Metal-insulator transition in CaCu$_3$Fe$_4$O$_{12}$

We study structurally-triggered metal-insulator transition in CaCu$_3$Fe$_4$O$_{12}$ by means of local density approximation (LDA) +$U$ and LDA+dynamical mean-field theory (DMFT). The ferrimagnetic insulating phase is essentially the same within both approaches. While LDA+$U$ describes the metal-insulator transition as a Peierls-like instability driven by Fermi surface nesting in the magnetically ordered phase, LDA+DMFT allows also the site-selective Mott transition without magnetic ordering as well as smooth crossover between the two pictures. We point out similarities and differences to rare-earth nickelates.