Band structures of generalized eigenvalue equation and conic section

TL;DR

This paper explores the band structures of metamaterials described by generalized eigenvalue equations, revealing how real and complex bands transition and relate to phenomena like Lifshitz transitions and Dirac cones, with implications for photonic systems.

Contribution

It provides a geometrical framework for understanding real-complex band transitions and links photonic band structures to electronic Fermi surfaces, highlighting the role of material parameters.

Findings

Real-complex transition of bands is analogous to Lifshitz transition.

Real and complex bands correspond to different types of Dirac cones.

Frequency dependence of permittivity and permeability induces exceptional points.

Abstract

Band structures of several metamaterials are described by generalized eigenvalue equations where complex bands emerge even if the involved matrices are Hermitian. In this paper, we provide a geometrical understanding of the real-complex transition of the band structures. Specifically, our analysis, based on auxiliary eigenvalues, elucidates the correspondence between the real-complex transition of the generalized eigenvalue equations and Lifshitz transition in electron systems. Furthermore, we elucidate that real (complex) bands of a photonic system correspond to the Fermi surfaces of type-II (type-I) Dirac cones in electron systems when the permittivity $\varepsilon$ and the permeability $\mu$ are independent of frequency. In addition, our analysis elucidates that EPs are induced by the frequency dependence of the permittivity $\varepsilon$ and the permeability $\mu$ in our photonic system.