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

TL;DR

This paper investigates how magnetic fields and chemical potential influence the RKKY interaction in the $ ext{alpha}$-${ ext{T}_3}$ lattice, revealing anisotropic effects and a phase transition at $ ext{alpha}=0$ through analytical and numerical methods.

Contribution

It provides a comprehensive analysis of the RKKY interaction in the $ ext{alpha}$-${ ext{T}_3}$ lattice, including effects of magnetic field, chemical potential, and sublattice placement, highlighting a phase transition at $ ext{alpha}=0$.

Findings

Demonstrates anisotropy of RKKY interaction depending on impurity placement.

Shows a phase transition at $ ext{alpha}=0$ compared to graphene.

Provides analytical and numerical results for various doping and magnetic field conditions.

Abstract

The interaction energy for the indirect-exchange or Ruderman-Kittel-Kasuva-Yosida (RKKY) interaction between magnetic spins localized on lattice sites of the $\alpha$-${\cal T}_3$ 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 $\alpha$ for the ratio of the hopping integral from hub-to-rim and that around the rim of the hexagonal lattice. A valley and $\alpha$-dependent retarded Greens function matrix is used to form the susceptibility. Analytic and numerical results are obtained for undoped $\alpha$-${\cal T}_3$, 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 $\alpha$-${\cal T}_3$ with graphene shows that there is a phase transition at $\alpha=0$.