Topological enhancement of non-normality in non-Hermitian skin effects

TL;DR

This paper introduces a topological measure of non-normality that accurately quantifies non-Hermitian skin effects, revealing their relation to phase transitions, symmetry protection, and dynamic sensitivity in non-Hermitian systems.

Contribution

It proposes a novel topological measure of non-normality under open boundary conditions that captures non-Hermitian skin effects and related phase transitions.

Findings

Non-normality is enhanced under OBC in skin effects.

The measure predicts phase transitions and symmetry-protected absence of skin effects.

Enhanced non-normality influences spectral sensitivity and dynamics.

Abstract

The non-Hermitian skin effects are representative phenomena intrinsic to non-Hermitian systems: the energy spectra and eigenstates under the open boundary condition (OBC) drastically differ from those under the periodic boundary condition (PBC). Whereas a non-trivial topology under the PBC characterizes the non-Hermitian skin effects, their proper measure under the OBC has not been clarified yet. This paper reveals that topological enhancement of non-normality under the OBC accurately quantifies the non-Hermitian skin effects. Correspondingly to spectrum and state changes of the skin effects, we introduce two scalar measures of non-normality and argue that the non-Hermitian skin effects enhance both macroscopically under the OBC. We also show that the enhanced non-normality correctly describes phase transitions causing the non-Hermitian skin effects and reveals the absence of non-Hermitian skin effects protected by average symmetry. The topological enhancement of non-normality governs the perturbation sensitivity of the OBC spectra and the anomalous time-evolution dynamics through the Bauer-Fike theorem.