Ferro-octupolar order and low-energy excitations in d$^2$ double perovskites of Osmium

Conflicting interpretations of experimental data preclude the understanding of the quantum magnetic state of spin-orbit coupled $d^2$ 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 $d^2$ double perovskite series Ba$_2M$OsO$_6$ ($M$= Ca, Mg, Zn). Using advanced many-body first principles methods we show that the ground state is formed by ferro-ordered octupoles coupled 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. Minuscule distortions of the parent cubic structure are shown to qualitatively modify the structure of gaped magnetic excitations.