Bridging two families of non-Hermiticity: from non-reciprocal couplings to an imaginary potential using the Lanczos transformation
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 HL 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 HL 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.
Related
- Phase structure of magnetized quark matter at imaginary chemical potential
- Non-minimal α-attractor magnetogenesis with potential dependent couplings
- Discrete Laplacian structure and kernel reduction of the gap equation in odd-dimensional large-N vector models at imaginary chemical potential
- Non-Hermiticity of an anomalous superradiant phase
- Imaginary-time-enhanced feedback-based quantum algorithms for ground-state preparation
BibTeX
@misc{ge2026bridging,
title = {{Bridging two families of non-Hermiticity: from non-reciprocal couplings to an imaginary potential using the Lanczos transformation}},
author = {Li Ge},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28326},
eprint = {2609.28326},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28326},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Bridging two families of non-Hermiticity: from non-reciprocal couplings to an imaginary potential using the Lanczos transformation
AU - Li Ge
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28326
UR - https://arxiv.org/abs/2609.28326
AB - 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.
ER -