The Tomonaga-Luttinger Liquid with Quantum Impurity Revisited: Critical Line and Phase Diagram

We revisit the $(1+1)$ dimensional field theoretical model, which describes the Tomonaga-Luttinger liquid (TLL), interacting with a static impurity at the origin of the half line. Applying the Fermi-Bose equivalence and finite conformal transformations only, we map the model onto the Schmid model. Some details of the bosonization procedure have been given. The critical line and the phase diagram of the model follow from the renormalization group analysis of the Schmid model. The obtained critical line of the model is a hyperbola in the parameter space of the two couplings of the TLL.