Fermi surface geometry and momentum dependent electron-phonon coupling drive the charge density wave in quasi-1D ZrTe$3$

ZrTe$_3$ is a prototypical quasi-one-dimensional compound undergoing a charge density wave transition via a very sharp Kohn anomaly in phonon momentum space. While Fermi surface geometry has long been considered the primary driver of the instability, a full understanding of the lattice dynamics and electron-phonon role has remained elusive. Our first principles calculations in the high-symmetry phase show that the Fermi surface is correctly reproduced only when the Hubbard interaction on the Te $5p$ orbitals is included, which in turn is essential for the appearance of a soft harmonic phonon mode at the CDW wavevector. Analyzing the mode and momentum dependence of the electron-phonon coupling, we find that its variations with phonon momentum dominate over electronic effects. These results identify unambiguously the CDW origin in ZrTe$_3$ as a cooperative effect of Fermi surface geometry and momentum-dependent electron-phonon coupling, with the latter playing the leading role. We further determine the atomic structure in the low-symmetry CDW phase, revealing a nonchiral modulation. The mechanisms revealed in our work are directly relevant to other quasi-1D systems, including trichalcogenides and compounds hosting Peierls-like chains.