Revisiting the symmetry-resolved entanglement for non-invertible symmetries in $1{+}1$d conformal field theories
Jared Heymann Thomas Quella


Defect Landscape Engineering to Tune Skyrmion-Antiskyrmion Systems in FeGe
    Serena Eley
    September 9, 2024
==========================================================================

§ ABSTRACT 
  Recently, a framework for computing the symmetry-resolved entanglement entropy for non-invertible symmetries in 1+1d conformal field theories has been proposed by Saura-Bastida, Das, Sierra and Molina-Vilaplana [https://doi.org/10.1103/PhysRevD.109.105026Phys. Rev. D109, 105026]. 
  We revisit their theoretical setup, paying particular attention to possible contributions from the conformal boundary conditions imposed at the entangling surface – a potential subtlety that was not addressed in the original proposal. We find that the presence of boundaries modifies the construction of projectors onto irreducible sectors, compared to what can be expected from a pure bulk approach. This is a direct consequence of the fusion algebra of non-invertible symmetries being different in the presence or absence of boundaries on which defects can end.

  We apply our formalism to the case of the Fibonacci category symmetry in the three-state Potts and tricritical Ising model and the Rep(S_3) fusion category symmetry in the SU(2)_4 Wess-Zumino-Witten conformal field theory. We numerically corroborate our findings by simulating critical anyonic chains with these symmetries as a finite lattice substitute for the expected entanglement Hamiltonian. Our predictions for the symmetry-resolved entanglement for non-invertible symmetries seem to disagree with the recent work by Saura-Bastida et al.
 








§ INTRODUCTION


Entanglement plays a key role in both understanding and simulating many-body systems. It is used to identify phenomena such as quantum phase transitions and critical points <cit.>, or to identify phases that lack local order parameters, such as topological phases <cit.>.
Not only is entanglement used to understand phenomena in many-body systems <cit.>, it is also used for efficient simulation of them.
In particular, the success of tensor network algorithms <cit.> hinges on the entanglement structure displayed by  ground states <cit.>: understanding entanglement allows more efficient simulations of quantum many-body systems <cit.>.

Recently, there has been a surge of interest in understanding the interplay between entanglement and another pillar of modern physics: symmetries.
Of particular interest has been the so-called symmetry resolved entanglement entropy (SREE), which quantifies the entanglement entropy in the different representations of a symmetry.
The idea of symmetry resolution dates back to Ref. <cit.>, where spin-resolution of critical quantum spin chains was studied. 
More recently, the SREE has been studied extensively in 1+1d conformal field theories (CFTs) due to the powerful analytical methods available <cit.>
(for a more comprehensive review see Ref. <cit.>).
The SREE has also found applications in studying strongly correlated models <cit.> where it can provide non-trivial information concerning the configuration of the ground state or serve as an order parameter <cit.>.
From the numerous studies of the SREE in 1+1d CFTs, a key result has emerged: the equipartition of the SREE.
At leading order in the ultraviolet cut off the SREE is equally distributed among the different representations of the theory.
Violations of the equipartition are observed at subleading order and are generally sensitive to the dimension of the representation in question <cit.>.

A powerful approach to symmetry resolution in 1+1d CFTs adopts the framework of boundary CFT (BCFT)
<cit.>.
The BCFT approach to the SREE arises because the spectra of the entanglement Hamiltonian in 1+1d CFTs is described by a BCFT <cit.>.
A BCFT is present because computing the entanglement entropy requires a bipartition of the Hilbert space, which comes with a choice of boundary conditions at the entangling surface <cit.>.
The choice of boundary conditions dictates the energy spectrum of the entanglement Hamiltonian <cit.> and which symmetries of the bulk are present in the BCFT <cit.>.
By using the BCFT approach to the SREE, when the symmetry in the BCFT is a finite group G, Ref. <cit.> found

    lim_q→ 1[S_n(q,r)- S_n(q)] = lnd_r^2/G ,

where d_r is the dimension of the representation r of G and G is the cardinality G.
Here, S_n(q,r) is the n-th Rényi entropy within the irreducible representation r, while S_n(q) is the n-th Rényi entropy.
The q → 1 limit corresponds to taking the ultraviolet cut off (lattice spacing) to 0 (or the width of the annulus in the BCFT to infinity).
When the symmetry of interest is a finite group G, <ref> indicates that the equipartition of the SREE is broken by a term that depends on the dimension of the r representation of G and the cardinality of G.

In recent years, after the seminal work of Ref. <cit.> the notion of a symmetry in quantum field theory and quantum lattice models has been generalised systematically and now extends beyond the concept of groups. 
As part of this endeavour, there have also been substantial efforts to abstract the concept of the symmetry (and its representations) from that of the underlying physical system in terms of what has become to be known as a “SymTFT” or sandwich construction. 
For recent reviews on these very dynamical areas of research and references to the original literature see Refs. <cit.>.
From the modern perspective that has emerged, symmetries should be identified with topological defects of various dimensions and composition of symmetries be realised as defect fusion. 
In this framework it is very natural to allow for symmetries that are not invertible.
Predating the more recent efforts, the consideration of topological defects and their role in terms of describing symmetries and dualities has a long history in the study of 1+1d rational CFTs (RCFTs), see e.g. <cit.>.

Recently, in Ref. <cit.> the SREE was analysed for non-invertible symmetries, dubbed the Cat-SREE, generated by Verlinde lines in 1+1d RCFTs.
The tricritical Ising model was studied and the entanglement entropy was resolved with respect to the Fibonacci category.
The Fibonacci category has two simple objects C= {1, W} with non-trivial fusion relation W × W = 1 + W and two irreducible representations, also labelled by C.
Analogous to the group-like case, Eq. (33) of Ref. <cit.> implies 

    lim_q → 1[ S_n(q,r) - S_n(q) ] = lnd_r^2/C^2

where d_r is the (quantum) dimension of the representation r ∈ C and C^2 = ∑_c ∈ C^ d_c^2.[We note that our convention for the total quantum dimension C differs from the one used in <cit.>.]
However, we find that the above breaking of equipartition eq:equipartition-wrong is incorrect.
Instead, for symmetries generated by Verlinde lines, we find that the equipartition of the SREE is 

    lim_q → 1[ S_n(q,r) - S_n(q) ]=lnd_r/√(C) ,

where √(C) = ∑_c ∈ C^ d_c and r ∈ C labels an irreducible representation of C.
When the Verlinde lines form a finite (abelian) group <ref> is in agreement with <ref> because  the quantum dimensions of invertible symmetries are one.
The deviation from Ref. <cit.> can be succinctly summarised as follows: the fusion algebra of non-invertible symmetries is sensitive to the presence of boundary conditions and is different than the fusion algebra of the bulk.
In this work, we find that if one wishes to understand the relation between entanglement measures and non-invertible symmetries, extra care due to the presence of boundaries is required – a issue that was not addressed in Ref. <cit.>.

Before proceeding to more technical considerations, we note that topological defects in the presence of boundaries find many other applications.
They have been utilised to analyse boundary renormalisation group flows <cit.> and open string field theory <cit.>.
Recently, they have also been explored in the representation theory of solitons <cit.>.

Before proceeding to more technical considerations, we note that topological defects in the presence of boundaries find numerous other applications.
In the CFT setting, they have been used to analyse boundary renormalisation group flows <cit.> and study open string field theory <cit.>.
The algebraic structure formed by considering topological defects terminating on two domain walls (or boundaries) is known as the “ladder algebra” in the mathematics literature <cit.>;
this has its roots in analysing gapped boundaries and domain walls for topological phases <cit.>.
However, the ladder algebra can also be viewed analogously to the tube algebra in the bulk <cit.>, providing the appropriate mathematical setting to analyse generalised symmetries in the presence of boundaries.
This has motivated a recent SymTFT perspective of the ladder algebra to study the representation theory of solitons <cit.>. We would also like to mention <cit.>, where the ladder algebra was discussed in the context of the three-state Potts model.

