Impurities coupled to superconductors offer a controlled platform to understand the interplay between superconductivity, many-body interactions, and nonequilibrium physics. In the equilibrium situation, local interactions at the impurity induce a transition from the spin-singlet to the spin-doublet ground state, resulting in a supercurrent sign reversal ($0\text{−}\pi$ transition). In this work, we apply the exact time-dependent density matrix renormalization group method to simulate the transient dynamics of such superconducting systems. We also use a perturbative approximation to analyze their properties at longer times. These two methods agree for a wide range of parameters. In a phase-biased situation, the system gets trapped in a metastable state characterized by a lower supercurrent compared to the equilibrium case. We show that local Coulomb interactions do not provide an effective relaxation mechanism for the initially trapped quasiparticles. In contrast, other relaxation mechanisms, such as coupling to a third normal lead, make the impurity spin relax for parameter values corresponding to the equilibrium 0 phase. For parameters corresponding to the equilibrium $\pi$ phase the impurity converges to a spin-polarized stationary state. Similar qualitative behavior is found for a voltage-biased junction, which provides an effective relaxation mechanism for the trapped quasiparticles in the junction.

We report an experimental and theoretical study of the electron-phonon coupling for $\alpha \text{−}\mathrm{Sn}/\mathrm{Ge}\left(111\right)$, a prototypical triangular lattice surface, closely related to $\mathrm{Sn}/\mathrm{Si}\left(111\right)\text{−}(\sqrt{3}\times \sqrt{3})$, where recent experimental evidence has found superconductivity [X. Wu et al., Phys. Rev. Lett. 125, 117001 (2020)]. We concentrate our study on the ($3\times 3$) phase of $\alpha \text{−}\mathrm{Sn}/\mathrm{Ge}\left(111\right)$ that appears between 150 and 120 K and has a well-known geometry with a half-filled electronic band around the Fermi energy. We show that this surface presents a giant electron-phonon interaction that can be considered at least partially responsible for the different phases that this system shows at very low temperature. Our theoretical results indicate that indeed the electron-phonon interaction in $\alpha \text{−}\mathrm{Sn}/\mathrm{Ge}\left(111\right)\text{−}(3\times 3)$ is unusually large, since we find that $\lambda$, the electron mass enhancement for the half-filled band, is $\lambda =1.3$. This result is in good agreement with the experimental value obtained from high-resolution angle-resolved photoemission spectroscopy measurements, which yield $\lambda =1.45\pm 0.1$.

In order to elucidate the mechanism creating superconductivity in the 2-dimensional layer of a p-doped Sn/Si(111) surface, we have analyzed the many-body effects associated with the electron-phonon (e-ph) coupling and the electron–electron interaction. First, we have calculated the DFT surface band of the system and the coupling associated with the different interactions. In our calculations we find a mean field (DFT) electron bandwidth of 0.54 eV, an attractive coupling $U^{neg}=0.32$ eV associated with the e-ph coupling and an effective electron–electron Hubbard repulsion of $U=0.83$ eV. Then, we analyze the Hubbard Hamiltonian, neglecting in this step the e-ph coupling that is much smaller than the Hubbard coupling, by considering a p-doping in this Hamiltonian of 10%; by means of a Dynamical Mean Field (DMF) approach combined with an interpolative calculation for the self-energy, we deduce the local density of states (DOS) and show that the quasi-particle DOS induced by the doping is not large enough to induce magnetism in the Sn-monolayer. This leads us to analyze the possibility of having superconductivity by considering the attractive interaction induced by the e-ph coupling within an appropriate BCS-Hamiltonian. Our calculations show that the quasiparticle metallic system has a superconductivity critical temperature of ≈7–9 K, in good agreement with experiments.