Document Type


Publication Date



Non-Hermitian singularities are ubiquitous in non-conservative open systems. Owing to their peculiar topology, they can remotely induce observable effects when encircled by closed trajectories in the parameter space. To date, a general formalism for describing this process beyond simple cases is still lacking. Here we develop a general approach for treating this problem by utilizing the power of permutation operators and representation theory. This in turn allows us to reveal a surprising result that has so far escaped attention: loops that enclose the same singularities in the parameter space starting from the same point and traveling in the same direction, do not necessarily share the same end outcome. Interestingly, we find that this equivalence can be formally established only by invoking the topological notion of homotopy. Our findings are general with far reaching implications in various fields ranging from photonics and atomic physics to microwaves and acoustics.

Publisher's Statement

© The Author(s) 2018. Article deposited here in compliance with publisher policy. Publisher's version of record:

Publication Title

Nature Communications

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.


Publisher's PDF

Included in

Physics Commons



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.