Antiferromagnetism, charge density wave, and d-wave superconductivity in   the extended $t$--$J$--$U$ model: role of intersite Coulomb interaction and a   critical overview of renormalized mean field theory
  In the first part of the paper, we study the stability of antiferromagnetic (AF), charge density wave (CDW), and superconducting (SC) states within the $t$-$J$-$U$-$V$ model of strongly correlated electrons by using the statistically consistent Gutzwiller approximation (SGA). We concentrate on the role of the intersite Coulomb interaction term $V$ in stabilizing the CDW phase. In particular, we show that the charge ordering appears only above a critical value of $V$ in a limited hole-doping range $\delta$. The effect of the $V$ term on SC and AF phases is that a strong interaction suppresses SC, whereas the AF order is not significantly influenced by its presence. In the second part, separate calculations for the case of pure SC phase have been carried out within an extended approach (the diagrammatic expansion for the Gutzwiller wave function, DE-GWF) in order to analyze the influence of the intersite Coulomb repulsion on the SC phase with the higher-order corrections included beyond the SGA method. In the Appendices we discuss the ambiguity connected with the choice of the Gutzwiller renormalization factors within the renormalized mean filed theory when either AF or CDW orders are considered. At the end we overview briefly the possible extensions of the current models to make description of the SC, AF, and CDW states on equal footing. 
Marcin Abram, Micha{\l} Zegrodnik, Jozef Spa{\l}ek
cond-mat.str-el