Realizing exceptional points of any order in the presence of symmetry

TL;DR

This paper provides a general framework for realizing exceptional points of any order in non-Hermitian systems, highlighting the role of symmetries and deriving conditions for their occurrence and dispersion.

Contribution

It introduces a determinant and trace-based approach to identify constraints for high-order exceptional points, accounting for symmetry effects and classifying their dispersions.

Findings

Constraints for EPs expressed via determinant and traces.

Symmetries can reduce the number of constraints needed.

Explicit classification of EP dispersions in low-band systems.

Abstract

Exceptional points~(EPs) appear as degeneracies in the spectrum of non-Hermitian matrices at which the eigenvectors coalesce. In general, an EP of order $n$ may find room to emerge if $2(n-1)$ real constraints are imposed. Our results show that these constraints can be expressed in terms of the determinant and traces of the non-Hermitian matrix. Our findings further reveal that the total number of constraints may reduce in the presence of unitary and antiunitary symmetries. Additionally, we draw generic conclusions for the low-energy dispersion of the EPs. Based on our calculations, we show that in odd dimensions the presence of sublattice or pseudo-chiral symmetry enforces $n$th order EPs to disperse with the $(n-1)$th root. For two-, three- and four-band systems, we explicitly present the constraints needed for the occurrence of EPs in terms of system parameters and classify EPs based on their low-energy dispersion relations.