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A nonzero-temperature technique for strongly correlated electron latticesystems, combining elements of both variational wave function (VWF) approachand expansion in the inverse number of fermionic flavors ($1/\mathcal{N}_f$),is developed. The departure point, VWF method, goes beyond the renormalizedmean-field theory and provides semi-quantitative description of principalequilibrium properties of high-$T_c$ superconducting cuprates. The developedhere scheme of VWF+$1/\mathcal{N}_f$, in the leading order provides dynamicalspin and charge responses around the VWF solution, generalizing theweak-coupling spin-fluctuation theory to the regime of strong correlations.Thermodynamic corrections to the correlated saddle-point state arisesystematically at consecutive orders. Explicitly, VWF+$1/\mathcal{N}_f$ isapplied to evaluate dynamical response functions for the hole-doped Hubbardmodel and compared with available determinant quantum-Monte-Carlo data,yielding a good overall agreement in the regime of coherent collective-modedynamics. The emergence of well-defined spin and charge excitations from theincoherent continua is explicitly demonstrated and a non-monotonic dependenceof the charge-excitation energy on the interaction magnitude is found. Thecharge-mode energy saturates slowly when approaching the strong-coupling limit,which calls for a reevaluation of the $t$-$J$-model approach to the chargedynamics in favor of more general $t$-$J$-$U$ and $t$-$J$-$U$-$V$ models. Theresults are also related to recent inelastic resonant $X$-ray and neutronscattering experiments for the high-$T_c$ cuprates.

Universal collective modes from strong electronic correlations: Modified $1/\mathcal{N}_f$ theory with application to high-$T_c$ cuprates