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

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 $SU\left(4\right)$ group to generate the appropriate parametrization of an $\mathbb{S}$ matrix representing small deviations from a given fixed point $\mathbb{S}$ matrix [obtained earlier in S. Das, S. Rao, and A. Saha, Phys. Rev. B 77, 155418 (2008)], and we then perform a comprehensive stability analysis. In particular, for the nontrivial 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 nonlinear functions of $V\left(0\right)$ and $V\left(2k_F\right)$ [where $V\left(k\right)$ represents the Fourier transform of the interelectron 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.