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