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

In this review, we single out selected universal features of high-$T_c$ and related systems, which can be compared with experiment. We start with the concept of real-space pairing, combined with strong correlations. The discussion of concrete properties relies on variational approach, based on renormalized mean-field theory (RMFT) in the form of statistically-consistent Gutzwiller approximation (SGA), and Diagrammatic Expansion of the Variational Wave Function (DE-GWF). Two energy scales appear, one involving quasiparticles close to the Fermi energy, and the other reflecting the correlated state. Those two regimes are separated by a kink in the dispersion relation, observed in photoemission. One obtains both the doping dependent properties and renormalized quasiparticles. The reviewed ground-state characteristics for high-$T_c$ systems encompass superconductivity, nematicity, charge- (and pair-) density-wave effects, as well as non-BCS kinetic energy gain in the paired state, all in quantitative manner. Calculated dynamic properties are: universal Fermi 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 is discussed. For heavy fermion systems we summarize superconducting, Kondo insulating, ferro- and anti-ferromagnetic states. We overview also coexistent ferromagnetic (spin-triplet) superconducting phases observed for $\mathrm{UGe_2}$. Finally, we extend our scheme to collective spin and charge fluctuations in high-$T_c$ systems, starting from variational approach, combined with $1/N$ expansion (beyond random phase approximation). Spectrum of collective spin and charge excitations is determined for the Hubbard and $t$-$J$-$U$ models, and compared quantitatively with recent experiments.