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Motivated by recent theoretical and experimental controversy, we present a theoretical study to clarify the orbital symmetry of the ground state of vanadium spinel oxides AV$_2$O$_4$ (A=Zn, Mg, Cd). The study is based on an effective Hamiltonian with spin-orbital superexchange interaction and a local spin-orbit coupling term. We construct a classical phase-diagram and prove the complex orbital nature of the ground state. Remarkably, with our new analysis we predict correctly also the coherent tetragonal flattening of oxygen octahedra. Finally, through analytical considerations as well as numerical ab-initio simulations, we propose how to detect the predicted complex orbital ordering through vanadium K edge resonant x-ray scattering.
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We study vanadium spinels $A$V$_2$O$_4$ ($A$ = Cd, Mg) in pulsed magnetic fields up to 65 T. A jump in magnetization at $mu_0 H approx$ 40 T is observed in the single-crystal MgV$_2$O$_4$, indicating a field induced quantum phase transition between two distinct magnetic orders. In the multiferroic CdV$_2$O$_4$, the field-induced transition is accompanied by a suppression of the electric polarization. By modeling the magnetic properties in the presence of strong spin-orbit coupling characteristic of vanadium spinels, we show that both features of the field-induced transition can be successfully explained by including the effects of the local trigonal crystal field.
We explore mechanisms of orbital order decay in doped Mott insulators $R_{1-x}$(Sr,Ca)$_x$VO$_3$ ($R=,$Pr,Y,La) caused by charged (Sr,Ca) defects. Our unrestricted Hartree-Fock analysis focuses on the combined effect of random, charged impurities and associated doped holes up to $x=0.5$. The study is based on a generalized multi-band Hubbard model for the relevant vanadium $t_{2g}$ electrons, and includes the long-range (i) Coulomb potentials of defects and (ii) electron-electron interactions. We show that the rotation of occupied $t_{2g}$ orbitals, induced by the electric field of defects, is a very efficient perturbation that largely controls the suppression of orbital order in these compounds. We investigate the inverse participation number spectra and find that electron states remain localized on few sites even in the regime where orbital order is collapsed. From the change of kinetic and superexchange energy we can conclude that the motion of doped holes, which is the dominant effect for the reduction of magnetic order in high-$T_c$ compounds, is of secondary importance here.
In strongly correlated multi-orbital systems, various ordered phases appear. In particular, the orbital order in iron-based superconductors attracts much attention since it is considered to be the origin of the nematic state. In order to clarify the essential condition for realizing orbital orders, we study simple two-orbital ($d_{xz}$, $d_{yz}$) Hubbard model. We find that the orbital order, which corresponds to the nematic order, appears due to the vertex corrections even in the two-orbital model. Thus, $d_{xy}$ orbital is not essential to realize the nematic orbital order. The obtained orbital order depends on the orbital dependence and the topology of fermi surfaces. We also find that another type of orbital order, which is rotated $45^circ$, appears in the heavily hole-doped case.
We demonstrate that the onset of complex spin orders in ACr$_2$O$_4$ spinels with magnetic A$=$Co, Fe and Cu ions lowers the lattice symmetry. This is clearly indicated by the emergence of anisotropic lattice dynamics -- as evidenced by the pronounced phonon splittings -- even when experiments probing static distortions fail. We show that the crystal symmetry in the magnetic phase is reduced from tetragonal to orthorhombic for FeCr$_2$O$_4$ and CuCr$_2$O$_4$ with Jahn-Teller active A-site ions. The conical spin structure in FeCr$_2$O$_4$ is also manifested in the phonon frequencies. In contrast, the multiferroic CoCr$_2$O$_4$ with no orbital degrees of freedom remains nearly cubic in its ground state.
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 $5d^2$ 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, $mu$SR, X-ray diffraction, and inelastic neutron scattering measurements on a range of cubic $5d^2$ double perovskite materials including Ba$_2$ZnOsO$_6$, Ba$_2$CaOsO$_6$, and Ba$_2$MgOsO$_6$. Our proposal for exciton condensation leading to type-I AFM order may be relevant to materials such as Sr$_2$MgOsO$_6$.
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