Geometric and nongeometric contributions to the surface anomalous Hall conductivity

A static electric field generates circulating currents at the surfaces of a magnetoelectric insulator. The anomalous Hall part of the surface conductivity tensor describing such bound currents can change by multiples of $e^2/h$ depending on the insulating surface preparation, and a bulk calculation does not fix its quantized part. To resolve this ambiguity, we develop a formalism for calculating the full surface anomalous Hall conductivity in a slab geometry. We identify a Berry-curvature term, closely related to the expression for the bulk anomalous Hall conductivity, whose value can change by quantized amounts by adjusting the surface Hamiltonian. In addition, the surface anomalous Hall conductivity contains a nongeometric part that does not depend on the surface preparation.

Figure 1

Coarse-grained layer-resolved AHC [Eq. (51)] for a 16-atom-thick slab of the cubic-lattice model with $\phi =\pi$.

Figure 2

AHC of the top surface (upper panel) and of the right surface (lower panel) of 16-layer slabs of the cubic-lattice model cut along $z$ and $x$, respectively, as a function of the cyclic parameter $\phi$. The solid (dashed) lines denote the total (CS) surface AHC. Circles represent the quantity appearing on the right-hand side of Eq. (9), with filled and empty circles denoting the total and the CS piece, respectively.

Figure 3

Cyclic evolution of the Hamiltonian of a 10-layer slab of the layered Haldane model, during which the isotropic ME coupling ${\alpha }_{\mathrm{CS}}$ of the bulk crystal changes by $e^2/h$. The AHC of the top surface is plotted as a solid line, and the black and gray circles denote two different branches of $-{\alpha }_{\mathrm{CS}}$. For a given choice of branch, Eq. (48) is satisfied throughout the cycle with the value of $m$ increasing by one at ${\varphi }_c=3\pi /2$.

Figure 4

The same quantities as in Fig. 3, but now for a cyclic evolution of the Hamiltonian of the surface layers only, keeping the Hamiltonian of the rest of the slab fixed.