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

In Mott insulators with a half-filled $t_{2g}$ shell the Hund's rule coupling induces a spin-3/2 orbital-singlet ground state. The spin-orbit interaction is not expected to qualitatively impact low-energy degrees of freedom in such systems. Indeed, $d^3$ cubic double perovskites (DP) of heavy transition metals are believed to exhibit conventional collinear magnetic orders. However, their inelastic neutron scattering spectra feature large gaps of unclear origin. Here we derive first-principles low-energy Hamiltonians for the cubic DP Ba$_2$Y$B'$O$_6$ ($B'=$ Os, Ru) and show that they include significant multipolar - dipole-octupolar - intersite exchange terms. These terms break continuous symmetry of the spin-3/2 Hamiltonian opening an excitation gap. The calculated gap magnitudes are in good agreement with experiment. The dipole-octupolar intersite exchange is induced due to excited states of the $t_{2g}^3$ manifold that are admixed by the spin-orbit interaction into the spin-3/2 ground state.