A Tropical Geometric Approach To Exceptional Points

TL;DR

This paper introduces a novel tropical geometric framework to analyze non-Hermitian systems, enabling characterization of exceptional points, prediction of phenomena like the skin effect, and understanding disorder effects, thus advancing the mathematical tools in non-Hermitian physics.

Contribution

It develops a unified tropical geometric approach to study non-Hermitian systems, connecting algebraic and polyhedral geometry with physical phenomena.

Findings

Able to select higher-order exceptional points in gain and loss models

Predicts the skin effect in the non-Hermitian Su-Schrieffer-Heeger model

Extracts universal properties in disordered Hatano-Nelson model

Abstract

Non-Hermitian systems have been widely explored in platforms ranging from photonics to electric circuits. A defining feature of non-Hermitian systems is exceptional points (EPs), where both eigenvalues and eigenvectors coalesce. Tropical geometry is an emerging field of mathematics at the interface between algebraic geometry and polyhedral geometry, with diverse applications to science. Here, we introduce and develop a unified tropical geometric framework to characterize different facets of non-Hermitian systems. We illustrate the versatility of our approach using several examples, and demonstrate that it can be used to select from a spectrum of higher-order EPs in gain and loss models, predict the skin effect in the non-Hermitian Su-Schrieffer-Heeger model, and extract universal properties in the presence of disorder in the Hatano-Nelson model. Our work puts forth a new framework for studying non-Hermitian physics and unveils a novel connection of tropical geometry to this field.