Local non-Hermitian Hamiltonian formalism for dissipative fermionic systems and loss-induced population increase in Fermi superfluids.

We examine a standard scheme to obtain the non-Hermitian Hamiltonian (NHH) from the Lindblad master equation by neglecting its jump term, and propose an alternative approach to address its limitations. The NHH obtained by the conventional scheme fails to provide a good approximation for fermionic many-body systems, even on short timescales. To resolve this issue, we present a framework called the local NHH formalism, which describes the loss process in each individual mode locally. This formalism is applicable to general dissipative fermionic systems and remains consistent with the underlying Lindblad master equation at the level of the equations of motion of local observables. The local NHH formalism also provides a convenient framework for spectral analysis, compared to the Lindblad master equation. As an illustration, we consider a fermionic superfluid subject to one-body loss and find the loss-induced population increase. The conventional NHH fails to capture these unique phenomena.