Magnetic Anisotropies and Skyrmion Lattice Related to Magnetic Quadrupole Interactions of the RKKY Mechanism in Frustrated Spin-Trimer System Gd${}_3$Ru${}_4$Al${}_{12}$ with a Breathing Kagome Structure

The origin of the magnetic quadrupole (MQ) interactions in Gd${}_3$Ru${}_4$Al${}_{12}$ which is known as a frustrated spin system and as a host material of skyrmion with a sentrosymmetric crystal structure are discussed. The MQ interactions between ferromagnetic (FM) spin trimers with imperfect FM directivity are deduced from synthesis of dipole Ruderman-Kittel-Kasuya-Yosida (RKKY) interactions. The Hamiltonian which includes both the MQ interactions and dipole interactions is proposed, and magnetic anisotropies, magnetic phase transitions and contribution of the MQ interactions to stabilize the skyrmion lattice (SkL) are discussed based on this Hamiltonian. Degrees of MQ freedom carried by the trimers contribute to stabilizing SkL which appears at finite temperatures.