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

Jesper Lykke Jacobsen, Hubert Saleur

► To cite this version:

Jesper Lykke Jacobsen, Hubert Saleur. Non-invertible symmetries and RG flows in the two-dimensional O(n) loop model. Journal of High Energy Physics, 2023, 2023 (12), pp.90. 10.1007/jhep12(2023)090. hal-04979626

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

Jesper Lykke Jacobsen,ª,ᵇ,ᶜ and Hubert Saleurª,ᵈ

ªInstitut de physique théorique, CEA, CNRS, Université Paris-Saclay,
France

ᵇLaboratoire de Physique de l'École Normale Supérieure, ENS, Université PSL, CNRS,
Sorbonne Université, Université de Paris,
France

ᶜSorbonne Université, École Normale Supérieure, CNRS, Laboratoire de Physique (LPENS),
France

ᵈDepartment of Physics and Astronomy, University of Southern California,
Los Angeles, U.S.A.

E-mail: jesper.jacobsen@ens.fr, hubert.saleur@cea.fr

**ABSTRACT:** 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.

**KEYWORDS:** Lattice Integrable Models, Scale and Conformal Symmetries

ARXIV ePRINT: 2305.05746
# Contents

1 Introduction 1

2 General idea 3

3 Non-invertible (topological) symmetry in the lattice model 6

3.1 Algebras and the defect 6

3.2 The dense case 8

3.3 The dilute case 11

4 The non-invertible symmetry at the dilute fixed point and in the RG flow 12

5 The defect Hilbert space 15

6 Conclusion 17

A Allowing crossings at the dilute critical point 18

A.1 Model 18

A.2 Phase diagram 19

A.3 Universality class in the dilute case 19

A.4 Critical exponents 21

B The non-invertible symmetry with open boundary conditions 25

C TDLs with crossings: the (dilute) Brauer algebra case 26

## 1 Introduction

In the conclusion to their paper, Gorbenko and Zan [1] raise a puzzling question about our understanding of the phase diagram of the $O(n)$ loop model. The puzzle occurs because, in the $O(n)$ model at the (dilute) critical point, geometrically well-identified fields with given, known conformal dimensions, come with multiplicities that correspond to sums of several $O(n)$ irreducible representations. For instance, the four-leg (fuseau, or watermelon) operator comes with the multiplicity space $[4] \oplus [22] \oplus [211] \oplus [2] \oplus []$, where we used the standard row-length notation $[\lambda]$ for (Young diagrams of) irreducible representations of $O(n)$ (see (3.10f) for the origin of this decomposition). This multiplicity means that there are copies of the four-leg operator coming with different $O(n)$ tensorial content.<fnref n="1" />

<footnote n="1">¹In most of this paper, we follow most of the literature and treat $O(n)$ as an “ordinary” symmetry, while in fact $n$ is not an integer but a real (or even complex) number. This point is further discussed in the conclusion.</footnote>
in the multiplicity of the four-leg operator would not be a scalar anymore. It is not clear, however, what this hidden symmetry might be: attempts to find the corresponding extra generators are difficult, and there is evidence that it is not possible to define the necessary associated tensor products.

Reference [1] makes two final suggestions to solve the puzzle: one involves integrability, the other the presence of a yet unidentified symmetry involving non-invertible topological defect lines. We shall now demonstrate that the latter suggestion is correct, and analyze it in detail.

The paper is organized as follows. The set-up and the general idea how to introduce the relevant topological defect lines in the loop model are given in section 2. Section 3 provides technical details of the construction for the lattice model, in the dense and dilute cases. Section 4 discusses the ensuing properties at the fixed points and along the RG flow. This concludes the goal of this paper proper. We find it useful, however, to discuss briefly further aspects in section 5, and to establish in particular a connection with the so-called Verlinde lines [8]. Obviously, our work opens the way to considering many other questions in loop models, some of which are mentioned in the conclusion.

The paper also contains three appendices. In appendix A we discuss the claim that crossings do not change the universality class at the dilute fixed point. In appendix B we consider what happens to our topological defect lines with open boundary conditions and in appendix C, what happens when one tries to build similar objects in models with crossings.

## 2 General idea

Topological defects are conveniently introduced in CFTs (see e.g. [9, 10] for early references) as operators<fnref n="2" /> acting on the Hilbert space $\mathcal{H}$ of the theory, which it itself a sum of representations of $\text{Vir} \otimes \bar{\text{Vir}}$. These operators, which we will denote generically by $\mathcal{D}$, must commute with $\text{Vir}$ and $\bar{\text{Vir}}$:

$$[\mathcal{D}, \mathcal{L}_p] = [\mathcal{D}, \bar{\mathcal{L}}_p] = 0, \quad \forall p \in \mathbb{Z} \qquad (2.1)$$

or, equivalently, commute independently with the left- and right-moving components of the stress-energy tensor.

The corresponding objects in the Euclidian description are topological defect-lines (TDL), which we denote by the symbol $\mathcal{D}$. When correlation functions are calculated in their presence, relations (2.1) mean that the lines can be deformed at will without changing the result, provided the field insertions are not crossed. This latter definition extends easily to the non-conformal case. For a more thorough set of definitions, see e.g. [8].

While these definitions are well known in the continuum limit, they can be extended to lattice models (see e.g. [11] for a discussion in the context of the Ising model). Radial quantization corresponds then to studying the model on a cylinder with periodic boundary conditions in the space direction, while the Hamiltonian (or, more generally, the transfer

<footnote n="2">²The name “defect operator” per se is used for a different object in the literature [8].</footnote>
matrix) propagates in the imaginary-time direction. It is natural to define a lattice equiv-
alent of D in this context as an operator that commutes with the local densities or energy
and momentum — this is discussed in more detail below. In the Euclidian description,
the lattice defect line should be — just like in the continuum — deformable at will with-
out changing correlation functions, provided it does not cross insertions of discrete lattice
observables.

Since we are interested in systems with global $O(n)$ symmetry — and forgetting for
now the fact that $n$ is in general non-integer in this paper — so-called invertible topological
lines (i.e. associated with the action of a group element $g$) can easily be constructed, and
were in fact part of the analysis in [2]. In the present paper we are however concerned
with a different type of lines which are *non-invertible* and not related to a group element in
any obvious way. We now claim that there exists a natural pair of such lines in the lattice
$O(n)$ model answering the suggestion in [1]: the presence of the associated operators $D, \bar{D}$
(or their lattice equivalent) will be called “non-invertible (topological) symmetry”. Before
dwelling into technicalities, let us give a qualitative idea of how this works.

We start by recalling that the partition function of the $O(n)$ model is given by an
expression of the type

$$
Z_{O(n)} = \sum_{C} K^{|A|(C)} n^{N(C)}, \qquad (2.2)
$$

where the sum is over self- and mutually-avoiding loop configurations $C$ (no crossings, no
overlaps), $N(C)$ is the number of loops so $n$ is interpreted as their fugacity, $|A|(C)$ is the
number of lattice edges ("monomers") involved in $C$, and $K$ is their fugacity. The $O(n)$
(dilute) fixed point is obtained for a particular critical value, $K = K_c$; the model is then
in the dense critical phase and flows to the dense critical fixed point (whose properties are
independent of $K$) for all $K > K_c$. We will consider more generally a model where the
monomer couplings are edge-dependent, with a generalized partition function

$$
Z_{O(n)} = \sum_C \left( \prod_{e \in A(C)} K_e \right) n^{N(C)}, \qquad (2.3)
$$

where now the product is taken over the set of monomers $A(C)$ in each configuration $C$.
The best known observables in this model are the fuseau operators, sources and sinks of $2r$
lines which cannot be contracted among themselves, but which can only connect to another
fuseau operator.

While the definition (2.2) is purely geometrical, we also recall that it arises from a
high-temperature expansion for a model of spins living on the edges or the vertices of the
lattice (depending on the particular underlying microscopic model). These spins belong
to the vector (fundamental) representation of $O(n)$, denoted [1]. This is in particular
what explains the fugacity of loops — the factor $n$ corresponds to the $n$ colors that can
“propagate” around it. The role of the $O(n)$ symmetry will become much clearer from the
further details given below. It is convenient in what follows to imagine that we have some
sort of continuum version of (2.2)–(2.3), where $K$ is coupled to a properly defined “length”
of the loops, and to forget for the time being about the lattice.
We now introduce into this system another type of geometrical object, representing a topological defect line. This line (which can be in particular a closed loop) goes “above” the loops of the $O(n)$ model. To give a precise meaning to this, we parameterize

$$n = q + q^{-1}, \qquad (2.4)$$

and we interpret over- and under-crossings as giving rise to the decompositions

$$\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[thick, red] (-1.5,0) -- (1.5,0);
\node[above] at (-1.5,0) {$q^{-1/2}$};
\node[below] at (1.5,0) {$q^{1/2}$};
\end{tikzpicture} + (-q)^{-1/2}, \qquad (2.5a)$$

$$\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[thick, red] (-1.5,0) -- (1.5,0);
\node[above] at (-1.5,0) {$q^{-1/2}$};
\node[below] at (1.5,0) {$q^{1/2}$};
\end{tikzpicture} + (-q)^{-1/2}, \qquad (2.5b)$$

where $O(n)$ lines are represented in blue and topological lines in red. After the expansion we obtain lines or loops with two colors. The colors do not imply any distinction by themselves — after all, loops of any color live in the same representation [1] — but they provide a nice visual means of tracking the role of the topological lines in the decompositions. The rule is that we give to every loop, irrespective of its (bi)color the weight $n$, while the variable $K$ is only coupled to the length of the blue parts,<fnref n="3" /> which is not affected by the topological line.

