Effective pairing interaction in a system with an incipient band

The nature and mechanism of superconductivity in the extremely electron-doped FeSe-based superconductors remains an outstanding problem. In these systems, the hole-like band has moved below the Fermi energy, and various spin-fluctuation theories involving pairing between states near the electron Fermi surface and states of this incipient band have been proposed. Here, using a nonperturbative dynamic cluster quantum Monte Carlo calculation for a bilayer Hubbard model, we show how spin-fluctuation scattering involving intermediate virtual states of the incipient band leads to an effective pairing interaction for the electrons on the remaining Fermi surface. The key feature of this interaction is that it is retarded due to the incipient nature of the hole band and the dynamics of the spin-fluctuation interaction. This allows the pairs to avoid the instantaneous repulsive Coulomb interaction and leads to a frequency dependence of the gap which should be observable in tunneling experiments. Our work provides a different perspective for the pairing mechanism in systems with an incipient band that can be tested in future experiments.

Figure 1

(a) Single-particle spectral weight $A(\mathit{k},\omega )$ at $T=0.2$ for $t_{\perp }=2.5, n=1.15$, and $U=8$ shows an electron pocket around the $(k_x=\pi ,k_y=\pi ) M$ point for $k_z=0$ and an incipient hole band lying below the Fermi energy and centered at the $(k_x=0,k_y=0) \mathrm{\Gamma }$ point for $k_z=\pi$. (b) Momentum occupation $\langle n\left(\mathit{k}\right)\rangle$ plotted for $\mathit{k}$ running from $\mathrm{\Gamma }$ to $M$ sharpens at the electron FS as the temperature is decreased and is suppressed at $\mathrm{\Gamma }$ as the $k_z=\pi$ band drops below the Fermi energy.

Figure 2

Intrinsic pair-field susceptibilities $P_0^0\left(T\right)$ and $P_{\pi }^0\left(T\right)$ vs $T$. $P_0^0\left(T\right)$ exhibits a $log(t/T)$ Cooper instability associated with the $k_z=0$ Fermi surface, while $P_{\pi }^0\left(T\right)$ for the incipient band remains flat as $T$ decreases.

Figure 3

Schematic illustration of the effective intermediate state contribution to the ${\mathrm{\Gamma }}_2$ pairing vertex. The Feynman diagram for ${\mathrm{\Gamma }}_2$ shown in the lower part of the figure involves the intermediate states illustrated above it. The thin interaction line denotes the ${\mathrm{\Gamma }}_{0\pi }^0$ fully irreducible vertex and the thicker interaction line the ${\mathrm{\Gamma }}_{\pi 0}$ vertex, which is only irreducible in the $k_z=0$ two-particle channel.

Figure 4

Two-particle scattering vertices for $T=0.125t$, normalized to $U$, vs the Matsubara energies ${\omega }_n$ and ${\omega }_{n^{\prime }}$. (a) ${\tilde{\mathrm{\Gamma }}}_1({\omega }_n,{\omega }_{n^{\prime }})$, (b) ${\tilde{\mathrm{\Gamma }}}_2({\omega }_n,{\omega }_{n^{\prime }})$, and (c) their sum $\tilde{\mathrm{\Gamma }}({\omega }_n,{\omega }_{n^{\prime }})$.

Figure 5

(a) Leading eigenvalue ${\lambda }_0\left(T\right)$ of the Bethe-Salpeter equation as $T$ decreases. (b) Momentum dependence of the gap function ${\varphi }_0(\mathit{k},{\omega }_n)$ of the leading eigenvalue for ${\omega }_0=\pi T$ for $\mathit{k}$ values near the FS as shown in the inset and (c) its frequency dependence for $\mathit{k}=(\pi /2,\pi /2)$ for $T=0.125t$. ${\varphi }_0(\mathit{k},{\omega }_n)$ changes sign leading to a reduction of the effect of the repulsive ${\mathrm{\Gamma }}_1$ interaction.