Towards a theory of surface orbital magnetization

Abstract

The theory of bulk orbital magnetization has been formulated both in reciprocal space based on Berry curvature and related quantities, and in real space in terms of the spatial average of a local marker. Here we consider a three-dimensional antiferromagnetic material having a vanishing bulk but a nonzero surface orbital magnetization. We ask whether the surface-normal component of the surface magnetization is well defined, and if so, how to compute it. As the physical observable corresponding to this quantity, we identify the macroscopic current running along a hinge shared by two facets. However, the hinge current only constrains the difference of the surface magnetizations on the adjoined facets, leaving a potential ambiguity. By performing a symmetry analysis, we find that only crystals exhibiting a pseudoscalar symmetry admit well-defined magnetizations at their surfaces. We then explore the possibility of computing surface magnetization via a coarse-graining procedure applied to a local marker. We show that multiple expressions for the local marker exist, and apply constraints to filter out potentially meaningful candidates. Using several tight-binding models as our theoretical test bed and several potential markers, we compute surface magnetizations for slab geometries and compare their predictions with explicit calculations of the macroscopic hinge currents of rod geometries. We find that only a particular form of the marker consistently predicts the correct hinge currents.

Orbital magnetic quadrupole moment in higher order topological phases

“…During the preparation of this manuscript we became aware of an independent work which also considers a similar method for calculating surface orbital magnetization [49]. In this section, we prove that for the hinge configuration in Fig.…”

“…We combine calculations of hinge currents with a new slab-geometry method to calculate surface magnetization to determine the MQM (see Ref. 49 for a recent work proposing a similar surface magnetization formalism). One of the key observable signatures of these MQMs is the presence of hinge currents localized at the intersection of two surfaces.…”