Fig. 1. Global torque in the 3Q triangular antiferromagnet.

a Sketch of the triangular lattice with 3Q non-coplanar configuration of the magnetic moments. The blue atoms are magnetic and the green atoms break the planar inversion symmetry. b Top view of the triangular lattice. The hopping parameters of the black and red bounds are t and $t^{\prime }$, respectively. c In-plane spin texture in momentum space at energy ε = −5 eV, corresponding to the red dashed line in panel (e). d, e Band structure for $t^{\prime }=t$ and $t^{\prime }=t/2$, respectively. In the absence of inversion symmetry breaking, $t^{\prime }=t$, the degenerate bands display compensating spin texture in momentum space. When $t^{\prime }\ne t$, the band degeneracy is lifted and the spin textures no longer compensate. The color scale indicates the value of S_z. f, g $\mathcal{T}$-even (f) $\mathcal{T}$-odd (g) contributions for $t^{\prime }=t/2$ and ε = −4 eV when rotating the electric field direction in the (x,y) plane. We set t = 1 eV, Γ = 0.1 eV and the exchange is J = −2 eV.