The rest of this paper is organised as follows.
In <ref> we briefly review the BCFT approach to computing the SREE.
In <ref> we provide an explicit counter example to the results in Ref. <cit.> by analysing the SREE with respect to the Fibonacci category in the three-state Potts model.
In <ref> we briefly review topological defect networks and the action of defects on states in the open string Hilbert space.
This allows us to construct projections on the open string Hilbert space and analyse the asymptotic limit of the SREE in <ref>.
A crucial ingredient is a detailed understanding of the fusion algebra of defects in the presence of boundaries. We then proceed to apply our framework to specific examples in <ref> and numerically corroborate our results by simulating anyonic chains.






§ BRIEF REVIEW OF THE BCFT APPROACH TO THE SREE
 

Defining the entanglement between regions A and B for a state described by the density matrix ρ requires a bipartition of the Hilbert space 

    ℋ = ℋ_A ⊗ℋ_B .

The reduced density matrix ρ_A = _B ρ corresponding to region A is obtained by tracing out the degrees of freedom in B, and the entanglement entropy can be computed.
However, as explained in Ref. <cit.>, a well defined bipartition of the total Hilbert space eq:bipartition-1 requires a choice of boundary conditions at the entangling surface (the regions separating A and B). 
The need for boundary conditions at the entangling surface is most explicit in theories whose Hilbert space does not admit a simple tensor product decomposition, such as anyonic chains <cit.> which in turn represent important examples of physical systems with non-invertible symmetries.
For a conformal field theory, it is natural to choose boundary conditions that preserve the (extended) chiral algebra.
Formally, this is achieved by a factorisation map ι_a,b:

    ι_a,b: ℋ→ℋ_A,ab⊗ℋ_B,ab ,

where ℋ_A,ab (ℋ_B,ab) is the Hilbert space in region A (B) with boundary conditions a and b at the entangling surface.
In the path integral, the factorisation map is implemented by inserting two disks of radius ε≪ 1 that serve as an ultraviolet (UV) cut off.
The region A of length ℓ is then mapped to an annulus of width w by a conformal transformation <cit.>, see <ref>.
The dependence of w on ℓ depends on the initial geometry of the problem: for instance, whether the Hilbert space is defined on an infinite line or finite periodic chain etc..
For a comprehensive list of different geometries see Ref. <cit.>.
For our purposes, we are interested in the BCFT defined on an annulus of width w and it is not important what the initial geometry of the problem was.


The reduced density matrix ρ_A,ab = _B,abρ is obtained by tracing out the degrees of freedom in ℋ_B,ab, where ρ is the density matrix of the ground state.
In terms of the parameters of the conformal field theory, the reduced density matrix is

    ρ_A,ab   = q^L_0 - c/24/Z_ab(q) ,

where q = e^-2 π^2/w is the modular parameter, c is the central charge,

    Z_ab(q) = _abq^L_0 - c/24 ,

is the open string partition function with boundaries a and b and _ab denotes the trace over the open string Hilbert space with boundaries a and b.
The normalisation factor of 1/Z_ab(q) in <ref> ensures _abρ_A,ab = 1.
 By a modular transformation, the open string partition function is written in the closed string channel as

    Z_ab(q) =  ⟨a|q̃^L_0 - c/24|b⟩ ,

where q̃ = e^-2 w is the dual modular parameter and |a⟩ and |b⟩ are conformal boundary states imposed at the entangling surface by the factorisation map ι_ab.
The Rényi entropies are 

    S_n(q)1/1-nlog_abρ_A^n = 1/1-nlogZ(q^n)/Z^n(q) ,

with the entanglement entropy being given by lim_n → 1 S_n(q).

Suppose we have some symmetry category C acting on the open string Hilbert space.
The SREE is computed by introducing 

    𝒵_ab(q^n, r) _abΠ_r ρ_A^n ,

where Π_r projects onto the representation r of C. 
The symmetry resolved Rényi entropy (SRRE) is given analogously to the Rényi entropies

    S_n(q,r) 1/1-nlog𝒵_ab(q^n,r)/𝒵_ab^n(q,r) ,

with the SREE being obtained in the limit n → 1.
It is useful to define the charged moments of the reduced density matrix 

    z_ab(q^n,c) _abℒ_c q^n(L_0 - c/24) ,

where ℒ_c is the operator corresponding to the object c ∈ C acting on states in the open string Hilbert space.
In the closed string channel the charged moments read 

    z_ab(q^n ,c) = c⟨a|q̃^1/n(L_0 - c/24)|b⟩_c ,

where |a⟩_c and |b⟩_c are boundary states in the Hilbert space of the c-twisted sector.
The charged moments eq:charged-moments correspond to computing the open string partition function with an insertion of the defect ℒ_c terminating on the boundaries a and b.
We require that ℒ_c can topologically end on both |a⟩ and |b⟩.
This requires that the fusion c × a contains a and, likewise, the fusion c × b contains b.
Using the terminology of Ref. <cit.>, the boundaries a and b must be weakly symmetric under ℒ_c for <ref> to be non-zero.
Moreover, we will assume that a and b are simple boundaries; they are not written as a superposition of elementary boundary states.

The charged moments in the closed string channel make it quite simple to derive the asymptotic behaviour of the SREE.
Generally, a projector onto the irreducible representation r of C is written as a linear combination of operators in C,

    Π_r^[a,b] = ∑_c∈ C^Λ_r c^[a,b]ℒ_c ,

for constants Λ_rs^[a,b] that ensure the projectors are orthogonal.
The superscripts [a,b] make it explicit that such projectors are defined in the presence of boundaries a and b.
If the boundaries are equal, a = b, we will only include one boundary in the superscript, [a]≡[a,a].
For the symmetry C to be present in the open string Hilbert space, all defects ℒ_c, for c ∈ C, must be able to topologically terminate on the boundaries a and b: the boundaries a and b must be weakly symmetric under C.
As a consequence, C is non-anomalous and be “gauged” in a generalised sense <cit.>.

Using the general form of a projector eq:projector-ansatz, in the closed string channel <ref> reads

    𝒵_ab(q^n, r)    = ∑_c ∈ C^Λ_rc^[a,b]c⟨a|q̃^1/n(L_0 - c/24)|b⟩_c , 
       q → 1∼Λ_r1^[a,b]q̃^-c/24n⟨a||0⟩⟨0||b⟩ ,

where we have assumed that there is a unique vacuum |0⟩ in the bulk CFT.
Therefore, in the asymptotic limit q → 1, the vacuum sector dominates in the closed string channel, leading to <ref>.
It will be useful to define the difference between the symmetry resolved Rényi entropies and the Rényi entropies

    Δ S_n(q,r)  S_n(q,r) - S_n(q) .

Plugging in the symmetry resolved Rényi entropies eq:SRRE and utilising the asymptotic limit eq:asymptotic-general, one finds that the equipartition of the SREE is broken by the term

    lim_q→ 1Δ S_n(q,r) = lnΛ_r1^[a,b] .

For a finite group G, the projectors onto the r irrep are 

    P^G_r    = χ_r(1)/G∑_g∈ G^χ_r(g) U(g) ,

where U(g) is the representation of the element g ∈ G and χ_r(g) is the group character.
From the above projector, one has Λ_r1^[a,b] = χ_r(1)^2/G, recovering <ref>.






§ SYMMETRY RESOLUTION FOR THE FIBONACCI CATEGORY
  

In this section we will revisit the Cat-SREE framework proposed in Ref. <cit.> and show it is insufficient to produce the correct SREE with respect to the Fibonacci category in the three-state Potts model.
Using the extended W_3 algebra, the three-state Potts model has a diagonal modular invariant.
There are six primary fields 1, ψ, ψ^†, ε, σ and σ^† with scaling dimension 0,  2/3, 2/3,  2/5,  1/15 and 1/15, respectively.
In the basis (1, ε, ψ, σ , ψ^†, σ^†) the modular S matrix reads <cit.>

    S    = 1/√(3)(
      s     s     s 
    
      s    ω s   ω^2 
    
      s    ω^2 s    ω s
      ) ,

with 

    s    = 2/√(5)(
      sinπ/5   sin2π/5
    sin2 π/5    -sinπ/5
      ) ,     ω = e^2π i/3 .

