Non-invertible symmetries and RG flows in the two-dimensional $O(n)$ loop model

In a recent paper, Gorbenko and Zan [arXiv:2005.07708] observed that $O(n)$ symmetry alone does not protect the well-known renormalization group flow from the dilute to the dense phase of the two-dimensional $O(n)$ model under thermal perturbations. We show in this paper that the required "extra protection" is topological in nature, and is related to the existence of certain non-invertible topological defect lines. We define these defect lines and discuss the ensuing topological protection, both in the context of the $O(n)$ lattice model and in its recently understood continuum limit, which takes the form of a conformal field theory governed by an interchiral algebra.