Instability of the engineered dark state in two-band fermions under number-conserving dissipative dynamics

Correlated quantum many-body states can be created and controlled by the dissipative protocols. Among these, particle number-conserving protocols are particularly appealing due to their ability to stabilize topologically nontrivial phases. Is there any fundamental limitation to their performance? We address this question by examining a general class of models involving a two-band fermion system subjected to dissipation designed to transfer fermions from the upper band to the lower band. By construction, these models have a guaranteed steady state -- a dark state -- with a completely filled lower band and an empty upper band. In the limit of weak dissipation, we derive equations governing the dynamics of the fermion densities over long length and time scales. These equations belong to the Fisher-Kolmogorov-Petrovsky-Piskunov reaction-diffusion universality class. Our analysis reveals that the engineered dark state is generically unstable, giving way to a new steady state with a finite density of particles in the upper band. Our results suggest that number-conserving dissipative protocols may not be a reliable universal tool for stabilizing dark states.