The fusion rule 

    ε×ε = 1 + ε

indicates that the Verlinde lines ℒ_c, for c∈ C = {1, ε}, generate a Fibonacci symmetry.
For the case of diagonal RCFTs, the primary operators, Verlinde lines and boundary states are in one-to-one correspondence.
Therefore, they can all share labels in some modular fusion category ℳ.
Thus, the Cardy state |ε⟩ is weakly symmetric  with respect to C <cit.>. The open string partition function with both boundaries a = b = ε is 

    Z_εε = χ_1(q) + χ_ε(q) .


Next, we require projectors onto the irreducible representations of C.
As C has two elements there are two irreducible representations.
In the bulk CFT, the action of the Verlinde line ℒ_ε on each conformal family <cit.> in the basis (1, ε, ψ, σ ,ψ^† ,σ^†) is

    ℒ_ε   = ( S_ε j/S_1j) =( φ,    1- φ,    φ,     1- φ,   φ,     1- φ) ,

where φ = (1+√(5))/2 is the golden ratio.
The action of the topological defect ℒ_c on the vacuum representation is given by the quantum dimension d_c of c.
This means that the conformal towers  1, ψ and  ψ^† transform trivially under ℒ_ε, while the conformal towers ε , σ and σ^† transform non-trivially under ℒ_ε.

For the bulk CFT with diagonal modular invariant, the Hilbert space decomposes as ℋ= ⊕_μ𝒱_μ⊗𝒱_μ.
In the bulk CFT, the projectors P_r^bulk: ℋ→𝒱_r ⊗𝒱_r onto the conformal tower r are <cit.>

    P_r^bulk   = ∑_b ∈ℳ^ S_1rS_b r ℒ_b .

The primary operators {1, ψ, ψ^†} transform in the trivial representation of C.
Therefore, the projector onto the trivial irrep of C is the sum of projectors onto the conformal towers {1, ψ, ψ^†},

    Π_1^bulk   = P_1^bulk + P_ψ^bulk + P_ψ^†^bulk .

Similarly, the primary operators {ε, σ, σ^†} transform in the non-trivial irrep of C; the projector onto the non-trivial irrep of C is the sum of projectors onto the conformal towers {ε, σ, σ^†},

    Π_ε^bulk   = P_ε^bulk + P_σ^bulk + P_σ^†^bulk .

Using the bulk modular fusion category projectors eq:MFC-projector we find

    Π_1^bulk   = 1/1+φ^2( ℒ_1 + φℒ_ε) 
    Π_ε^bulk   = φ/1+φ^2(φℒ_1 - ℒ_ε) .

As observed in Ref. <cit.>, these are the projectors one would obtain when using <ref> with the modular S matrix for the Fibonacci category,

    S_Fib   = 1/√(1+φ^2)(1    φ
    φ    -1) .


We now need to compute the charged moment z_εε(q, ε), which requires computing the overlap of twisted boundary states. 
The Hilbert space of the c-twisted sector decomposes as <cit.>

    ℋ_ℒ_c   = ⊕_μ,ν N_c ν^μ𝒱_μ⊗𝒱_ν .

This implies that the diagonal sectors in ℋ_ℒ_ε are of the form 𝒱_ρ⊗𝒱_ρ with ρ running over the set {ε, σ ,σ^†}.
We denote the boundary state |ε⟩ in the ε-twisted sector by 

    |ε⟩_ε   =a_1 ε_ε + a_2 σ_ε + a_3 σ^†_ε ,

  where  h_ε are the Ishibashi states in the ε-twisted sector and a_i are undetermined constants.
Using a vector notation for the characters, this implies the charged moment z_εε(q, ε) = _εεℒ_ερ_A,ε, ε
is  

    z_εε(q, ε)    = α_1 χ_ε(q̃) + α_2 χ_σ(q̃) + α_3 χ_σ^†(q̃) =   (
      μ (α_1 + α_2 + α_3) 
    ν (α_1+α_2 +α_3) 
    μ(α_1 + ω^2 α_2 + ωα_3) 
    ν (α_1 + ω^2 α_2 + ωα_3) 
    μ(α_1 + ωα_2 + ω^2 α_3) 
    ν(α_1 + ωα_2 + ω^2 α_3)
      )_(q) ,

where we use the basis  {χ_1(q), χ_ε(q), χ_ψ(q), χ_σ(q), χ_ψ^†(q), χ_σ^†(q) },   α_i = a_i^2 and

    μ   = √(5 + √(5)/30) ,     ν = - √(5 - √(5)/30) .

Note that Z_εε(q) only contains the characters χ_1(q) and χ_ε(q); therefore the final four rows in z_εε(q, ε) must be zero.[This is because, by construction, the Verlinde lines commute with the extended chiral algebra and therefore, by Schur's lemma, act as scalar multiples of the identity on irreducible representations of the chiral algebra <cit.>.
]
This implies that α_1 + ω^2 α_2 + ωα_3 =0 and α_1 + ωα_2 + ω^2 α_3 = 0.
Together, these constraints imply that α_1 = α_3 and α_2 = -ω(α_1 + ωα_3) = -ω(1+ω)α_1.
Plugging these into the charged-moment eq:pott-charged-moment yields

    z_εε(q, ε)    = -(1+ ω) α_1( μχ_1(q) + νχ_ε(q)) .

Finally, we need a constraint to fix α_1.
This is determined by the action of ℒ_ε on the boundary identity or ε conformal tower.
At this stage, we do not know what that action is;
for now, we will let ℒ_ε's action on the boundary identity conformal tower be some non-zero scalar λ.
This implies

    -(1+ω) α_1 μ = λα_1 = λ√(5 + √(5)/30) ,

resulting in 
 
    z_εε(q, ε)    =   λχ_1(q) + λ (1 - φ) χ_ε(q) .

As the coefficients in front  of χ_1(q) and χ_ε(q) are different in <ref>, we conclude that the boundary identity conformal tower and the boundary ε conformal tower correspond to different irreps of C.
 As a consequence, for <ref> to be projectors onto the irreducible representations of C in the presence of boundaries, we must have _εεΠ_1^bulkρ_A,εε!=χ_1(q) or χ_ε(q) and vice versa for Π_ε^bulk.
 
Using the  charged moment z_εε(q, ε) defined in <ref> we compute 

    _εεΠ_1^bulkρ_A,εε   = 1/1+φ^2( (1 + λφ) χ_1 + (1 - λφ^2 + λφ)χ_ε) ,

and assert

    _εεΠ_1^bulkρ_A !=χ_1 (q) .

However, no choice of λ satisfies the above, <ref>.
Similarly, no choice of λ can satisfy 

    _εεΠ_1^bulkρ_A,εε!=χ_ε(q) .

We conclude that the original bulk projectors eq:Fib-projectors are incorrect in the presence of boundary conditions.

The Fibonacci projectors in the bulk eq:Fib-projectors leading to incorrect results in the presence of boundaries could be established on different, simpler grounds.
Namely, the fusion algebra of defect lines terminating on a boundary is generally different than the bulk fusion algebra <cit.>.
Following the prescription in Ref. <cit.> we find that the boundary fusion rules for C  = {1, ε} with boundary condition ε on both sides is 

    ℒ_ε^[ε]×ℒ_ε^[ε]   = 1/φℒ_ε^[ε] + φ - 1/φℒ_ε^[ε] ,

where ℒ_c^[ε] indicates that the defect ℒ_c terminates on the two boundaries labelled by ε.
From the above boundary fusion rules eq:Potts-boundary-fusion, it is clear that the bulk projectors eq:Fib-projectors do not satisfy Π_r^bulkΠ_h^ bulk = δ_rhΠ_r^bulk when the defects terminate on the boundary ε.

