A Geometric Design Principle for $\mathbb{Z}_2$ Topological Phases in Twisted Triangular-Lattice Bilayers

Abstract

Twisted van der Waals bilayers provide a versatile platform for moiré electronic states, yet a transferable symmetry-based principle for time-reversal-invariant $\mathbb{Z}_2$ moiré bands has remained largely missing. Here we show that triangular-lattice bilayers with symmetry-related stacking minima provide a geometric route to an emergent honeycomb moiré lattice. Band-edge states derived from the untwisted $Γ$ valley are trapped by the reconstructed stacking landscape, forming A/B moiré orbitals whose inter-domain coupling generates Dirac crossings. Spin--orbit coupling opens a topological gap, yielding an effective Kane--Mele description and a quantum spin Hall phase characterized by a nontrivial $\mathbb{Z}_2$ invariant. First-principles calculations for Janus BiTeBr confirm the robustness of this phase over a broad twist-angle range and demonstrate an electric-field-driven topological transition. Representative triangular-lattice bilayers further establish this symmetry-based design principle as a broadly applicable route to tunable moiré quantum spin Hall materials.