  In this work we study many-body 'steady states' that arise in the
non-Hermitian generalisation of the non-interacting Su-Schrieffer-Heeger model
at a finite density of fermions. We find that the hitherto known phase diagrams
for this system, derived from the single-particle gap closings, in fact
correspond to distinct non-equilibrium phases, which either carry finite
currents or are dynamical insulators where particles are entrapped. Each of
these have distinct quasi-particle excitations and steady state correlations
and entanglement properties. Looking at finite-sized systems, we further
modulate the boundary to uncover the topological features in such steady states
-- in particular the emergence of leaky boundary modes. Using a variety of
analytical and numerical methods we develop a theoretical understanding of the
various phases and their transitions, and uncover the rich interplay of
non-equilibrium many-body physics, quantum entanglement and topology in a
simple looking, yet a rich model system.
