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.


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] where we used the standard row-length notation [λ] 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 1 .
While having representations with four boxes (one per leg) in this content is fully expected, the emergence of the [2] and the [ ] representations is more surprising.As discussed in [2], this is related with the fact (to be discussed in more detail) that lines cannot cross, and is interpreted Figure 1: With only O(n) symmetry "protection", the 4-leg operator shold be generated under the RG, and the flow go to the σ-model fixed point [5] instead of the dense one.
mathematically in terms of branching rules from the Brauer algebra down to the (so-called "unoriented Jones") Temperley-Lieb algebra.What matters for the puzzle raised in [1] is that there exists a copy of the four-leg operator coming in the trivial representation-i.e. an O(n) scalar.Now imagine perturbing the critical O(n) model by the energy operator, itself an O(n) scalar with conformal weights ∆ = ∆ = ∆ (1,3) (see below for conventions).This perturbation is, for the sign that corresponds to increasing the fugacity of monomers, known to drive the model into the dense critical phase [3,4].With respect to this renormalization group (RG) flow, the four-leg operator is dangerously irrelevant: while it is irrelevant at the (dilute) ultraviolet (UV) critical point, it is relevant at the dense infrared (IR) critical point.For instance, the case of selfavoiding walks (SAW) corresponds to the n → 0 limit, where the corresponding scaling dimensions x = ∆ + ∆ at either end of the RG flow take the values x UV = 35  12 > 2 and x IR = 3 4 < 2. Since there is a copy of the four-leg operator that comes with the scalar O(n) representation, the O(n) symmetry by itself is not sufficient to prevent this operator from "interfering" with the RG flow induced by the energy-operator.With only O(n) symmetry, there is therefore no reason why the dense fixed point should ever be reached: rather, one should expect an RG trajectory where models get close enough to that fixed point, but eventually reach the more generic O(n) low-temperature phase studied in [5,6,7].See the cartoon in figure 1.
Yet, there is large numerical evidence over more than 30 years of research that this is not what happens, and that in all cases (in particular, irrespectively of the lattice) the dense fixed point is indeed reached by the RG flow when the monomer fugacity is increased beyond the critical value, without undue interference from the dangerously irrelevant four-leg operator.This is the puzzle.
Of course, the fact that the four-leg operator comes with a sum of O(n) representations suggests that there is a hidden symmetry at play-a symmetry glueing together these various representations, and for which the copy of the trivial O(n) representation [ ] appearing 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.

General idea
Topological defects are conveniently introduced in CFTs (see e.g.[9,10] for early references) as operators2 acting on the Hilbert space H of the theory, which it itself a sum of representations of Vir ⊗ Vir.These operators, which we will denote generically by D, must commute with Vir and Vir: or, equivalently, commute independently with the left-and right-moving components of the stressenergy tensor.The corresponding objects in the Euclidian description are topological defect-lines (TDL), which we denote by the symbol 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 matrix) propagates in the imaginary-time direction.It is natural to define a lattice equivalent 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 without 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, 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 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 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 hightemperature 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 and we interpret over-and under-crossings as giving rise to the decompositions 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, 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: The final result means that red loops can be moved around without affecting the partition function, as summarized in the graphical identity 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 topological loop simply multiplies the partition function by a factor n. We shall denote in what follows statistical averages associated with the partition function by • , leading to We can of course write a similar statement on the cylinder.Denoting the (normalized) ground state of the model by |1 we have then It is of course possible to have the defect line go under instead of over the loops.The rules will be the same as before (see figure 2.5), but with red and blue colors swapped.We will refer to the corresponding objects as D, D.
While seemingly trivial when calculating the partition function, this TDL is non-trivial when inserted into correlation functions.To illustrate this point we may consider the two-point function of the one-leg operator in the presence of a TDL.This gives rise to the calculation4 On the cylinder, denoting by |(r, 0) the ground state of the sector with 2r non-contractible linesthe notation is due to the operator inserting these lines having the Kac labels (r, 0)-we will show later that (r, 0)|D|(r, 0) ≡ D (r,0) = (−q) r + (−q) −r . (2.11) Note that for q generic, these numbers are different for different values or r, showing that the associated fields belong to different topological sectors.In a nutshell, the rest of the argument will be that the thermal perturbation of the dilute fixed point corresponding to taking K > K c preserves the non-invertible symmetry.Since the four-leg operator is not the in the topological sector of the identity, it cannot be generated under RG.This explains the "protection" enjoyed by the model, and the possibility of having a flow to the dense fixed point in the IR.
We now get into technicalities to put these ideas on firmer ground.In particular, while we have been a bit cavalier in the foregoing discussion about distinguishing the lattice and the continuum properties, we shall soon see that the non-invertible symmetry can be properly defined for the lattice model itself, whether critical or not.
3 Non-invertible (topological) symmetry in the lattice model

