  Odd-parity magnetic and magnetic toroidal multipoles in the absence of both
spatial-inversion and time-reversal symmetries are sources of multiferroic and
nonreciprocal optical phenomena. We investigate electronic states caused by an
emergent odd-parity magnetic quadrupole (MQ) as a representative example of
magnetic odd-parity multipoles. It is shown that spontaneous ordering of the MQ
leads to an antisymmetric spin-orbital polarization in momentum space, which
corresponds to a spin-orbital momentum locking at each wave vector. By symmetry
argument, we show that the orbital or sublattice degree of freedom is
indispensable to give rise to the spin-orbital momentum locking. We demonstrate
how the electronic band structures are modulated by the MQ ordering in the
three-orbital system, in which the MQ is activated by the spin-dependent
hybridization between the orbitals with different spatial parities. The
spin-orbital momentum locking is related with the microscopic origin of
cross-correlated phenomena, e.g., the magnetic-field-induced symmetric and
antisymmetric spin polarization in the band structure, the current-induced
distortion, and the magnetoelectric effect. We also discuss similar
spin-orbital momentum locking in antiferromagnet where the MQ degree of freedom
is activated through the antiferromagnetic spin structure in a sublattice
system.
