Phonons and the coherence scale of models of heavy fermions

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.