  A one-dimensional Bose Hubbard model with unidirectional hopping is shown to
be exactly solvable. Applying the algebraic Bethe ansatz method, we prove the
integrability of the model and derive the Bethe ansatz equations. The exact
eigenvalue spectrum can be obtained by solving these equations. The
distribution of Bethe roots reveals the presence of a superfluid-Mott insulator
transition at the ground state, and the critical point is determined. By
adjusting the boundary parameter, we demonstrate the existence of non-Hermitian
skin effect even in the presence of interaction, but it is completely
suppressed for the Mott insulator state in the thermodynamical limit. Our
result represents a new class of exactly solvable non-Hermitian many-body
systems, which have no Hermitian correspondence and can be used as a benchmark
for various numerical techniques developed for non-Hermitian many-body systems.