The choice of the coefficients guarantees that the topological line(s) can be moved at will without changing partition or correlation functions, justifying the graphical representation that hints at invariance under topological (Reidemeister) moves. A short calculation to illustrate this point goes as follows:

$$\begin{align}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[thick, red] (-1.5,0) circle (0.5);
\node[above] at (-1.5,0) {$q^{-1/2}$};
\draw[thick, blue] (-1.5,0) -- (-1.5,0.5);
\node[above] at (-1.5,0.5) {$q^{1/2}$};
\end{tikzpicture}
&= (-q) + \begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[thick, red] (-1.5,0) circle (0.5);
\node[above] at (-1.5,0) {$q^{-1/2}$};
\draw[thick, blue] (-1.5,0) -- (-1.5,0.5);
\node[above] at (-1.5,0.5) {$q^{1/2}$};
\end{tikzpicture} + \begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[thick, red] (-1.5,0) circle (0.5);
\node[above] at (-1.5,0) {$q^{-1/2}$};
\draw[thick, blue] (-1.5,0) -- (-1.5,0.5);
\node[above] at (-1.5,0.5) {$q^{1/2}$};
\end{tikzpicture} + (-q)^{-1} \begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[thick, red] (-1.5,0) circle (0.5);
\node[above] at (-1.5,0) {$q^{-1/2}$};
\draw[thick, blue] (-1.5,0) -- (-1.5,0.5);
\node[above] at (-1.5,0.5) {$q^{1/2}$};
\end{tikzpicture} \nonumber \\
&= (q + q^{-1}) \begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[thick, red] (-1.5,0) circle (0.5);
\node[above] at (-1.5,0) {$q + q^{-1}$};
\draw[thick, blue] (-1.5,0) circle (0.5);
\node[above] at (-1.5,0) {$q - q^{-1}$};
\end{tikzpicture} \tag{2.6}
\end{align}$$

In the first step we have used (2.5) to dissolve crossings, and in the second step any second loop has been replaced by $q + q^{-1}$ according to (2.4). The final result means that red loops can be moved around without affecting the partition function, as summarized in the graphical identity

$$\begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[thick, red] (-1.5,0) circle (0.5);
\node[above] at (-1.5,0) {$q^{-1/2}$};
\draw[thick, blue] (-1.5,0) circle (0.5);
\node[above] at (-1.5,0) {$q^{1/2}$};
\end{tikzpicture} = \begin{tikzpicture}[baseline=(current bounding box.center)]
\draw[thick, red] (-1.5,0) circle (0.5);
\node[above] at (-1.5,0) {$q^{-1/2}$};
\draw[thick, blue] (-1.5,0) circle (0.5);
\node[above] at (-1.5,0) {$q^{1/2}$};
\end{tikzpicture} \qquad (2.7)$$

The possibility of moving the TDL around arises — as discussed below — from the fact that the combination on the right-hand side of equation (2.5) is a solution of the spectral-parameter independent Yang-Baxter equation. Without field insertions, adding a

<footnote n="3">³This is only a question of convention: since the topological defect line is not a fluctuating object, giving its monomers a non-trivial weight can easily be taken into account by adjusting normalizations.</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 2. The defect line is introduced.</figcaption></figure>

but actually they differ only in the way we draw them, namely by the alternating inclination of their bottom and top boundaries with respect to horizontal. More importantly, their algebraic expansions are identical, with the two terms on the right-hand side fitting, respectively, ordinary tiles with and without monomers that can be placed below and above the defect tile so as to touch its grey points. Nothing happens if the defect tile is empty of monomers (second term), otherwise monomers can go below $D$ (first term) — the meaning of “going below” being defined by the decomposition (2.5). Instead of below, we could decide that monomers go above the defect instead, giving rise to a different defect line $\bar{D}$.

## 3.2 The dense case

The fully dense (also called “completely packed” in some works) case<fnref n="6" /> occurs when the monomer fugacity is very large ($K \to \infty$), so all edges of the lattice $S$ are occupied.<fnref n="7" /> In this case, the relevant algebra is the unoriented Jones Temperley-Lieb algebra $uJTL_N(n)$. Correspondingly, the ordinary tiles are restricted to the last two diagrams of (3.1), and the defect tile corresponds to only the first term on the right-hand sides of (3.4).

The fully dense case has in fact already been studied in [17]. It was shown in this paper that the two defects $D$ and $\bar{D}$ corresponding to passing a loop over or under the other lines commute with the whole algebra. It is easy, in fact, to write expressions for these defects

<footnote n="6">⁶We call here “fully dense” the case where the monomer fugacity is very large, so that all edges of the lattice $S$ are occupied. The model then coincides with the model of “surrounding lattice decompositions” that appears, e.g., in the reformulation of the Potts model [13]. It is widely believed that our lattice model for all $K > K_c$ is in the same universality class as this fully dense limit. We use the name fully dense in contrast with the often used “fully packed” to stress the fact that loop models with maximum occupancy of the edges and/or vertices enjoy less universality than their dense but not maximally so counterparts. On some lattices, the thermodynamic limit and the $K \to \infty$ limits do not commute [14, 15]. We shall not be concerned with this subtlety here.</footnote>

<footnote n="7">⁷Note that in that limit, it does not matter whether all fugacities are the same or not: provided they are taken to be large, the “monomer” weight of all configurations is the same — the monomer fugacity couples to a redundant operator.</footnote>
by systematically splitting the crossings. One finds

$$ \begin{aligned} \mathcal{D} &= (-q)^{-N/2}\tau(1-qe_{N-1})\cdots(1-qe_2)(1-qe_1)+(-q)^{N/2}(1-q^{-1}e_1)\cdots(1-q^{-1}e_{N-1})\tau^{-1}, \\ \bar{\mathcal{D}} &= (-q)^{N/2}\tau(1-q^{-1}e_{N-1})\cdots(1-q^{-1}e_2)(1-q^{-1}e_1)+(-q)^{-N/2}(1-qe_1)\cdots(1-qe_{N-1})\tau^{-1}, \end{aligned} $$

where N is the width of the system, and $\tau$ translates every site one step to the right. The main result of [17] can then be formulated as<fnref n="8" />

$$ [\mathcal{D}, u\mathcal{J}\mathcal{L}_N(\mathbf{n})] = [\bar{\mathcal{D}}, u\mathcal{J}\mathcal{L}_N(\mathbf{n})] = 0. \qquad (3.5) $$

These commutation relations mean that the defect operators commute with all Hamiltonians built out of elements of the algebra. They also of course commute with local (as well as row-to-row) transfer matrices, implying in particular invariance of the partition function of the model under arbitrary deformations of the associated TDL $\mathcal{D}$. We insist on the fact that this result is true on the lattice, before any kind of continuum limit is taken.

While it is convenient to think of this problem geometrically, it is important to realize that all we said has a strict equivalent, when $n$ is an integer, for the original $O(n)$ spin model. The Hamiltonian or transfer matrix acts on the tensor product of fundamental representations $\mathcal{H} = [1]^{\otimes N}$, with the well-known actions (here and later the Roman labels $a, b$ etc. take values 1, 2, ..., $n$ and all sums run over this set of values):

$$ e_i|a_1, \dots, a_N\rangle = \delta_{a_i a_{i+1}} \sum_b |a_1, \dots, a_{i-1}, b, b, a_{i+2}, \dots a_N\rangle, \qquad (3.6a) $$

$$ \tau|a_1, \dots, a_N\rangle = |a_N, a_1, \dots, a_{N-1}\rangle. \qquad (3.6b) $$

These formulas give rise to an explicit form for the operators $\mathcal{D}, \bar{\mathcal{D}}$. On two ($N=2$) and three ($N=3$) sites, for instance, we have

$$ \mathcal{D}|ab\rangle = -(q+q^{-1})|ba\rangle + 2\delta_{ab}\sum_c |cc\rangle, \qquad (3.7a) $$

$$ \begin{aligned} \mathcal{D}|abc\rangle = & (-q)^{-3/2}|cab\rangle + (-q)^{3/2}|bca\rangle + \delta_{ab} \left( (-q)^{1/2} \sum_d |dcd\rangle + (-q)^{-1/2} \sum_d |cdd\rangle \right) \\ & + \delta_{bc} \left( (-q)^{1/2} \sum_d |dda\rangle + (-q)^{-1/2} \sum_d |dad\rangle \right) \\ & + \delta_{ac} \left( (-q)^{1/2} \sum_d |bdd\rangle + (-q)^{-1/2} \sum_d |ddb\rangle \right). \end{aligned} \qquad (3.7b) $$

