Pairing interaction near a nematic quantum critical point of a three-band ${\mathrm{CuO}}_2$ model

Here we calculate the pairing interaction and the $k$ dependence of the gap function associated with the nematic charge fluctuations of a ${\mathrm{CuO}}_2$ model. We find that the nematic pairing interaction is attractive for small momentum transfer and that it gives rise to $d$-wave pairing. As the doping $p$ approaches a quantum critical point, the strength of this pairing increases and higher $d$-wave harmonics contribute to the $k$ dependence of the superconducting gap function, reflecting the longer range nature of the nematic fluctuations.

Figure 1

(a) The Fermi surface for $t_{pd}=1.0,t_{pp}=0.5$ at a hole doping $p=0.20.$ The inset shows the hopping and Coulomb interactions parameters. (b) The orbital weights $|a_{\nu }^l\left(k\right)|$ on the Fermi surface with the angle $\Theta$ measured from $k=(\pi ,0)$.

Figure 2

The RPA nematic charge susceptibility ${\chi }_N\left(q\right)$ vs $q$ for $p=0.205$. (b) ${\chi }_N(q=0)$ vs $p$. The inset illustrates the nematic mode charge fluctuation on the oxygens.

Figure 3

The $d$-wave pairing strength ${\lambda }_d$ associated with the exchange of charge fluctuations vs the hole doping $p$. The inset shows a diagrammatic representation of the contribution of the charge vertex ${\Gamma }_c$ to the gap equation.

Figure 4

The gap function ${\varphi }_d\left(k\right)$ (solid) vs $\theta$ for $k$ on the Fermi surface normalized to its value at $\theta =0$. The dashed curve is $cos{k}_x-cos{k}_y$. (a) $p=0.205$ and (b) $p=0.22$.