Intersite Coulomb Repulsion Driven Quadrupole Instability and Magnetic Ordering in the Orbital Frustrated Ba₂MgReO₆

In order to calculate the multipoles in real materials with considerable intersite Coulomb interaction V, we develop a self-consistent program which starts from the first-principles calculations to solve the tight-binding Hamiltonian including onsite Coulomb repulsion U, V, and spin-orbital coupling \lambda. The program is applied to Ba_2MgReO_6 to figure out the mechanism of structural instability and magnetic ordering. A comprehensive quadrupole phase diagram versus U and V with \lambda=0.28 eV is calculated. Our results demonstrate that the easy-plane anisotropy and the intersite Coulomb repulsion V must be considered to remove the orbital frustration. The increase of V to >20 meV would arrange quadrupole Q_x^2-y^2 antiparallelly, accompanied by small parallel Q_3z^2-r^2, and stabilize Ba_2MgReO_6 into the body-centered tetragonal structure. Such antiparallel Q_x^2-y^2 provides a new mechanism for the Dzyaloshinskii-Moriya interaction and gives rise to the canted antiferromagnetic (CAF) state along the 110 axis. Moreover, sizable octupoles such as O_21^31, O_21^33, O_21^34 and O_21^36 are discovered for the first time in the CAF state. Our study not only provides a comprehensive understanding of the experimental results in Ba_2MgReO_6, but also serves as a general and useful tool for the study of multipole physics in 5d compounds.