Since $\mathcal{D}$ is expressed in terms of the Temperley-Lieb generator and the shift operator, it commutes with $O(n)$ in the sense of quantum Schur-Weyl duality:

$$ [\mathcal{D}, O(n)] = 0. \qquad (3.8) $$

<footnote n="8">⁸In fact, reference [17] also addresses the question of whether these are the only operators satisfying this property — in more precise terms, the center of the algebra. We will not discuss this aspect here.</footnote>
Indeed, the space of states of the fully dense $O(n)$ model can be written as a bimodule over $uJTL \otimes O(n)$:

$$ \mathcal{H}_{O(n)} = W_{\langle 1,1 \rangle} \otimes [\ ] \oplus \bigoplus_{r \in \frac{1}{2}\mathbb{N}^*} \bigoplus_{\substack{s \in \frac{1}{r}\mathbb{Z} \\ -1 < s \le 1}} W_{(r,s)} \otimes \Lambda_{(r,s)}. \qquad (3.9) $$

Here the $\Lambda_{(r,s)}$ are sums of $O(n)$ representations, a few of which are [2]

$$ \Lambda_{(\frac{1}{2},0)} = [1], \qquad (3.10a) $$

$$ \Lambda_{(1,0)} = [2], \qquad (3.10b) $$

$$ \Lambda_{(1,1)} = [1^2], \qquad (3.10c) $$

$$ \Lambda_{(\frac{3}{2},0)} = [3] + [1^3], \qquad (3.10d) $$

$$ \Lambda_{(\frac{3}{2},\frac{2}{3})} = [21], \qquad (3.10e) $$

$$ \Lambda_{(2,0)} = [4] + [2^2] + [21^2] + [2] + [\ ], \qquad (3.10f) $$

$$ \Lambda_{(2,\frac{1}{2})} = [31] + [21^2] + [1^2], \qquad (3.10g) $$

$$ \Lambda_{(2,1)} = [31] + [2^2] + [1^4] + [2]. \qquad (3.10h) $$

These result still hold for $n \in \mathbb{C}$, with a proper definition of the $O(n)$ symmetry for $n$ non integer, as discussed in [2, 16]. The objects $W_{\langle 1,1 \rangle}$ and $W_{(r,s)}$ appearing in (3.9) are irreducible modules of $uJTL$. Here, the label $r$ is one-half the number of non-contractible lines, while the label $s$ is a pseudomomentum that arises from representations of the cyclic group acting on these lines. Geometrically, it can be understood from phase terms $z$ that are gained by each individual line when it winds around the cylinder. It is customary to set $z = e^{i\pi s}$, with the condition $z^{2r} = e^{2i\pi rs} = 1$ that a simultaneous winding of all the lines produces no phase. The eigenstate with the smallest eigenvalues in $W_{(r,s)}$ gives the leading term in the two-point function of the corresponding generalized fuseau operators. The module $W_{\langle 1,1 \rangle}$ (sometimes denoted by $\bar{W}_{0,q\pm 2}$ in other work) is the module with no throughlines, where non-contractible loops can occur. Since they are given the same weight $n$ as contractible ones, the way two points are connected in this representation of the algebra does not matter. In this setup, $\Lambda_{(2,0)}$ is then the $O(n)$ multiplicity space of the four-leg operator without winding phases which was mentioned in the Introduction.

Every term in the sum (3.9) is an eigenspace (for $q$ not a root of unity$^9$) of the operators $D, \bar{D}$. The corresponding eigenvalues are

$$ D_{\langle 11 \rangle} = q + q^{-1}, \qquad (3.11a) $$

$$ D_{(r,s)} = e^{i\pi s}(-q)^r + e^{-i\pi s}(-q)^{-r}, \qquad (3.11b) $$

and the same with $q \to q^{-1}$ for $\bar{D}$. For $q$ generic, it is easy to see that the eigenvalues of the $D$ operators determine the $(r, s)$ uniquely. Note that we can think of the different

<footnote>$^9$Subtleties arise when $q$ is a root of unity, where some representations might not be fully reducible, and operators not diagonalizable.</footnote>
$O(n)$ representations involved in a given $\Lambda_{(r,s)}$ as giving rise to the same eigenvalue of $D, \bar{D}$. Going back to $n$ integer and taking $a \neq b \neq c$ we have, for instance,

$$|abc\rangle_{[3]} \equiv |abc\rangle + |acb\rangle + |bac\rangle + |bca\rangle + |cba\rangle + |cab\rangle, \qquad (3.12a)$$

$$|abc\rangle_{[1]^3} \equiv |abc\rangle - |acb\rangle - |bac\rangle + |bca\rangle - |cba\rangle + |cab\rangle, \qquad (3.12b)$$

and

$$D|abc\rangle_{[3]} = ((-q)^{-3/2} + (-q)^{3/2})|abc\rangle_{[3]}, \qquad (3.13a)$$

$$D|abc\rangle_{[1]^3} = ((-q)^{-3/2} + (-q)^{3/2})|abc\rangle_{[111]}, \qquad (3.13b)$$

with the same eigenvalue — a result compatible with the identity $\Lambda_{(\frac{3}{2},0)} = [3] + [1]^3$ from (3.10d).

We note that it is possible to build “higher-spin” defects by fusion. In the integrable construction, the defect line can be considered as carrying a spin-1/2 representation of $U_q sl(2)$: it is perfectly possible to equip it with a spin-$j$ representation instead (for any $2j \in \mathbb{N}^*$). This can be done in practice by introducing $2j$ defect lines, and projecting them onto spin $j$ using a Jones-Wenzl projector. The construction has been discussed in detail in [17] and leads to the straightforward identity

$$D^{(j)}D = D^{(j+1/2)} + D^{(j-1/2)}, \qquad (3.14)$$

with $D = D^{(1/2)}$. Hence we have in particular

$$D^{(1)} = D^2 - 1, \qquad (3.15a)$$

$$D^{(3/2)} = D^3 - 2D, \qquad (3.15b)$$

$$D^{(2)} = D^4 - 3D^2 + 1. \qquad (3.15c)$$

In practice we shall not need these higher defects in our analysis: although their existence is important for some other purposes, we shall not consider them further here.

### 3.3 The dilute case

The construction obviously generalizes to the case where not every edge of the lattice $S$ is covered by a monomer: in this case, there is no interaction with the defect line, while, if there is a monomer, the interaction is as before. Here is an explicit example:

$$ \begin{aligned} D^{\text{dilute}} &= \frac{1}{2} \left( \begin{array}{cccc} \text{red} & \text{red} & \text{red} & \text{red} \\ \text{blue} & \text{blue} & \text{blue} & \text{blue} \end{array} \right) \\ &= (-q) \frac{1}{2} \left( \begin{array}{cccc} \text{red} & \text{red} & \text{red} & \text{red} \\ \text{blue} & \text{blue} & \text{blue} & \text{blue} \end{array} \right) + (-q)^{-1} \frac{1}{2} \left( \begin{array}{cccc} \text{red} & \text{red} & \text{red} & \text{red} \\ \text{blue} & \text{blue} & \text{blue} & \text{blue} \end{array} \right) \\ &\quad + \frac{1}{2} \left( \begin{array}{cccc} \text{red} & \text{red} & \text{red} & \text{red} \\ \text{blue} & \text{blue} & \text{blue} & \text{blue} \end{array} \right) + \frac{1}{2} \left( \begin{array}{cccc} \text{red} & \text{red} & \text{red} & \text{red} \\ \text{blue} & \text{blue} & \text{blue} & \text{blue} \end{array} \right) \end{aligned} \qquad (3.16) $$

In terms of $O(n)$ variables, the defect operator acts on states now in $([1] + [1])^{\otimes N}$ by being simply transparent to the variables in the trivial representation $[]$ (geometrically an empty
edge), while acting on those in the fundamental [1] (an edge carrying a blue line) as before. So, for instance, corresponding to (3.16) we have

$$ \mathcal{D}|0a0b\rangle = -(q+q^{-1})|0b0a\rangle + 2\delta_{ab}\sum_c |0c0c\rangle, \qquad (3.17) $$

to be compared with (3.7a). The Roman labels here take the same values 1, 2, ..., n as before, whereas 0 refers to the empty state in the trivial representation []. The Hilbert space still decomposes as in (3.9), but now with representations of the dilute algebra $d\mathcal{J}\mathcal{T}\mathcal{L}$:

$$ \mathcal{H}_{O(n)} = \tilde{W}_{(1,1)} \otimes [] \oplus \bigoplus_{r \in \frac{1}{2}\mathbb{N}^*} \bigoplus_{s \in \frac{1}{r}\mathbb{Z}} \tilde{W}_{(r,s)} \otimes \Lambda_{(r,s)}. \qquad (3.18) $$

In contrast with the dense case, the monomer fugacity affects non-trivially the properties (partition function and correlation functions) of the model. But it does *not* affect the formal structure of the Hilbert space decomposition (3.18). The defect operators are still diagonal on each term in the sum, with eigenvalues still given by (3.11).

