  We investigate the onset of parity-time ($\mathcal{PT}$) symmetry breaking in
non-Hermitian tight-binding lattices with spatially-extended loss/gain regions
in presence of an advective term. Similarly to the instability properties of
hydrodynamic open flows, it is shown that $\mathcal{PT}$ symmetry breaking can
be either absolute or convective. In the former case, an initially-localized
wave packet shows a secular growth with time at any given spatial position,
whereas in the latter case the growth is observed in a reference frame moving
at some drift velocity while decay occurs at any fixed spatial position. In the
convective unstable regime, $\mathcal{PT}$ symmetry is restored when the
spatial region of gain/loss in the lattice is limited (rather than extended).
We consider specifically a non-Hermitian extension of the Rice-Mele tight
binding lattice model, and show the existence of a transition from absolute to
convective symmetry breaking when the advective term is large enough. An
extension of the analysis to ac-dc-driven lattices is also presented, and an
optical implementation of the non-Hermitian Rice-Mele model is suggested, which
is based on light transport in an array of evanescently-coupled optical
waveguides with a periodically-bent axis and alternating regions of optical
gain and loss.
