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In Mott insulators with a half-filled $t_{2g}$ shell the Hund's rule couplinginduces a spin-3/2 orbital-singlet ground state. The spin-orbit interaction isnot expected to qualitatively impact low-energy degrees of freedom in suchsystems. Indeed, $d^3$ cubic double perovskites (DP) of heavy transition metalsare believed to exhibit conventional collinear magnetic orders. However, theirinelastic neutron scattering spectra feature large gaps of unclear origin. Herewe derive first-principles low-energy Hamiltonians for the cubic DPBa$_2$Y$B'$O$_6$ ($B'=$ Os, Ru) and show that they include significantmultipolar - dipole-octupolar - intersite exchange terms. These terms breakcontinuous symmetry of the spin-3/2 Hamiltonian opening an excitation gap. Thecalculated gap magnitudes are in good agreement with experiment. Thedipole-octupolar intersite exchange is induced due to excited states of the$t_{2g}^3$ manifold that are admixed by the spin-orbit interaction into thespin-3/2 ground state.

Multipolar interactions and magnetic excitation gap in d$^3$ spin-orbit Mott insulators