Non-abelian nature of systems with multiple exceptional points

TL;DR

This paper explores the non-abelian, non-commutative behavior of multiple exceptional points in non-Hermitian systems, providing a topological framework and proposing an experimental setup to demonstrate these effects.

Contribution

It introduces a topological approach using fundamental groups to analyze multiple EPs and suggests an experimental implementation to observe non-abelian effects.

Findings

Theoretical framework for composing effects of multiple EPs using fundamental groups.

Extension of the theory to exceptional lines in 3D parameter space.

Proposal for experimental demonstration of non-abelian behavior in PT-symmetric waveguides.

Abstract

The defining characteristic of an exceptional point (EP) in the parameter space of a family of operators is that upon encircling the EP eigenstates are permuted. In case one encircles multiple EPs, the question arises how to properly compose the effects of the individual EPs. This was thought to be ambiguous. We show that one can solve this problem by considering based loops and their deformations. The theory of fundamental groups allows to generalize this technique to arbitrary degeneracy structures like exceptional lines in a three-dimensional parameter space. As permutations of three or more objects form a non-abelian group, the next question that arises is whether one can experimentally demonstrate this non-commutative behavior. This requires at least two EPs of a family of operators that have at least 3 eigenstates. A concrete implementation in a recently proposed $\mathcal{PT}$ symmetric waveguide system is suggested as an example of how to experimentally check the composition law and show the non-abelian nature of non-hermitian systems with multiple EPs.