Characterizing and Tuning Exceptional Points Using Newton Polygons

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, and develop its connection to Puiseux expansions. 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. As an 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.