Hardening of Ni-O bond-stretching phonons in LaNiO$_2$

We demonstrate that dynamical electron correlation and fluctuating local magnetic moments are crucial for the phonon spectra of the infinite-layer nickelate superconductor, LaNiO$_2$, using DFT plus dynamical mean-field theory (DFT+DMFT) calculations. We find significant hardening of optical Ni-O bond-stretching phonons when going from non-magnetic to paramagnetic state, and increasing Coulomb interaction will make them even harder. The electron correlation is found to be sensitive to the Ni-O bond-stretching distortions, indicating strong interplay between electron correlation and lattice. We find that the strong local electron correlation will not favor charge orders that couple to the Ni-O bond-stretching phonons, in support of the recent experiment that a $3a_0$ charge order is absent in the infinite-layer nickelates. Our results emphasize that the effects of local magnetic fluctuations should be fully taken into account when describing the lattice dynamics of the infinite-layer nickelates without long-range magnetic orders, and also provide evidence to ascribe the kink observed in the recent angle-resolved photoemission experiment to possible strong electron coupling to the Ni-O bond-stretching phonons.