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

TL;DR

This work identifies non-invertible topological defect lines as the hidden protection that prevents the $O(n)$ symmetry from generically obstructing the renormalization group flow from the dilute to the dense phase in the 2D loop model. By constructing the lines on the lattice within the ${d\mathcal{J\!T\!L}}$ framework and connecting them to the interchiral algebra in the continuum, the authors show that RG perturbations expressible in the defect algebra cannot generate the dangerous four-leg operator, thereby enabling the observed flow to the dense fixed point. The analysis includes explicit lattice realizations, dense and dilute limits, and the defect Hilbert space, revealing a Verlinde-line–like structure that constrains operator content and RG trajectories. The results illuminate how non-invertible symmetries can govern critical behavior in loop models and hint at broader implications for interchiral algebras, defect classification, and non-invertible symmetries in statistical and conformal field theories.

Abstract

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.

Figures (5)

• Figure 1: With only $O(n)$ symmetry "protection", the 4-leg operator shold be generated under the RG, and the flow go to the $\sigma$-model fixed point JReadS instead of the dense one.
• Figure 2: The defect line is introduced.
• Figure 3: The insertion of a pair of disorder fields affects loops encircling the extremities of the corresponding defect line.
• Figure 4: Effective central charge $c_{\rm eff}(K)$ of the dilute Brauer model. The left panels show two-point fits, while the right panels show three-point fits (with a $1/L^4$ term). The three rows of figures correspond to $w=0, 0.8, 1.6$. The horizontal lines indicate the analytically known $c$ in the dilute phase (upper line), dense phase (lower line) and Goldstone phase (middle line). All figures are for the generic loop weight $n=1/\sqrt{2}$. System sizes are $L=6,\ldots,11$, with the purple curves corresponding to the largest sizes.
• Figure 5: The $\mathsf{d}$-operator in the open case.