Algebras and the defect
There are different lattice versions of the O(n) model.While we believe their qualitative features and phase diagrams are universal (if crossings are not allowed), although some details might differ.We will consider here a version where the monomers live on the edges of the square lattice S. The interactions at one vertex of S can be represented by drawing the possible states of a tile containing the given vertex and its adjacent half-edges: For clarity we have drawn only the monomers (in blue color), omitting the lattice edges themselves.This model results from the usual high-temperature expansion of a model with O(n) spins living at the mid-points of the edges of S, represented as grey dots in (3.1),where we have also represented (still in grey color) the edges of the dual lattice S * that form the boundaries of the tile.A straight line or a corner on S is counted as one monomer, so the first vertex has no monomers, the next six vertices have one, and the last two vertices have two.
It is convenient in what follows to consider a "natural propagation" for the loops, corresponding to a diagonal propagation for the initial O(n) spins-as in figure 2. The transfer matrix is then an element of the dilute Jones Temperley-Lieb algebra dJTL, itself a periodicized version of the dilute Temperley-Lieb algebra (isomorphic to the Motzkin algebra), denoted dTL in [2].
In this diagram algebra, each fundamental vertex in (3.1) corresponds to a different generator.These generators can be depicted in a more conventional way by omitting the tile boundaries and deforming the monomer lines as follows: (3.3) (in fact, not all these generators are independent but this will not matter here [2]).On top of this we have the usual rules that loops get the fugacity n, and also that lines cannot stop-end points are not allowed.With periodic boundary conditions, we allow lines to connect around the system, and non-contractible loops are also given the fugacity n.It is important to understand that the algebra encodes all of the geometrical features, and is not dependent on the monomer fugacity K-the latter simply appears in the expression for the transfer matrix (or Hamiltonian) in the form of coefficients of words in the algebra, as in (3.2).
The defect line D is then introduced as shown in figure 2: we add a line of defect tiles carrying a red line (or non-contractible loop on the cylinder).There are two types of defect tiles that alternate throughout a horizontal row of the system: 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 D.

The dense case
The fully dense (also called "completely packed" in some works) case5 occurs when the monomer fugacity is very large (K → ∞), so all edges of the lattice S are occupied. 6In 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 [16].It was shown in this paper that the two defects D and 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 by systematically splitting the crossings.One finds where N is the width of the system, and τ translates every site one step to the right.The main result of [16] can then be formulated as7 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 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 H = [1] ⊗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): τ |a 1 , . . ., a N = |a N , a 1 , . . ., a N −1 . (3.6b) These formulas give rise to an explicit form for the operators D, D. On two (N = 2) and three (N = 3) sites, for instance, we have ) Since D is expressed in terms of the Temperley-Lieb generator and the shift operator, it commutes with O(n) in the sense of quantum Shuhr-Weyl duality: Indeed, the space of states of the fully dense O(n) model can be written as a bimodule over uJTL ⊗ O(n): (3.9) Here the Λ (r,s) are sums of O(n) representations, a few of which are [2] Λ These result still hold for n ∈ C, with a proper definition of the O(n) symmetry for n non integer, as discussed in [2,15].The objects W 1,1 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 iπs , with the condition z 2r = e 2iπrs = 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 1,1 (sometimes denoted by W 0,q ±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, Λ (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 unity8 ) of the operators D, D. The corresponding eigenvalues are and the same with q → q −1 for 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 O(n) representations involved in a given Λ (r,s) as giving rise to the same eigenvalue of D, D. Going back to n integer and taking a = b = c we have, for instance, and ) with the same eigenvalue-a result compatible with the identity Λ 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 ∈ 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 [16] and leads to the straightforward identity with D = D (1/2) .Hence we have in particular 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.

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: In terms of O(n) variables, the defect operator acts on states now in ([ ] + [1]) ⊗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 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 dJTL: 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 dJTL module becomes a sum over representations of Vir ⊗ Vir: where the notations for the representations and their content of primaries are as follows [2]: 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): where η(τ ) is the Dedekind eta function.We have then Recall also the central charge together with the parametrization (β 2 is often denoted by the Coulomb gas coupling constant g in the O(n)-model literature).Finally we have set so the conformal weights are given by ∆ (r,s) = P 2 (r,s) − P 2 (1,1) .(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 dJTL algebra becomes the interchiral algebra C β 2 , an algebra containing Vir ⊗ Vir (denoted C c in [2]) but extended by the degenerate field V 1,3 : We see that the topological defects in fact commute not only with C c but with this extended algebra as well: 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 pert .As long as this interaction can be expressed locally in terms of the elements in dJTL (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 dJTL identity module-which is what we have denoted R 1,s .This is because every word w in generators of the (dilute) algebra dJTL acting on the ground state of the theory will produce a state in W 1,1 : w|1 = α |w α -since this is a representation space of the algebra, and since it contains the ground state.By the state-operator correspondence, each state |w α will then correspond to an operator within R 1,s .Hence we have where the sum is over fields of the CFT in R 1,s -i.e.descendants under Vir ⊗ Vir of the primary fields Φ 1,s -and the c α 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 dJTL, the resulting Hamiltonian still commutes with the topological defect, [H pert , D] = 0. Now the point is that the four-leg operator does not belong to R 1,s .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 dJTL module is equivalent to classifying them in terms of their D eigenvalues.Hence, once we know that the perturbation is expressed in terms of dJTL, this guarantees that the four-leg operator cannot appear in the continuum limit of H 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 R 1,s 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|1 , and thus again decomposes onto fields in the identity sector.Hence the four-leg operator never appears when doing the RG for H pert , and cannot drive the flow away from the known, topological symmetrypreserving, IR dense fixed point.
We note that a similar argument was made in [20] to argue topological protection of "topological phases" in anyonic chains (see also [21]).In fact, the anyonic operator Y of [20] coincides formally with D once formulated in terms of the Temperley-Lieb algebra.In this correspondence, however, the critical point considered in [20] 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 dJTL algebra, and, in its presence, the defect line cannot be moved around any longer.Here is an example (where we call Π 12 the operator crossing lines 1 and 2): we have With crossings allowed (giving rise to what is sometimes called self-avoiding trails [23] 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.
Figure 3: The insertion of a pair of disorder fields affects loops encircling the extremities of the corresponding defect line.

