The mechanisms of these processes are often captured qualitatively by treating spin as a conserved quantity. However, in materials that exhibit strong spintronics phenomena, such an approximation is frequently questionable. This dilemma leads us to pose the following question: Can conduction electrons carry physical observables other than spin, even if these observables are not strictly conserved, and give rise to new measurable phenomena? In this talk, I will introduce the language and tools for defining and calculating spin magnetic multipole moments of Bloch electrons in a gauge-invariant manner [1]. These magnetic multipole moments are inherently local in both real and momentum spaces, achieved through a wave-packet approach. One can then introduce a semiclassical Boltzmann theory to describe the transport of these multipole moments. Using monolayer phosphorene as a prototypical example, I will elucidate some physical consequences of spin magnetic multipole moments generated by electric fields and currents, in particular current-induced spin accumulation with staggered signs at corners of a square sample. Additionally, I will discuss how the spin Hall effect can be understood as an electrical response of spin magnetic quadrupole moment or spin density polarization [2]. In the end I will briefly mention our ongoing work on the modern theory of equilibrium higher-order spin magnetic multipole moments [3].

[1] M. Tahir and HC, Phys. Rev. Lett. 131, 106701 (2023)
[2] HC, Q. Niu, and A. H. MacDonald, arXiv:1803.01294
[3] HC, G.-Y. Guo, D. Xiao, in preparation.