Colour symmetry and non-collinear altermagnetism

We present a formalism based on colour symmetry to analyse the momentum-space spin textures of non-collinear magnets. We show that, out of the spin textures allowed by the magnetic point group, one can extract an `altermagnetic' component that is invariant by general rotations in spin space, and can exist in the absence of spin-orbit coupling. We demonstrate this approach in the case of three complex, non-collinear magnets, Mn$_3$Ir(Ge,Si), Pb$_2$MnO$_4$ and Mn$_3$GaN. For Mn$_3$GaN, we also show that the predictions of colour-symmetry analysis are consistent with density functional theory calculations performed on the same system both with and without spin-orbit coupling.