Title: Theory of Mobility Rings in Non-Hermitian Systems

Abstract: Through Avila global theorem, we analytically study the non-Hermitian mobility edge. The results show that the mobility edge in non-Hermitian systems has a ring structure, which we named as "mobility ring". Furthermore, we carry out numerical analysis of the eigenenergy spectra in several typical cases, and the consistence of the numerical results with the analytical expression proves the correctness and universality of the mobility ring theory. Further, based on the analytical expression, we discuss the properties of multiple mobility rings. Finally, we compare the results of mobility rings with that of dual transformations, and find that although the self-dual method can give the interval of real eigenvalues corresponding to the extended states, it can not fully display the mobility edge information in the complex plane. The mobility ring theory proposed in this paper is universal for all non-Hermitian systems.