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

We develop a microscopic theory of multipole interactions and orderings in $5{d}^{2}$ transition metal ion compounds. In a cubic environment, the ground state of $5{d}^{2}$ 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 ${180}^{\ensuremath{\circ}}$ bonding, we find nearly equal mixture of Heisenberg and ${e}_{g}$ orbital compass-type couplings. The ${90}^{\ensuremath{\circ}}$ 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.