Modulated, three-directional, and polar structural instability in layered $d^1$ NaTiO$_2$

I study the experimentally observed metal-to-metal structural phase transition in NaTiO$_2$ using density functional calculations. I do not find the previously proposed low-temperature structure energetically favorable with respect to the high-temperature rhombohedral structure. The calculated phonon dispersions of the rhombohedral phase show dynamical instabilities at several inequivalent parts of the Brillouin zone, including at the wavevector $(\frac{1}{2},\frac{1}{5},\frac{1}{5})$. These instabilities lead to monoclinic structures without inversion symmetry that are modulated along all three directions. The calculated electronic structures show that a local bonding instability of the Ti $3d$ states is associated with the structural transition.