A tropical geometric approach to exceptional points

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 framework for studying non-Hermitian physics and unveils a connection of tropical geometry to this field.

Automated summary

In this study, a unified tropical geometric framework is introduced and developed to characterize different aspects of non-Hermitian systems. The versatility of this approach is demonstrated through several examples, including selecting from a spectrum of higher-order exceptional points (EPs) in gain and loss models, predicting the skin effect in the non-Hermitian Su-Schrieffer-Heeger model, and extracting universal properties in the presence of disorder in the Hatano-Nelson model. This work provides a framework for studying non-Hermitian physics and reveals a connection between tropical geometry and this field.