Interaction effects on Andreev states from an electromagnetic environment

Here we present a detailed theoretical investigation of the environment-induced interaction correction to the ground states and addition energies in a short superconducting junction. We derive general formulas for these corrections linking them to the frequency-dependent admittance of the junction and the environment impedance. We specify two environmental models: that of an external impedance and the Mattis-Bardeen model. In both cases we assume $G_Q Z \ll 1$, where $G_Q$ is the conductance quantum and $Z$ the typical environmental resistance. While an expectation is that the typical scale of the relative correction is of the order $G_Q Z$, we have found that in many cases the dimensionless scale $L G_Q Δ$ provides a better estimation, where $Δ$ is the superconducting energy gap and $L$ an external inductance. We discuss the logarithmic renormalizations of the energies by low- and high-frequency phase fluctuations and single out a non-superconducting contribution related to the renormalization of transmission eigenvalues. We investigate in detail the peculiarities of the corrections at: i. small phases, where the interaction causes a current jump, ii. Andreev bound state energies close to the gap edge, iii. Andreev bound state energies close to Fermi level. In all these cases the correction may become comparable with the unperturbed energy, even for $G_Q Z \ll 1$, and we briefly sketch non-perturbative models relevant for the situations.