Superconductivity with and without glue and the role of the double-occupancy forbidding constraint in the $t\text{−}J\text{−}V$ model

The occurrence of retarded (with glue) and unretarded (without glue) pairing is thoroughly discussed in cuprates. We analyze some aspects of this problem in the context of the $t\text{−}J\text{−}V$ model in a large-$N$ approximation. When $1/N$ renormalizations are neglected the mean-field result is recovered, where the unretarded $d$-wave superconducting pairing triggered by the spin-exchange interaction $J$ is obtained. However, the presence of a nonnegligible nearest-neighbors Coulomb interaction $V\left(\mathbf{q}\right)$ kills superconductivity. If the non-double-occupancy constraint and its fluctuations are considered, the situation changes drastically. In this case, $V\left(\mathbf{q}\right)$ is screened making $d$-wave superconductivity very robust. In addition, we show that the early proposal for the presence of an unretarded pairing contribution triggered by the spin-exchange interaction $J$ can be discussed in this context.

Figure 1

(a) Summary of the Feynman rules. Solid line represents the fermionic propagator $G_{p{p}^{\prime }}^{\left(0\right)}$. Dashed line represents the $6\times 6$ bosonic propagator $D_{ab}^{\left(0\right)}$. ${\mathrm{\Lambda }}_a^{p{p}^{\prime }}$ and ${\mathrm{\Lambda }}_{ab}^{p{p}^{\prime }}$ represent the interaction between two fermions and one and two bosons, respectively. (b) Diagrammatic representation of the Dyson equation. (c) The two different contributions to the irreducible boson self-energy. (d) Effective interaction between fermions. Looking at the order of the propagators and vertices we see that $V_{\mathrm{eff}}$ is $O(1/N)$, thus superconductivity arises at $O(1/N)$ in this large-$N$ scheme.

Figure 2

The superconducting coupling $\lambda$ and ${\lambda }^{\left(0\right)}$ versus doping $\delta$ for $V=0$ and $V=0.3$. Inset: The superconducting coupling $\lambda$ and ${\lambda }^{\left(0\right)}$ versus $V$ for $\delta =0.25$. For this $\delta , V_c\sim 1.9$.

Figure 3

(a) ${\lambda }^{\mathrm{Ch}-\mathrm{FP}}$ versus doping for $V=0$ and $V=0.3$. $\lambda$ from Fig. 2 is included for comparison. (b) ${\lambda }^{\mathrm{Ch}}$ for $V=0$ and $V=0.3$, and ${\lambda }^{\mathrm{J}}$ versus doping. (c) ${\lambda }^{\mathrm{FP}}$ and ${\lambda }^{\mathrm{J}}$ versus $\delta$. Inset: ${\lambda }^{\mathrm{FP}}$ and ${\lambda }^{\mathrm{J}}$ versus $\delta$ for $\mathrm{\Gamma }=0.05$.