  We study the quench dynamics of noninteracting fermionic quantum many-body
systems that are subjected to Markovian drive and dissipation and are described
by a quadratic Liouvillian which has parity-time (PT) symmetry. In recent work,
we have shown that such systems relax locally to a maximum entropy ensemble
that we have dubbed the PT-symmetric generalized Gibbs ensemble (PTGGE), in
analogy to the generalized Gibbs ensemble that describes the steady state of
isolated integrable quantum many-body systems after a quench. Here, using
driven-dissipative versions of the Su-Schrieffer-Heeger (SSH) model and the
Kitaev chain as paradigmatic model systems, we corroborate and substantially
expand upon our previous results. In particular, we confirm the validity of a
dissipative quasiparticle picture at finite dissipation by demonstrating light
cone spreading of correlations and the linear growth and saturation to the
PTGGE prediction of the quasiparticle-pair contribution to the subsystem
entropy in the PT-symmetric phase. Further, we introduce the concept of
directional pumping phases, which is related to the non-Hermitian topology of
the Liouvillian and based upon qualitatively different dynamics of the dual
string order parameter and the subsystem fermion parity in the SSH model and
the Kitaev chain, respectively: Depending on the postquench parameters, there
can be pumping of string order and fermion parity through both ends of a
subsystem corresponding to a finite segment of the one-dimensional lattice,
through only one end, or there can be no pumping at all. We show that
transitions between dynamical pumping phases give rise to a new and independent
type of dynamical critical behavior of the rates of directional pumping, which
are determined by the soft modes of the PTGGE.
