Collective excitations of dynamic Fermi surface deformations in BaFe$_2$(As$_{0.5}$P$_{0.5}$)$_2$

Recent transport studies of the overdoped BaFe$_2$(As$_{0.5}$P$_{0.5}$)$_2$ superconductor reported a crossover from non-Fermi liquid(NFL) to Fermi liquid(FL) behavior at around 100 K\footnote{J. Analytis et al. Nat.Phys.10,194(2014)}. Here we elucidate the spectroscopic signatures in this crossover regimes using polarization resolved Raman spectroscopy. In the FL regime below 100 K, we detect long-lived collective quadrupole symmetry excitations at about 32 meV in the B$_{1g}$ and B$_{2g}$ channels. We assign them to Pomeranchuk oscillations related to dynamical Fermi surface (FS) deformations. The Pomeranchuk oscillations show a similarity for the B$_{1g}$ and B$_{2g}$ quadrupole channels, which we explain by the large As/P disorder. We explain that the small Fermi energy is an essential condition for Pomeranchuk oscillations to be underdamped at low temperature\footnote{S.Wu et al. arXiv:1607.06575}.