Transport of Spin Magnetic Multipole Moments Carried by Bloch Quasiparticles

Magnetic ordering beyond the standard dipolar order has attracted significant attention in recent years, but it remains an open question how to effectively manipulate such nontrivial order parameters using external perturbations such as electric currents or fields. In particular, it is desirable to have a conceptual tool similar to nonequilibrium spin currents in spintronics to describe the creation and transport of multipole moments. In this context, we present a theory for Cartesian spin magnetic multipole moments of Bloch quasiparticles and their transport based on a general gauge-invariant formula obtained using the wave packet approach. As a concrete example, we point out that the low-energy Hamiltonian of phosphorene subject to a perpendicular electric field has a valley structure that hosts magnetic octupole moments. The magnetic octupole moments can be exhibited by an in-plane electric current and lead to accumulation of staggered spin densities at the corners of a rectangular sample. Our Letter paves the way for systematically seeking and utilizing quasiparticles with higher-order magnetic multipole moments in crystal materials towards the emergence of multipoletronics.

Figure 1

Schematic illustration of a wave packet carrying ${\mathcal{M}}_{xy}^x$ and the corresponding corner spin pattern in a macroscopic monolayer phosphorene sample with Rashba spin-orbit coupling.

Figure 2

(a) Structure of monolayer phosphorene. (b) Contour plot of the eigenenergy (in units of $\mathrm{\mu }\mathrm{eV}$) for the lower conduction band of monolayer phosphorene minus $\mathrm{\Delta }$.

Figure 3

(a) ${\mathcal{M}}_{xy}^x$ of the lower conduction band of phosphorene in units of ${\AA }^2·(\hslash /2)$. The regions where its values become too large due to the vanishing spin-orbit splitting between the two conduction bands are crossed out for better illustration. (b) Integrand of Eq. (7) for ${\mathcal{M}}_{xy}^x$ with $E_F-\mathrm{\Delta }=0.1\mathrm{eV}$, $k_{B}T=0.01\mathrm{eV}$, and $\mathbf{E}\parallel \widehat{x}$.