Anisotropic Multipolar Exchange Interactions in Systems with Strong Spin-Orbit Coupling

We introduce a theoretical framework for computaions of anisotropic multipolar exchange interactions found in many spin--orbit coupled magnetic systems and propose a method to extract these coupling constants using a density functional total energy calculation. This method is developed using a multipolar expansion of local density matrices for correlated orbitals that are responsible for magnetic degrees of freedom. Within the mean--field approximation, we show that each coupling constant can be recovered from a series of total energy calculations via what we call the ``pair--flip'' technique. This technique flips the relative phase of a pair of multipoles and computes corresponding total energy cost associated with the given exchange constant. To test it, we apply our method to Uranium Dioxide, which is a system known to have pseudospin $J=1$ superexchange induced dipolar, and superexchange plus spin--lattice induced quadrupolar orderings. Our calculation reveals that the superexchange and spin--lattice contributions to the quadrupolar exchange interactions are about the same order with ferro-- and antiferro--magnetic contributions, respectively. This highlights a competition rather than a cooperation between them. Our method could be a promising tool to explore magnetic properties of rare--earth compounds and hidden--order materials.