Localized RG flows on composite defects and $\mathcal{C}$-theorem

TL;DR

This paper investigates RG flows on composite defects in a free O(N) model, identifying fixed points and confirming a localized -theorem through perturbative analysis.

Contribution

It constructs a composite defect system in a -dimensional free O(N) model and analyzes localized RG flows and fixed points, including symmetry-breaking cases.

Findings

Existence of -symmetric fixed point for all N.

Emergence of symmetry-breaking fixed points for N  23.

Validation of the -theorem for localized RG flows.

Abstract

We consider a composite defect system where a lower-dimensional defect (sub-defect) is embedded to a higher-dimensional one, and examine renormalization group (RG) flows localized on the defect. A composite defect is constructed in the $(3-\epsilon)$-dimensional free $\text{O}(N)$ vector model with line and surface interactions by triggering localized RG flows to non-trivial IR fixed points. Focusing on the case where the symmetry group $\text{O}(N)$ is broken to a subgroup $\text{O}(m)\times\text{O}(N-m)$ on the line defect, there is an $\text{O}(N)$ symmetric fixed point for all $N$, while two additional $\text{O}(N)$ symmetry breaking ones appear for $N\ge 23$. We also examine a $\mathcal{C}$-theorem for localized RG flows along the sub-defect and show that the $\mathcal{C}$-theorem holds in our model by perturbative calculations.