Spin-triplet $s$-wave local pairing induced by Hund’s rule coupling

We show within the dynamical mean field theory that local multiplet interactions such as Hund’s rule coupling produce local pairing superconductivity in the strongly correlated regime. Spin-triplet superconductivity driven by the Hund’s rule coupling emerges from the pairing mediated by local fluctuations in pair exchange. In contrast to the conventional spin-triplet theories, the local orbital degrees of freedom has the antisymmetric part of the exchange symmetry, leaving the spatial part as fully gapped and symmetric $s$-wave.

Figure 1

Pair correlation energy $E_p$ for two-site model Eq. (5) with the Hund’s rule coupling. $E_p$ is attractive for a wide range of hopping integral to the quasi-particle bath, $t_q$. Coupling to quasi-particle bath (thick line) produced a good agreement with the perturbation theory, $-30t_q^4∕K{U}^{*2}\left(U^{*}=U+2K\right)$. The optimal interaction is at $4t_q^2∕U\sim K$ for the maximum pair attraction $E_p\propto -K$.

Figure 2

Single site energy level scheme for two on-site electrons $\left(N_d=2\right)$ with Hund’s rule and Jahn-Teller couplings. The degeneracy of 6 is lifted by the multiplet interaction $H_{\mathrm{mult}}$. The ground states are spin-triplet $\left(S=1\right)$ and spin-singlet $\left(S=0\right)$ for Hund’s rule. The Jahn-Teller multiplets have inverted level structure of the Hund’s rule coupling.

Figure 3

One of the contributions in $E(N_d+2)$ for the pair correlation energy $E_p$ in Eq. (20). Two extra electrons created on the interacting site dissolve into the quasiparticle bath to avoid Coulomb repulsion, which produces the ground state configurations shown on the left and right panel. The two unperturbed degenerate ground states couple via exchange of the spin-triplet electron pairs in the fourth order perturbation of $t_q$ (from the left configuration to the right). In the intermediate state, the spins on the interacting site are in a locally excited state (see Fig. 2).

Figure 4

Expansion of spin-triplet pairing susceptibility $\chi$. The local index $m\sigma \left(m=1,2\right)$ is for orbital and spin labels.

Figure 5

The effective interaction $V_{\mathrm{eff}}({\omega }_n,{\omega }_m)\left[\equiv -\Gamma ({\omega }_n,{\omega }_m)∕T\right]$. The attractive interaction is for small energy transfer $\left({\omega }_n\approx {\omega }_m\right)$. The interaction at higher frequencies is repulsive.

Figure 6

(a) Superconducting transition temperature $T_c$ vs Coulomb repulsion $U$ with Hund’s rule coupling. For the Hund’s rule (HR) coupled system, $T_c$ monotonically increases with $U$ showing local pairing mechanism. Gray symbol is an extrapolated estimate from the superconducting kernel ${\chi }_0\Gamma$. (b) Quasiparticle renormalization factor $z$ (left axis) and effective pairing interaction $V_{\mathrm{eff}}$ at lowest Matsubara frequency (right axis). $z$ decreases as $U$ approaches $U_c$. The product $z{V}_{\mathrm{eff}}$ for the HR system increased as $U\to U_c$, consistent with the increasing $T_c$. $W$ is the noninteracting bandwidth.

Figure 7

(a) Superconducting transition temperature $T_c$ vs Coulomb repulsion $U$ with Jahn-Teller electron-phonon coupling. The Jahn-Teller (JT) phonon coupling produces superconductivity with a crossover from the conventional superconductivity (small $U$) to the local pairing regime (large $U$) for $2\Delta ∕W=0.1$. For a larger electron-phonon coupling $\left(2\Delta ∕W=0.15\right)$, the Mott insulator transition happened at smaller $U$. (b) Renormalization factor $z$ as in Fig. 6.