Exchange interactions, Jahn-Teller coupling, and multipole orders in pseudospin one-half $5d^2$ Mott insulators

We develop a microscopic theory of multipole interactions and orderings in $5d^2$ transition metal ion compounds. In a cubic environment, the ground state of $5d^2$ ions is a non-Kramers $E_g$ doublet, which is nonmagnetic but hosts quadrupole and octupole moments. We derive low-energy pseudospin one-half Hamiltonians describing various spin-orbital exchange processes between these ions. Direct overlap of the $t_{2g}$ orbitals results in bond-dependent pseudospin interactions similar to those for $e_g$ orbitals in manganites, except for different orientations of the pseudospin easy axes. On the other hand, the superexchange process, where two different $t_{2g}$ orbitals communicate via oxygen ions, generates new types of pairwise interactions. In perovskites with ${180}^{\circ }$ bonding, we find nearly equal mixture of Heisenberg and $e_g$ orbital compass-type couplings. The ${90}^{\circ }$ superexchange in compounds with edge-shared octahedra is most unusual: Despite highly anisotropic shapes of the $E_g$ wave functions, the pseudospin interactions have no bond dependence. We consider the $E_g$ pseudospin models on various lattices and obtain their ground state properties using analytical and numerical methods. On the honeycomb lattice, we observe a duality with the extended Kitaev model, and use it to uncover a critical point where the quadrupole and octupole states are exactly degenerate. On the triangular lattice, an exotic pseudospin state, corresponding to the coherent superposition of vortex-type quadrupole and ferri-type octupole orders, is realized due to geometrical frustration. This state breaks both spatial and time-reversal symmetries, but possesses no dipolar magnetism. We also consider Jahn-Teller coupling effects and lattice mediated interactions between $E_g$ pseudospins, and find that they support quadrupole order. Possible implications of the results for recent experiments on double perovskite osmates are discussed, including effects of local distortions on the pseudospin wave functions and interactions.

Figure 1

(a) Cubic splitting ${\mathrm{\Delta }}_c$ of the $J=2$ level, and the spatial shapes of $E_g$-doublet wave functions (see also Fig. 2 of Ref. [4]). Right panel shows the pseudospin ($s^x,s^y,s^z$) coordinate axes with respect to oxygen octahedra. (b) Direct hopping between $xy$ orbitals in honeycomb (left) and double perovskite (right) lattices, resulting in bond-dependent pseudospin $\tau$ interactions Eq. (10) between spin-orbit entangled $E_g$ states. (c) Two-orbital superexchange via ${180}^{\circ }$ Me-O-Me bonding geometry. Hopping is orbital conserving: $xz\leftrightarrow xz$ and $yz\leftrightarrow yz$. This process results in pseudospin interactions Eq. (13), comprising isotropic Heisenberg and bond-dependent compass-type couplings. (d) Two-orbital superexchange via ${90}^{\circ }$ bonding geometry. Hopping interchanges the orbital labels: $xz\leftrightarrow yz$. This process leads to the interactions Eq. (15), which are anisotropic in pseudospin space but have no bond dependence.

Figure 2

(a) Phase diagram of the Hamiltonian (17) on honeycomb lattice as a function of $\lambda =2J/J_{\tau }$. There are two states, separated by a first-order spin-flop transition at the hidden SU(2) symmetric point $\lambda =1$. Left and right insets show the ordered patterns of the vortex-type quadrupole and AF-octupole phases, respectively. (b) Quantum zero-point energy $\delta E$ as a function of ordered moment orientation $\phi$ within the honeycomb plane, calculated at $\lambda =0.5$. A ground-state pattern with $\phi =\pi /6$ is shown in panel (a) left. (c) The in-plane (${\mathrm{\Delta }}_{\parallel }$) and out-of-plane (${\mathrm{\Delta }}_{\perp }$) magnon gaps in the vortex-quadrupole phase, and the magnon gap ($\mathrm{\Delta }$) in the uniaxial AF-octupole ordered state. Dashed line shows the pseudospin order parameter length $2\langle s\rangle$ near the transition point, where the model is dual to a fluctuation free Heisenberg FM.

Figure 3

(a) Phase diagram of Hamiltonian (17) on the triangular lattice as a function of $\lambda =2J/J_{\tau }$. The phases are separated by spin-flop transitions at ${\lambda }_1$ and ${\lambda }_2$. Insets: The ordering patterns of phases I and III. (b) Magnetic unit cell in the quadrupole-octupole mixed phase II. Triangular sublattice (red circles) is occupied by down-oriented octupole moments. On a hexagon, the in-plane components of spins form a vortex. Left and right patterns are related by in-plane $\phi$ rotations, in opposite directions on $A$ and $B$ sites; quantum effects slightly favor the right pattern. (c) Upper part: Down-oriented octupole moment at the center, and out-of-plane canted spins (blue arrows) on a hexagon. Lower part: Decomposition of a canted spin into octupole and quadrupole moments.

Figure 4

(a) Double perovskite lattice, where the transition metal ions (filled circles) reside on the fcc sublattice. Arrows show the ordered pattern of quadrupole moments. It can be viewed as a stack of triangular lattice [111] planes (shaded), with stripy-AF quadrupole order within each plane. (b) Temperature dependence of the order parameter $\langle n\rangle$ for classical $J_{\tau }$ model in DP (solid line) and triangular (dashed line) lattices, obtained by classical Monte Carlo simulation.

Figure 5

(a) Antiferro correlated $Q_3$ distortions on $xy$ plane of the DP lattice, leading to $V$ coupling (28) between quadrupole moments of ${\mathrm{\Gamma }}_3$ symmetry. (b) Antiferro correlated $Q_{xy}$ distortions on $xy$ plane, leading to $V^{\prime }$ coupling (29) in the ${\mathrm{\Gamma }}_5$ quadrupolar channel.