Scaling rule for the critical non-Hermitian skin effect

Non-Hermitian systems show a non-Hermitian skin effect, where the bulk states are localized at boundary of with open conditions. In this paper, we study dependence localization length eigenstates on system size in specific model critical energy spectrum undergoes discontinuous transition thermodynamic limit. We analytically that exhibit remarkable localization, known as scale-free is proportional to size. Our result gives theoretical support for which has been proposed only numerically previous works.