Topology plays a central role in nearly all disciplines of 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 recent generalization of the Chern number to non-Hermitian Hamiltonians initiated this reexamination; however, there is so far no established connection between a non-Hermitian topological invariant and the quantization of an observable. In this work, we show that no such relationship exists between the Chern number of non-Hermitian bands and the quantization of the Hall conductivity. Using field theoretical techniques, we calculate the longitudinal and Hall conductivities of a non-Hermitian Hamiltonian with a finite Chern number to explicitly demonstrate the physics of a non-quantized Hall conductivity despite an invariable Chern number. These results demonstrate that the Chern number does not provide a physically meaningful classification of non-Hermitian Hamiltonians.
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