Topological Phase Transition and Edge States with Tunable Localization in the Cyclic Four-Mode Optical System

A cyclic four-mode optical system is investigated with anti-parity-time symmetry. The energy spectra is analyzed and reveal the topological phase transition under different parameter regimes. In the topological phase regions, it is found that the degree of localization of the edge states can be tuned by modulating the vertical hopping strength. Moreover, it is observed that the emergence of nonzero energy edge states and the expansion of the topological phase regions. Meanwhile, it is found that the nonzero energy edge states still adhere to the bulk boundary correspondence, and which can be demonstrated by the bulk bandgap. Zero and nonzero energy edge states are also characterized by Zak phase and hidden Chern number. These results enrich the study of topological phases and edge states in non-Hermitian optical systems.