  Non-Hermitian Hamiltonian (NHH) is an effective formalism for open quantum
systems. In common wisdom, when the system is described by the Lindblad master
equation, the NHH obtained by neglecting its jump term is believed to be a good
approximation for a timescale sufficiently shorter than the inverse of the
dissipation rate. We challenge this common wisdom and develop a scheme to
obtain an appropriate NHH from the original master equation for dissipative
fermionic systems. This NHH, called the local NHH, describes the loss process
in each individual mode locally. As a concrete example, we justify our new
scheme using fermionic superfluid under one-body loss. Furthermore, we find
loss-induced population increase in the long time evolution due to the
dissipation-induced phase locking between the pairing gap and the anomalous
field.
