We consider models of heavy fermions in the strong coupling or local moment limit and include phonon degrees of freedom on the conduction electrons. Due to the large mass or low coherence temperature of the heavy fermion state, it is shown that such
a regime is dominated by vertex corrections which leads to the complete failure of the Migdal theorem. Even at weak electron-phonon couplings, binding of the conduction electrons competes with the Kondo effect and substantially reduces the coherence temperature, ultimately leading to the Kondo breakdown. Those results are obtained using a combination of the slave boson method and Migdal-Eliashberg approximation as well as the dynamical mean-field theory approximation.
