Symmetry, topology, and geometry: The many faces of the topological magnetoelectric effect

TL;DR

This paper explores the topological magnetoelectric effect in 3D $ ext{Z}_2$ topological insulators, proposing a modified interpretation of theoretical responses, emphasizing the role of magnetic surface dopants, and discussing its experimental observability and robustness.

Contribution

It introduces a new interpretation of bulk response calculations in $ ext{Z}_2$ TIs and proposes a microscopic bulk-boundary correspondence related to the TME's activation.

Findings

Magnetic surface dopants are necessary for TME in non-magnetic $ ext{Z}_2$ TIs.

Charge response to magnetic fields is localized at the surface.

TME is less robust against disorder than the quantum Hall effect.

Abstract

A delicate tension complicates the relationship between the topological magnetoelectric effect (TME) in three-dimensional (3D) $\mathbb{Z}_2$ topological insulators (TIs) and time-reversal symmetry (TRS). TRS underlies a particular $\mathbb{Z}_2$ topological classification of the electronic ground state of crystalline band insulators and the associated quantization of the magnetoelectric response coefficient calculated using bulk linear response theory but, according to standard symmetry arguments, simultaneously forbids a nonzero magnetoelectric coefficient in any physical finite-size system. This contrast between theories of magnetoelectric response in formal bulk models and in real finite-sized materials originates from the distinct approaches required to introduce notions of (electronic) polarization and orbital magnetization in these fundamentally different environments. In this work we argue for a modified interpretation of the bulk linear response calculations in non-magnetic $\mathbb{Z}_2$ 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. Motivated by analytical results, we conjecture a type of microscopic bulk-boundary correspondence: a bulk insulator with (generalized) TRS supports a magnetoelectric coefficient that is purely itinerant (which is generically related to the geometry of the ground state) if and only if magnetic surface dopants are required for the TME to manifest in finite samples thereof. We conclude that in non-magnetic $\mathbb{Z}_2$ TIs the TME is activated by magnetic surface dopants, that the charge density response to a uniform dc magnetic field is localized at the surface and specified by the configuration of those dopants, and that the TME is qualitatively less robust against disorder than the integer quantum Hall effect.