Glue function of optimally and overdoped cuprates from inversion of the Raman spectra

We address the issue of identifying the mediators of effective interactions in cuprates superconductors. Specifically, we use inversion theory to analyze Raman spectra of optimally and over-doped La$_{2-x}$Sr$_x$CuO$_4$ samples. This allows us to extract the so-called glue function without making any a priori assumption based on any specific model. We use instead two different techniques, namely the singular value decomposition and a multi-rectangle decomposition. With both techniques we find consistent results showing that: i) two distinct excitations are responsible for the glue function, which have completely different doping dependence. One excitation becomes weak above optimal doping, where on the contrary the other keeps (or even slightly increases) its strength; ii) there is a marked temperature dependence on the weight and spectral distribution of these excitation, which therefore must have a somewhat critical character. It is quite natural to identify and characterize these two distinct excitations as damped antiferromagnetic spin waves and damped charge density waves, respectively. This sets the stage for a scenario in which superconductivity is concomitant and competing with a charge ordering instability.