Slave-spin method for broken symmetry states: Néel antiferromagnetism and its phase separation in multi-orbital Hubbard models

We introduce the generalization of the Slave-Spin Mean-Field method to broken symmetry phases. Through a variational approach we derive the single-particle energy shift in the mean-field equations which generates the appropriate self-consistent field responsible for the stabilization of the broken symmetry. With this correction the different flavours of the slave-spin mean-field are actually the same method, and they give identical results to Kotliar-Ruckenstein slave-bosons and to the Gutzwiller approximation. We apply our formalism to the Néel antiferromagnetic state and study it in multi-orbital models as a function of the number of orbitals and Hund's coupling strength, providing phase-diagrams in the interaction-doping plane. We show that the doped antiferromagnet in proximity of half-filling is typically unstable towards phase separation into antiferromagnetic insulating and metallic zones or between antiferromagnetic and paramagnetic zones. Hund's coupling favors this instability.