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In this review, we single out selected universal features of high-$T_c$ andrelated systems, which can be compared with experiment. We start with theconcept of real-space pairing, combined with strong correlations. Thediscussion of concrete properties relies on variational approach, based onrenormalized mean-field theory (RMFT) in the form of statistically-consistentGutzwiller approximation (SGA), and Diagrammatic Expansion of the VariationalWave Function (DE-GWF). Two energy scales appear, one involving quasiparticlesclose to the Fermi energy, and the other reflecting the correlated state. Thosetwo regimes are separated by a kink in the dispersion relation, observed inphotoemission. One obtains both the doping dependent properties andrenormalized quasiparticles. The reviewed ground-state characteristics forhigh-$T_c$ systems encompass superconductivity, nematicity, charge- (and pair-)density-wave effects, as well as non-BCS kinetic energy gain in the pairedstate, all in quantitative manner. Calculated dynamic properties are: universalFermi velocity, Fermi wave-vector, effective mass enhancement, pseudogap, and$d$-wave gap magnitude. The minimal realistic model is represented by the$t$-$J$-$U$ Hamiltonian. Inadequacy of the $t$-$J$ and Hubbard models isdiscussed. For heavy fermion systems we summarize superconducting, Kondoinsulating, ferro- and anti-ferromagnetic states. We overview also coexistentferromagnetic (spin-triplet) superconducting phases observed for$\mathrm{UGe_2}$. Finally, we extend our scheme to collective spin and chargefluctuations in high-$T_c$ systems, starting from variational approach,combined with $1/N$ expansion (beyond random phase approximation). Spectrum ofcollective spin and charge excitations is determined for the Hubbard and$t$-$J$-$U$ models, and compared quantitatively with recent experiments.

Superconductivity in high-$T_c$ and related strongly correlated systems from variational perspective: Beyond mean field theory