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We study the Hatano-Nelson model, i.e., a one-dimensional non-Hermitian chainof spinless fermions with nearest-neighbour nonreciprocal hopping, in thepresence of repulsive nearest-neighbour interactions. At half filling, we findtwo $\mathcal{PT}$ transitions, as the interaction strength increases. Thefirst transition is marked by an exceptional point between the first and thesecond excited state in a finite-size system and is a first-ordersymmetry-breaking transition into a charge-density wave regime. Persistentcurrents characteristic of the Hatano-Nelson model abruptly vanish at thetransition. The second transition happens at a critical interaction strengththat scales with the system size and can thus only be observed in finite-sizesystems. It is characterized by a collapse of all energy eigenvalues onto thereal axis. We further show that in a strong interaction regime, but away fromhalf filling, the many-body spectrum shows point gaps with nontrivial windingnumbers, akin to the topological properties of the single-particle spectrum ofthe Hatano-Nelson chain, which indicates the skin effect of extensive many-bodyeigenstates under open boundary conditions. Our results can be applied to othermodels such as the non-Hermitian Su-Schrieffer-Heeger-type model and contributeto an understanding of fermionic many-body systems with non-HermitianHamiltonians.

Symmetry breaking and spectral structure of the interacting Hatano-Nelson model