## 4 The non-invertible symmetry at the dilute fixed point and in the RG flow

At the dilute critical point, the decomposition (3.9) carries over to the continuum limit, and hence holds for the fixed-point CFT as well. Each $d\mathcal{J}\mathcal{T}\mathcal{L}$ module becomes a sum over representations of $\text{Vir} \otimes \bar{\text{Vir}}$:

$$ \tilde{W}_{(1,1)} \mapsto \bigoplus_{s \in 2\mathbb{N}+1} \mathcal{R}_{(1,s)}, \qquad (4.1a) $$

$$ \tilde{W}_{(r,s)} \mapsto \bigoplus_{s' \in 2\mathbb{Z}+s} W_{(r,s)}, \qquad (4.1b) $$

where the notations for the representations and their content of primaries are as follows [2]:

<table><thead><tr><th>Notation</th><th>Indices</th><th>Name</th><th>Primary states</th></tr></thead><tbody><tr><td>R<sub>(1,s)</sub></td><td>s ∈ N*</td><td>Degenerate</td><td>(Δ<sub>(1,s)</sub>, Δ<sub>(1,s)</sub>)</td></tr><tr><td>W<sub>(r,s)</sub></td><td>r ∉ Z* or s ∉ Z*</td><td>Verma module</td><td>(Δ<sub>(r,s)</sub>, Δ<sub>(r,-s)</sub>)</td></tr><tr><td>W<sub>(r,s)</sub></td><td>r, s ∈ N*</td><td>Logarithmic</td><td>(Δ<sub>(r,s)</sub>, Δ<sub>(r,-s)</sub>)<br>(Δ<sub>(r,-s)</sub>, Δ<sub>(r,s)</sub>)<br>(Δ<sub>(r,-s)</sub>, Δ<sub>(r,-s)</sub>)</td></tr><tr><td>W<sub>(r,s)</sub></td><td>r, -s ∈ N*</td><td>0</td><td></td></tr></tbody></table>

Rather than dwell on the (irrelevant for us) details of these representations, let us give the corresponding characters (the superscript N stands for non-diagonal):

$$ \chi_{(1,s)}(\tau) = \left| \frac{e^{2\pi i \tau P_{(1,s)}^2} - e^{2\pi i \tau P_{(1,-s)}^2}}{\eta(\tau)} \right|^2, \qquad (4.3a) $$

$$ \chi_{(r,s)}^{N}(\tau) = \frac{e^{2\pi i \tau P_{(r,s)}^2} e^{2\pi i \tau P_{(r,-s)}^2}}{|\eta(\tau)|^2}, \qquad (4.3b) $$
where $\eta(\tau)$ is the Dedekind eta function. We have then

$$ \mathrm{Tr}_{\mathcal{R}_{(1,s)}} \left( e^{2\pi i \tau (L_0 - \frac{c}{24})} e^{2\pi i \bar{\tau} (\bar{L}_0 - \frac{c}{24})} \right) = \chi_{(1,s)}(\tau), \qquad (4.4a) $$

$$ \mathrm{Tr}_{W_{(r,s)}} \left( e^{2\pi i \tau (L_0 - \frac{c}{24})} e^{2\pi i \bar{\tau} (\bar{L}_0 - \frac{c}{24})} \right) = \begin{cases} \chi_{(r,s)}^N(\tau) & \text{if } r \notin \mathbb{Z}^* \text{ or } s \notin \mathbb{Z}^*, \\ \chi_{(r,s)}^N(\tau) + \chi_{(r,-s)}^N(\tau) & \text{if } r, s \in \mathbb{N}^*, \\ 0 & \text{if } r, -s \in \mathbb{N}^*. \end{cases} \qquad (4.4b) $$

Recall also the central charge

$$ c = 13 - 6\beta^2 - \frac{6}{\beta^2} \qquad (4.5) $$

together with the parametrization

$$ n = -2 \cos(\pi \beta^2), \quad \beta^2 \in [1, 2] \qquad (4.6) $$

($\beta^2$ is often denoted by the Coulomb gas coupling constant $g$ in the $O(n)$-model literature). Finally we have set

$$ P_{(r,s)} = \frac{1}{2}(\beta r - \beta^{-1}s), \qquad (4.7) $$

so the conformal weights are given by

$$ \Delta_{(r,s)} = P_{(r,s)}^2 - P_{(1,1)}^2. \qquad (4.8) $$

Since the topological nature of the defect line is exact on the lattice for all finite sizes, it holds as well in the continuum limit. In this limit, the $d\mathcal{J}\mathcal{P}\mathcal{L}$ algebra becomes the interchiral algebra $\tilde{\mathcal{C}}_{\beta^2}$, an algebra containing $\mathrm{Vir} \otimes \overline{\mathrm{Vir}}$ (denoted $\mathcal{C}_c$ in [2]) but extended by the degenerate field $V_{(1,3)}$:

$$ \mathcal{C}_c = \mathrm{Span} (T, \bar{T}) \subset \tilde{\mathcal{C}}_{\beta^2} = \mathrm{Span} (T, \bar{T}, V_{(1,3)}), \qquad (4.9) $$

$$ \lim_{\text{critical}} d\mathcal{J}\mathcal{P}\mathcal{L}(n) = \tilde{\mathcal{C}}_{\beta^2}. \qquad (4.10) $$

We see that the topological defects in fact commute not only with $\mathcal{C}_c$ but with this extended algebra as well:

$$ [D, \tilde{\mathcal{C}}_{\beta^2}] = 0, \qquad (4.11) $$

where we have denoted by the same symbol $D$ the defect on the lattice and in the continuum limit of the dilute critical point.

Now let us imagine perturbing this critical point by some interaction on the lattice, leading to a new Hamiltonian $H_{\mathrm{pert}}$. As long as this interaction can be expressed locally in terms of the elements in $d\mathcal{J}\mathcal{P}\mathcal{L}$ (which is the case for the thermal perturbation), it will give rise, in the continuum limit, to an expansion in terms of conformal fields that occur in the continuum limit of the $d\mathcal{J}\mathcal{P}\mathcal{L}$ identity module — which is what we have denoted $\mathcal{R}_{(1,s)}$. This is because every word $w$ in generators of the (dilute) algebra $d\mathcal{J}\mathcal{P}\mathcal{L}$ acting on the ground state of the theory will produce a state in $\tilde{W}_{(1,1)}$: $w|1\rangle = \sum_{\alpha} |w_{\alpha}\rangle$ — since this is a representation space of the algebra, and since it contains the ground state. By the
state-operator correspondence, each state $|w_\alpha\rangle$ will then correspond to an operator within $\mathcal{R}_{\langle 1,s \rangle}$. Hence we have

$$w \approx \sum_{\alpha} c_{\alpha} \Phi_{\alpha}, \qquad (4.12)$$

where the sum is over fields of the CFT in $\mathcal{R}_{\langle 1,s \rangle}$ — i.e. descendants under $\text{Vir} \otimes \overline{\text{Vir}}$ of the primary fields $\Phi_{\langle 1,s \rangle}$ — and the $c_\alpha$ are non-universal constants depending on the choice of lattice and the particular choice of microscopic model (as well as on the lattice spacing). It follows that, for any such perturbation expressed locally in terms of $d\mathcal{J}\mathcal{T}\mathcal{L}$, the resulting Hamiltonian still commutes with the topological defect, $[H_{\text{pert}}, \mathcal{D}] = 0$.

Now the point is that the four-leg operator does not belong to $\mathcal{R}_{\langle 1,s \rangle}$. This is of course obvious from the fact that its eigenvalue (3.11) under the defect operator is different from the one of the ground state (for $q$ generic). In fact, classifying fields using their associated $d\mathcal{J}\mathcal{T}\mathcal{L}$ module is equivalent to classifying them in terms of their $\mathcal{D}$ eigenvalues. Hence, once we know that the perturbation is expressed in terms of $d\mathcal{J}\mathcal{T}\mathcal{L}$, this guarantees that the four-leg operator cannot appear in the continuum limit of $H_{\text{pert}}$. Moreover, fields in the identity module are stable under operator product expansion — this is known explicitly from the fact that all conformal fields in the $\mathcal{R}_{\langle 1,s \rangle}$ are exactly degenerate. It also follows from the topological argument: insertion of several fields in the identity sector can always be represented on the lattice as a word $w|\mathbf{1}\rangle$, and thus again decomposes onto fields in the identity sector. Hence the four-leg operator never appears when doing the RG for $H_{\text{pert}}$, and cannot drive the flow away from the known, topological symmetry-preserving, IR dense fixed point.

