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The Chern topological numbers of a material platform are usually written interms of the Berry curvature, which depends on the normal modes of the system.Here, we use a gauge invariant Green's function method to determine from firstprinciples the topological invariants of photonic crystals. The proposedformalism does not require the calculation of the photonic band-structure, andcan be easily implemented using the operators obtained with a standardplane-wave expansion. Furthermore, it is shown that the theory can be readilyapplied to the classification of topological phases of non-Hermitian photoniccrystals with lossy or gainy materials, e.g., parity-time symmetric photoniccrystals.

First Principles Calculation of Topological Invariants of non-Hermitian Photonic Crystals