  We study the finite-energy density phase diagram of spinless fermions with
attractive interactions in one dimension in the presence of uncorrelated
diagonal disorder. Unlike the case of repulsive interactions, a delocalized
Luttinger-liquid phase persists at weak disorder in the ground state, which is
a well-known result. We revisit the ground-state phase diagram and show that
the recently introduced occupation-spectrum discontinuity computed from the
eigenspectrum of one-particle density matrices is noticeably smaller in the
Luttinger liquid compared to the localized regions. Moreover, we use the
functional renormalization scheme to study the finite-size dependence of the
conductance, which resolves the existence of the Luttinger liquid as well and
is computationally cheap. Our main results concern the finite-energy density
case. Using exact diagonalization and by computing various established measures
of the many-body localization-delocalization transition, we argue that the
zero-temperature Luttinger liquid smoothly evolves into a finite-energy density
ergodic phase without any intermediate phase transition.
