Dynamics of the Pairing Interaction in the Hubbard and $t\mathrm{\text{−}}J$ Models of High-Temperature Superconductors

The question of whether one should speak of a “pairing glue” in the Hubbard and $t\mathrm{\text{−}}J$ models is basically a question about the dynamics of the pairing interaction. If the dynamics of the pairing interaction arises from virtual states, whose energies correspond to the Mott gap, and give rise to the exchange coupling $J$, the interaction is instantaneous on the relative time scales of interest. In this case, while one might speak of an “instantaneous glue,” this interaction differs from the traditional picture of a retarded pairing interaction. However, as we will show, the dominant contribution to the pairing interaction for both of these models arises from energies reflecting the spectrum seen in the dynamic spin susceptibility. In this case, the basic interaction is retarded, and one speaks of a spin-fluctuation glue which mediates the $d$-wave pairing.

Figure 1

(a) The imaginary part of the Pb gap function ${\varphi }_2(\omega )$ versus $\omega$ (solid curve). The peaks in ${\varphi }_2(\omega )$ occur at the transverse ${\omega }_T$ and longitudinal ${\omega }_L$ peaks of ${\alpha }^{2}F(\omega )$ (dashed curve) shifted up by the gap ${\Delta }_0$. (b) The pairing interaction spectral weight $I(\Omega )$ versus $\Omega$ for Pb. $I(\Omega )$ increases as $\Omega$ passes through ${\omega }_T+{\Delta }_0$ and ${\omega }_L+{\Delta }_0$ reflecting the transverse and longitudinal phonon contributions to the pairing. At larger values of $\Omega$, $I(\Omega )$ exceeds unity because ${\varphi }_1(0)$ is reduced from the value that it would have just due to the phonons by the presence of the nonretarded screened Coulomb pseudopotential ${\mu }^{*}$.

Figure 2

(a) The real (solid curve) and imaginary (dashed curve) parts of ${\varphi }_2(k,\omega )$ versus $\omega /t$ obtained for a 32-site cluster. Here $k=(0,\pi )$, $J/t=0.3$, and the doping $x\simeq 3\%$. (b) $I(k,\Omega )$ versus $\Omega /t$ for $J/t=0.3$ (solid curve) and $J/t=0.5$ (dashed curve) for $k=(0,\pi )$ and $x\simeq 3\%$. Here a broadening $\eta =0.05$ was used for (a) and $\eta =0.005$ for (b).

Figure 3

(a) $I(k_A,\Omega )$ versus $\Omega /t$ for the $2\times 2$ DCA-NCA Hubbard calculation for $T/T_c\simeq 0.95$, $⟨n⟩=0.8$, and $U/t=8$, 10, and 12. Here $k_A=(0,\pi )$. (b) The $d$-wave projected ${\chi }_d^{\prime \prime }(\Omega )$ versus $\Omega /t$ for the same parameters.