Generalized $η$-pairing theory and anomalous localization in non-Hermitian systems

TL;DR

This work develops a comprehensive non-Hermitian eta-pairing theory for Hubbard models on arbitrary lattices, providing necessary and sufficient conditions for eta^ abla to be an eigenoperator and revealing a family of exotic phenomena tied to non-Hermiticity, including site-dependent pairing amplitudes and anomalous localization (skin effects). It uncovers deep symmetry unification, showing that SU(2) spin, SU(2) pseudospin, and SO(4) symmetries, along with discrete particle-hole and Z2 Shiba transformations, are interconnected within a single algebraic framework even when bulk translation symmetry is broken. The authors construct explicit non-Hermitian and Hermitian-relevant models—the Hatano-Nelson-Hubbard chain, its 2D generalization, and a versatile two-sublattice model—to illustrate right/left eta-pairing eigenstates that localize at opposite boundaries or corners and to demonstrate ODLRO in certain regimes. This framework provides rigorous tools for understanding interacting non-Hermitian physics in any dimension and without bulk translational invariance, with potential implications for skin-effect phenomenology and beyond.

Abstract

By generalizing the eta-pairing theory to non-Hermitian Hubbard models on arbitrary lattices, we obtain the sufficient and necessary condition for the eta-pairing operator to be an eigenoperator of the Hamiltonian $H$, and find unique eta-pairing phenomena without Hermitian analogs. For instance, the Hermitian conjugate of an eta-pairing eigenoperator may not be an eigenoperator, eta-pairing eigenoperators can be spatially modulated, and the $SU(2)$ pseudospin symmetry may not be possible even if $H$ commutes with the eta-pairing operators. Remarkably, these novel non-Hermitian phenomena are closely related to each other by several theorems we establish and can lead to, for example, new types of eta-pairing operators (e.g., the notion of non-Hermitian angular-momentum operators) and the anomalous localization (e.g., the skin effect) of eta-pairing eigenstates. Some issues on the $SO(4)$ and particle-hole symmetries are clarified. Our general eta-pairing theory also reveals a previously unnoticed unification of these symmetries of the Hubbard model. To exemplify these findings, we first propose the Hatano-Nelson-Hubbard model. In this interacting non-Hermitian system without even the bulk translation invariance, the right and left two-particle eta-pairing eigenstates are exponentially localized at opposite boundaries of the chain. Then, we generalize this model to two dimensions and find that the eta-pairing eigenstates can exhibit the first- or second-order skin effect. Finally, a general two-sublattice model is constructed to realize all of the non-Hermitian eta-pairing phenomena. This work provides a new and rigorous theoretical framework for studying novel physical phenomena (e.g., the skin effect) in interacting non-Hermitian systems, even in arbitrary spatial dimensions and without the bulk translation symmetry.