Backfolding at q but nesting at 2q in Ta$_2$NiSe$_7$

Charge Density Waves (CDW) are commonly associated with the presence of near-Fermi level states which are separated from others, or "nested", by a wavector of $\mathbf{q}$. Here we use Angle-Resolved Photo Emission Spectroscopy (ARPES) on the CDW material Ta$_2$NiSe$_7$ and identify a total absence of any plausible nesting of states at the primary CDW wavevector $\mathbf{q}$. Nevertheless we observe spectral intensity on replicas of the hole-like valence bands, shifted by a wavevector of $\mathbf{q}$, which appears with the CDW transition. In contrast, we find that there is a possible nesting at $\mathbf{2q}$, and associate the characters of these bands with the known atomic modulations at $\mathbf{2q}$. Our comprehensive electronic structure perspective shows that the CDW-like transition of Ta$_2$NiSe$_7$ is unique, with the primary wavevector $\mathbf{q}$ being unrelated to any low-energy states, but suggests that the modulation at $\mathbf{2q}$, which does plausibly connect low-energy states, might be more important for the overall energetics of the problem.