Before concluding this section, note that if one used the projectors 

    Π_1^[ε]   = 1/1+φ(ℒ_1  + φℒ_ε) , 
    Π_ε^[ε]   = φ/1+φ(ℒ_1 - ℒ_ε) ,

then for λ = 1 we would find 

    _εεΠ_1^[ε]ρ_A    = χ_1(q) , 
    _εεΠ_ε^[ε]ρ_A    = χ_ε(q) .

We will show how to construct these projectors in <ref>.






§ TOPOLOGICAL DEFECT NETWORKS
  
In <ref>, the charged moments eq:charged-moments were computed by taking overlaps of twisted boundary states.
In Ref. <cit.> the authors noted the similarity between computing these overlaps and orbifold CFTs, developing a method to compute the SREE by utilising boundary states in orbifold CFTs.
The orbifold construction of the SREE makes it explicit why the symmetry of interest, C, needs to be anomaly free: constructing the orbifold CFT is only possible if C is free of a 't Hooft anomaly.
In contrast to Refs. <cit.>, we wish to compute the SREE strictly from the open string channel and only use the closed string channel to compute the breaking of equipartition in the asymptotic limit, <ref>.
This requires knowledge of the action of defects on states in the open string Hilbert space without appealing to the closed string channel and twisted boundary states.

When working in the open string channel, it is natural to use the graphical calculus afforded by fusion categories; 
this requires performing manipulations of topological defect networks by utilising the appropriate F-symbols.
In the presence of boundaries, as is the case here, these manipulations can become difficult because one needs to keep track of the F-symbols associated with the symmetry category of interest and the F̃-symbols associated with the module category of interest (the boundary states).
However, with simplifying assumptions, only a single set of F-symbols is required.
We will focus on diagonal RCFTs in which charge conjugation acts trivially.
As the CFT has a diagonal modular invariant, primary operators, Cardy boundary states, and Verlinde lines are in one to one correspondence and share the same labels in a modular fusion category ℳ.
Invariance under charge conjugation means that all Verlinde lines are self dual i.e. ℒ_c = ℒ_c̅ for c ∈ℳ. In other words, it is not necessary to specify the orientation of defects in diagrammatic manipulations. 
For simplicity, we also restrict ourselves to situations where the fusion of Verlinde lines

    ℒ_c ×ℒ_d = ∑_e^ N_cd^e ℒ_e

   is multiplicity free. This means that the fusion coefficients N_cd^e are either 0 or 1 for all c,d,e ∈ℳ. The fusion coefficients N_cd^e can be determined from the modular S matrix with the Verlinde formula <cit.>.
   Finally, we assume parity invariance of our defects and CFTs.
With these assumptions, we review the relevant formalism presented in Ref. <cit.> to compute the action of defects on states in the open string Hilbert space.





 §.§ Defect networks
 


Manipulations of networks of topological defects is done by utilising the F-symbols.
Our convention for the F-symbols are shown in <ref>.
Moreover, if any of the incoming defects a, b, c or d are the identity defect, denoted by 1, then the corresponding F-symbol is trivial.
  For consistency of defect network manipulations, the F-symbols must satisfy the pentagon identity, <ref>.
 
 Equating both directions in <ref> results in the celebrated pentagon equation

    ∑_s^ F_ps[ b c; a q ]
      F_qt[ s d; a e ]
      F_sr[ c d; b t ]   = F_qr[ c d; p e ]
      F_pt[ b r; a e ] .


As emphasised in Ref. <cit.> (see also Ref. <cit.>), the pentagon identity has a large symmetry, sometimes referred to as a gauge symmetry.
From a set of F-symbols that satisfy the pentagon identity, applying an appropriate gauge transformation to these F-symbols results in another equivalent solution of the pentagon identity.
Of course, after manipulating defect networks the final result obtained should be independent of the gauge choice (assuming the result corresponds to a physical observable).
For our purposes, it is useful for our diagrammatic rules to be fully isotopy invariant; the defects can be freely moved and bent without incurring additional negative signs or phase factors.
This condition corresponds to a specific choice of gauge for the F-symbols which we will use in this work.
In this gauge some of the F-symbols simplify <cit.> 

    F_1c[ a b; a b ]= F_c 1[ a a; b b ] = √(d_c/d_a d_b) .

Due to the assumption of parity invariance, we have F-symbol identities such as 

    F_pq[ a b; c d ]
      = F_pq[ c d; a b ]
      = F_pq[ b a; d c ] .






 §.§ Open topological defects
 

As explained in Ref. <cit.>, attaching defects to a boundary comes with a choice of normalisation for the junction field.
A natural choice of normalisation arises from using the F-symbols to attach a parallel defect to a boundary and assigning a factor of √(F_1b[
    d  a; d a ]) to each junction.
This junction factor is denoted by a filled circle, see <ref>.
These junction factors have some convenient properties <cit.>.
One particularly useful property we will utilise is the fact that, with these junction factors, we can shrink a defect on a  boundary to the identity, <ref>.




Next we require the action of defects on boundary condition changing operators.
By the operator-state correspondence in CFT, boundary condition changing operators correspond to states in the open string Hilbert space <cit.>.
Determining the action of a defect on boundary condition changing operators is then equivalent to determining the defect's action on states in the open string Hilbert space.
Following Ref. <cit.>, manipulations of boundaries with boundary condition changing operators can be lifted to defect manipulations by representing the boundary condition by the corresponding defect label.
This assumes that there is some identity boundary that can be used to generate all other boundaries by fusion with an appropriate defect.
In our case, this assumption is trivially satisfied by the vacuum Cardy state.
Then, the boundary condition changing operator with representation label i can be traded for a defect labelled by i and a defect ending field on the identity boundary, <ref>.
The constants α_i^ab associated with this manipulation were computed in Ref. <cit.>.
However, the constants α_i^ab will not be needed for our purposes.
This manoeuvre, <ref>, also has an interpretation in terms of the topological field theory formulation of CFTs <cit.>.






 §.§ Action of defects on boundary fields
  



To compute the SREE one needs to compute charged moments z_a b(q,d) = _a bℒ_d ρ_A,ab.
The insertion of the defect ℒ_d in this trace
corresponds to connecting a defect to both boundaries, a and b, of the annulus, <ref>(left).
By a conformal transformation, the annulus can be mapped to the upper-half plane and the defect then traverses a semi-circle around the origin, where a possible boundary condition changing operator is located, <ref>(right).
By the operator-state correspondence, a state |i, M⟩ in the open string Hilbert space,  where i labels a boundary conformal tower and M enumerates its states, is created by a local boundary condition changing operator ψ_i, M <cit.>.
The action of the defect operator ℒ_d on states in the open string Hilbert space is determined by the action of ℒ_d on boundary condition changing operators.
With the assumptions specified at the start of <ref> – primary operators, boundary states and defect lines are in one-to-one correspondence – this action has been derived in Ref. <cit.>.
This action is illustrated in <ref> under the assumption that the boundaries at the entangling surface are described by simple boundary conditions.
For the generalisation to non-simple boundary conditions see Ref. <cit.>.
If the boundary conditions a and b are not simple then the action of defect operators is more complicated.
This is because the defect operator can now intertwine different boundary condition changing operators in the same representation.


In the BCFT approach to the SREE, <ref>, we required our boundary states to be weakly symmetric under our fusion category of interest.
We now see how this requirement manifests itself in the open string Hilbert space.
The requirement of the defect d being able to topologically end on a and b is encoded in the F-symbol F_ba[ ad; ib], see <ref>, which specifies the action of ℒ_d on states in the open string Hilbert space.
If a and b are not weakly symmetric under d then F_ba[ad; ib] = 0 for all i, implying that ℒ_d = 0 in the open string Hilbert space.

The fusion of two defects in the presence of boundaries a and b is denoted by

    ℒ_c^[a,b]×ℒ_d^[a,b] = ∑_ e ∈ c × d^Ñ_cd^[a,b] eℒ_e^[a,b] ,

where Ñ_cd^[a,b] e are generally non-integer coefficients <cit.>.
By using the F-symbols to fuse two defects in the presence of boundaries, as in <ref>, and utilising <ref>, the boundary fusion coefficients are given by

    Ñ_cd^[a,b]e = F_eb[ d c; b b ]
      F_ea[ d c; a a ] .


