Ferro-octupolar order in d² double perovskites of Osmium from first principles

Conflicting interpretations of experimental data preclude the understanding of the quantum magnetic state of spin-orbit coupled d² double perovskites. Whether the ground state is a Janh-Teller-distorted order of quadrupoles or the hitherto elusive octupolar order remains debated. We resolve this uncertainty through direct calculations of all-rank inter-site exchange interactions and inelastic neutron scattering cross-section for the d² double perovskite series Ba₂MOsO₆  (M= Ca, Mg, Zn). Using advanced many-body first principles methods we show that the ground state is formed by ferro-ordered octupoles and is dominated by superexchange interactions within the ground-state E_g doublet. Computed ordering temperature of the single second-order phase transition is consistent with experimentally observed material-dependent trends. We further investigate the electronic, structural and magnetic properties of such compounds by purely Density Functional Theory (DFT) calculations with a new approach that consists in the constrain of the onsite density matrix, as obtained via DFT +  Dynamical Mean Field Theory calculations. We prove that this method is able to reproduce the ferro-octupolar order and we compare this result to conventional DFT dipolar solutions.