Flat phonon soft modes and unconventional charge density wave formation in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>ScV</mml:mi><mml:mn>6</mml:mn></mml:msub><mml:msub><mml:mi>Sn</mml:mi><mml:mn>6</mml:mn></mml:msub></mml:mrow></mml:math>: Microscopic and effective theory

Kagome materials with flat bands exhibit wildly different physical properties depending on symmetry group and electron number. We analyze the case of the kagome 166 material ${\mathrm{ScV}}_{6}{\mathrm{Sn}}_{6}$ in symmetry group 191, with the Fermi level away from the flat bands. Experimentally, an $\ensuremath{\sim}95$ K charge density wave (CDW) at vector $\overline{K}=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ exists, with no nesting or peaks in the electron susceptibility at $\overline{K}$. We show that ${\mathrm{ScV}}_{6}{\mathrm{Sn}}_{6}$ has a collapsed phonon mode at $H=(\frac{1}{3},\frac{1}{3},\frac{1}{2})$ and an imaginary flat phonon band in the $H$ vicinity. The soft phonon is supported on the triangular Sn (${\mathrm{Sn}}^{T}$) $z$-directed mirror-even vibration, which is faithfully described by a three-degree-of-freedom simple force constant model. We show that the phonon softening comes from the electron-phonon coupling between mirror-even electron orbitals at ${\mathrm{Sn}}^{T}$ and the mirror-even vibration of ${\mathrm{Sn}}^{T}$ atoms. We model it using a (Gaussian) approximation of the hopping parameter Yu et al., Nat. Phys. 20, 1262 (2024) and show that the resulting field-theoretical renormalization of the phonon frequency reproduces the collapse of the $H$ phonon and induces small in-plane dispersion away from $H$. To explain the appearance of the $\overline{K}$ CDW we build an effective model of two order parameters (OPs): one at the collapsed phonon $H$ and one at the CDW $\overline{K}$. Comparing our results with experimental data [Korshunov et al., Nat. Commun. 14, 6646 (2023)], we show that the $H$ OP undergoes a second-order phase transition; however, its flatness around $H$ induces large fluctuations and stabilizes a $\overline{K}$ CDW via a first-order transition. We construct CDW OPs in the electron and phonon fields that match the ab initio calculations. Our results not only explain the CDW in ${\mathrm{ScV}}_{6}{\mathrm{Sn}}_{6}$ but also show an unprecedented level of modeling of complex electronic systems that open different collaboration between ab initio calculations and analytics.