Magnetic Quadrupole Moment in Higher-Order Topological Phases

We study 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 MQM manifests as surface-localized magnetization and hinge currents. The surface magnetization is anomalous in the sense that it is generally not equal to the hinge current surrounding the same surface. 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. The formalism is then applied to several higher-order topological phases, and we show that the MQM can distinguish some intrinsic and boundary-obstructed higher-order topological insulators. Our work allows for new studies of electromagnetic responses in higher-order topological phases and beyond.