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We investigate how topological Chern numbers can be defined whensingle-particle states hybridize with continua. We do so exemplarily in abosonic Haldane model at zero temperature modified by an additional on-sitedecay of one boson into two and the conjugate fusion of two bosons into one.Restricting the Hilbert space to two bosons at maximum, the exact self-energyis accessible. We use the bilinear Hamiltonian $H_0$ corrected by theself-energy $\Sigma$ to compute Chern numbers by two different approaches. Theresults are gauged against a full many-body calculation in the restrictedHilbert space. We find evidence that the effective Hamiltonian $H_0(\vec k)+\Sigma(\omega,\vec k)$ reproduces the correct many-body topology even if theconsidered band overlaps with the continuum. However, in the latter case thebulk-boundary correspondence appears to be no longer valid and the edge modesdelocalize.

Topological Properties of Single-Particle States Decaying into a Continuum due to Interaction