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.