Speaker
Description
Exceptional points (EPs), at which two or more eigenvalues and eigenvectors coalesce, are ubiquitous and unique features of non-Hermitian (NH) systems. Second-order EPs (EP2s) are by far the most studied due to their abundance, requiring only the tuning of two real parameters, which is less than the three parameters needed to generically find ordinary Hermitian eigenvalue degeneracies. Higher-order EPs generically require more fine- tuning, and are thus assumed to play a much less prominent role. However, I will demonstrate that physically-relevant symmetries make higher-order EPs dramatically more abundant and conceptually richer. Additionally, by expressing symmetry-preserving perturbations in the Jordan-normal basis of the defective matrix at an EP, I will show that the upper-k Hessenberg structure of the perturbation directly dictates the leading-order eigenvalue- and eigenvector-splitting to be ∝ϵ^{1/k}, when expanded in a Puiseux series. Applying this to three-band NH models invariant under parity (P), charge-conjugation (C), or parity-time-reversal (PT), one finds that third-order EPs (EP3s) in P- and C-symmetric systems are restricted to at most ∼ϵ^{1/2} branch points, while PT-symmetric systems generically support EP3s with the strongest possible singularities (viz. ∼ϵ^{1/3}).