Spin-orbital-momentum locking under odd-parity magnetic quadrupole ordering

TL;DR

This paper explores how odd-parity magnetic quadrupole ordering induces spin-orbital momentum locking in electronic states, revealing implications for multiferroic and nonreciprocal optical phenomena.

Contribution

It demonstrates the emergence of spin-orbital momentum locking due to magnetic quadrupole ordering and elucidates its microscopic origin and effects in multi-orbital systems.

Findings

Magnetic quadrupole ordering causes antisymmetric spin-orbital polarization.

Spin-orbital momentum locking influences band structure modulation.

Related phenomena include magnetoelectric effects and current-induced distortions.

Abstract

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.