  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.
