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

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