  We study the Hatano-Nelson model, i.e., a one-dimensional non-Hermitian chain
of spinless fermions with nearest-neighbour nonreciprocal hopping, in the
presence of repulsive nearest-neighbour interactions. At half filling, we find
two $\mathcal{PT}$ transitions, as the interaction strength increases. The
first transition is marked by an exceptional point between the first and the
second excited state in a finite-size system and is a first-order
symmetry-breaking transition into a charge-density wave regime. Persistent
currents characteristic of the Hatano-Nelson model abruptly vanish at the
transition. The second transition happens at a critical interaction strength
that scales with the system size and can thus only be observed in finite-size
systems. It is characterized by a collapse of all energy eigenvalues onto the
real axis. We further show that in a strong interaction regime, but away from
half filling, the many-body spectrum shows point gaps with nontrivial winding
numbers, akin to the topological properties of the single-particle spectrum of
the Hatano-Nelson chain, which indicates the skin effect of extensive many-body
eigenstates under open boundary conditions. Our results can be applied to other
models such as the non-Hermitian Su-Schrieffer-Heeger-type model and contribute
to an understanding of fermionic many-body systems with non-Hermitian
Hamiltonians.
