Spontaneous symmetry breaking on surface defects

TL;DR

This paper investigates whether surface defects in conformal field theories can spontaneously break continuous symmetries, concluding that under typical conditions, such symmetry breaking is not possible, extending known results to higher-dimensional defects.

Contribution

The work generalizes Coleman's theorem to surface defects in higher-dimensional CFTs, showing spontaneous symmetry breaking is generally forbidden under standard RG flows.

Findings

Spontaneous symmetry breaking is impossible for surface defects under reasonable RG flow assumptions.

Counterexamples exist only for exotic RG flows that do not reach fixed points.

Identifies a defect universality class with no SSB where correlations decay logarithmically.

Abstract

Coleman's theorem states that continuous internal symmetries cannot be spontaneously broken in two-dimensional quantum field theories (QFTs). In this work we consider surface (i.e. two-dimensional) defects in $d$-dimensional conformal field theories (CFTs) invariant under a continuous internal symmetry group $G$. We study under which conditions it is possible for a surface defect to break spontaneously a continuous internal symmetry. We find that spontaneous symmetry breaking (SSB) is impossible under reasonable assumptions on the defect Renormalization Group (RG) flow. Counterexamples are possible only for exotic RG flows, that do not terminate at a fixed-point. We discuss an example of this kind. We also illustrate our no-go result with an effective field theory analysis of generic defect RG flows. We find a generic weakly coupled defect universality class (with no SSB), where correlation functions decay logarithmically. Our analysis generalizes the recent discovery by Metlitski of the extraordinary-log boundary universality class in the $O(N)$ model.