Anti-parity-time topologically undefined state

We constructed an anti-parity-time-symmetric photonic lattice by using perturbations. The results show the topological state will appear when the waveguide coupling constants $kappa_1<kappa_2$; Interestingly, a state with undefined winding numbers occurs when $kappa_1=kappa_2$, in which the light distributes only in the wide waveguides with equal magnitude distribution. Further studies show that the edge state will be strengthened by introducing defect for the topologically non-trivial case, while it will not affect the equal intensity transmission for the topologically undefined state. Our work provides a new way to realize the topological state and equally divided light transmission and might be applicable in optical circuits and optical interconnect.