Topology of non-Hermitian systems from the exceptional point

We 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. Additionally, we reveal the relation between the spectral topology originating from complex eigenvalues in non-Hermitian systems and wavefunction topology related to the additional geometric phases. Finally, we discuss the topology of one-dimensional multi-bands systems based on exceptional points.