Notes on a Surface Defect in the $O(N)$ Model

TL;DR

This paper investigates a surface defect in the $O(N)$ model, analyzing its RG flow and conformal properties using epsilon and large $N$ expansions, and confirms the defect $b$-theorem.

Contribution

It introduces a quadratic perturbation defect in the $O(N)$ model and computes its RG flow and physical quantities using multiple expansion methods, providing new insights into defect conformal field theories.

Findings

The defect flows to a nontrivial DCFT at long distances.

Agreement between epsilon and large $N$ expansion results.

Consistency with the $b$-theorem for defect RG flows.

Abstract

We study a surface defect in the free and critical $O(N)$ vector models, defined by adding a quadratic perturbation localized on a two-dimensional subspace of the $d$-dimensional CFT. We compute the beta function for the corresponding defect renormalization group (RG) flow, and provide evidence that at long distances the system flows to a nontrivial defect conformal field theory (DCFT). We use epsilon and large $N$ expansions to compute several physical quantities in the DCFT, finding agreement across different expansion methods. We also compute the defect free energy, and check consistency with the so-called $b$-theorem for RG flows on surface defects.