We note that a similar argument was made in [21] to argue topological protection of “topological phases” in anyonic chains (see also [22]). In fact, the anyonic operator $Y$ of [21] coincides formally with $\mathcal{D}$ once formulated in terms of the Temperley-Lieb algebra. In this correspondence, however, the critical point considered in [21] coincides with the dense critical point of the $O(n)$ model — the situation is thus physically different from the one we are interested in here.

The four-leg operator is prevented from appearing not by $O(n)$ symmetry (since it can occur within the singlet representation, as argued in the Introduction) but by the “non-invertible symmetry” which is a more elaborate way to say that lattice loops are prevented from crossing. We notice that it is generally believed that allowing crossings at the critical point would not change the universality class — in agreement with the fact that the four-leg operator at this fixed point is irrelevant (this is discussed more fully in appendix A).

In the case with crossings, however, the non-invertible symmetry would not be present on the lattice, since the four-leg operator does not commute with the $d\mathcal{J}\mathcal{T}\mathcal{L}$ algebra, and, in its presence, the defect line cannot be moved around any longer. Here is an example (where we call $\Pi_{12}$ the operator crossing lines 1 and 2): we have

$$\begin{aligned} \mathrm{D}\Pi_{12}|abc\rangle = & (-q)^{-3/2}|cba\rangle + (-q)^{3/2}|acb\rangle + \delta_{ab}\left( (-q)^{1/2}\sum_d|dcd\rangle + (-q)^{-1/2}\sum_d|cdd\rangle \right) \\ & + \delta_{ac}\left( (-q)^{1/2}\sum_d|ddb\rangle + (-q)^{-1/2}\sum_d|dbd\rangle \right) + \delta_{bc}\left( (-q)^{1/2}\sum_d|add\rangle + (-q)^{-1/2}\sum_d|dda\rangle \right) \end{aligned}$$
<figure><img src="image_4.png" /><figcaption>**Figure 3.** The insertion of a pair of disorder fields affects loops encircling the extremities of the corresponding defect line.</figcaption></figure>

and<fnref n="10" />

$$ \Pi_{12}D|abc\rangle = (-q)^{-3/2}|acb\rangle + (-q)^{3/2}|cba\rangle + \delta_{ab} \left( (-q)^{1/2} \sum_d |cdd\rangle + (-q)^{-1/2} \sum_d |dcd\rangle \right) \\ + \delta_{bc} \left( (-q)^{1/2} \sum_d |dda\rangle + (-q)^{-1/2} \sum_d |add\rangle \right) + \delta_{ac} \left( (-q)^{1/2} \sum_d |dbd\rangle + (-q)^{-1/2} \sum_d |ddb\rangle \right). $$

With crossings allowed (giving rise to what is sometimes called self-avoiding trails [24] when $n = 0$), the dilute critical point remains the same. In this case, therefore, the degeneracies in the partition function appear only in the continuum limit, and so does the topological symmetry. Increasing the fugacity of monomers is now a perturbation that will generically involve a component on the four-leg operator, since the lattice model is not topologically protected: the flow will then look as in figure 1.

# 5 The defect Hilbert space

Consider now the situation where we have an open version of our defect line, which we take to start at the origin and end at the point $(z, \bar{z})$. By definition, the $O(n)$ model loops will then pass “under” this defect line, as shown in figure 3. This means that only loops encircling one (and only one) of the two end-points of the defect line will be affected by its presence. By decomposing the corresponding crossings as usual, using (2.5), we see that we obtain essentially a one leg-operator, but care must be taken. On the one hand, any given loop encircling one of the extremities has, before decomposition of the crossings, a fugacity $n$ that disappears after decomposition; on the other hand, each term in the decomposition comes with a factor $(-q)^{\pm 1/2}$.

To handle these two facts properly, it is convenient to map the system onto the cylinder. We are then in a situation where the TDL, instead of acting as the operator $D$, runs parallel to the imaginary time direction and gives rise to a modified “defect” Hilbert space [8]. In the absence of through lines and of defects, the weight $n$ for non-contractible loops can be obtained by orienting these loops, summing over the two possible orientations, and giving oriented lines that go around the system a factor $z$ or $z^{-1}$. To get the correct loop fugacity $n = q + q^{-1}$, we should set $z = q$.

<footnote n="10">¹⁰The reader may notice that these two expressions only differ by $q \to q^{-1}$, implying that $D\Pi_{12} = \Pi_{12}\bar{D}$ in this example. This identity is however a coincidence for this small size, and does not hold for larger $N$, as can be verified by going through the same computation for $N = 4$.</footnote>
In the presence of the TDL we decompose the crossings and see now that the problem splits into two sectors, one with a new parameter $z_1 = (-q)^{1/2}z$ and the other with $z_2 = (-q)^{-1/2}z$. Moreover, there is now one throughline traversing the system.

Choosing then $z = q$, we see that we get $z_1 = -(-q)^{3/2}$ and $z_2 = -(-q)^{1/2}$. For a general value $z = e^{i\pi e_\phi}$, the associated conformal weights are well known from the analysis of the continuum limit of modified loop models, and read, in the Kac-table notation of (4.8),

$$ (\Delta, \bar{\Delta}) = (\Delta_{(1/2, -e_\phi)}, \Delta_{(1/2, e_\phi)}). \qquad (5.1) $$

The two special choices of z-factors give, after reshuffling the labels, exponents $(\Delta_1, \bar{\Delta}_1) = (\Delta_{(2,1)}, 0)$ and $(\Delta_2, \bar{\Delta}_2) = (\Delta_{(0,1)}, 0)$.

A more thorough study (to appear in the sequel of this paper) would show that the second field disappears from the spectrum because the identity itself is degenerate<fnref n="11" /> (and contributes as the field $\langle 1, 1 \rangle$) in the CFT. The defect Hilbert space $(\mathcal{H}_{O(n)})_D$ [8] therefore has $(\Delta_{(2,1)}, 0)$ as its lowest excitation.<fnref n="12" /> It turns out, in fact, that the whole $(\mathcal{H}_{O(n)})_D$ can obtained by fusion with a degenerate field with those weights, so one has

$$ (\mathcal{H}_{O(n)})_D = \bigoplus_{s \in 2\mathbb{N}+1} \mathcal{R}_{\langle 2,1;s \rangle} \otimes [] \oplus \dots, \qquad (5.2) $$

where for simplicity we have refrained to get into the details of the whole field content. The characters for the corresponding representations indicated in (5.2) are

$$ \chi_{\langle 2,1;s \rangle}(\tau) = \frac{e^{2\pi i \tau P_{(2,s)}^2} - e^{2\pi i \tau P_{(2,-s)}^2}}{\eta(\tau)} \times \frac{e^{2\pi i \bar{\tau} P_{(1,s)}^2} - e^{2\pi i \bar{\tau} P_{(1,-s)}^2}}{\eta(\bar{\tau})} \qquad (5.3) $$

Of course, the defect $\bar{D}$ would give identical results with the chiral and antichiral sectors exchanged.

Notice that the geometry of a cylinder with the TDL running parallel to the imaginary time direction should be related with the one where the TDL acts as the operator $\mathcal{D}$ by modular transformation. This leads to the interpretation of, for instance, $\langle 1|\mathcal{D}|1 \rangle$ as a ratio of modular transformation matrix elements, something we will discuss in the sequel. The important point for now is that our TDLs are a close relative of Verlinde lines [8], although of course the underlying CFT is not rational. The same would hold for the higher-spin defects (3.14), which are associated with a degenerate field with weights $(\Delta_{(2j+1,1)}, 0)$. Known fusion rules for these fields (which mimic (3.14)) guarantee that the defect lines are not invertible, as claimed as the beginning of this paper.

As an aside, we notice that the field with weight $(\Delta_{(2,1)}, 0)$ is mutually local with all the fields in the identity sector, in particular the diagonal primaries $\langle 1, s \rangle$, $s \in 2\mathbb{N} + 1$. This extends to all “higher” disorder fields with conformal weights $(\Delta_{(r,1)}, 0)$ thanks to the identity

$$ \Delta_{(r,1)} + \Delta_{(1,s)} - \Delta_{(r,s)} = \frac{(r-1)(s-1)}{2}, \qquad (5.4) $$

which is an integer for $r$ integer, and $s \in 2\mathbb{N} + 1$.<fnref n="13" />

<footnote n="11">¹¹Meaning, its null descendent is zero indeed.</footnote>

<footnote n="12">¹²This calculation holds in the dilute as well as in the dense case. In the latter case, however, our conventions where the Kac formula is parametrized by $\beta^2$ corresponds, in the conventions of [17], to $(\Delta_{(1,2)}, 0)$, as found (using the Bethe ansatz) in this reference.</footnote>

<footnote n="13">¹³Similar properties were noticed in [18], although the relationship with our work is not clear to us.</footnote>
# 6 Conclusion

We conclude this paper with some general comments and possible generalizations.

