Non-abelian nature of systems with multiple exceptional points

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.