We relate observables in open quantum systems with the topology of non-Hermitian models using the Keldysh path-integral method. This allows to extract an effective Hamiltonian from the Green's function which contains all the relevant topological information and produces $\omega$-dependent topological invariants, linked to the response functions at a given frequency. Then, we show how to detect a transition between different topological phases by measuring the response to local perturbations. Our formalism is exemplified in a one-dimensional Hatano-Nelson model, highlighting the difference between the bosonic and the fermionic case.

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• Bosons
• Fermions
• Photonics
• Quantum optics
• Skin effect
• Topological insulators
• Topological materials
• Topological phase transition
• Topological phases of matter