Generalized $\eta$-pairing approach to interacting non-Hermitian systems in arbitrary dimensions

We generalize the eta-pairing theory to very general non-Hermitian Hubbard models and find many novel 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 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 one-dimensional Hatano-Nelson-Hubbard model (with or without the bulk translation invariance) and show that 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, to realize all of the non-Hermitian eta-pairing phenomena, we construct a general two-sublattice model that is defined on an arbitrary lattice; this model can also reveal the eta-pairing structure in systems with Hermitian hoppings, including the original eta-pairing theory for square lattice, the extension to triangular lattice, and some topological systems. Our results establish a new and rigorous theoretical framework for studying novel quantum phenomena in interacting non-Hermitian many-body systems, even in arbitrary spatial dimensions and without the bulk translation symmetry.