  We develop a stochastic description of the topological properties in an
interacting Chern insulator. We confirm the Mott transition's first-order
nature in the interacting Haldane model on the honeycomb geometry, from a
mean-field variational approach, supported by density matrix renormalization
group results and Ginzburg-Landau arguments. From the Bloch sphere, we make
predictions for circular dichroism of light related to the quantum Hall
conductivity on the lattice and in the presence of interactions. This analysis
shows that the topological number can be measured from the light response at
the Dirac points. Electron-electron interactions can also produce a substantial
number of particle-hole pairs above the band gap, which leads us to propose a
"stochastic Chern number" as an interacting measure of the topology. The
stochastic Chern number can describe disordered situations with a fluctuating
staggered potential and we build an analogy between interaction-induced
particle-hole pairs and temperature effects. Our stochastic approach is
physically intuitive, easy to implement and leads the way to further studies of
interaction effects.
