  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.
