On the equivalence between sine-Gordon model and Thirring model in the chirally broken phase of the Thirring model

TL;DR

This paper demonstrates the equivalence between the Thirring and sine-Gordon models in the chirally broken phase, revealing new relations between their parameters and insights into their vacuum structure and symmetries.

Contribution

It extends the bosonization correspondence to the chirally broken phase of the Thirring model, providing a new relation between coupling constants and analyzing the vacuum and current algebra.

Findings

Bosonized massless Thirring model described by a free scalar field.

Massive Thirring model bosonizes to sine-Gordon with a new coupling relation.

Chirally broken phase vacuum resembles BCS ground state.

Abstract

We investigate the equivalence between Thirring model and sine-Gordon model in the chirally broken phase of the Thirring model. This is unlike all other available approaches where the fermion fields of the Thirring model were quantized in the chiral symmetric phase. In the path integral approach we show that the bosonized version of the massless Thirring model is described by a quantum field theory of a massless scalar field and exactly solvable, and the massive Thirring model bosonizes to the sine-Gordon model with a new relation between coupling constants. We show that the non-perturbative vacuum of the chirally broken phase in the massless Thirring model can be described in complete analogy with the BCS ground state of superconductivity. The Mermin-Wagner theorem and Coleman's statement concerning the absence of Goldstone bosons in the 1+1-dimensional quantum field theories are discussed. We investigate the current algebra in the massless Thirring model and give a new value of the Schwinger term. We show that the topological current in the sine-Gordon model coincides with the Noether current responsible for the conservation of the fermion number in the Thirring model. This allows to identify the topological charge in the sine-Gordon model with the fermion number.