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

In a recent paper, Gorbenko and Zan [1] 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.