We present an alternative approach to studying topology in open quantum systems, relying directly on Greens functions and avoiding the need to construct an effective non-Hermitian Hamiltonian. We define an energy-dependent Chern number based on the eigenstates of the inverse Greens function matrix of the system which contains, within the self-energy, all the information about the influence of the environment, interactions, gain or losses. We explicitly calculate this topological invariant for a system consisting of a single 2D Dirac cone and find that it is half-integer quantized when certain assumptions over the damping are made. Away from these conditions, which cannot or are not usually considered within the formalism of non-Hermitian Hamiltonians, we find that such a quantization is usually lost and the Chern number vanishes, and that in special cases, it can change to integer quantization.
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