Topology and observables of the non-Hermitian Chern insulator
Topology now plays a central role in physics, yet its applications have so far been restricted to closed, lossless systems in thermodynamic equilibrium. Given that many physical systems are open and may include gain and loss mechanisms, there is an eminent need to reexamine topology within the context of non-Hermitian theories that describe open, lossy systems. The generalization of the Chern number to non-Hermitian Hamiltonians initiated this reexamination; however, there is no established connection between a non-Hermitian topological invariant and the quantization of an observable. Using field-theoretical techniques, we show that no such relationship exists between the non-Hermitian Chern number and the Hall conductivity, a consequence of the discontinuous nature of Green's functions of non-Hermitian Hamiltonians. Furthermore, we derive an exact formula for the Chern-Simons Hall response of a generic two-level non-Hermitian Hamiltonian and present an illustrative calculation for a non-Hermitian massive Dirac Hamiltonian in (2+1) dimensions. We conclude by clarifying how these results extend to higher-dimensional systems and detailing their implications for recent experiments.