**Crossings versus self-attractions.** First, the reader might be confused by the fact that an operator that “looks like the four-leg operator” is often considered in the self-avoiding walk problem in the context of self-attracting chains. It is possible indeed to consider a general $O(n)$ loop model where neighboring segments are favored with some fugacity $\mu$ greater than one. At the level of the transfer matrix, this corresponds to multiplying by $\mu$ the last two terms in (3.2). As long as $\mu$ is not too large, the universality class remains the same (although the value of the critical coupling depends on $\mu$), but for a critical value $\mu_c$ it becomes different and, in the case $n=0$, corresponds to “polymers at the theta point”. It is important to understand that the operator $\tilde{\epsilon}$ that couples to the parameter $\mu$ is, despite the rough similarity of the geometry, not the four-leg operator — precisely because it does not break the topological symmetry. On the lattice, the four-leg operator is represented instead by the last diagram in (A.1), corresponding to the crossing of two lines or the operator denoted $\Pi$ in the $O(n)$ language. The operator $\tilde{\epsilon}$, by contrast, is in the sector of the identity, and can be obtained by fusion of the energy operator with itself. In the dilute $O(n)$ model the energy operator corresponds to $\Phi_{\langle 1,3 \rangle}$, a field which remains exactly degenerate at level 3 (despite the theory being in general logarithmic): it follows that $\tilde{\epsilon} \propto \Phi_{\langle 1,5 \rangle}$. Related aspects are discussed in [31].

**Classification of topological defects.** As mentioned at the beginning of this paper, the loop model admits also invertible defects [8] where the topological lines are now associated with a group element of $O(n)$. Such defects were already considered in [2] and obtained in the lattice formulation by modifying the basic spin-spin coupling for bonds intersecting the defect line: $\vec{S}_x \cdot \vec{S}_y \mapsto \vec{S}_x \cdot g\vec{S}_y$ where $g \in O(n)$. For this to make sense, a convention is needed — say, after orienting the defect line, that this applies for $x$ to the left of the line and $y$ to the right. In the plane, loops intersecting the defect line an even number of times do not get their weight modified, as a result of $O(n)$ symmetry. For instance, a loop intersecting the defect line twice will get a weight $\sum_{a,b} g_{ab}^2 = \text{Tr } gg^t = n$, where $g^t$ denotes the transpose of $g$, and we used $gg^t = 1$.

Denoting such defects by $D(g)$, the opposite orientation of the defect line would define similarly $\bar{D}(g) := D(g^t)$. In the plane, note that when such a defect line is inserted at a given point, loops surrounding this point get a modified weight $n \mapsto \text{Tr } g$. Hence it seems possible to think of the diagonal operators of [23] as operators sitting at the end of defect lines.<fnref n="14" /> This aspect, how our non-invertible lines can be considered as Verlinde lines, and the more general question of classifying topological defects for the $O(n)$ CFT, will be discussed in the sequel.

We also notice that in the recent paper [30], correlation functions in loop models were closely associated with combinatorial maps. This suggests that there might yet be another, “combinatorial” type of defect in this problem.<fnref n="15" />

<footnote n="14">¹⁴They are called “defect operators” standard terminology of [8], not to be confused with the operator version of the TDL such as D.</footnote>

<footnote n="15">¹⁵S. Ribault, private communication.</footnote>
Finally, we mention that it is an outstanding question whether one can make rigorous
sense of TDLs in terms of Deligne categories [16]. Here and in [2], we have pursued the
pragmatic approach of making sense of computations with topological defect lines (be they
invertible or non-invertible) for non-integer *n* in terms of naive diagrammatic calculus and
analytic continuation (in particular, when discussing RG flows), leaving for future work
the question of whether this can be lifted to a more rigorous setting. It is encouraging in
this respect that the current in the *O(n)* model — in some sense the infinitesimal form of
a defect lines — has already been given a categorial sense in [16]. We shall have more to
say about the *O(n)* current in other future work.

Acknowledgments

We thank L. Grans-Samuelsson, R. Nivesvivat and S. Ribault for related collaborations, many interesting discussions, and a careful reading of the manuscript. We are also grateful to V. Gorbenko, S. Rychkov and B. Zan for remarks and correspondence. This work was supported by the French Agence Nationale de la Recherche (ANR) under grant ANR-21-CE40-0003 (project CONFICA).

A Allowing crossings at the dilute critical point

In this appendix we present a numerical study of the stability of the *O(n)* model towards
the inclusion of some amount of crossing. We first examine the global phase diagram,
giving evidence that crossings are irrelevant at the dilute critical point, whereas they are
relevant in the dense phase and drive the system to the *σ*-model fixed point shown in
figure 1. Next we investigate in more details the dilute critical point, finding notably that
while the crossings lead to a decrease of the critical monomer fugacity *K*<sub>c</sub>, the universality
class, as measured by the central charge, is the same as in the model without crossings.
And finally we study the same issue from the point of view of critical exponents, finding
that the crossings nevertheless lead to more subtle and non-trivial algebraic effects that
can be appreciated by comparing with the branching rules from the (dilute) Brauer to the
dJTL algebra.

A.1 Model

We define a *dilute Brauer model* on the square lattice with the following allowed vertex
diagrams and weights:

<figure><img src="image_5.png" /></figure>

The meaning of the last diagram is that the two strands cross. The weights, distributed
over half-edges in the usual way, can be summarised by saying that there is a weight *K*
per monomer and *w* per crossing, and in addition a factor *n* per loop.
Notice that even when $w = 0$, this model does not quite simplify to that of (3.1). We have omitted for simplicity the last two diagrams in (3.1) in which two distinct loop strands come into close contact at the vertex. We have also rotated the vertices to make clear that we study the model on an axially oriented square lattice (auxiliary-space geometry), wrapped onto a cylinder of circumference $L$ lattice sites. The numerical technique is the exact diagonalization of the corresponding transfer matrix.

The dilute Brauer model with $w = 0$ reduces in the $n \to 0$ limit to a much-studied self-avoiding walk (SAW) model on the square lattice. In this case the location of the critical point is known very precisely [25] from the method of graph polynomials [26, 27],

$$K_c = 0.379052277755161(5), \qquad (A.1)$$

and exact enumeration studies have provided very compelling evidence [28] that the universality class is that of the dilute $O(n \to 0)$ model.

For $n = 1$ the loops are in bijection with the domain walls of an Ising model defined on the dual lattice. Then $K$ is the weight of a piece of domain wall, and the range $w > 1$ (resp. $0 \le w < 1$) corresponds to a four-spin interaction that favours (resp. penalises) the alternating arrangement $+ - + -$ of spins around a vertex. The case $w = 1$ is just the usual Ising model without such a four-spin interaction, and the critical point in then

$$K_c = \sqrt{2} - 1. \qquad (A.2)$$

## A.2 Phase diagram

For convenience we pick a generic value of $n$, far from the root of unity cases. We take $n = 1/\sqrt{2}$. From the finite-size free energy per vertex $f(L)$ on cylinders of circumference $L$, $L-1$, $L-2$ we fit for the bulk free energy $f(\infty)$, the effective central charge $c_{\text{eff}}$ and the non-universal amplitude $A$ related to $T\bar{T}$, via the formula

$$f(L) = f(\infty) - \frac{\pi c_{\text{eff}}}{6L^2} + \frac{A}{L^4}. \qquad (A.3)$$

We can also omit the $1/L^4$ term and study fits based on only two sizes $L, L-1$.

We first study the global phase diagram and the flow from the dilute to the dense phase. Effective central charges $c_{\text{eff}}(K)$ are shown in figure 4. The first row of figures corresponds to the case without crossings ($w = 0$), in which the usual flow from the dilute to the dense $O(n)$ fixed point is observed. The upper (resp. lower) horizontal line shows for comparison the analytical value for the dilute (resp. dense) $O(n)$ model, namely $c = 0.357946...$ (resp. $c = -0.445833...$).

The next rows include crossings and show the cases $w = 0.8, 1.6$. They appear to be compatible with a flow to the Goldstone phase with central charge $c = n-1$ (shown for comparison by the middle horizontal line). The convergence is slow, even for the three-point fits (right panels), as was previously observed in the fully-dense case [5].

