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Dzyaloshinskii-Moriya Interaction between Multipolar Moments in $5d^1$ Systems

We propose a new type of Dzyaloshinskii-Moriya (DM) interactions which act on high-rank multipolar moments such as quadrupolar and octupolar moments. Here we consider 5d1 systems with broken spatial inversion symmetry, where the interplay of electron correlation, the spin-orbit coupling, and inversion symmetry breaking plays a crucial role. Using a numerical diagonalization on a two-site multiorbital Hubbard model, we reveal that anti-symmetric products of multipole operators have finite expectation values, indicating the existence of DM interactions for multipoles. We also find that the spin-orbit coupling dependences of DM interactions for multipoles are significantly different depending on the lattice structure. Finally, we discuss the numerical results for small and large spin-orbit coupling region by using perturbative analysis.