Peculiarly, even if e appears in the bulk fusion c × d, the boundary fusion coefficient Ñ_cd^[a,b] e eq:boundary-fusion-rules may be zero, resulting in ℒ_e not appearing on the right hand side of <ref>.
We will see an explicit example of this when we study Rep(S_3) in the presence of boundaries in <ref>.
With our choice of junction normalisation, and specialising to the case a = b,  we note some identities of the boundary fusion rules <cit.>:

    N_1d^[a]d = 1 ,      N_dd^[a] 1 = 1/d_d ,     ∑_e ∈ c × d^ N_cd^[a] e = 1 .

The last identity arises due to the fact that defects can be shrunk on a boundary with no additional factors incurred, <ref>.






§ BOUNDARY PROJECTORS
  





 §.§ Projector construction


We are now in a position to use Verlinde lines to project onto sectors of the open string Hilbert space.
First, we will construct projectors that project onto the boundary conformal towers in the open string Hilbert space.
Before we proceed, recall that all boundary states, Verlinde lines and primary operators share labels in the modular fusion category ℳ and the fusion of defects is multiplicity free.

The open string Hilbert space decomposes as ℋ_ab = ⊕_r N_ab^r 𝒱_r.
We want to find operators P_r^[a,b] that implement the projections P_r^[a,b]: ℋ_ab→ N_ab^r 𝒱_r, i.e.

    _abΠ_r^[a,b]ρ_A,ab = N_ab^rχ_r(q) .

These projectors were constructed on a formal level in Ref. <cit.>,

    P_r = ∑_M^|r,M⟩⟨r,M| ,

where M is a descendant label.
Ref. <cit.> used <ref> to analyse the SREE with respect to conformal symmetry.

Firstly, let WS^[a,b]⊆ℳ denote the subset of Verlindes lines that both boundaries a and b are weakly symmetric under.
We propose that the projector onto 𝒱_r is 

    P_r^[a,b]   = 1/F_1b[a     r 
    a    r ]∑_s∈WS^[a,b]^ F_1s[a     a 
     a     a] F_sr[a     b 
     a     b] ℒ_s  .

By using the following F-symbol identity, which can be derived from the pentagon identity ( see <ref>)

    1/F_1b[ a r; a r ]∑_s ∈WS^[a,b]^ F_1s[ a a; a a ] F_sr[ a b; a b ]F_b a[ a s; i b ]  = δ_i r ,

we have

    _ab P_r^[a,b]ρ_A,ab   = ∑_i^ N_ab^i 1/F_1b[ a r; a r ]∑_s ∈WS^[a,b]^ F_1s[ a a; a a ] F_sr[ a b; a b ]F_b a[ a s; i b ]χ_i(q) 
       = N_ab^r χ_r(q) ,

as required.
Using <ref>, it is easy to show that {P_r^[a,b]}_r∈ a × b are orthogonal projectors when acting on states in the open string Hilbert space,

    P_h^[a,b]P_r^[a,b]|i,M⟩= δ_rh P_r^[a,b]|i,M⟩ .

Similarly, using <ref> and the pentagon identity (<ref>), one can show

    ∑_r ∈WS^[a,b] P_r^[a,b] = ℒ_1 .






 §.§ Asymptotic limit
  
Utilising the projectors (<ref>), the asymptotic limit of the SREE can be computed.
In the asymptotic limit, the untwisted sector in the closed string channel dominates. 
Therefore 

    𝒵_ab(q,r) =  _ab P_r^[a,b]ρ_A,ab   q→1∼F_11[ a a; a a ] F_1 r[ a b; a b ]/F_1b[ a r; a r ]⟨a||0⟩⟨0||b⟩q̃^c/24 , 
       = d_r/d_a d_b⟨a||0⟩⟨0||b⟩q̃^c/24 , 
       = S_1rq̃^c/24 ,

where we have explicitly expanded the Cardy states in terms of the modular S matrix to go from the second last to the final line.
Unsurprisingly, the final, simplified result is identical to results found in Ref. <cit.>, which utilised the projectors in <ref>.
This serves as a useful check of the validity of the BCFT projectors in <ref>.
Using <ref>, we find that the equipartition of the SREE is broken by the term ln d_r/d_a d_b when projecting onto the conformal family  r.

Currently, the projectors <ref> do not provide any insight on the asymptotic limit of the SREE for a fusion category symmetry because it is not obvious how the coefficients d_r/d_a d_b in <ref> relate to WS^[a,b].
More insight can be gained with some additional assumptions.
Firstly, we will assume that both boundaries are identical, a = b.
Next, suppose that there is a sub fusion category C ⊆ℳ; we wish to compute the SREE with respect to C.
Additionally, assume there exists a boundary state a_C ∈ℳ such that a_C × a_C = ∑_c∈ C^ c.
As the quantum dimensions furnish a representation of the fusion ring, it follows that

    √(C) d_a^2    = ∑_c ∈ C ^ d_c .

Using <ref>, we construct the projectors onto the irreps of C

    P_r^[a_C]   = 1/F_1a_C[ a_C   r; a_C   r ]∑_d ∈ C ^ F_1d[ a_C a_C; a_C a_C ]
      F_dr[ a_C a_C; a_C a_C ]ℒ_d ,

which can be written in the nicer form

    P_r^[a_C]  = d_r/√(C)∑_d ∈ C^ d_d  F_a_C a_C[    d a _C;  a_C    r ]ℒ_d .

The boundary fusion rules (<ref>) can be used to explicitly show that {P_r^[a_C]}_r ∈ C are orthogonal projectors,

    P_r^[a_C] P_h^[a_C]   = δ_rh P_r^[a_C] .

This is a tedious calculation, so we differ it to <ref>.
Additionally, from <ref> we have

    ∑_r ∈ C P_r^[a_C] = ℒ_1.


Using <ref>, we see that the equipartition of the SREE is broken by the term

    Δ S_n(q,r) = lnd_r/√(C) .

This is in contrast with the results obtained in Ref. <cit.>, <ref>, which suggest that the equipartition should be broken by ln d_r^2/C^2 where C^2  = ∑_c ∈ C^ d_c^2.
<ref> is in agreement with Ref. <cit.>, <ref>, when C is a finite abelian group 
because irreducible representations of finite abelian groups are one dimensional.
Strictly speaking, due to the assumption of self-duality of representations, C must be a product of ℤ_2-factors.
However, we believe this assumption can easily by relaxed without changing the outcome, <ref>.
While <ref> disagrees with the results of Ref. <cit.>, <ref>, for non-abelian finite groups, this disagreement is superficial: C can never be a non-abelian group because the fusion of Verlinde lines is commutative.

A natural concern one may have is the generality of the projectors, <ref>.
Indeed, the construction of <ref> required a careful choice of boundary conditions.
The boundary conditions are referenced explicitly in the boundary fusion rules (<ref>) so it would appear that the projectors constructed in <ref> and all computations based on them are true specifically for the chosen boundary condition a_C.
This is a valid concern, and we cannot definitively establish at the time being that <ref> are the general projectors that project onto irreducible representations of C in the presence of more general boundary conditions.
 Currently, <ref> can be used on a case-by-case basis.
Suppose that one is studying a theory with boundary conditions a' and b' that are weakly symmetric under C.
Due to the orthogonality condition eq:orthogonal-projectors, a sufficient condition for the projectors with boundaries a_C eq:MFC-general? to be projectors in the presence of boundaries a' and b' is requiring that the fusion of defects in C does not depend on whether the boundaries are labelled by a_C or a' and b':

    Ñ_dc^[a_C] e = Ñ_dc^[a',b'] e ,

for all d,c,e ∈ C.
If <ref> is true, then <ref> will project onto the irreps of C in the theory with boundaries a' and b' even though they were constructed with the boundary a_C.
We will show an example of this when we study the tricritical Ising model in <ref>.

