Algebraic States in Continuum in $ d\gt 1$ Dimensional Non-Hermitian Systems

TL;DR

This paper discovers algebraically localized eigenstates within the continuum spectrum of 2D non-Hermitian systems with a single impurity, highlighting their unique existence in higher dimensions and proposing a local density of states for detection.

Contribution

It analytically derives the conditions for algebraic states in continuum (AICs) in 2D non-Hermitian systems and shows they are absent in Hermitian and 1D non-Hermitian systems, emphasizing their novelty.

Findings

AICs decay as 1/|r| from the impurity site.

Energies of AICs lie within the bulk continuum spectrum.

A local density of states can be used to detect AICs experimentally.

Abstract

We report the existence of algebraically localized eigenstates embedded within the continuum spectrum of 2D non-Hermitian systems with a single impurity. These modes, which we term algebraic states in continuum (AICs), decay algebraically as $1/|r|$ from the impurity site, and their energies lie within the bulk continuum spectrum under periodic boundary conditions. We analytically derive the threshold condition for the impurity strength required to generate such states. Remarkably, AICs are forbidden in Hermitian systems and in 1D non-Hermitian systems, making them unique to non-Hermitian systems in two and higher dimensions. To detect AICs, we introduce a local density of states as an experimental observable, which is readily accessible in photonic/acoustic platforms.