Derivation of RKKY Interaction between Multipole Moments in CeB$_6$ by the Effective Wannier Model based on the Bandstructure Calculation

We have investigated the electronic states of CeB$_6$ and have directly calculated the RKKY interaction on the basis of the 74-orbital effective Wannier model which includes 14 Ce-$f$ orbitals and 60 conduction ($c$) orbitals of Ce-$d,s$ and B-$p,s$ derived from the density-functional theory bandstructure calculation. By using not only the $c$-band dispersion but also the $f$-$c$ mixing matrix elements of the Wannier model, the realistic couplings for all 15 active multipole moments in $Gamma_8$ quartet subspace are obtained in the wavevector $q$-space and real-space. Both of the $Gamma_{5g}$ quadrupoles $(O_{yz},O_{zx},O_{xy})$ and the $Gamma_{2u}$ octupole $T_{xyz}$ couplings are maximally enhanced with $q=(pi,pi,pi)$ which naturally explains the phase II of the antiferro-quadrupolar ordering at $T_{Q}=3.2$ K, and are also enhanced with $q=(0,0,0)$ corresponding to the elastic softening of $C_{44}$. Also the couplings of the $Gamma_{5u}$ octupoles $T_{x}^{beta}$, $T_{y}^{beta}$ and $T_{z}^{beta}$ are quite large for $q=(pi,0,0)$, $(0,pi,0)$ and $(0,0,pi)$, which yields the antiferro-octupolar ordering of a possible candidate for phase IV of Ce$_{x}$La$_{1-x}$B$_6$. The intersite vector dependence of the RKKY couplings exhibit different long-range, oscillating, isotropic and anisotropic behaviors depending on the types of the multipole moments. The present approach enables us to provide the information about the possible multipole ordering in an unbiased way and is easily available for other localized $f$ electron materials once the $c$ states and $f$-$c$ mixing elements are given from the bandstructure calculation.