A Criterion for localization phenomena inherent in non-Hermitian systems

When a Hamiltonian is effectively non-Hermitian, its eigenstates can be localized at the boundary of the system, which is known as the non-Hermitian skin effect(NHSE). The NHSE was initially characterized as a phenomenon in which all the eigenstates of an open chain are localized at the end. However, this counting is not obvious in higher dimensions. A question of interest is how many localized eigenstates are needed to characterize the NHSE.

In this talk, we propose a criterion for localization phenomena inherent in non-Hermitian systems just using some eigenstates. It is applicable to one- or higher-dimensional systems and even the generic geometry systems. For this purpose, we define ``localization'' and ``localization length'' of state vectors in our manner. We rigorously show that if the number of eigenstates localized at the same location exceeds a certain threshold, the corresponding Hamiltonian must be non-Hermitian.