We investigate the localization properties of non-Hermitian bound states in the nonreciprocal two-particle Hubbard model. Remarkably, under the conditions of finite-size truncation, non-Hermitian skin effect (NHSE), bound-state localization (BSL), and topological edge localization will compete with each other, giving rise to diverse density profiles. When BSL and NHSE favor different direction of localization, bound states can either be destroyed or survive depending on the strength of nonreciprocal tunneling and two-particle interaction. For the unbound scattering states, unlike the exponential localization predicted by the Hatano-Nelson model, interaction can get the total localization split into multiple centers. Meanwhile, apart from non-Hermitian bound states out of continuum, non-Hermitian bound states in continuum can also be achieved as a by-product of the multiple independent tunneling factors. Besides, we not only consider the bosonic but also the fermionic interacting model, where in the latter case, localization properties are greatly enriched by the configuration space boundary caused by Pauli exclusion rule, and the non-Hermitian near-bound states are proposed. Finally, to achieve nontrivial topology, we include terms taking the form of two-photon tunneling, and the competition among all three localization mechanisms is analyzed via the transfer matrix approach, where the unique phenomenon, NHSE oriented by two-particle interaction, is demonstrated.

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• Edge states
• Many-body localization
• Open quantum systems
• Skin effect
• Non-Hermitian systems