Four-fermion deformations of the massless Schwinger model and confinement

TL;DR

This paper shows that adding specific four-fermion operators to the massless Schwinger model can induce confinement and chiral symmetry breaking, contrary to the usual deconfined nature of the model.

Contribution

It demonstrates that four-fermion deformations can turn the massless Schwinger model into a confining theory, revealing new mechanisms for confinement in 2D gauge theories.

Findings

Four-fermion operators induce confinement in the massless Schwinger model.

Confinement is driven by fractional instantons on .

The model exhibits spontaneous chiral symmetry breaking with these deformations.

Abstract

We consider the massless charge-$N$ Schwinger model and its deformation with two four-fermion operators. Without the deformations, this model exhibits chiral symmetry breaking without confinement. It is usually asserted that the massless Schwinger model is always deconfined and a string tension emerges only when a mass for the fermion field is turned on. We show that in the presence of these four-fermion operators, the massless theory can in fact confine. One of the four-fermion deformations is chirally neutral, and is a marginal deformation. The other operator can be relevant or irrelevant, and respects a $\mathbb{Z}_2$ subgroup of chiral symmetry for even $N$, hence forbidding a mass term. When it is relevant, even the exactly massless theory exhibits both confinement and spontaneous chiral symmetry breaking. The construction is analogous to QCD(adj) in 2d. While the theory without four-fermion deformations is deconfined, the theory with these deformations is generically in a confining phase. We study the model on $\mathbb{R}^2$ using bosonization, and also analyze the mechanism of confinement on $\mathbb{R}\times S^1$, where we find that confinement is driven by fractional instantons.