Duality and metal-insulator transition in imperfect Luttinger liquids coupled to phonons

We study the effect of an embedded weak scatterer (WS) or a weak link (WL) on the electronic transport in a Luttinger liquid (LL) with interacting electrons coupled to massless bosons (e.g., acoustic phonons). We find that a well-known (for the standard LL) duality relation between scaling dimensions of the electron backscattering in the WS and WL limits, D_ws D_wl = 1, holds in the presence of the additional coupling for an any fixed strength of boson scattering from the impurity. This means that at low temperatures such a system remains either an ideal insulator or an ideal metal, regardless of the scattering strength. On the other hand, when fermion and boson scatterings from the impurity are correlated, the system has a rich phase diagram that includes a metal-insulator transition at some intermediate values of the scattering. The results are translated to a 1D flow of fermion-boson mixture of cold atoms through a constriction.