Octupolar order in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>d</mml:mi></mml:math>-orbital Mott insulators

Abstract

Motivated by experimental and theoretical interest in realizing multipolar orders in $d$-orbital materials, we discuss the quantum magnetism of $J=2$ ions which can be realized in spin-orbit coupled oxides with $5{d}^{2}$ transition metal ions. Based on the crystal-field environment, we argue for a splitting of the $J=2$ multiplet, leading to a low-lying non-Kramers doublet which hosts quadrupolar and octupolar moments. We discuss a microscopic mechanism whereby the combined perturbative effects of orbital repulsion and antiferromagnetic Heisenberg spin interactions leads to ferro-octupolar coupling between neighboring sites, and stabilizes ferro-octupolar order for a face-centered cubic lattice. This same mechanism is also shown to disfavor quadrupolar ordering. We show that studying crystal field levels via Raman scattering in a magnetic field provides a probe of octupolar order. We study spin dynamics in the ferro-octupolar state using a slave-boson approach, uncovering a gapped and dispersive magnetic exciton. For sufficiently strong magnetic exchange, the dispersive exciton can condense, leading to conventional type-I antiferromagnetic (AFM) order which can preempt octupolar order. Our proposal for ferrooctupolar order, with specific results in the context of a model Hamiltonian, provides a comprehensive understanding of thermodynamics, $\ensuremath{\mu}\mathrm{SR}$, x-ray diffraction, and inelastic neutron-scattering measurements on a range of cubic $5{d}^{2}$ double perovskite materials including ${\mathrm{Ba}}_{2}{\mathrm{ZnOsO}}_{6}$, ${\mathrm{Ba}}_{2}{\mathrm{CaOsO}}_{6}$, and ${\mathrm{Ba}}_{2}{\mathrm{MgOsO}}_{6}$. Our proposal for exciton condensation leading to type-I AFM order may be relevant to materials such as ${\mathrm{Sr}}_{2}{\mathrm{MgOsO}}_{6}$.

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