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The introduction of non-Hermiticity has greatly enriched the research fieldof traditional condensed matter physics, and eventually led to a series ofdiscoveries of exotic phenomena. We investigate the effect of non-Hermitianimaginary hoppings on the attractive Hubbard model. The exact bound-pairsolution shows that the electron-electron correlation suppresses thenon-Hermiticity, resulting in off-diagonal long-range order (ODLRO) groundstate. In a large negative $U $ limit, the ODLRO ground state corresponds to$\eta $-spin ferromagnetic states. We also study the system with mixed hoppingconfiguration. The numerical result indicates the existence of the transitionfrom normal to $\eta $-pairing ground states by increasing the imaginaryhopping strength. Our results provide a promising approach for thenon-Hermitian strongly correlated system.

$\eta $-pairing ground states in the non-Hermitian Hubbard model