## A.3 Universality class in the dilute case

In the dilute case with $w \ge 0$, we observe a peak in $c_{\text{eff}} = c_{\text{eff}}(K)$, whose position gives a finite-size estimate $K_c(L)$ of the critical monomer weight and a finite-size estimate $c(L) =$
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Effective central charge $c_{\text{eff}}(K)$ 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 $L = 6, \dots, 11$, with the purple curves corresponding to the largest sizes.</figcaption></figure>

$c_{\text{eff}}(K_c(L))$ of the central charge. Both of these estimates can be well extrapolated to the thermodynamic limit $L \to \infty$ by using a polynomial fit in the variable $1/L^2$, and error bars can be judged by comparing different orders of the fit. Using data up to size $L = 13$ we
obtain the following values (the estimated error bars affect that last digit shown):

<table><thead><tr><th>w</th><th>K<sub>c</sub></th><th>c</th></tr></thead><tbody><tr><td>0.0</td><td>0.406178</td><td>0.35784</td></tr><tr><td>0.1</td><td>0.405558</td><td>0.35787</td></tr><tr><td>0.2</td><td>0.404935</td><td>0.35790</td></tr><tr><td>0.4</td><td>0.403686</td><td>0.35794</td></tr><tr><td>0.8</td><td>0.401165</td><td>0.35798</td></tr><tr><td>1.6</td><td>0.396004</td><td>0.35802</td></tr><tr><td>3.2</td><td>0.385022</td><td>0.35789</td></tr></tbody></table>

The value of $K_c$ decreases with $w$, as it should. Moreover, $c$ agrees with the analytical value for the dilute $O(n)$ model to four significant digits, for any of the $w$ considered. Based on this evidence we conjecture that the dilute Brauer model for any $0 \le n < 2$ and any finite $w \ge 0$ flows to the dilute $O(n)$ fixed point.

## A.4 Critical exponents

We now turn to the study of critical exponents, still at the dilute critical point. In the dilute Brauer model we impose the propagation of $2j$ throughlines (with $j \in \mathbb{Z}/2$) carrying an irrep $\lambda$ of the symmetric group $S_{2j}$. Experience teaches us that such a study is very sensitive to the precise determination of $K_c$. We therefore focus on the Ising case, $n = 1$, for which the critical point $(w_c, K_c) = (1, \sqrt{2}-1)$ is known analytically.

The methodology for obtaining precise estimates of the critical exponents has been explained in [29, appendix A.5]. The quality of the fits varies considerably with the number of available sizes. The preliminary results below are based on data up to $L = 11$ in all cases, $L = 12$ in most cases, and $L = 13$ for the $\lambda = []$ sector.

The critical exponents for the Jones-Temperley-Lieb (JTL) standard modules $\Lambda_{(r,s)}$ to be compared with are denoted

$$x_{(r,s)} = \Delta_{(r,s)} + \Delta_{(r,-s)} \qquad (A.5)$$

with the Kac table for the Ising model being

$$\Delta_{(r,s)} = \frac{(4r - 3s)^2 - 1}{48}. \qquad (A.6)$$

We list here for reference the first few exponents for JTL primaries:

$$\begin{align*}
x_{(\frac{1}{2},0)} &= \frac{1}{8} = 0.125, \\
x_{(1,0)} &= \frac{5}{8} = 0.625, & x_{(1,1)} &= 1, \\
x_{(\frac{3}{2},0)} &= \frac{35}{24} = 1.45833, & x_{(\frac{3}{2},\frac{2}{3})} &= \frac{13}{8} = 1.625, \tag{A.7} \\
x_{(2,0)} &= \frac{21}{8} = 2.625, & x_{(2,\frac{1}{2})} &= \frac{87}{32} = 2.71875, & x_{(2,1)} &= 3, \\
x_{(\frac{5}{2},0)} &= \frac{33}{8} = 4.125, & x_{(\frac{5}{2},\frac{2}{5})} &= \frac{837}{200} = 4.185.
\end{align*}$$
For the eigenvalues in the $\lambda = []$ sector (no throughlines) we obtain the following results:

$$ \begin{array}{|c|c|c|c|} \hline \text{No.} & \text{Mult.} & x \text{( numerics )} & x \text{( exact )} \\ \hline 1 & 1 & 0 & 0 \\ 2 & 1 & 0.99999999 & 1 \\ 3 & 4 & 2.0002 & 2 \\ 4 & 4 & 2.99995 & 3 \\ 5 & 1 & 2.66 & 2.625 \\ 6 & 1 & 3.0003 & 3 \\ 7 & 2 & 3.0006 & 3 \\ 8 & 4 & 3.97 & 4 \\ 9 & 2 & 3.64 & 3.625 \\ 10 & 4 & 4.0001 & 4 \\ 11 & 2 & ? & ? \\ 12 & 2 & ? & ? \\ 13 & 4 & 4.97 & 5 \\ \hline \end{array} \qquad (A.8) $$

where the columns denote the eigenvalue number, its multiplicity (valid for sufficiently large $L$, and in most cases for all $L$), the corresponding numerically determined exponent (with a number a digits roughly reflecting the quality of the fit), and finally the conjectured exact value (with question marks indicating uncertain cases). These results seem consistent with the Brauer $\downarrow$ JTL branching rule [2, eq. (B.4a)]

$$ B_{[1]} \downarrow \overline{W}_{0} + W_{(2,0)} + \cdots . \qquad (A.9) $$

Notice that we also observe a descendent state in the $W_{2,0}$ module. Although multiplicities would have to be explained, the results (A.8) confirm that the inclusion of crossings at the dilute critical point preserves not only the central charge but also the critical exponents, up to the expected reshuffling of sectors implied by the branching rules.

For eigenvalues in the $\lambda = [1]$ sector we obtain:

$$ \begin{array}{|c|c|c|c|} \hline \text{No.} & \text{Mult.} & x \text{( numerics )} & x \text{( exact )} \\ \hline 1 & 1 & 0.124999 & 0.125 \\ 2 & 2 & 1.12497 & 1.125 \\ 3 & 2 & 1.633 & 1.625 \\ 4 & 2 & 2.124 & 2.125 \\ 5 & 2 & 2.15 & 2.125 \\ 6 & 1 & 2.124998 & 2.125 \\ 7 & 2 & 2.6 & 2.625 \\ 8 & 1 & 2.62 & 2.625 \\ 9 & 2 & 3.11 & 3.125 \\ 10 & 2 & 2.7 ? & 2.625 \\ 11 & 1 & 2.64 & 2.625 \\ \hline \end{array} \qquad (A.10) $$

This already appears more surprising. The branching rule is here [2, eq. (B.4b)]

$$ B_{[1]} \downarrow W_{(\frac{1}{2},0)} + W_{(\frac{5}{2},0)} + \cdots . \qquad (A.11) $$
But we see here with a good precision an exponent $x = 1.625$, which must correspond to a primary (since there is no $x = 0.625$), and hence it can only be $x(\frac{3}{2}, \frac{2}{3})$, although the representation $W(\frac{3}{2}, \frac{2}{3})$ does not appear on the right-hand side of the branching rule. We also observe several descendents of that primary.

It should be noticed that if we had diagonalised a transfer matrix for a loop model without crossings (i.e., an element of the $dJTL$ algebra) within a given standard module of the dilute Brauer algebra, we would *by definition* obtain the critical exponents of the dilute model up to the reshuffling implied by the branching rules. As a matter of fact, this is precisely how the branching rules were obtained in [2, appendix B], albeit in the fully-dense incarnations of both models (but the results should carry over unchanged by Morita equivalence). However, this is not quite what is done here, since we use a transfer matrix in a loop model where crossings are now allowed. Even if those crossings are not expected to change the universality class, they could — and as we have just observed, do — change the identification of critical exponents within each representation $\lambda$.

We leave for future work the analytical understanding of exactly how critical exponents are reshuffled within this scenario, and here content ourselves with reporting the numerical results for a few more sectors. For eigenvalues in the $\lambda = [2]$ sector we obtain:

<table><thead><tr><th>No.</th><th>Mult.</th><th>x ( numerics )</th><th>x ( exact )</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>0.622</td><td>0.625</td></tr><tr><td>2</td><td>2</td><td>1.620</td><td>1.625</td></tr><tr><td>3</td><td>2</td><td>2.12</td><td>?</td></tr><tr><td>4</td><td>2</td><td>2.63</td><td>2.625</td></tr><tr><td>5</td><td>1</td><td>2.61</td><td>2.625</td></tr><tr><td>6</td><td>2</td><td>2.6</td><td>2.625</td></tr><tr><td>7</td><td>2</td><td>3.2 ?</td><td>?</td></tr></tbody></table>
(A.12)

The branching rule is now $B_{[2]} \downarrow W_{(1,0)} + W_{(2,0)} + W_{(2,1)} + \cdots$. The third exponent in the table is surprising and it is not yet clear what it corresponds to. The seventh exponent is possibly its descendent, but the quality of the fit is not very convincing. We would need more eigenvalues to confirm the presence of $W_{2,1}$.

