solve the RPA equation for the $\rho-\omega$ mixing. In
this case, it is found that the symmetry energy is not only significantly
softened by the $\rho-\omega$ mixing at supra-saturation densities, similar to
the lowest-order $\rho-\omega$ mixing, but interestingly also softened at
subsaturation densities. In addition, the softening of the symmetry energy at
subsaturation densities can be partly suppressed by the nonlinear
self-interaction of the $\sigma$ meson.
