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Relating the quantized Hall response of correlated insulators to many-bodytopological invariants is a key challenge in topological quantum matter. Here,we use Streda's formula to derive an expression for the many-body Chern numberin terms of the single-particle interacting Green's function and its derivativewith respect to a magnetic field. In this approach, we find that this many-bodytopological invariant can be decomposed in terms of two contributions, $N_3[G]+ \Delta N_3[G]$, where $N_3[G]$ is known as the Ishikawa-Matsuyama invariantand where the second term involves derivatives of Green's function and the selfenergy with respect to the magnetic perturbation. As a by-product, theinvariant $N_3[G]$ is shown to stem from the derivative of Luttinger's theoremwith respect to the probe magnetic field. These results reveal under whichconditions the quantized Hall conductivity of correlated topological insulatorsis solely dictated by the invariant $N_3[G]$, providing new insight on theorigin of fractionalization in strongly-correlated topological phases.

Connecting the Many-Body Chern Number to Luttinger's Theorem through St\v{r}eda's Formula