Given the explicit dependence of the boundary fusion rules on the choice of boundary conditions, <ref>, it would be quite surprising if <ref> turned out to be true in full generality. 
In the next section, after studying explicit examples of the boundary projectors (<ref>), we will briefly discuss the validity of <ref> in the context of C = Fib and C = Rep(S_3) from the perspective of anyonic chains.

Before we study examples of the BCFT projectors, <ref> and <ref>, we note that it is impossible to construct the projectors onto the irreducible representations of C (<ref>) by only using the F-symbols of C and making no reference to a boundary condition.[We restrict the data to only the F-symbols of C because C need not be a modular fusion category.] 
We will comment on this further when we study the Ising model in <ref>.
It may be possible to write the projectors eq:MFC-general? in terms of other data of C, such as the characters of C in the presence of boundaries.
The boundary characters χ̃_i^[a,b](c) should form a one-dimensional representation of the boundary fusion algebra

    χ̃_i^[a,b](c) χ̃_j^[a,b](c) = ∑_k ∈ i × jÑ_ij^[a,b] kχ̃_k(c)^[a,b] ,

where c ∈ C and irreps of C are labelled by  i,j,k ∈ C.
Moreover, they should obey relevant orthogonality relations and reduce to suitable (quantum) dimensions under appropriate circumstances, analogous to the finite group characters, to befit being called a character.[In the bulk the natural characters χ_j(i)=S_ij/S_i1 would be quotients of modular S-matrix entries.]
Again, it seems likely that one would also have to make an explicit reference to boundary conditions to define the characters χ̃_i^[a,b](c) because defining the boundary fusion rules rules requires a choice of boundaries. 
Therefore, from a mathematical perspective, it is unclear whether formulas for fusion category projectors in the presence of boundaries can be constructed without explicit reference to the boundaries.
Despite this, in <ref> we appeal to physical arguments to suggest that such projectors are independent of the choice of boundary conditions.






§ EXAMPLES
  

  After our general discussion of the theoretical underpinnings we now apply the above framework to concrete examples. The F-symbols used in this section have been obtained utilising Refs. <cit.>.





 §.§ Revisiting the three-state Potts model.


Our goal is to find the projectors (<ref>) using <ref>. The action of ℒ_1 on the boundary conformal towers  1 and ε is trivial.
The action of ℒ _ε is trivial on the boundary conformal tower  1, while the action of ℒ_ε  on the boundary conformal tower  ε is given by

    F_εε[ ε ε; ε ε ]  = 1- φ .

This is in agreement with taking λ = 1 in <ref>.
Finally, using the boundary fusion rules eq:boundary-fusion-rules, the non-trivial fusion rules are[We note that boundary fusion rules for the Fibonacci category appear in Refs. <cit.>, however, their choice of junction normalisation differs from ours.]

    ℒ_ε^[ε]×ℒ_ε^[ε] = 1/φℒ_1^[ε] + φ -1/φℒ_ε^[ε] .

Using these fusion relations, it can be checked that the Fibonacci projectors in the three-state Potts model (<ref>) are orthogonal.





 §.§ ℤ_2: Ising Model
 

The Ising model has a ℤ_2 symmetry generated by ε; this ℤ_2 fusion category is a subcategory of the Ising modular fusion category and can thus be used within our formalism.
The Cardy state |σ⟩, corresponding to the primary spin operator, is weakly symmetric with respect to the ℤ_2 symmetry.
The open string partition function with boundaries σ is 

    Z_σσ(q) = _σσρ_A,σσ = χ_1(q) + χ_ε(q) .

From Ref. <cit.>, we know that χ_1(q) is the ℤ_2 even sector and χ_ε(q) is the ℤ_2 odd sector.
Using <ref>, we find the following projectors

    P_1^[σ]   = 1/2(ℒ_1 + ℒ_η) , 
    
      P_ε^[σ]   = 1/2(ℒ_1 - ℒ_ε) ,

which are nothing other than ℤ_2 projectors!
The action of ℒ_ε on the ε boundary conformal tower is given by the F-symbol

    F_σσ[ σ   ε
    ε   σ] = -1 .

From this, it follows that

    _σσ P_1^[σ]ρ_A,σσ = χ_1(q) , 
    _σσ P_ε^[σ]ρ_A, σσ = χ_ε(q) ,

in agreement with <cit.>.

Finally, note that the ℤ_2 projectors (<ref>) cannot be constructed in terms of the F-symbols of the (non-anomalous)[For a category C to kinematically allow a (weakly) symmetric boundary state, it needs to be anomaly free <cit.>.] ℤ_2 fusion category.
All the F-symbols in ℤ_2 are trivial, so the coefficient of -1/2 in <ref> could not possibly be expressed in terms of these F-symbols. 
To obtain this negative sign with the F-symbols, we need to make reference to a boundary condition, in this case σ.





 §.§ Tricritical Ising Model
 
To make contact with the original Cat-SREE paper <cit.>, we will analyse the tricritical Ising model in greater depth.
Following conventions in the literature <cit.>, the primary operators are labelled as 

    1 ,     η ,     𝒩 ,      W ,     η W ,     𝒩 W .

The tricritical Ising model is a diagonal RCFT: there is a one-to-one correspondence between boundary states and Verlinde lines.
One finds that the three (Cardy) boundary states |W⟩, |η W⟩ and |𝒩 W⟩ are weakly symmetric under C:

    W × W= W + 1 ,      W ×η W = η W + η ,      W ×𝒩 W =  𝒩 W + 𝒩  .

In the basis  {1, η W, W, η, 𝒩  W,  𝒩} 
the modular S matrix is given by <cit.>

    S=([       s_2       s_1       s_1       s_2  √(2) s_1  √(2) s_2;       s_1      -s_2      -s_2       s_1  √(2) s_2 -√(2) s_1;       s_1      -s_2      -s_2       s_1 -√(2) s_2  √(2) s_1;       s_2       s_1       s_1       s_2 -√(2) s_1 -√(2) s_2;  √(2) s_1  √(2) s_2 -√(2) s_2 -√(2) s_1         0         0;  √(2) s_2 -√(2) s_1  √(2) s_1 -√(2) s_2         0         0 ])

with

    s_1 = sin(2π /5)      s_2 = sin(4π /5) .

The Verlinde lines labelled by C = {1, W } form a Fibonacci subcategory; we with to resolve the entanglement entropy with respect to C.
To compute the SREE, Ref. <cit.> considered the case when the boundaries at the entangling surface were a = b =  𝒩 W.
With these boundaries, the open string partition function is 

    Z_𝒩W | 𝒩W(q) = χ_1(q) + χ_η W(q) + χ_W(q) + χ_η(q) .

To construct projectors onto the irreps of C, we first need to determine how C acts on each boundary conformal tower, <ref>; we require the F-symbols

    F_𝒩W| 𝒩W[ 𝒩W     d 
     i    𝒩W] ,

with d ∈ C and  i ∈{1, η W, W, η}.
If d or i is 1 then this action is simply the identity.
Therefore, the relevant F-symbols are

    F_𝒩W| 𝒩W[ 𝒩W     W 
     W    𝒩W ]    = 1 - √(5)/2 = 1 -φ ,   
    F_𝒩W| 𝒩W[ 𝒩W    η W 
     W    𝒩W ]  = 1 -φ ,  
    
    F_𝒩W| 𝒩W[ 𝒩W    η
     W    𝒩W ]     = 1 .

We conclude that the 1 and η boundary conformal towers transform trivially under C, while the η W and W boundary conformal towers transform non-trivial under C.
The projectors onto the trivial, 1, and non-trivial, W, irreducible representation of C are then given by

    Π_1^[𝒩W]   = P_1^[𝒩 W] + P_η^[𝒩W] , 
    Π_W^[𝒩W]   = P_W^[𝒩 W] + P_η W^[𝒩W] .

Using <ref> we find 

    Π_1^[𝒩W]   = 1/1+φ(ℒ_1 + φℒ_W) , 
    Π_W^[𝒩W]   = φ/1+φ(ℒ_1 - ℒ_W) .

