Abstract
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.
Highlights
Non-Hermitian singularities are ubiquitous in non-conservative open systems
Our approach utilizes the power of group theory together with group representations to decompose the final action of any loop into more elementary exchange processes across the relevant branch cuts (BCs)
This formalism simplifies the analysis significantly, which in turn allows us to gain an insight into the problem at hand and unravel a number of intriguing results: trajectories that encircle the same exceptional points (EPs) starting from the same initial point and having the same direction do not necessarily lead to an identical exchange between the eigenstates; establishing such equivalence between the loops is guaranteed only by invoking the topological notion of homotopy
Summary
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. An intriguing aspect of non-Hermitian systems is the eigenstate exchange along loops that trace closed trajectories around EPs. In this regard, stroboscopic encircling of EP2 (EP of order 2) has been studied theoretically[22,23] and demonstrated experimentally in various platforms such as microwave resonators[24,25] and exciton-polariton setups[26].
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