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, 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) ±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.
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 iπe φ , 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), (∆, ∆) = (∆ (1/2,−e φ ) , ∆ (1/2,e φ ) ) . ( The two special choices of z-factors give, after reshuffling the labels, exponents (∆ 1 , ∆1 ) = (∆ (2,1) , 0) and (∆ 2 , ∆2 ) = (∆ (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 degenerate10 (and contributes as the field 1, 1 ) in the CFT.The defect Hilbert space H O(n) D [8] therefore has (∆ (2,1) , 0) as its lowest excitation. 11It turns out, in fact, that the whole H O(n) D can obtained by fusion with a degenerate field with those weights, so one has 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 Of course, the defect 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 D by modular transformation.This leads to the interpretation of, for instance, 1|D|1 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 (∆ (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 (∆ (2,1) , 0) is mutually local with all the fields in the identity sector, in particular the diagonal primaries 1, s , s ∈ 2N + 1.This extends to all "higher" disorder fields with conformal weights (∆ (r,1) , 0) thanks to the identity which is an integer for r integer, and s ∈ 2N + 1. 12

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 µ greater than one.At the level of the transfer matrix, this corresponds to multiplying by µ the last two terms in (3.2).As long as µ is not too large, the universality class remains the same (although the value of the critical coupling depends on µ), but for a critical value µ 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 ˜ that couples to the parameter µ is, despite the rough similarity of the geometry, not the four-leg operatorprecisely 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 Π in the O(n) language.The operator ˜ , 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 Φ 1,3 , a field which remains exactly degenerate at level 3 (despite the theory being in general logarithmic): it follows that ˜ ∝ Φ 1,5 .Related aspects are discussed in [30].
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: S x • S y → S x • g S y where g ∈ 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 a,b g 2 ab = 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 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 → Tr g.Hence it seems possible to think of the diagonal operators of [22] as operators sitting at the end of defect lines. 13This 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 [29], 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. 14inally, we mention that it is an outstanding question whether one can make rigorous sense of TDLs in terms of Deligne categories [15].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 [15].We shall have more to say about the O(n) current in other future work.
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: 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 at 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 → 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 [24] from the method of graph polynomials [25,26], and exact enumeration studies have provided very compelling evidence [27] that the universality class is that of the dilute O(n → 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 ≤ w < 1) corresponds to a four-spin interaction that favours (resp.penalises) the alternating arrangement + − +− of spins arond a vertex.The case w = 1 is just the usual Ising model without such a four-spin interaction, and the critical point in then

A.2 Phase diagram
For convenience we pick a generic value of n, far from the root of unity cases.We take n = 1/ √ 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 (∞), the effective central charge c eff and the non-universal amplitude A related to T T , via the formula 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 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 = 0.357 946 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 ≥ 0, we observe a peak in c eff = c eff (K), whose position gives a finitesize estimate K c (L) of the critical monomer weight and a finite-size estimate c(L) = c eff (K c (L)) of the central charge.Both of these estimates can be well extrapolated to the thermodynamic limit L → ∞ 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): 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 ≤ n < 2 and any finite w ≥ 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 ∈ Z/2) carrying an irrep λ 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, √ 2 − 1) is known analytically.The methodology for obtaining precise estimates of the critical exponents has been explained in [28,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 λ = [ ] sector.
The critical exponents for the Jones-Temperley-Lieb (JTL) standard modules Λ (r,s) to be compared with are denoted A very encouraging fact is that this resembles the results for λ = [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 ( 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 q sl(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 τ .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 [19], where it is shown that this defect (denoted d here) is diagonal on every TL module W r15 with eigenvalue 1 r |d|1 r = q 2r+1 + q −2r−1 , (B.1) that is, d acts in fact as the Casimir of U q sl(2).We can illustrate this using O(n) variables: for instance on two sites, with n = q + q −1 .We see that d is equal to the U q sl(2) Casimir, with eigenvalue q 3 + q −3 on [2] and q + q −1 on [ ].
In general we can write schematically d = r (q 2r+1 + q −2r−1 )P r , (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.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.[18]).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] ⊗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.

Figure 2 :
Figure 2: The defect line is introduced.

Figure 5 :
Figure 5: The d-operator in the open case.