  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 quantum mechanical 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 at
the classical level. We then explore the possibility of computing surface
magnetization via a coarse-graining procedure applied to a quantum 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.
