Fig. 1: Illustration of the switching effect and Berry phase after an encircling of Exceptional Points (EPs) in parameter space.

The figures depict the evolution of the topological winding on the real energy band across parameter space (α, β), as indicated by the Berry phase color bar on each energy band (see Supplementary Note 2). Each set consists of three representations: (i) a parameter space mapping, (ii) an unfolded band structure, and (iii) a corresponding schematic of a strip. The color coding within each figure notes the energy band index, with the normal strip symbolizing a trivial phase and the Möbius strip symbolizing a non-trivial phase. For a No encircling of EPs results in no state switching, indicating a zero phase change and a trivial topology. For b Encircling one EP leads to state switching, denoted by a π phase change, signifying a non-trivial topology. For c Encircling two EPs, despite no state switching, results in a π phase change due to the cumulative topological impact.