  In the context of many-body interacting systems described by a topological
Hamiltonian, we investigate the robustness of the Chern number with respect to
different sources of error in the self-energy. In particular, we analyze the
importance of non-local (momentum dependent) vs. local contributions to the
self-energy and show that the local self-energy provides a qualitative
description of the topological phase diagrams of many-body interacting systems,
whereas the explicit momentum-dependence constitutes a correction to the exact
location of the phase transition. For the latter, we propose a statistical
analysis, on the basis of which we develop a stochastic upper bound for the
uncertainty of the Chern number as a function of the amount of
momentum-dependence of the self-energy. We apply this analysis to the
Haldane-Hubbard model and discuss the implications of our results for a general
class of many-body interacting systems.
