We provide a unified interpretation of both paramagnon and plasmon modes in high-$T_c$ copper-oxides, and verify it quantitatively against available resonant inelastic $x$-ray scattering (RIXS) data across the hole-doped phase diagram. The three-dimensional extended Hubbard model, with included long-range Coulomb interactions and doping-independent microscopic parameters for both classes of quantum fluctuations, is used. Collective modes are studied using $\mathrm{VWF}+1/{\mathcal{N}}_f$ approach which extends variational wave function (VWF) scheme by means of an expansion in inverse number of fermionic flavors $(1/{\mathcal{N}}_f)$. We show that intense paramagnons persist along the antinodal line from the underdoped to the overdoped regime and undergo rapid overdamping in the nodal direction. Plasmons exhibit a three-dimensional character, with minimal energy corresponding to antiphase oscillations on neighboring ${\mathrm{CuO}}_2$ planes. The theoretical spin- and charge excitation energies reproduce semiquantitatively RIXS data for ${(\mathrm{Bi},\mathrm{Pb})}_2{(\mathrm{Sr},\mathrm{La})}_2{\mathrm{CuO}}_{6+\delta }$. The present $\mathrm{VWF}+1/{\mathcal{N}}_f$ analysis of dynamics and former VWF results for static quantities combine into a consistent description of the principal properties of hole-doped high-$T_c$ cuprates as strongly correlated systems.

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• Electronic structure
• Impurities in superconductors
• Plasmons
• Pseudogap
• Spin fluctuations
• Superconducting gap
• Superconducting phase transition
• Cuprates
• High-temperature superconductors
• Strongly correlated systems
• Density functional calculations
• Methods in superconductivity
• X-ray scattering