Connecting the Many-Body Chern Number to Luttinger's Theorem through Středa's Formula

Relating the quantized Hall response of correlated insulators to many-body topological invariants is a key challenge in topological quantum matter. Here, we use Středa's formula to derive an expression for the many-body Chern number in terms of the single-particle interacting Green's function and its derivative with respect to a magnetic field. In this approach, we find that this many-body topological invariant can be decomposed in terms of two contributions, N_{3}[G]+ΔN_{3}[G], where N_{3}[G] is known as the Ishikawa-Matsuyama invariant and where the second term involves derivatives of Green's function and the self-energy with respect to the magnetic perturbation. As a by-product, the invariant N_{3}[G] is shown to stem from the derivative of Luttinger's theorem with respect to the probe magnetic field. These results reveal under which conditions the quantized Hall conductivity of correlated topological insulators is solely dictated by the invariant N_{3}[G], providing new insight on the origin of fractionalization in strongly correlated topological phases.