Giant phonon anomaly associated with strong superconducting fluctuations in the pseudogap phase of underdoped cuprates

Phonons with wavevectors connecting Fermi-arc ends are strongly damped in the pseudogap phase as $T \rightarrow T_c$ from above. Below $T_c$ the anomaly abruptly switches to a dip in the phonon dispersions at the same wavevectors. Based on our proposed 2-subband model of superconductivity, we analyze the coupling between these phonons and the Leggett mode. We consider the phonon self energy arising from a forward scattering of the phonon accompanied by Cooper-pair transfers between the subbands. The intermediate state of this process involves 2 Leggett modes and a forward scattered phonon. Above $T_c$, the Leggett mode is overdamped giving rise to the giant phonon damping, while below $T_c$, the Leggett mode develops a finite energy turning the phonon anomaly into a dip in the energy dispersion.