Using the action of C on each boundary conformal tower eq:Tricrit-F-symbols, one can verify that the above projectors eq:Tricrit-Project implement 

    _𝒩W|𝒩WΠ_1^[𝒩 W]ρ_A,𝒩W|𝒩W = χ_1(q) + χ_η(q) , 
    _𝒩W|𝒩WΠ_W^[𝒩 W]ρ_A,𝒩W|𝒩W = χ_η W(q) + χ_W(q) .

Note that the projectors we found, <ref>, are identical to those found in the three state Potts model, <ref>.
Indeed, as both boundaries are identical, a = b = 𝒩 W, the boundary fusion rule identities eq:boundary-fusion-identities can be used to show the non-trivial boundary fusion rules of C are identical to those of the three-state Potts model:

    ℒ_W^[𝒩 W]×ℒ_W^[𝒩 W] = 1/φℒ_1^[𝒩 W] + φ -1/φℒ_W^[𝒩  W] .

The breaking of the equipartition of the SREE eq:SREE-equipartition is then

    Δ S_n(q, 1) = ln1/1+ φ ,     Δ S_n(q, W) = lnφ/1+φ .

This is in contradiction with Ref. <cit.>, which suggest that Δ S_n(q,1) and Δ S_n(q,W) are ln 1/(1+φ^2) and lnφ/(1+φ^2), respectively.

Next, let us consider the case of two different boundaries, a = η W and b = 𝒩W.
The open string partition function is 

    Z_η W| 𝒩W(q) = χ_𝒩(q) + χ_𝒩W(q) .

Before we appeal to the projector formula eq:BCFT-projector, let us illustrate the logic surrounding <ref>.
In the previous example, we have constructed the projectors onto the irreps of C when both boundaries are 𝒩 W.
These projectors can be used with the new boundary conditions, a = η W and b = 𝒩W, if the boundary fusion rules of C are unchanged with these new boundaries.
Using <ref> we find

    Ñ_dc^[𝒩 W] e =   Ñ_dc^[η W, 𝒩W]  ,

for d, c, e ∈ C.
Therefore, the projectors onto the irreps of C with boundaries a = η W and b = 𝒩 W are also given by <ref>.

To know which boundary conformal towers transform in the trivial and non-trivial irreps of C we can: appeal to explicit F-symbols, as we did in <ref>,  construct the projectors onto each conformal tower with <ref>, or analyse the asymptotic SREE by computing eq:asymptotic-1 d_r/d_η W d_𝒩W for r ∈η W ×𝒩 W.
Using the latter option, we find d_𝒩/d_η W d_𝒩W  = 1/(1 + φ) and d_r/d_η W d_𝒩W  = φ/(1+φ),
indicating that the 𝒩 and 𝒩W boundary conformal towers transform in the trivial and non-trivial irrep of C, respectively.




  §.§.§ Numerical check: golden chain



We corroborate our results numerically by exact diagonalisation of the golden chain.
The golden chain is an anyonic lattice model describing the tricritical Ising CFT <cit.> (see also <cit.>).
We consider an open golden chain with L_A external anyons or L = L_A - 1 links as illustrated in <ref>.
The local link variables take values in the Fibonacci category, x_i ∈{1, W} for i = 1, …, L.
The neighbouring links are subject to the constraints of the Fibonacci fusion rules and we fix the boundary anyons to be x_0 = x_L+1 = W.
The Hamiltonian of the golden chain projects neigbouring anyons to fuse into the trivial channel.
Using the local basis {1, W } for the link variables, the Hamiltonian is <cit.> 

    H_Golden = - ∑_i^ h_i     with     h_i = n_i-1 + n_i+1 -n_i-1n_i+1(φ^-3/2+ σ_i^x + φ^-3 n_i +1 + φ^-2) ,

where n_i counts the number of W particles at link i and σ^x_i is the Pauli matrix x matrix.
The spectrum of the open golden chain is described by the tricritical Ising model BCFT.
For an even (odd) number of anyons, the spectrum is described by a BCFT with boundary conditions a = W and b = W (η W) <cit.>.

On general grounds, the entanglement spectrum of a critical 1+1d system is described a BCFT <cit.>.
We are therefore permitted to interpret the open boundary condition Hamiltonian as a possible entanglement Hamiltonian: the reduced density matrix is ρ_A = e^- H.
We determined the spectrum of H using exact diagonalization for chain lengths L ranging from 3 to L=19 and organized it in irreducible representations with respect to the Fibonacci symmetry.

To achieve this, we constructed the Fibonacci projectors (<ref>) on the lattice.
This requires the lattice analogs of the Verlinde lines ℒ_1 and ℒ_W.

The defects on the lattice, known as the topological symmetry <cit.>, can be described pictorially as illustrated in <ref>.
The general action of the topological symmetry ℒ_i for boundary anyons x_0 = x_L+1 = ℬ and external anyons j is

    ℒ_i |x_1,…,x_L⟩ = ∑_y_1,…,y_L^∏_n = 0^L F_x_n y_l+1[   y_i     j;     i x_n+1 ]|y_1, …, y_L⟩ .

Here, x_0 = x_L+1 = y_0 = y_L+1 = ℬ and i labels the defect ℒ_i.
For the golden chain, the boundary anyons and external anyons are ℬ   = j= W.
Note that <ref> differs from the topological symmetry in Ref. <cit.> by a factor of √(d_i) due to the presence of junction factors in <ref>.

Both boundaries in the golden chain are labelled by W and therefore the non-trivial boundary fusion rule is given by <ref> no matter the chain length.
However, the above statement is not obvious from the BCFT perspective.
This is because the boundary conditions in the BCFT describing the open golden chain differs depending on whether the number of external anyons is odd or even <cit.>.
It is not obvious that the boundary fusion rules in <ref> should be identical when there is an even number of external anyons, a = b = W, and when there is an odd number of external anyons, a = W and b = η W.
For this to happen we must have

    Ñ_cd^[W] e = Ñ_cd^[W, η W] e ,

for c, d, e ∈ C {1, W}.
This example is a lattice realisation showing that the boundary fusion rules of C can be identical for different boundary conditions in the BCFT.

The projectors onto the two irreps of the Fibonacci category are given by <ref>.
Using the asymptotic equipartition of the SREE, <ref>, we expect

    Δ S_2(q, 1) =  ln1/1+φ ,     Δ S_2(q, W) =  lnφ/1+φ ,

where 1 is the trivial irrep and W is the non-trivial irrep.
Our numeric results plotting Δ S_2(q, r), r ∈{1, W }, are shown in <ref>.
We find that both the 1 and W irreducible representation are approaching their predicted asymptotic value.
This provides compelling evidence that the appropriate asymptotic values of the SREE for the Fibonacci category are given by <ref>.
However, the anyonic chains we could simulate were not long enough to observe saturation of the asymptotic value.






 §.§ Rep(S_3)
 

When constructing the BCFT projectors (<ref>), we did not need to assume that the symmetry category of interest, C, was modular.
If one can find an appropriate embedding C ⊂ℳ of the symmetry category C into a modular fusion category ℳ, then projectors for the fusion category C can also be constructed.
We will illustrate this procedure by constructing projectors for the Rep(S_3) fusion category in the presence of boundaries.

To construct these in the context of a CFT, we consider the diagonal, A-type modular invariant for the SU(2)_4 Wess-Zumino-Witten model.
This theory contains fields with spin 0, 1/2, 1, 3/2 and 2.
As mentioned in <cit.>, the boundary state |k/4 = 1⟩ is weakly symmetric under a C =Rep(S_3) category that is generated by three simple lines: ℒ_j, for j =0, 1 and 2.
 ℒ_0 is the identity line and ℒ_1 and ℒ_2  satisfy the fusion rules
 
    2 × 2    = 0 
    
       1 × 1    = 0 + 1 + 2 
    
       1 × 2 = 2× 1    = 1 .

