  In this paper, based on a one-dimensional non-Hermitian spin model with
$\mathcal{RT}$-invariant term, we study the non-Hermitian physics for the two
(nearly) degenerate ground states. By using the high-order perturbation method,
an effective pseudo-spin model is obtained to describe non-Hermitian physics
for the two (nearly) degenerate ground states, which are precisely consistent
with the numerical calculations. We found that there may exist effective (anti)
$\mathcal{PT}$ symmetry for the effective pseudo-spin model of the two (nearly)
degenerate ground states. In particular, there exists spontaneous (anti)
$\mathcal{PT}$ -symmetry breaking for the topological degenerate ground states
with tunable parameters in external fields. We also found that even a very tiny
imaginary external field applied will drive $\mathcal{PT}$ phase transition.
