Winding around Non-Hermitian Singularities: General Theory and Topological Features

TL;DR

This paper develops a general theoretical framework using permutation operators and topology to understand how non-Hermitian singularities influence system behavior when parameters are varied, revealing surprising non-uniqueness in their effects.

Contribution

It introduces a comprehensive formalism for analyzing non-Hermitian singularities and their topological effects, challenging previous assumptions about their behavior in parameter space.

Findings

Loops enclosing the same singularities can have different outcomes

Homotopy determines the equivalence of parameter loops

The theory applies across multiple physical systems

Abstract

Non-Hermitian singularities are ubiquitous in non-conservative open systems. These singularities are often points of measure zero in the eigenspectrum of the system which make them difficult to access without careful engineering. Despite that, they can remotely induce observable effects when some of the system's parameters are varied along closed trajectories in the parameter space. To date, a general formalism for describing this process beyond simple cases is still lacking. Here, we bridge this gap and 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 the following surprising result which contradicts the common belief in the field: loops that enclose the same singularities starting from the same initial 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.