  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.
