Universal properties of high-temperature superconductors from real-space pairing: Role of correlated hopping and intersite Coulomb interaction within the $t\text{−}J\text{−}U$ model

We study the effect of the correlated hopping term and the intersite Coulomb interaction term on principal features of the $d$-wave superconducting (SC) state, in both the electron- and hole-doped regimes within the $t\text{−}J\text{−}U$ model. In our analysis, we use the approach based on the diagrammatic expansion of the Gutzwiller wave function (DE-GWF), which allows us to go beyond the renormalized mean-field theory (RMFT). We show that the correlated hopping term enhances the pairing at the electron-doped side of the phase diagram. Moreover, the so-called non-BCS regime (which manifests itself by the negative kinetic energy gain at the transition to the SC phase) is narrowed down with the increasing magnitude of the correlated hopping $\sim K$. Also, the doping dependencies of the nodal Fermi velocity and Fermi momentum, as well as the average number of double occupancies, are analyzed with reference to the experimental data for selected values of the parameter $K$. For the sake of completeness, the influence of the intersite Coulomb repulsion on the obtained results is provided. Additionally, selected results concerning the Hubbard-model case are also presented. A complete model with all two-site interactions is briefly discussed in Appendix for reference.

Figure 1

(a) Correlated gap (${\mathrm{\Delta }}_G$) as a function of intrasite Coulomb repulsion $U$ and doping $\delta$, for $J=0.25$ and $V=0$. (b) Correlated gap as a function of correlated hopping integral $K$ and $\delta$, for $J=0.25$, $U=21$, and $V=0$. In both figures, the borders between the BCS-like and non-BCS regimes are marked by the dashed lines.

Figure 2

Doping dependencies of Fermi velocity in the nodal direction (a), nodal Fermi momentum (b), effective chemical potential (c), and double occupancy of electrons $d_e^2$ (at the hole-doping side) and holes $d_h^2$ (at the electron-doping side) (d), for selected values of the correlated-hopping integral $K$. In (a) we show also the measured values of $v_F$ which are taken from Refs. [30, 42, 43, 44, 45]. For the case of electron doping, the values of $v_F$ have been extracted by us from the measured dispersion relations close to $E_F$ presented in Refs. [44, 45]. In Ref. [45], the doping of the sample has not been explicitly provided so the corresponding $v_F$ value is marked by us by an arrow on the vertical scale. All the presented experimental values of $v_F$ correspond to the region below the energy kink reported at $50\text{–}70$ meV and above the low energy kink measured in Ref. [42]. In analogy to (a), we show in (b) also the Fermi momentum values at the electron doped side extracted from Refs. [44, 45]. In (c), we show the measured values of the chemical potential shift for LSCO compound taken from Ref. [33]. The calculations have been carried out for $J=0.25$, $U=21$, and $V=0$.

Figure 3

Doping dependencies of Fermi velocity in the nodal direction (a) and Fermi momentum (b) for $K=0$ and for selected values of the next-nearest-neighbor hopping.

Figure 4

(a) and (b) Doping dependence of the effective hopping parameters $t_{ij}^{\mathrm{eff}}$ between subsequent neighboring sites defined by the vectors $\mathrm{\Delta }{\mathbf{R}}_{ij}={\mathbf{R}}_i-{\mathbf{R}}_j=(X,Y)$. The vector components provided in the figure are given in the units of lattice constant $a$. (c) and (d) The corresponding effective gap magnitudes ${\mathrm{\Delta }}^{\mathrm{eff}}$. The results correspond to $J=0.25$, $U=21$, and $K=0$.

Figure 5

(a) Correlated gap as a function of intersite Coulomb repulsion $V$ and doping $\delta$, for $J=0.25$, $U=21$, and $K=0.35$. The borders between the BCS-like and non-BCS regimes are marked by the dashed lines. (b) Correlated gap as a function of doping for two selected sets of model parameters. The vertical arrows show the dopings for which the crossover between the BCS-like and non-BCS states appear for the corresponding model parameters.

Figure 6

(a) Correlated gap as a function of both $U$ and $\delta$ for the Hubbard model. The border between the BCS-like and non-BCS regimes is marked by the dashed line. In (b) and (c), we show the kinetic energy gain at the transition to the superconducting phase as a function of doping for $U=21$ and 11, respectively. Note the sign change of $\mathrm{\Delta }{E}_{\mathrm{kin}}$ for $U=11$ when we set nonzero values of $K$ parameter. In (d) and (e), we exhibit also the correlated gap as a function of doping for a set of $K$ values. The discontinuity of the gap signals the appearance of phase separation.

Figure 7

Number of electrons per atomic site, per spin vs chemical potential for the case of the Hubbard model with $U=21$ and $K=0.15$. Two superconducting solutions can be found what indicates the appearance of a metastable state and phase separation into regions with different electron densities. For ${\mu }_G\approx 2.06$, the thermodynamic potential values $F=E-{\mu }_{G}N$, corresponding to the two solutions, are equal.

Figure 8

(a) and (b) Doping dependence of the effective hopping parameters $t_{ij}^{\mathrm{eff}}$ between subsequent neighboring sites defined by the vectors $\mathrm{\Delta }{\mathbf{R}}_{ij}={\mathbf{R}}_i-{\mathbf{R}}_j=(X,Y)$. The vector components provided in the figure are given in the units of lattice constant $a$. (c) and (d) The same for the effective superconducting gap parameters ${\mathrm{\Delta }}_{ij}^{\mathrm{eff}}$. The results correspond to the Hubbard model for $U=21$, and $K=0.15$. Note the discontinuity appearing for $\delta \approx 0.147$, which is also seen in the doping dependence of the correlated gap (Fig. 6) and is a manifestation of phase separation. Note that for $\delta \gtrsim 0.147$ the effective gap in (c) is still nonzero, however, very small.

Figure 9

Double occupancy as a function of doping for selected interaction parameters $U$ and $K$ in the case of the Hubbard model ($J=0$, $V=0$).