Bridging two families of non-Hermiticity: from non-reciprocal couplings to an imaginary potential using the Lanczos transformation

arXiv:2609.28326v1 Announce Type: new Abstract: Two types of non-Hermiticity found in physical systems, i.e., non-reciprocal couplings and complex potentials, have gained considerable interest recently. The former leads to the non-Hermitian skin effect, while the latter gives rise to a range of non-Hermitian symmetries such as parity-time ($PT$) and particle-hole symmetries. Although these two non-Hermitian forms have been incorporated in the same models, so far they have been treated as distinct families of non-Hermiticity. In this work, we establish an isospectral mapping between them using the Lanczos transformation. More specifically, we show that a generalized Hatano-Nelson model $H$ with non-reciprocal and spatially varying couplings can be mapped to a tight-binding model $H_L$ with real symmetric couplings and an imaginary on-site potential. This result holds even when the original $H$ already has an imaginary potential and exceptional points. Our work provides a refreshed understanding of the connection between different forms of non-Hermiticity. It also offers an approach to obtaining an intuitive physical understanding of exceptional points in $H_L$ that are elusive in $H$. Finally, our findings also warrant a versatile approach to constructing non-Hermitian systems with a complex potential and a real spectrum, without $PT$ symmetry.

Abstract:Two types of non-Hermiticity found in physical systems, i.e., non-reciprocal couplings and complex potentials, have gained considerable interest recently. The former leads to the non-Hermitian skin effect, while the latter gives rise to a range of non-Hermitian symmetries such as parity-time ($PT$) and particle-hole symmetries. Although these two non-Hermitian forms have been incorporated in the same models, so far they have been treated as distinct families of non-Hermiticity. In this work, we establish an isospectral mapping between them using the Lanczos transformation. More specifically, we show that a generalized Hatano-Nelson model $H$ with non-reciprocal and spatially varying couplings can be mapped to a tight-binding model $H_L$ with real symmetric couplings and an imaginary on-site potential. This result holds even when the original $H$ already has an imaginary potential and exceptional points. Our work provides a refreshed understanding of the connection between different forms of non-Hermiticity. It also offers an approach to obtaining an intuitive physical understanding of exceptional points in $H_L$ that are elusive in $H$. Finally, our findings also warrant a versatile approach to constructing non-Hermitian systems with a complex potential and a real spectrum, without $PT$ symmetry.

Quantum Physics (quant-ph); Optics (physics.optics)

arXiv:2609.28326 [quant-ph]

(or arXiv:2609.28326v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.28326 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Bridging two families of non-Hermiticity: from non-reciprocal couplings to an imaginary potential using the Lanczos transformation

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