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

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