Topological phases of commensurate or incommensurate non-Hermitian Su-Schrieffer-Heeger lattices

We theoretically investigate topological features of a one-dimensional Su-Schrieffer-Heeger lattice with modulating non-Hermitian on-site potentials containing four sublattices per unit cell. The lattice can be either commensurate or incommensurate. In the former case, the entire lattice can be mapped by supercells completely. While in the latter case, there are two extra lattice points, thereby making the last cell incomplete. We find that an anti-PT transition occurs at exceptional points of edge states at certain parameters, which does not coincide with the conventional topological phase transition characterized by the Berry phase, provided the imaginary on-site potential is large enough. Interestingly, when the potential exceeds a critical value, edge states appear even in the regime with a trivial Berry phase. To characterize these novel edge states we present topological invariants associated with the system's parity. Finally, we analyze the dynamics for initial states with different spatial distributions, which exhibit distinct dynamics for the commensurate and incommensurate cases, depending on the imaginary part of edge state energy.