Scaling rule in critical non-Hermitian skin effect

TL;DR

This paper analytically demonstrates scale-free localization in a critical non-Hermitian skin effect model, showing that eigenstate localization length scales with system size, thus providing theoretical support for previously numerically observed phenomena.

Contribution

It provides the first analytical proof of scale-free localization in a non-Hermitian skin effect model, confirming numerical observations and elucidating the localization behavior.

Findings

Eigenstates exhibit scale-free localization with localization length proportional to system size.

The energy spectrum undergoes a discontinuous transition in the thermodynamic limit.

Analytical results support previous numerical findings of scale-free localization.

Abstract

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