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Hermiticity is a fundamental aspect of isolated quantum systems. Nevertheless, non-Hermitian effects are ubiquitous in both classical and quantum systems. In the classical realm this is manifested e.g. as friction in mechanical systems, resistivity in electrical circuits, and losses in optics. In the quantum realm it reflects the dynamics of open systems, as well as decay, scattering, resonances and broadening due to e.g. interactions and disorder. During the past few decades notions of topology have revolutionised the understanding of matter as recognised by several Nobel prizes. This understanding is however based on the topology and stability of Hermitian matrices. In contrast, in the past few years, intense theoretical and experimental research has revealed that non-Hermitian effects dramatically enrich the phenomenology of topological physics—providing a cross-disciplinary frontier that is rapidly expanding [1]. Using simple examples, I will describe the essence of these developments starting with the non-Hermitian concept of exceptional degeneracies at which both eigenvalues and eigenvectors coalesce. I will also discuss how the bulk-boundary correspondence is radically modified in non-Hermitian systems and how this might be harnessed in novel sensor devices [2].
[1] E.J. Bergholtz, J.C. Budich, and F.K. Kunst, Reviews of Modern Physics 93, 15005 (2021).
[2] J.C. Budich and E.J. Bergholtz, Physical Review Letters 125, 180403 (2020).