Exchange interactions, Jahn-Teller coupling, and multipole orders in pseudospin one-half $5\boldsymbol{d}^\mathbf{2}$ 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 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. The superexchange process 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 couplings. The 90$^\circ$ superexchange in compounds with edge-shared octahedra is most unusual: despite highly anisotropic shapes of the $E_g$ wavefunctions, the pseudospin interactions have no bond dependence and show instead a hidden SU(2) symmetry, which equally supports quadrupole and octupole orders. We consider the $E_g$ pseudospin models on various lattices and obtain their ground state properties using analytical, classical Monte Carlo, and exact diagonalization methods. On the honeycomb lattice, we observe a duality with the extended Kitaev model, and 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. We also consider Jahn-Teller coupling effects and lattice mediated interactions between $E_g$ pseudospins. Possible implications of the results for recent experiments on double perovskite osmates are discussed, including effects of local distortions on the pseudospin wavefunctions and interactions.