We study the influence of electron-phonon coupling on electron transport through a Luttinger liquid with an embedded weak scatterer or weak link. We derive the renormalization group (RG) equations, which indicate that the directions of RG flows can change upon varying either the relative strength of the electron-electron and electron-phonon coupling or the ratio of Fermi to sound velocities. This results in a rich phase diagram with up to three fixed points: an unstable one with a finite value of conductance and two stable ones, corresponding to an ideal metal or insulator.
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Galda, A., Yurkevich, I., & Lerner, I. V. (2011). Impurity scattering in a Luttinger liquid with electron-phonon coupling. Physical Review B, 83(4), . https://doi.org/10.1103/PhysRevB.83.041106