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RKKY interactions of CeB6 based on effective Wannier model

We examine the RKKY interactions of CeB$_6$ between multipole moments based on the effective Wannier model obtained from the bandstructure calculation including 14 Ce-$f$ orbitals and 60 conduction orbitals of Ce-$d,s$ and B-$p,s$. By using the $f$-$c$ mixing matrix elements of the Wannier model together with the conduction band dispersion, the multipole couplings with the RKKY oscillation are obtained for the active moments in $Γ_{8}$ subspace. Both of the $Γ_{5g}$ quadrupole $O_{xy}$ and the $Γ_{2u}$ octupole $T_{xyz}$ couplings are largely enhanced with $\bm{q}=(π,π,π)$ which naturally explains the antiferro-quadrupolar phase of the phase II, and are also enhanced with $\bm{q}=(0,0,0)$ corresponding to the elastic softening of $C_{44}$. Also the couplings of the $Γ_{5u}$ octupole $T_{z}^β$ is quite large for $\bm{q}=(0,0,π)$ which is related to the antiferro-octupolar ordering of a possible candidate for the phase IV of Ce$_{x}$La$_{1-x}$B$_6$.