# Comment on 'Winding around non-Hermitian singularities' by Zhong et al., Nat. Commun. 9, 4808 (2018)

## TL;DR

This paper critiques a recent formalism for calculating eigenstate permutations around non-Hermitian singularities, highlighting a flaw and proposing a more reliable method based on fundamental loops, with experimental verification potential.

## Contribution

It identifies a flaw in Zhong et al.'s approach and introduces a more robust method using fundamental loops for analyzing eigenstate permutations.

## Key findings

- The existing method can give incorrect results for certain loops.
- The proposed fundamental loop method is more accurate and reliable.
- Experimental verification can be performed in a three wave-guide system.

## Abstract

In a recent paper entitled "Winding around non-Hermitian singularities" by Zhong et al., published in Nat. Commun. 9, 4808 (2018), a formalism is proposed for calculating the permutations of eigenstates that arise upon encircling (multiple) exceptional points (EPs) in the complex parameter plane of an analytic non-Hermitian Hamiltonian. The authors suggest that upon encircling EPs one should track the eigenvalue branch cuts that are traversed, and multiply the associated permutation matrices accordingly. In this comment we point out a serious shortcoming of this approach, illustrated by an explicit example that yields the wrong result for a specific loop. A more general method that has been published earlier by us and that does not suffer from this problem, is based on using fundamental loops. We briefly explain the method and list its various advantages. In addition, we argue that this method can be verified in a three wave-guide system, which then also unambiguously establishes the noncommutativity associated with encircling multiple EPs.