Unveiling the Interplay of Charge and Magnetic Excitations in HgBa$_2$Ca$_2$Cu$_3$O$_{8+δ}$
Unraveling the mechanism that binds electrons into Cooper pairs in cuprate high-temperature superconductors remains one of the most fundamental challenges in condensed-matter physics. While both magnetic interactions and lattice vibrations are known to govern key electronic properties, their possible cooperation has never been directly observed. We investigate HgBa$_2$Ca$_2$Cu$_3$O$_{8+δ}$ (Hg1223) - the cuprate with the highest $T_{\mathrm{c}}$ at ambient pressure - as a magnifying glass to probe the possible entwining of the excitations at the core of the pairing. Using resonant inelastic X-ray scattering, we find that the charge response is dominated by dynamic charge density fluctuations (CDF) extending up to several hundred meV, where magnetic excitations reside. At the same momentum where CDF are most intense, the paramagnon energy exhibits a pronounced softening, revealing a strong interplay among charge, lattice, and spin excitations. Our results point to a cooperative mechanism in which dynamic charge fluctuations mediate the coupling between lattice, charge and spin degrees of freedom-shedding new light on the fundamental origin of high-$T_{\mathrm{c}}$ superconductivity.