Incorporation of charge- and pair-density-wave states into the one-band model of d-wave superconductivity

We study the coexistence of pair- (PDW) and charge-density-wave (CDW) states within the single-band $t$-$J$-$U$ and Hubbard models of $d$-$wave$ superconductivity and discuss our results in the context of the experimental observations for the copper-based compounds. In order to take into account the correlation effects with a proper precision, we use the approach based on the diagrammatic expansion of the Gutzwiller wave function (DE-GWF), that goes beyond the renormalized mean field theory (RMFT) in a systematic manner. According to our analysis of the $t$-$J$-$U$ model, the transition between the pure $d$-$wave$ superconducting phase (SC) and the coexistent CDW+PDW phase takes place at $\delta\approx 0.18$ (close to the optimal doping), with the modulated phase located in the underdoped regime. The situation is slightly different for the case of the Hubbard model, where a narrow stability regime of a precursor nematic phase sets in preceding the formation of the modulated CDW+PDW state, with the decreasing hole doping. The results complete our discussion of the standard phase diagram for high-T$_C$ superconducting compounds within the DE-GWF variational approach in the single narrow-band case.