Optical and acoustic plasmons in the layered material Sr$_2$RuO$_4$

We use momentum-dependent electron energy-loss spectroscopy in transmission to study collective charge excitations in the layer metal Sr$_2$RuO$_4$. This metal has a transition from a perfect Fermi liquid below $T\approx30\,$K into a "strange" metal phase above $T\approx800\,$K. We cover a complete range between in-phase and out-of-phase oscillations. Outside the classical range of electron-hole excitations, leading to a Landau damping, we observe well-defined plasmons. The optical (acoustic) plasmon due to an in-phase (out-of-phase) charge oscillation of neighbouring layers exhibits a quadratic (linear) positive dispersion. Using a model for the Coulomb interaction of the charges in a layered system, it is possible to describe the range of optical plasmon excitations at high energies in a mean-field random phase approximation without taking correlation effects into account. In contrast, resonant inelastic X-ray scattering data show at low energies an enhancement of the acoustic plasmon velocity due to correlation effects. This difference can be explained by an energy dependent effective mass which changes from $\approx$ 3.5 at low energy to 1 at high energy near the optical plasmon energy. There are no signs of over-damped plasmons predicted by holographic theories.