Regular Article

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

¹ Department of Physics and Astronomy, Aarhus University, 8000 Aarhus C, Denmark
² Shenzhen Institute for Quantum Science and Engineering (SIQSE) and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, P.R. China
³ Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen 518055, P.R. China
⁴ Center for Theoretical Physics and Department of Physics, Capital Normal University, Beijing 100048, P.R. China

^a e-mail: midtgaard@phys.au.dk

Received: 7 August 2019
Published online: 25 November 2019

Abstract

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 originates from very general physical considerations. Using this construction we extract the real-space behaviour of the non-hermitian terms. 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.