Quadrupolar superexchange interactions, multipolar order, and magnetic phase transition in ${\mathrm{UO}}_2$

The origin of noncollinear magnetic order in ${\mathrm{UO}}_2$ is studied by an ab initio dynamical-mean-field-theory framework in conjunction with a linear-response approach for evaluating intersite superexchange interactions between U $5f^2$ 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 noncollinear 3k-magnetic order. Therefore, the exotic 3k-antiferromagnetic order in ${\mathrm{UO}}_2$ 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 $T_N$.

Figure 1

1k (upper panel), 2k (middle panel), and 3k (lower panel) antiferromagnetic orders in the unit cell of ${\mathrm{UO}}_2$. Uranium and oxygen sites are shown as large gray and small cyan balls, respectively. The quadrupole moments on inequivalent simple-cubic sublattices of the magnetic cell obtained by the mean-field solution of the ab initio SE Hamiltonian, Eqs. (3) and (4), at $T=0$ for each antiferromagnetic structure are displayed on the right-hand side and colored to indicate the corresponding U site in the unit cell.

Figure 2

The DFT+DMFT spectral function of ${\mathrm{UO}}_2$ within the Hubbard-I approximation. The black, red, and green lines are the total, partial U $5f$, and O $2p$ spectral functions, respectively. The experimental emission and bremsstrahlung isochromat spectra of Ref. [46] are displayed by blue circles.

Figure 3

(a) The expectation values of dipole and $t_{2g}$ quadrupole tensors as a function of temperature. A phase transition at $T=56$ K is clearly seen. (b) The mean-field magnetic energy $E_{\text{mag}}$ at zero temperature as a function of the anisotropy parameter $r$ of the QQ SE, see text.

Figure 4

The dipole-dipole and quadrupole-quadrupole nearest-neighbor pair correlation functions above $T_N$. The quadrupole-quadrupole pair correlation functions are multiplied by 10.