Bulk versus surface: Nonuniversal partitioning of the topological magnetoelectric effect

The electronic ground state of a three-dimensional (3D) band insulator with time-reversal ($\mathrm{\Theta }$) symmetry or time-reversal times a discrete translation ($\mathrm{\Theta }{T}_{1\mathrm{/}2}$) symmetry is classified by a ${\mathbb{Z}}_2$-valued topological invariant and characterized by quantized magnetoelectric response. Here we demonstrate by explicit calculation in model ${\mathbb{Z}}_2$ topological insulator thin-films that whereas the magnetoelectric response is localized at the surface in the $\mathrm{\Theta }$ symmetry (nonmagnetic) case, it is nonuniversally partitioned between surface and interior contributions in the $\mathrm{\Theta }{T}_{1\mathrm{/}2}$ (antiferromagnetic) case, while remaining quantized. Within our model the magnetic field induced polarization arises entirely from an anomalous $\mathcal{N}=0$ Landau level subspace within which the projected Hamiltonian is a generalized Su-Schrieffer-Heeger model whose topological properties are consistent with those of the starting 3D model. We identify a new connection between the ground-state geometry of that 3D model and surface-interior partitioning in thin films.