Fig. 4
Numerical illustration of our approach. a The branches of Riemann surface of the real part of eigenvalues of H in Eq. (6) are distinguished by different colors according to the magnitude of Re[λ]. b The exceptional points (EPs) (black dots) and their corresponding branch cuts (BCs) (red lines) are illustrated. Each BC is related with a permutation matrix M1,2,3 in Eq. (7). One closed loop (blue line) encircles EP1 and EP2 counterclockwise (CCW), starting from κ0 or \(\kappa _0^\prime\) (the solid or hollow gray points) on the loop. Loops intersecting with BCs would lead to eigenvalues moving from one branch to another, and result in the swap of eigenstates finally. c The stroboscopic evolution of complex eigenvalues are plotted as a parametric function of κ when it moves along the loop CCW. The eigenvalues at the starting point are labeled as gray points (solid or hollow) on their trajectory. The colors in the eigenvalue trajectory represent which branch the eigenvalues are located at instantaneously. The joints of two colors are where the κ crosses the BCs. The gray points (solid or hollow) and arrows illustrate the evolution of eigenvalues for starting from κ0 or κ′0, and therefore the evolution of eigenstates is {s1, s2, s3, s4} → {s3, s1, s4, s2} and →{s2, s4, s1, s3}, respectively
