Spin-resolved topology and partial axion angles in three-dimensional insulators

Abstract

Symmetry-protected topological crystalline insulators (TCIs) have primarily been characterized by their gapless boundary states. However, in time-reversal- ($${{{{{{{\mathcal{T}}}}}}}}$$ T -) invariant (helical) 3D TCIs—termed higher-order TCIs (HOTIs)—the boundary signatures can manifest as a sample-dependent network of 1D hinge states. We here introduce nested spin-resolved Wilson loops and layer constructions as tools to characterize the intrinsic bulk topological properties of spinful 3D insulators. We discover that helical HOTIs realize one of three spin-resolved phases with distinct responses that are quantitatively robust to large deformations of the bulk spin-orbital texture: 3D quantum spin Hall insulators (QSHIs), “spin-Weyl” semimetals, and $${{{{{{{\mathcal{T}}}}}}}}$$ T -doubled axion insulator (T-DAXI) states with nontrivial partial axion angles indicative of a 3D spin-magnetoelectric bulk response and half-quantized 2D TI surface states originating from a partial parity anomaly. Using ab-initio calculations, we demonstrate that β-MoTe2 realizes a spin-Weyl state and that α-BiBr hosts both 3D QSHI and T-DAXI regimes.

High spin axion insulator

“…In addition to the time-reversal ( ) symmetry, the quantized axion field θ = π can also be protected by combined lattice and time-reversal symmetry (for example with τ 1/2 the half translation operator) 12 , as the quantity E ⋅ B undergoes the same sign reversal. This kind of materials, termed axion insulator 13 – 24 , holds significant potential in layertronics 25 – 29 and quantum computing 30 , 31 . MnBi 2 Te 4 and its family have recently been proposed as axion insulators in the antiferromagnetic state 13 , 32 – 36 , which finds a concise description with an effective Hamiltonian defined on the orbital and spin-1/2 spaces 33 .…”