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Effect of magnetic field and chemical potential on the RKKY interaction in the $α$-${\cal T}_3$ lattice

The interaction energy for the indirect-exchange or Ruderman-Kittel-Kasuva-Yosida (RKKY) interaction between magnetic spins localized on lattice sites of the $α$-${\cal T}_3$ model is calculated using linear response theory. In this model, the $\texttt{AB}$-honeycomb lattice structure is supplemented with $\texttt{C}$ atoms at the centers of the hexagonal lattice. This introduces a parameter $α$ for the ratio of the hopping integral from hub-to-rim and that around the rim of the hexagonal lattice. A valley and $α$-dependent retarded Greens function matrix is used to form the susceptibility. Analytic and numerical results are obtained for undoped $α$-${\cal T}_3$, when the chemical potential is finite and also in the presence of an applied magnetic field. We demonstrate the anisotropy of these results when the magnetic impurities are placed on the $\texttt{A,B}$ and $\texttt{C}$ sublattice sites. Additionally, comparison of the behavior of the susceptibility of $α$-${\cal T}_3$ with graphene shows that there is a phase transition at $α=0$.