  Over the past few years bulk pyrochlore iridates of the form $A_2$Ir$_2$O$_7$
(where $A$ is a rare earth element, Ir is iridium, and O is oxygen) have been
studied as model systems for investigating the interplay of electronic
correlations and strong spin-orbit coupling, particularly with the aim of
finding correlation-driven topological phases. In this work, we use cellular
dynamical mean field theory (CDMFT) to study effects of electronic correlations
beyond Hartree-Fock theory in thin films of pyrochlore irradiates grown along
the $[111]$ direction. We focus on the bilayer and trilayer systems, and
compute the phase diagrams of these systems as a function of electron-electron
interaction strength, which is modeled by an on-site Hubbard interaction. By
evaluating the $Z_2$ invariant and Chern number using formulas based on the
single-particle Green's function and the quasi-particle effective Hamiltonian,
we show that on-site correlations can drive an interaction-induced topological
phase transition, turning a time-reversal invariant topological insulator and a
nearly flat band metal to a correlated Chern insulator (CI) in bilayer and
trilayer systems, respectively. By comparing with the Hartree-Fock results, the
CDMFT results show that quantum fluctuations enhance the robustness of the
interaction-driven CI phase in the thin films. Furthermore, our numerical
analysis of the quasiparticle spectrum reveals that the topological phases we
find in our many-body calculations are adiabatically connected to those in the
single-particle picture.
