We study the RKKY interaction between two magnetic impurities located on same layer (intralayer case) or on different layers (interlayer case) in undoped bilayer graphene in the four-bands model, by directly calculating the Green functions in the eigenvalues and eigenvectors representation. Our results show that both intra- and interlayer RKKY interactions between two magnetic impurities located on same (opposite) sublattice are always ferromagnetic (antiferromagnetic). Furthermore we find unusual long-distance decay of the RKKY interaction in BLG. The intralyer RKKY interactions between two magnetic impurities located on same sublattice, $J^{A_{n}A_{n}}(\mathbf{R})$ and $J^{B_{n}B_{n}}(\mathbf{R)}$, decay closely as $1/R^{6}$ and $1/R^{2}$ at large impurity distances respectively, but when they are located on opposite sublattices the RKKY interactions exhibit $1/R^{4}$ decays approximately. In the interlayer case, the RKKY interactions between two magnetic impurities located on same sublattice show a decay close to $1/R^{4}$ at large impurity distances, but if two magnetic impurities be on opposite sublattices the RKKY interactions, $J^{A_{1}B_{2}}(\mathbf{R})$ and $J^{B_{1}A_{2}}(\mathbf{R)}$, decay closely as $1/R^{6}$ and $1/R^{2}$ respectively. Both intra- and interlayer RKKY interactions have anisotropic oscillatory factors which for intralayer case is equal to that for single layer graphene. Our results at weak and strong interlayer coupling limits reduce to the RKKY interaction of SLG and that of BLG in the two-bands approximation respectively.

翻译：我们通过直接计算位于同一层(内层)或不同层(内层)的两种磁性杂质之间的RKKY相互作用。我们通过直接计算位于同一层(内层)和外层代表的绿色函数,研究位于同一层(外层)的两种磁性杂质之间的RKY相互作用。此外,我们发现在BLG中,RKY相互作用在解析双层(内层)双层(内层)的异质(内层)变异(内层)的异质。我们发现,RKKY相互作用在平层(内层)的异质(内层)和内部(内层)的异质(内层/内层)相互作用在(内层)内(内层)和内层(内层)的(内层/内层(内层)相互作用在(内层/内层)内(内层(内层),在内层(内层)内(内层)内(内层)和内层(内层(内)内(内),内层(内层)内(内层)的对内(内层(内层)即层)的相互作用在内(内(内)内(内)内(内)内(内)即B(内)内),即基(内(内)即(内)即(内)即(内)即(内)内)内)内(内)即(内)即(内)内)内(内)和内(内(内)内)即层(内)内)内),即基(内)内(内(内)的和内),在(内),在(内)内(内)内)内(内)内)内(内(内(内)即(内)即(内)内)内)即(内)内),即(内)内)内(内(内),即(内),即(内)内)内)内)内)内(内(内)内(内(内)内)内)内(内(内(内)和内(内(内)内)内)内)和内(内)内),在内(内(内(内)即,即,即,即,即(内)内