Topological transitions with an imaginary Aubry-André-Harper potential

We study one-dimensional lattices with imaginary-valued Aubry-Andr\'e-Harper (AAH) potentials. Such lattices can host edge states with purely imaginary eigenenergies, which differ from the edge states of the Hermitian AAH model and are stabilized by a non-Hermitian particle-hole symmetry. The edge states arise when the period of the imaginary potential is a multiple of four lattice constants. They are topological in origin, and can manifest on domain walls between lattices with different modulation periods and phases, as predicted by a bulk polarization invariant. Interestingly, the edge states persist and remain localized even if the gap in the real spectrum closes. These features can be used in laser arrays to select topological lasing modes under spatially extended pumping.