Constraints on the energy spectrum of non-Hermitian models in open environments

Motivated by recent progress on non-Hermitian topological band theories, we study the energy spectrum of a generic two-band non-Hermitian Hamiltonian. We prove rigorously that the complex energy spectrum of such a non-Hermitian Hamiltonian is restricted to the lower complex plane, provided that the parameters of the Hamiltonian satisfy a certain constraint. Furthermore, we consider one specific scenario where such a non-Hermitian Hamiltonian can arise, namely a two-band model coupled to an environment, and show that this aforementioned constraint orignates from very general physical considerations. Our findings are relevant in the definition of the energy gap in non-Hermitian topological band theories and also have implications on simulations of such theories using quantum systems.Comment: 7 pages, 1 figur