A Tropical Geometric Approach To Exceptional Points

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.

arXiv commentPublished version

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Cited by in corpus (9)

• Emergence of tunable exceptional points in altermagnet-ferromagnet junctions
• Constraints of internal symmetry on the non-Hermitian skin effect and bidirectional skin effect under the action of the Hermitian conjugate of time-reversal symmetry
• General theory for infernal points in non-Hermitian systems
• Graph theory approach to exceptional points in wave scattering
• Néel vector controlled exceptional contours in $p$-wave magnet-ferromagnet junctions
• Theory for the spectral splitting exponent of exceptional points
• Critical properties in the non-Hermitian Aubry-Andre-Stark model
• Hexagonal Warping Control of Exceptional Points in Topological Insulator--Ferromagnetic Heterojunctions
• Characterizing Liouvillian Exceptional Points Through Newton Polygons and Tropical Geometry