Convective and absolute PT symmetry breaking in tight-binding lattices

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.