Fig. 3: Influence of 180^∘ twisting and electric fields on band structure and topology.

a Band structure for 180^∘ twisted CCDW 1T-TaSe₂ bilayer without (solid red) and with (dashed blue) SOC included. The degeneracy of the bands is lifted due to the absence of the inversion symmetry after twisting. b Influence of vertical electric fields on the low-energy band structure for 180^∘ twisted CCDW 1T-TaSe₂ bilayer. The gap at K closes and a band touching point occurs when the Semenoff mass term is ∣M∣ = ∣M₀ + ΔM∣ = 1.87 meV at E_z = –4.2 mV Å⁻¹. c Non-interacting λ_R vs. ΔM topological phase diagrams for non-twisted, 180^∘ twisted CCDW bilayer and two cases in between, where the spin–valley coupling B of the non-twisted CCDW 1T-TaSe₂ bilayer is varied. For B = 0 the system shows the symmetric onion-like shape similar to ref. ³¹. When B increases, the QSH region shrinks and disappears when B = λ_SOC. For B > λ_SOC, the system behaves as a trivial band insulator. The yellow arrows in the leftmost and rightmost panels indicate the path in phase space accessible by varying the vertical electric field E_z.