Renormalization group study of transport through a superconducting junction of multiple one-dimensional quantum wires

Abstract

We investigate transport properties of a superconducting junction of many $(N\ensuremath{\geqslant}2)$ one-dimensional quantum wires. We include the effect of electron-electron interaction within the one-dimensional quantum wire using a weak interaction renormalization group procedure. Due to the proximity effect, transport across the junction occurs via direct tunneling as well as via the crossed Andreev channel. We find that the fixed point structure of this system is far more rich than the fixed point structure of a normal metal--superconductor junction $(N=1)$, where we only have two fixed points---the fully insulating fixed point or the Andreev fixed point. Even a two-wire $(N=2)$ system with a superconducting junction, i.e., a normal metal--superconductor--normal metal structure, has nontrivial fixed points with intermediate transmissions and reflections. We also include electron-electron interaction induced backscattering in the quantum wires in our study and hence obtain non-Luttinger liquid behavior. It is interesting to note that (a) effects due to inclusion of electron-electron interaction induced backscattering in the wire, and (b) competition between the charge transport via the electron and hole channels across the junction give rise to a nonmonotonic behavior of conductance as a function of temperature. We also find that transport across the junction depends on two independent interaction parameters. The first one is due to the usual correlations coming from Friedel oscillations for spin-full electrons giving rise to the well-known interaction parameter $[\ensuremath{\alpha}=({g}_{2}\ensuremath{-}2{g}_{1})∕2\ensuremath{\pi}\ensuremath{\hbar}{v}_{F}]$. The second one arises due to the scattering induced by the proximity of the superconductor and is given by $[{\ensuremath{\alpha}}^{\ensuremath{'}}=({g}_{2}+{g}_{1})∕2\ensuremath{\pi}\ensuremath{\hbar}{v}_{F}]$. The nonmonotonic conductance and the identification of this new interaction parameter are two of our main results. In both the expressions ${g}_{1}=V(2{k}_{F})$ and ${g}_{2}=V(0)$, where $V(k)$ is the interelectron interaction potential.