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

TL;DR

This paper revisits the Tomonaga-Luttinger liquid with a static impurity, mapping it to the Schmid model to analyze its critical line and phase diagram using conformal transformations and RG analysis.

Contribution

It provides a detailed bosonization and RG analysis of the TLL with impurity, revealing the critical line as a hyperbola in coupling space.

Findings

Critical line is a hyperbola in coupling parameter space.

Mapping of TLL with impurity to Schmid model via conformal transformations.

Phase diagram derived from RG analysis of the mapped model.

Abstract

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.