Bridging two families of non-Hermiticity: from non-reciprocal couplings to an imaginary potential using the Lanczos transformation — E8 Intelligence Research

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 for Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com