Multipolar ordering from dynamical mean field theory with application to CeB6

Magnetic and multipolar ordering in $f$ electron systems takes place at low temperatures of order 1--10 Kelvin. Combinations of first-principles with many-body calculations for such low-energy properties of correlated materials are challenging problems. We address multipolar ordering in $f$ electron systems based on the dynamical mean-field theory (DMFT) combined with density functional theory. We derive the momentum-dependent multipolar susceptibilities and interactions in two ways: by solving the Bethe-Salpeter (BS) equation of the two-particle Green's function and by using a recently developed approximate strong-coupling formula. We apply the formalism to the prototypical example of multipolar ordering in ${\mathrm{CeB}}_{6}$ using the Hubbard-I solver, and demonstrate that the experimental quadrupole transition is correctly reproduced. The results by the approximate formula show good agreement with those by the BS equation. This first-principles formalism for multipolar ordering based on DMFT has applications which are beyond the reach of the traditional RKKY formula. In particular, more itinerant electron systems including $5f$, $4d$, and $5d$ electrons can be addressed.