Conductivity spectrum for impurity induced spin-charge coupled Luttinger liquid in an SNS junction

The problem of interacting electrons in a one dimensional metal is well described by the Luttinger liquid (LL) theory at low energies around the Fermi level. The density-density interactions can be treated exactly with a non-interacting bosonic Hamiltonian which is separated into charge and spin density modes of excitations. However, presence of back-scattering potential in the system couples the unperturbed spin and charge modes. We study the effects of impurity induced spin-charge coupling on the conductivity spectrum of a metal as the normal region in an SNS junction where the normal region is much longer than the superconducting coherence length. Treating the impurity with a disorder potential perturbatively, we show that the resultant conductivity spectrum not only contains a peak at the energy of the charge mode excitation but also contains peaks corresponding to spin density wave excitations at neighboring energies which is a manifestation of the impurity induced spin charge coupling. With varying interaction strengths of the LL the locations of the peaks on the energy axis shift due to the re-normalization of the spin and charge velocities. Thus, in presence of impurities the spin mode excitations can be probed via conductivity measurement setup with a long SNS junction.