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 ജ 2͒ 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 =1͒, where we only have two fixed points-the fully insulating fixed point or the Andreev fixed point. Even a two-wire ͑N =2͒ 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 electronelectron interaction induced backscattering in the quantum wires in our study and hence obtain non-Luttinger liquid behavior. It is interesting to note that ͑a͒ effects due to inclusion of electron-electron interaction induced backscattering in the wire, and ͑b͒ 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 ͓␣ = ͑g 2 −2g 1 ͒ / 2បv F ͔. The second one arises due to the scattering induced by the proximity of the superconductor and is given by ͓␣Ј = ͑g 2 + g 1 ͒ / 2បv F ͔. The nonmonotonic conductance and the identification of this new interaction parameter are two of our main results. In both the expressions g 1 = V͑2k F ͒ and g 2 = V͑0͒, where V͑k͒ is the interelectron interaction potential.

Junctions of anyonic Luttinger wires

“…The critical point η = ∞ has an opposite behavior, preserving theŨ (1) transformation (24), and breaking (23). For generic finite η > 0, it is easy to see that theŨ (1) symmetry is always conserved while U (1) is broken 34 .…”

“…The normalization of the charge densities ρ ± is fixed by requiring that they generate the transformations (24) and (23) in infinitesimal form, namely…”

Quantized charge pumping in a superconducting double-barrier structure: Nontrivial correlations due to proximity effect

“…We include the effects due to proximity of superconductor and e − e interaction in the wire via a RG approach developed very recently 37 for the case of 1-D normal metal−superconductor−normal metal (NSN) junction. As we are only interested in the coherent regime (L T >> L, L T is the thermal length), we can effectively treat the SDB system (NSNSN junction) as a single barrier (NSN junction) as far as RG is concerned.…”

Systematic stability analysis of the renormalization group flow for the normal-superconductor-normal junction of Luttinger liquid wires

“…Analogously, the NSN junction can be described in terms of an S-matrix with elements describing transmission of both electrons and holes and their mixing at the junction. The corresponding RG equation for the Smatrix was obtained by the present authors 22,23 which was an extension of Eq. 1 to the superconducting case.…”

“…In this article, we carry out a systematic stability analysis for each of the fixed points obtained earlier in Ref. 23 for the NSN junction and we predict the power laws associated with all possible independent perturba-tions that can be switched on around these fixed points. Our analysis provides us with renormalized values of the various transmission and reflection amplitudes around these fixed point values which can then be used to obtain the Landauer-Büttiker conductances.…”