A systematic stability analysis of the renormalisation group flow for the normal-superconductor-normal junction of Luttinger liquid wires

TL;DR

This paper performs a detailed stability analysis of the renormalization group flow for a superconducting junction of two Luttinger liquid wires, revealing complex power law behaviors and multiple perturbation directions.

Contribution

It introduces a comprehensive stability analysis of the RG flow around fixed points using SU(4) generators, including non-linear power laws dependent on interaction potentials.

Findings

Identifies eleven relevant or irrelevant perturbation directions at a non-trivial fixed point.

Derives power laws for conductance dependence on voltage and temperature.

Shows non-linear power laws involving Fourier components of interaction potential.

Abstract

We study the renormalization group flows of the two terminal conductance of a superconducting junction of two Luttinger liquid wires. We compute the power laws associated with the renormalization group flow around the various fixed points of this system using the generators of the SU(4) group to generate the appropriate parameterization of a S-matrix representing small deviations from a given fixed point S-matrix (obtained earlier in Phys. Rev. B 77, 155418 (2008)), and we then perform a comprehensive stability analysis. In particular, for the non-trivial fixed point which has intermediate values of transmission, reflection, Andreev reflection and crossed Andreev reflection, we show that there are eleven independent directions in which the system can be perturbed, which are relevant or irrelevant, and five directions which are marginal. We obtain power laws associated with these relevant and irrelevant perturbations. Unlike the case of the two-wire charge-conserving junction, here we show that there are power laws which are non-linear functions of V(0) and V(2k_{F}) (where V(k) represents the Fourier transform of the inter-electron interaction potential at momentum k). We also obtain the power law dependence of linear response conductance on voltage bias or temperature around this fixed point.