Fractional quantum transport and staggered topological transition in a lossy trimerized lattice

Transport in a non-Hermitian trimerized lattice with one lossy site per unit cell exhibits a quantized mean displacement before decay characterized by a two-step phase transition from zero to unity through a fractional value. Such a peculiar behavior, sensitive to both the lattice parameters and the initial state, is shown to be closely related to the presence of two nondegenerate dark states decoupled from the lossy sites and correspond to two staggered band closing points. We further check that distinct topological phases can also be characterized by the total Zak phase and by the number of midgap edge states. Moreover, we demonstrate that the edge states are topologically protected and their localization can be controlled via the couplings imbalance. Our results are expected to be relevant for implementations of topological photonics such as those based on arrays of evanescently coupled waveguides.