Characterizing and Tuning Exceptional Points Using Newton Polygons

TL;DR

This paper introduces the Newton polygon method as a novel algebraic framework to characterize, predict, and tune exceptional points in non-Hermitian systems across various physical models.

Contribution

It establishes a new connection between Newton polygons and non-Hermitian physics, enabling analysis and control of exceptional points in complex systems.

Findings

Predicts higher-order exceptional points in optical systems

Demonstrates the non-Hermitian skin effect prediction using Newton polygons

Shows tunable exceptional points in PT-symmetric models

Abstract

The study of non-Hermitian degeneracies -- called exceptional points -- has become an exciting frontier at the crossroads of optics, photonics, acoustics, and quantum physics. Here, we introduce the Newton polygon method as a general algebraic framework for characterizing and tuning exceptional points. These polygons were first described by Isaac Newton in 1676 and are conventionally used in algebraic geometry, with deep roots in various topics in modern mathematics. We have found their surprising connection to non-Hermitian physics. We propose and illustrate how the Newton polygon method can enable the prediction of higher-order exceptional points, using a recently experimentally realized optical system. Using the paradigmatic Hatano-Nelson model, we demonstrate how our Newton Polygon method can be used to predict the presence of the non-Hermitian skin effect. As further application of our framework, we show the presence of tunable exceptional points of various orders in $PT$-symmetric one-dimensional models. We further extend our method to study exceptional points in higher number of variables and demonstrate that it can reveal rich anisotropic behaviour around such degeneracies. Our work provides an analytic recipe to understand and tune exceptional physics.