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Conflicting interpretations of experimental data preclude the understandingof 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 thehitherto elusive octupolar order remains debated. We resolve this uncertaintythrough direct calculations of all-rank inter-site exchange interactions andinelastic neutron scattering cross-section for the $d^2$ double perovskiteseries Ba$_2M$OsO$_6$ ($M$= Ca, Mg, Zn). Using advanced many-body firstprinciples methods we show that the ground state is formed by ferro-orderedoctupoles coupled by superexchange interactions within the ground-state $E_g$doublet. Computed ordering temperature of the single second-orderphase-transition is consistent with experimentally observed material-dependenttrends. Minuscule distortions of the parent cubic structure are shown toqualitatively modify the structure of gaped magnetic excitations.

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