Quadrupolar Superexchange Interactions, Multipolar Order and Magnetic Phase Transition in UO$_2$

The origin of non-collinear magnetic order in UO$_{2}$ is studied by an ab initio dynamical-mean-field-theory framework in conjunction with a linear-response approach for evaluating inter-site superexchange interactions between U 5$f^{2}$ shells. The calculated quadrupole-quadruple superexchange interactions are found to unambiguously resolve the frustration of face-centered-cubic U sublattice toward stabilization of the experimentally observed non-collinear 3k-magnetic order. Therefore, the exotic 3k antiferromagnetic order in UO$_{2}$ can be accounted for by a purely electronic exchange mechanism acting in the undistorted cubic lattice structure. The quadrupolar short-range order above magnetic ordering temperature $T_N$ is found to qualitatively differ from the long-range order below $T_N$.