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The cooperation between non-Hermiticity and interaction brings about a lot ofcounterintuitive behaviors, which are impossible to exist in the framework ofthe Hermitian system. We study the effect of a non-Hermitian impurity on theHubbard model in the context of $\eta $ symmetry. We show that thenon-Hermitian Hubbard Hamiltonian can respect a full real spectrum even if alocal non-Hermitian impurity is applied to. The balance between dissipation ofsingle fermion and on-site pair fluctuation results in a highest-ordercoalescing state with off-diagonal long-range order (ODLRO). Based on thecharacteristic of High-order EP, the critical non-Hermitian Hubbard modelallows the generation of such a steady superconducting-like state through thetime evolution from an arbitrary initial state, including the vacuum state.Remarkably, this dynamic scheme is insensitive to the on-site interaction andentirely independent of the locations of particle dissipation and pairfluctuation. Our results lay the groundwork for the dynamical generation of asteady ODLRO state through the critical non-Hermitian strongly correlatedsystem.

Dynamical preparation of a steady ODLRO state in the Hubbard model with local non-Hermitian impurity