Multipolar magnetism in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-orbital systems: Crystal field levels, octupolar order, and orbital loop currents

Quantum magnets with spin $J=2$, which arise in spin-orbit coupled Mott insulators, can potentially display multipolar orders. Motivated by gaining a better microscopic understanding of the local physics of such $d$-orbital quantum magnets, we carry out an exact diagonalization study of a simple octahedral crystal field Hamiltonian for two electrons, incorporating spin-orbit coupling (SOC) and interactions. While the rotationally invariant Kanamori interaction in the ${t}_{2g}$ sector leads to a fivefold degenerate $J=2$ manifold, we find that either explicitly including the ${e}_{g}$ orbitals, or going beyond the rotationally invariant Coulomb interaction within the ${t}_{2g}$ sector, causes a degeneracy breaking of the $J=2$ levels. This can lead to a low-lying non-Kramers doublet carrying quadrupolar and octupolar moments and an excited triplet which supports magnetic dipole moments, bolstering our previous phenomenological proposal for the stabilization of ferro-octupolar order in heavy transition metal oxides. We show that the spontaneous time-reversal symmetry breaking due to ferro-octupolar ordering within the non-Kramers doublet leads to electronic orbital loop currents. The resulting internal magnetic fields can potentially explain the small fields inferred from muon-spin relaxation ($\ensuremath{\mu}\mathrm{SR}$) experiments on cubic $5{d}^{2}$ osmate double perovskites ${\mathrm{Ba}}_{2}{\mathrm{ZnOsO}}_{6}$, ${\mathrm{Ba}}_{2}{\mathrm{CaOsO}}_{6}$, and ${\mathrm{Ba}}_{2}{\mathrm{MgOsO}}_{6}$, which were previously attributed to weak dipolar magnetism. We make further predictions for oxygen NMR experiments on these materials. We also study the reversed level scheme, where the $J=2$ multiplet splits into a low-lying magnetic triplet and excited non-Kramers doublet, presenting single-ion results for the magnetic susceptibility in this case, and pointing out its possible relevance for the rhenate ${\mathrm{Ba}}_{2}{\mathrm{YReO}}_{6}$. Our work highlights the intimate connection between the physics of heavy transition metal oxides and that of $f$-electron based heavy fermion compounds.

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