  Much has been learned about universal properties of the eigenstate
entanglement entropy for one-dimensional lattice models, which is described by
a Hermitian Hamiltonian. While very less of it has been understood for
non-Hermitian systems. In the present work we study a non-Hermitian,
non-interacting model of fermions which is invariant under combined $PT$
transformation. Our models show a phase transition from $PT$ unbroken phase to
broken phase as we tune the hermiticity breaking parameter. Entanglement
entropy of such systems can be defined in two different ways, depending on
whether we consider only right (or equivalently only left) eigenstates or a
combination of both left and right eigenstates which form a complete set of
bi-orthonormal eigenstates. We demonstrate that the entanglement entropy of the
ground state and also of the typical excited states show some unique features
in both of these phases of the system. Most strikingly, entanglement entropy
obtained taking a combination of both left and right eigenstates shows an
exponential divergence with system size at the transition point. While in the
$PT$-unbroken phase, the entanglement entropy obtained from only the right (or
equivalently left) eigenstates shows identical behavior as of an equivalent
Hermitian system which is connected to the non-Hermitian system by a similarity
transformation.
