Fig. 6
Riemann surface and homotopy between loops. Two different perspectives for the four-sheet Riemann surface (associated with the real parts of the eigenvalues) that corresponds to Fig. 5d are depicted in a and b. The two loops \({\bigcirc{{\hskip -6.7pt}{\scriptstyle{3}}}}{\hskip 2pt}\) and \({\bigcirc{{\hskip -6.7pt}{\scriptstyle{4}}}}{\hskip 2pt}\) (blue lines) that encircle EP1 (white point) in the two-dimensional (2D) parameter space are also shown. As explained in the text, the homotopy test (performed parameter space) for these two loops shows that they are not equivalent, which results in different stroboscopic and dynamic features. On the Riemann surface, this property becomes even more evident by noting that the two loops span different sheets. The red point stands for eigenvalue λ1 (corresponding to eigenstate s1) at the initial parameter point. This state will evolve to itself or to the orange point along loops \({\bigcirc{{\hskip -6.7pt}{\scriptstyle{3}}}}{\hskip 2pt}\) and \({\bigcirc{{\hskip -6.7pt}{\scriptstyle{4}}}}{\hskip 2pt}\), respectively. The dashed white lines are vertical lines emanating from the exceptional points (EPs) to illustrate the fact that the projections of the two loops considered here encircle EP1 but not EP2. The white dotted lines illustrate the eigenvalue bifurcation across the EPs on the Riemann surface
