Characterizing Exceptional Topology Through Tropical Geometry*

Non-Hermitian (NH) Hamiltonians describing open quantum systems have been widely explored in platforms ranging from photonics to electric circuits [1]. A defining feature of NH systems is exceptional points (EPs), where both eigenvalues and eigenvectors coalesce [3]. The study of EPs has become an exciting frontier at the crossroads of optics, photonics, acoustics, and quantum physics [4]. Tropical geometry is an emerging field of mathematics at the interface between algebraic geometry and polyhedral geometry, with diverse applications to science [5]. Here, we introduce Newton's polygon method and adopt the notion of a geometrical object known as amoeba in developing a unified tropical geometric framework to characterize different facets of NH systems [6]. 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 NH 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 NH physics and unveils a novel connection of tropical geometry to this field.

[1] Bergholtz et al., Rev. Mod. Phys. 93, 015005 (2021).

[2] T. Kato, Vol. 132 (Springer Science & Business Media, 2013).

[3] Miri et al.,  Science 363, eaar7709 (2019).

[4] Maclagan et al., Graduate Stud. Math.161, 75–91(2009).

[5] Banerjee et al., Proc. Natl. Acad. Sci. U.S.A. 120, e2302572120 (2023).