Towards a theory of surface orbital magnetization

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.

Figure 1

Schematic depicting a hinge and the surface facets that compose it. The black arrowhead on the hinge depicts the direction of flow for the hinge current $I^{\text{LR}}$ to be positive. The maroon and blue arrows denote the surface-normal magnetization vectors $M_{\perp }^{\text{R}}$ and $M_{\perp }^{\text{L}}$, respectively. The hinge current is $I^{\text{LR}}=M_{\perp }^{\text{L}}-M_{\perp }^{\text{R}}$.

Figure 2

Schematic depiction of a hinge formed by two surface facets, one of which has a magnetization ${\mathbf{M}}^{\text{surf}}$ that is not normal to the surface, having components $M_{\perp }$ along $\widehat{\mathbf{z}}$ and $M_{\parallel }$ along $\widehat{\mathbf{y}}$. $M_{\parallel }$ is observable in principle by the presence of a concentration of magnetic field $B_y$ in the surface region. The hinge current is observable via the integral $\oint \mathbf{M}·d\mathbf{l}$ around the dashed blue Amperian loop, which remains normal to the surface while passing through the surface region delineated by the dashed black line.

Figure 3

Schematic of a slab geometry for a bulk material with an $R_{\widehat{\mathbf{z}}}^{-}$ symmetry. An additional facet of arbitrary orientation appears at right. The $R_{\widehat{\mathbf{z}}}^{-}$ symmetry ensures that the magnetizations on the top and bottom facets are equal and parallel, as indicated by the maroon arrows. All three facet magnetizations may be found using the known hinge currents $I_1$ and $I_2$, as discussed in the text.

Figure 4

Symmetry constraints arising from $3_m^{\prime }, 4_m^{\prime }$, and $6_m^{\prime }$ symmetries. Arrows indicate the directions of the surface-normal magnetizations relative to a global Cartesian frame. A hinge current flowing into or out of the page is shown with a cross or dot respectively. (a) Triangular prism construction consistent with $3_m^{\prime }$ symmetry. Surfaces have no magnetization. (b) Square prism construction consistent with $4_m^{\prime }$ symmetry. Surface magnetizations form a two-in two-out configuration with $M_{\perp }=I/2$. (c) Hexagonal prism construction consistent with $6_m^{\prime }$ symmetry. Surface magnetizations form a three-in three-out configuration with $M_{\perp }=I/2$.

Figure 5

Symmetry constraints arising from $3_m, 4_m$, and $6_m$ symmetries. Conventions are the same as in Fig. 4. (a) Triangular prism constructed using $3_m$ symmetry. (b) Square prism constructed using $4_m$ symmetry. (c) Hexagonal prism constructed using $6_m$ symmetry. In each case all facets have the same surface magnetization $M_{\perp }$, whose magnitude cannot be determined since hinge currents are absent.

Figure 6

Schematic of a slab geometry respecting an $m_x^{\prime }$ symmetry, with the mirror plane illustrated by the thatched gray plane bisecting the slab. An additional facet of arbitrary orientation appears at the top. The $m_x^{\prime }$ symmetry ensures that the magnetizations on the left and right facets are equal and antiparallel. In this case the knowledge of the hinge currents is insufficient to determine the $M_{\perp }$ values, as discussed in the text.

Figure 7

Insulating rod geometries utilized in computing macroscopic hinge currents. Each rod is cut out from the bulk, and satisfies periodic boundary conditions along one of the three Cartesian directions. The surface magnetizations $M_{\perp }^{\text{L}}, M_{\perp }^{\text{B}}$, and $M_{\perp }^{\text{L}}$ are determined from calculations described in Sec. 4b. The macroscopic hinge currents $I^{\text{FB}}, I^{\text{BL}}$, and $I^{\text{LF}}$ that are computed in the text are indicated by the black arrows.

Figure 8

(a) Visualization of a single Haldane layer. The different sublattices are indicated by filled and empty circles, respectively, and feature on-site potentials $\mathrm{\Delta }$ and $-\mathrm{\Delta }$. The nearest-neighbor hopping $t_1$ is indicated by the double-sided solid arrow, while the complex next nearest-neighbor hopping $t_2{e}^{i\varphi }$ is indicated by one-sided dashed arrows. (b) A side view of the bulk unit cell along the $-x$ direction for the model of Sec 5a. The vertical interlayer hoppings connect identical sublattices and are indicated by solid double-sided arrows that are labeled with the corresponding hopping amplitude. Crimson arrows indicate the magnetizations of the layers in the decoupled regime. To the right are the phases of the complex intralayer hoppings.

Figure 9

(a) Primitive cell of the 2D square plaquette model. The hoppings between different lattice sites, marked by solid straight lines with arrows, are such that the total flux through the cell is an integer multiple of $2\pi$. (b) Two-layer plaquette model. Only A-A and C-C interlayer hoppings are included, and each layer is related to the one below by a $C_{2z}^{\prime }$ rotation through the $C$ sites. (c) Four-layer plaquette model. Same as (b), except that the rotation is now $C_{4z}^{\prime }$ instead.

Figure 10

Plots of the (a) $I^{\text{FB}}$, (b) $I^{\text{BL}}$, (c) $I^{\text{LF}}$ hinge currents for the two-layer square plaquette model with zigzag edges versus $t_3^{\prime }$. The plots compare the hinge current predicted from the different local markers to the directly computed hinge current. When $t_3^{\prime }=0.1$ (axion-odd regime), all the markers yield the correct values of the hinge currents, as indicated by the intersections of the dashed horizontal and vertical gray lines. Generally, only the surface magnetizations found from the “lin” marker always match the directly computed hinge currents.

Figure 11

Plot of the hinge current $I^{\text{LF}}$ as a function of $t_3^{\prime }$ for the four-layer square plaquette model with straight edges. In the axion-odd regime when $t_3^{\prime }=0.1$, only the “lin” marker correctly predicts the hinge current; this is also the case even when no axion-odd symmetry is present.

Figure 12

Illustration of the structure, labeling, and on-site energies of the cubic molecule model discussed in Sec. 5d.