  We develop a non-Hermitian effective theory for a repulsively interacting
Fermi gas in the excited branch. The on-shell $T$-matrix is employed as a
complex-valued interaction term, which describes a repulsive interaction
between atoms in the excited branch and a two-body inelastic decay to the
attractive branch. To see the feature of this model, we have addressed, in the
weak coupling regime, the excitation properties of a repulsive Fermi polaron as
well as the time-dependent number density. The analytic expressions obtained
for these quantities qualitatively show a good agreement with recent
experiments. By calculating the dynamical transverse spin susceptibility in the
random phase approximation, we show that a ferromagnetic system with nonzero
polarization undergoes a dynamical instability and tends towards a
heterogeneous phase.
