Sine-Gordon Model - Renormalization Group Solutions and Applications

TL;DR

This paper applies renormalization group techniques to analyze the sine-Gordon model, deriving flow equations, phase diagrams, and exploring its connections to one-dimensional electron models, providing a comprehensive pedagogical overview.

Contribution

It introduces a detailed perturbative renormalization group analysis of the sine-Gordon model and discusses its applications to Luttinger liquids and related electron models.

Findings

Derivation of the Kosterlitz-Thouless phase diagram

Estimation of the model's energy gap

Mapping to interacting electron models like Hubbard and g-ology

Abstract

The sine-Gordon model is discussed and analyzed within the framework of the renormalization group theory. A perturbative renormalization group procedure is carried out through a decomposition of the sine-Gordon field in slow and fast modes. An effective slow modes's theory is derived and re-scaled to obtain the model's flow equations. The resulting Kosterlitz-Thouless phase diagram is obtained and discussed in detail. The theory's gap is estimated in terms of the sine-Gordon model paramaters. The mapping between the sine-Gordon model and models for interacting electrons in one dimension, such as the g-ology model and Hubbard model, is discussed and the previous renormalization group results, obtained for the sine-Gordon model, are thus borrowed to describe different aspects of Luttinger liquid systems, such as the nature of its excitations and phase transitions. The calculations are carried out in a thorough and pedagogical manner, aiming the reader with no previous experience with the sine-Gordon model or the renormalization group approach.