Classification based on exceptional points in non-Hermitian systems

We introduce non-Hermitian phenomena and non-Hermitian degeneracy, i.e., exceptional points, which are ubiquitous in various physical systems. We also discuss classifications of multiple arbitrary-order exceptional points by invoking the permutation group and its conjugacy classes. We classify topological structures of Riemann surfaces generated by multiple states around multiple arbitrary-order exceptional points, using the permutation properties of stroboscopic encircling exceptional points. The results are realized in non-Hermitian effective Hamiltonian based on Jordan normal forms and fully desymmetrized optical microcavities. Finally, we classify the multiple energy bands based on exceptional points.