Orbital magnetic quadrupole moment in higher order topological phases

Abstract

We study the orbital magnetic quadrupole moment (MQM) in three dimensional higher-order topological phases. Much like electric quadrupole moment, which is associated with a charge response on the boundaries of a finite sample, the diagonal components of the MQM manifest as surface-localized magnetization and hinge currents. The hinge current is generally not equal to the difference of surface magnetizations that intersect at the hinge, and we show this mismatch is precisely quantified by the bulk MQM. We derive a quantum mechanical formula for the layer-resolved magnetization in slab geometries and use it to define the MQM of systems with gapped boundaries. Our formalism is then applied to several higher-order topological phases, and we show that the MQM can distinguish phases in some intrinsic and boundary-obstructed higher-order topological insulators. We then show that derivatives of the MQM with respect to the chemical potential can act as quantized topological invariants, similar to obtaining the 2D Chern number as a derivative of the magnetization with respect to the chemical potential. These invariants provide a new way to characterize 3D time-reversal breaking insulators that have vanishing magnetization.

Theory of electric, magnetic, and toroidal polarizations in crystalline solids with applications to hexagonal lonsdaleite and cubic diamond

“…Conversely, the lowest order of a magnetic multipole density permitted by the group D 6h (C 6v ) has = 6 whereas the same group permits a magnetotoroidal multipole with = 1. This is similar to the fact discussed above that, under crystallographic point groups G, the order of a multipole density m need not equal the polynomial degree of the tensor operators K G α associated with m in the invariant expansion (6).…”

“…In crystalline solids, the non-uniqueness of higherorder multipoles takes a distinct twist. Within each family of polarizations (even-electric, odd-electric, evenmagnetic, and odd-magnetic), the respective invariants appearing in the expansion (6) are not uniquely defined, as any linear combination of invariants in a family is again an invariant in that family, see footnote 35 in Ref. [70].…”

“…However, for systems showing a spontaneous electric dipolarization, the interpretation and observability of this quantity have remained ambiguous. Also, it is a significant challenge to extend the modern-theory approaches to multipole densities of higher multipole order > 1 [6].…”

Connecting Higher-Order Topology with the Orbital Hall Effect in Monolayers of Transition Metal Dichalcogenides

“…Recent works start to uncover the role of orbital hybridization in HOTIs and the advent of orbital effects [77][78][79]. Here, we connect the HOTI phase to the existence of pseudo-time-reversal operators and associated pseudospinors.…”