Reconciling magnetoelectric response and time-reversal symmetry in non-magnetic $\mathbb{Z}_2$ topological insulators

A delicate tension complicates the relationship between the topological magnetoelectric effect in three-dimensional $\mathbb{Z}_2$ topological insulators (TIs) and time-reversal symmetry (TRS). TRS underlies a particular $\mathbb{Z}_2$ topological classification of the electronic ground state of a bulk insulator and the associated quantization of the magnetoelectric coefficient calculated using linear response theory, but according to standard symmetry arguments simultaneously forbids any physically meaningful magnetoelectric response. This tension between theories of magnetoelectric response in bulk and finite-sized materials originates from the distinct approaches required to introduce notions of polarization and orbital magnetization in those fundamentally different environments. In this work we argue for a modified interpretation of the bulk linear response calculations in non-magnetic TIs that is more plainly consistent with TRS, and use this interpretation to discuss the effect's observation - still absent over a decade after its prediction. Our analysis is reinforced by microscopic bulk and thin film calculations carried out using a simplified but still realistic model for the well established V$_2$VI$_3$ (V $=$ (Sb,Bi) and VI $=$ (Se,Te)) family of non-magnetic $\mathbb{Z}_2$ TIs. We conclude that the topological magnetoelectric effect in non-magnetic $\mathbb{Z}_2$ TIs is activated by magnetic surface dopants, and that the charge density response to magnetic fields and the orbital magnetization response to electric fields in a given sample are controlled in part by the configuration of those dopants.