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

• Published in: Physical review research 2024-06, Vol.6 (2), Article 023289
• Main Authors: Mahon, Perry T., Lei, Chao, MacDonald, Allan H.
• Format: Article
• Language: English
• ISSN: 2643-1564; 2643-1564
• DOI: 10.1103/PhysRevResearch.6.023289

A delicate tension complicates the relationship between the topological magnetoelectric effect (TME) in three-dimensional (3D) Z 2 topological insulators (TIs) and time-reversal symmetry (TRS). TRS underlies a particular 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 nonmagnetic 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. Our analysis is reinforced by microscopic bulk and thin-film calculations carried out using a simplified but still realistic effective model for the well established V 2 VI 3   [ V = (Sb, Bi) and VI = (Se, Te) ] family of nonmagnetic Z 2 TIs. When a uniform dc magnetic field is included in this model, the anomalous n = 0 Landau levels (LLs) play the central role, both in thin films and in bulk. In the former case, only the n = 0 LL eigenfunctions can support a dipole moment, which vanishes if there are no magnetic surface dopants and is quantized in the thick-film limit if magnetic dopants at the top and bottom surfaces have opposite orientation. In the latter case, the Hamiltonian projected into the n = 0 LL subspace is a one-dimensional Su-Schrieffer-Heeger model with ground-state polarization that is quantized in accordance with the bulk linear-response coefficient calculated for (a lattice regularization of) the starting 3D model. 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 nonmagnetic Z 2 TIs