For eigenvalues in the $\lambda = [11]$ sector we obtain:

<table><thead><tr><th>No.</th><th>Mult.</th><th>x ( numerics )</th><th>x ( exact )</th></tr></thead><tbody><tr><td>1</td><td>2</td><td>1.002</td><td>1</td></tr><tr><td>2</td><td>2</td><td>1.99</td><td>2</td></tr><tr><td>3</td><td>2</td><td>3.08</td><td>3 ?</td></tr><tr><td>4</td><td>2 ?</td><td>4.06</td><td>4</td></tr></tbody></table>
(A.13)

There are many complex eigenvalues in this sector which we have omitted from the fits. These do *not* appear to move deeper into the spectrum upon increasing $L$. In particular, the first complex eigenvalues (part of a complex conjugate doublet) is number 8, 9, 7, 5 (counting multiplicities in the numbering) in the spectrum sorted by largest real part for the sizes $L = 8, 9, 10, 11$. The branching rule is here $B_{[1,1]} \downarrow W_{(1,1)} + W_{(2,\pm\frac{1}{2})} + \cdots$. More eigenvalues would be needed to confirm the presence of $W_{(2,\frac{1}{2})}$.
For eigenvalues in the $\lambda = [3]$ sector we obtain:

<table><thead><tr><th>No.</th><th>Mult.</th><th>x ( numerics )</th><th>x ( exact )</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1.449</td><td>1.45833</td></tr><tr><td>2</td><td>2</td><td>2.447</td><td>2.45833</td></tr><tr><td>3</td><td>2</td><td>2.95</td><td>?</td></tr><tr><td>4</td><td>2</td><td>3.44</td><td>3.45833</td></tr><tr><td>5</td><td>2</td><td>3.46</td><td>3.45833</td></tr><tr><td>6</td><td>1 ?</td><td>3.42</td><td>3.45833</td></tr></tbody></table>

The branching rule is $B_{[3]} \downarrow W_{(\frac{3}{2},0)} + W_{(\frac{5}{2},0)} + \cdots$. The third exponent in the table is surprising. The quality of the corresponding fit is reasonable, but not excellent. More eigenvalues would be needed to confirm the presence of $W_{(\frac{5}{2},0)}$.

For eigenvalues in the $\lambda = [21]$ sector we obtain:

<table><thead><tr><th>No.</th><th>Mult.</th><th>x ( numerics )</th><th>x ( exact )</th></tr></thead><tbody><tr><td>1</td><td>2</td><td>1.633</td><td>1.625</td></tr><tr><td>2</td><td>2</td><td>2.15</td><td>?</td></tr><tr><td>3</td><td>2</td><td>2.59</td><td>2.625</td></tr><tr><td>4</td><td>1</td><td>2.638</td><td>2.625</td></tr><tr><td>5</td><td>1</td><td>?</td><td>?</td></tr><tr><td>6</td><td>2 ?</td><td>?</td><td>?</td></tr></tbody></table>

The branching rule is $B_{[2,1]} \downarrow 2W_{(\frac{3}{2},\pm\frac{2}{3})} + 2W_{(\frac{5}{2},0)} + 4W_{(\frac{5}{2},\pm\frac{2}{5})} + \cdots$. The second exponent is surprising, but the fit is well behaved. The last two fits are uncommonly capricious, even with $L = 12$ data being available, so I refrain from reporting any value of the corresponding exponents.

Finally, for eigenvalues in the $\lambda = [111]$ sector we obtain:

<table><thead><tr><th>No.</th><th>Mult.</th><th>x ( numerics )</th><th>x ( exact )</th></tr></thead><tbody><tr><td>1</td><td>1</td><td>1.48</td><td>1.45833</td></tr><tr><td>2</td><td>2</td><td>2.456</td><td>2.45833</td></tr><tr><td>3</td><td>2</td><td>3.0 ?</td><td>?</td></tr><tr><td>4</td><td>2</td><td>?</td><td>?</td></tr><tr><td>5</td><td>2</td><td>3.4 ?</td><td>3.45833 ?</td></tr><tr><td>6</td><td>2</td><td>3.9 ?</td><td>?</td></tr></tbody></table>

A very encouraging fact is that this resembles the results for $\lambda = [3]$. Indeed, the representations $B_{[3]}$ and $B_{[111]}$ have the same branching rules with respect to JTL. It should be stressed that the finite-size eigenvalues are completely different for the two cases when (as here) $w > 0$. In particular, the first eigenvalue in the [3] sector gives rise to an exponent that converges to $x_{(\frac{3}{2},0)}$ from below, whereas the first exponent from the [111] sector converges to that same value from above.
## B The non-invertible symmetry with open boundary conditions

The study of the $O(n)$ model on a strip (i.e. with open boundary conditions) reveals a pattern of degeneracies which is similar to — but different from — the pattern on the cylinder. In this case, a genuine symmetry has been identified to explain these degeneracies, and involves mixing of $O(n)$ representations under an algebra equivalent in a certain (Morita) sense to $U_qsl(2)$: in particular, for this symmetry, tensor products can be defined, and the multiplicities be written as $q$-dimensions.

It is tempting nevertheless to see whether the consideration of a possible topological symmetry in this case might lead to further insight. In the open case, we cannot have a loop going around the system since the algebra does not contain the shift operator $\tau$. Instead, what we can have is a loop going over the lines, bouncing on the boundary, then going under and bouncing again. This is in fact described in detail in [20], where it is shown that this defect (denoted d here) is diagonal on every $\mathcal{T}\mathcal{L}$ module $W_r^{16}$ with eigenvalue

$$ \langle 1_r | d | 1_r \rangle = q^{2r+1} + q^{-2r-1}, \qquad (B.1) $$

that is, d acts in fact as the Casimir of $U_qsl(2)$. We can illustrate this using $O(n)$ variables: for instance on two sites,

$$ d|ab\rangle = \sum_c I \otimes I \otimes \langle c|g_2g_1^2g_2|abc\rangle, \qquad (B.2) $$

as illustrated in figure 5, where we have rewritten (2.5) as

$$ g_i = (-q)^{1/2} + (-q)^{-1/2}e_i. \qquad (B.3) $$

A short calculation gives

$$ d|ab\rangle = (q^3 + q^{-3}) \left( |ab\rangle - \frac{\delta_{ab}}{n} \sum_c |cc\rangle \right) + n \frac{\delta_{ab}}{n} \sum_c |cc\rangle \qquad (B.4) $$

with $n = q + q^{-1}$. We see that d is equal to the $U_qsl(2)$ Casimir, with eigenvalue $q^3 + q^{-3}$ on [2] and $q + q^{-1}$ on [].

In general we can write schematically

$$ d = \sum_r (q^{2r+1} + q^{-2r-1}) P_r, \qquad (B.5) $$

where $P_r$ is the Jones-Wenzl projector onto the sector with $2r$ non-contractible lines. In this case, therefore, the potential “topological symmetry” simply reveals the underlying quantum-group structure of the problem. Since in the periodic case there is no such known structure, it is likely that the topological defect, in a sense, replaces it.

<footnote n="16">¹⁶Recall that these are modules with $2r$ throughlines.</footnote>
<figure><img src="image_7.png" /><figcaption>Figure 5. The d-operator in the open case.</figcaption></figure>

## C TDLs with crossings: the (dilute) Brauer algebra case

As commented earlier, the defect operator D is built out of the solution of the spectral-parameter independent Yang-Baxter equation in the Temperley-Lieb algebra. It is tempting to ask what would happen if we considered the same question in the case of the Brauer algebra.

Consider therefore the $O(n)$ integrable $\tilde{R}$ matrix acting in $[1]^{\otimes 2}$:

$$ \tilde{R}(u) = P_{[1]} - \frac{u - i\pi}{u + i\pi} P_{[11]} + \frac{u - i\pi (n - 2)u - i\pi}{u + i\pi (n - 2)u + i\pi} P_{[2]} \quad (C.1) $$

Solutions of the spectral-parameter independent Yang-Baxter equation are obtained by considering the limit $u \to \infty$, where we obtain

$$ \tilde{R}(u = \infty) = P_{[1]} - P_{[11]} + P_{[2]} = \Pi. \quad (C.2) $$

This is, in fact, the simple permutation operator, represented by two lines crossing. Hence, for Brauer, our construction is not relevant: the “topological loop” is just a loop crossing all the other lines, and it can be eliminated with a factor $n$, due to the general rules of the Brauer algebra.

The knowledgeable reader might also point out that potential topological defect operators can be obtained by considering the center of the algebra underlying the transfer matrix. This object for the Brauer algebra has of course been considered in the mathematics literature, and gives rise to the consideration of Jucys-Murphy elements (see e.g. [19]). In the end, the consideration of the center allows one to determine separately every irreducible $O(n)$ representation that appears in the tensor product $[1]^{\otimes N}$. Like in the open case, it thus reveals in a certain sense the underlying $O(n)$ structure potentially hidden beyond the loop model, but does not provide additional information.

In contrast, when crossings are not allowed, we get an algebra (the $uJTL$ or $dJTL$ algebra) which is smaller than Brauer, and thus the centralizer has to be bigger — leading to the multiplicities observed in the $O(n)$ model partition functions.

**Open Access.** This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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