A tropical geometric approach to exceptional points

Area of Science:

• Theoretical Physics
• Mathematical Physics
• Non-Hermitian Systems

Background:

• Non-Hermitian systems are crucial in diverse fields like photonics and electric circuits.
• Exceptional points (EPs), where eigenvalues and eigenvectors merge, are a key characteristic of these systems.
• Tropical geometry, an emerging mathematical field, offers novel analytical tools.

Purpose of the Study:

• To develop a unified tropical geometric framework for characterizing non-Hermitian systems.
• To demonstrate the framework's versatility in analyzing various aspects of non-Hermitian physics.
• To establish a novel connection between tropical geometry and non-Hermitian physics.

Main Methods:

• Development of a unified tropical geometric framework.
• Application of the framework to analyze gain and loss models, the Su-Schrieffer-Heeger model, and the Hatano-Nelson model.
• Characterization of exceptional points (EPs) and prediction of physical phenomena.

Main Results:

• The tropical geometric framework successfully characterizes diverse facets of non-Hermitian systems.
• The framework enables selection of higher-order exceptional points (EPs) in gain and loss models.
• It predicts the skin effect in the non-Hermitian Su-Schrieffer-Heeger model and extracts universal properties in disordered Hatano-Nelson models.

Conclusions:

• Tropical geometry provides a powerful and unified framework for studying non-Hermitian physics.
• This research unveils a significant and previously unrecognized connection between tropical geometry and non-Hermitian systems.
• The developed framework offers new insights into exceptional points, skin effects, and disordered systems.