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

Emergent Mott insulators and non-Hermitian conservation laws in an interacting bosonic chain with noninteger filling and nonreciprocal hopping