Coexistence between Kondo effect and ferromagnetism in a two orbital itinerant model for 5f compounds

In this work, we study a high-order Green's function decoupling in the Under-screened Anderson Lattice Model (UALM) to analyze the competition of Kondo effect and ferromagnetism. Here, we consider the model which describes $f$-electrons that can occupy two orbitals per site that hybridize with a band of conduction electrons. The $f$-electrons are not completely localized but have a small dispersion and interact via a local screened Coulomb interaction and an Ising Hund's rule exchange with strengths $U$ and $J$, respectively. We do not consider the spin-flip part of the Hund's rule exchange interaction. To simplify the calculation of the Green's functions, we assume that $U=J$. We examine the temperature vs $W$ phase diagram, where $W$ is the conduction band width. The phase diagram depends on two characteristic energy scales $T_K$ and $T_c$. For temperatures $T>T_K,$ the system is in a paramagnetic phase. We find that for $T_c<T<T_K$, the Kondo effect appears, and below $T_c$ the Kondo effect coexists with ferromagnetism. As $W$ increases, both energy scales $T_K$ and $T_c$ are reduced and vanish at the same Quantum Critical Point (QCP). Part of this work is based on unpublished notes of Bernard Coqblin.