  We investigate the ground state and quantum dynamics of an interacting
bosonic chain with the nonreciprocal hopping. In sharp contrast to its
Hermitian counterpart, the ground state can support Mott insulators in systems
with noninteger filling due to the competition between nonreciprocal hopping
and the on-site interaction. For the quantum dynamics, conservation laws for
non-Hermitian systems manifest a stark difference from their Hermitian
counterpart. In particular, for any Hermitian operator that commutes with the
Hamiltonian operator, its expectation value is guaranteed to be nonconserved in
the non-Hermitian quantum dynamics. To systematically identify the
non-Hermitian conservation law, we establish a generic approach for
constructing the conserved quantities in non-Hermitian many-body quantum
systems with completely real spectra, and illustrate it concretely by the
system under study. The direct experimental observation of Mott insulators in
systems with noninteger filling and non-Hermitian conservation laws can be
performed by ultracold atoms in optical lattices with the engineered
nonreciprocal hopping.
