Phase Transitions in the Two Chain Hatano-Nelson Model

The study of non-Hermitian physics has garnered significant recent interest due to predictions of novel phenomena such as the non-Hermitian skin effect. However, the tendency for non-Hermitian Hamiltonians to produce complex-valued spectra poses questions about stability and the general physicality of such an effective description for these systems. In this study, we investigate a paradigmatic non-Hermitian Hamiltonian, the Hatano-Nelson model, generalized for a ladder geometry in the presence of onsite interactions for spinful fermions. Such a model exhibits rich behavior in the real-complex transitions of its spectrum controlled by a few parameters, including the hybridization between the chains in the ladder. Notably, our exact diagonalization results suggest a boundary in the parameter space separating the regions of real and complex spectra as the thermodynamic limit is approached. Contrary to the case of analogous strictly one-dimensional models, our case demonstrates similar behavior regardless of boundary conditions. Motivated by the robustness of the real spectrum in the model, we compute physical observables such as the one that tracks the formation of a spin-density wave that could potentially be used to signal the real-complex transition and, more generally, demonstrate the system's stability, which would lead to further applications.