Giant Kohn anomaly and chiral phonons in the charge density wave phase of 1H-NbSe${}_2$

Despite extensive investigations, many aspects of charge density waves (CDWs) remain elusive, especially the relative roles of electron-phonon coupling and Fermi surface nesting as the underlying driving mechanisms responsible for the emergence of the CDW vector $Q_{CDW}$. It is puzzling that even though electrons interact strongly with optical phonons in many correlated systems, the actual mode softening is of an acoustic mode. Here we consider monolayer 1H-NbSe${}_2$ as an exemplar system, and through an accurate computation of the phonon self-energy, including its off-diagonal components, we provide compelling evidence that the relevant mode is a longitudinal optical phonon that softens by anti-crossing several intervening phonon bands, i.e. a Kohn ladder which has been only observed previously in high temperature superconductors. We also show that $Q_{CDW}$ is fixed by the convolution of the susceptibility and electron-phonon coupling, and that the softened phonons are circularly polarized.