 As expected, the above fusion rules show that the Cardy state | 1⟩ is weakly symmetric with respect to C.
 The open string partition function with both boundaries fixed as |1⟩ is
 
    Z_11(q)    = χ_0(q) + χ_1(q) + χ_2(q) .

The action of C on the boundary operators is described by the F-symbol
 
    F_11[1      d 
    i     1] ,

 for d,i ∈C.
 The relevant, non-trivial F-symbols are
 
    F_11[1     1 
    2     1] = -1 ,    F_11[1     2 
    1     1] = -1 ,     F_11[1     1 
    1     1] = 0 .

 Using <ref>, we find the three projectors 
 
    Π_0^[1]   = P_0^[1] = 1/4( ℒ_0 + 2 ℒ_1 + ℒ_2) 
    Π_1^[1]   = P_1^[1]= 1/4(2 ℒ_0  - 2 ℒ_2) 
    Π_2^[1]   = P_2^[1]= 1/4(ℒ_0 - 2 ℒ_1 + ℒ_2) .

 These projectors imply that for a Rep(S_3) symmetry the equipartition of the SREE is broken by, <ref>,
 
    Δ S_n(q, 0) = Δ S_n(q,2) = ln1/4 ,     Δ S_n(q,1) = ln1/2 .

 Using the projectors (<ref>) along with the non-trivial F-symbols (<ref>), a quick calculation shows
 
    _22Π_0^[1]ρ_A,11 = χ_0(q) ,     _22Π_1^[1]ρ_A,11 = χ_1(q) ,     _22Π_2^[1]ρ_A,11 = χ_2(q) .

  The boundary fusion rules of Rep(S_3) are identical to the bulk fusion rules, with the peculiar exception
 
    ℒ_1^[1]×ℒ_1^[1] = 1/2ℒ_0^[1] + 1/2ℒ_2^[1] .

 Note that no non-invertible symmetry appears on the right hand side of <ref>; this is to be contrasted with the bulk fusion rule 1 × 1 = 0 + 1 + 2. 
 Using these fusion rules, one can show that the projectors (<ref>) are orthogonal projectors.
 



  §.§.§ Numerical check: SU(2)_4 spin-1/2 anyon chain
 

To study the Rep(S_3) projectors (<ref>), we numerically simulate the SU(2)_4 spin-1/2 antiferromagnetic anyonic chain.
This chain describes the Tetracritical Ising model <cit.>, which has a Rep(S_3) symmetry.
We study an open chain with spin-1/2 external anyons and spin-1 anyons as boundaries, see <ref> for an illustration.
The Hamiltonian favours neighbouring spin-1/2 anyons to fuse into the trivial channel.
Due to the fusion rules, the local basis alternates between integer labels and half-integer labels.
Therefore, the number of links L on the anyon chain must be odd because we have fixed spin-1 anyons on the boundary.
In the local basis x_i∈{1/2, 3/2 }, x_i+1∈{0, 1, 2 } where i (i+1) is odd (even), the Hamiltonian is 

    H_SU(2)_4   = - ∑_i^ h_i   
    
      h_i    = n_i-1^0 n_i+1^0 + n_i-1^1 n^1_i+1[  1 -1; -1  1 ]_i
      + 1/3 n_i^1/2 n_i^1/2[     1 -√(2); -√(2)     2 ]_i+2
      +1/3 n_i^3/2 n_i^3/2[      2  -√(2); - √(2)      1 ]_i+2 .


A lattice realisation of the topological symmetry is analogous to the case of the golden chain in <ref>.
The lattice realisation of the defects is given by <ref> with the boundaries x_0 = x_L+1 = y_0 = y_L+1 = ℬ = 1, and external anyons j = 1/2.
Via arguments similar to the case of the golden chain, it is clear that the fusion rules of the ℒ_i are identical to those in (and above) <ref>.
Therefore, we can expect the following equipartition of the SREE:

    Δ S_n(q,0) = Δ S_n(q,2) = ln1/4 ,     Δ S_n (q, 1) = ln1/2 .

Similar to the golden chain, we interpret the energy levels of the open chain as an entanglement spectrum and define the reduced density matrix as ρ_A e^-H_SU(2)_4.
We then compute Δ S_2(q, r) for various chain lengths via exact diagonalisation and the Rep(S_3) projectors eq:Rep(S3)-projectors.
<ref>(left) shows Δ S_2(q,r) for r ∈{0,1,2} for various chain lengths. 
While the spin-2  representation appears to be converging to ln 1/2, the spin-0 and spin-1 representations are far from the expected value of ln 1/4.
We suspect that the difference in the rate of convergence between the spin-2 representation and the spin-0 and spin-1 representation is due to the number of states in each representation.
In the anyonic chain Hilbert space, approximately half of the states transform in the 1 representation, while only one quarter of the states transform in the 0 or 2 representation. This follows from investigating the rank of the respective projection operators but it also matches the corresponding ratios of quantum dimensions, d_1/d_i = 2 for i = 0 and 2.
Thus one would expect Δ S_n(q,2) to converge faster on the lattice as it is, in a sense, closer to the thermodynamic limit.

With this in mind, observe that the number of states in the combined 0 ⊕ 2 representation is equal to the number of states in the 1 representation.
We then expect that Δ S_2(q, 0 ⊕ 2) will converge at approximately the same rate as Δ S_2(q, 1) and the expected limit will be
Δ S_2(q, 0 ⊕ 2 ) = ln 1/2.
Indeed, as shown in <ref>(right), Δ S_2(q, 0⊕ 2) and  Δ S_2(q, 1) approach the expected value of ln 1/2 at similar rates as the chain length is increased.
However, similar to the case of the golden chain, the length of the chain L is not long enough for this asymptotic value to be saturated.






 §.§ Additional observations
 

After studying numerous examples in the previous sections, we have arrived at the following observation concerning the asymptotic limit of the SREE, <ref>.
Firstly, when we used the BCFT projectors P_r^[a,b] eq:BCFT-projector, recall that the asymptotic symmetry resolved partition function eq:asymptotic-1 contained the prefactor d_r/d_a d_b, where a and b are the (simple) boundaries at the entangling surface and r ∈ a × b labels a boundary conformal tower.
At first glance, this factor appears to have no relation to the set of Verlinde lines that the boundaries a and b are weakly symmetric under, WS^[a,b].
However, we believe that 

    d_r/d_a d_b = d_c/√(WS^[a,b ]) ,

for some c ∈WS^[a,b] and √(WS^[a,b]) = ∑_c ∈WS^[a,b] d_c.
In other words, the projectors onto the conformal families, P_r^[a,b], project onto the irreps of WS^[a,b].
This provides an interpretation of the asymptotic limit eq:asymptotic-1 in terms of the Verlinde lines present in the open string Hilbert space.
For the case where a = b, <ref> is easily seen to be true, with c = r and d_a^2 = √(WS^[a,b]).
More generally, for a ≠ b it is not obvious that <ref> is true, although we have checked it for some specific minimal models, such as the tricritical and tetracritical Ising model.

Let us briefly discuss the reasoning behind <ref> in the general case, a ≠ b.
Firstly, by using the Verlinde formula, one can show that

    ∑_d N_ad^a N_bd^b = ∑_r N_ab^r ,

which implies that the number of boundary conformal towers in the partition function Z_ab(q) is equal to the number of simple objects in WS^[a,b].
Then, if the irreducible representations of WS^[a,b] are in one-to-one correspondence with the boundary conformal towers i.e. for every r ∈ a × b there is a corresponding irrep c ∈WS^[a,b],
the projector onto the boundary conformal tower r, P_r^[a,b], then projects onto the irreducible representation c ∈WS^[a,b].
Appealing to the specific case of both boundaries being equal eq:MFC-SREE-equipartition, we then suggest the relation in <ref>.





 §.§ Boundary conditions and projectors
 

It is worth noting that the projectors we found for the Fibonacci symmetry in the three-state Potts model, <ref>, are identical to the Fibonacci projectors for the tricritical Ising model, <ref>.
Naturally, this leads one to ask: do the projectors defined in <ref> actually depend on the choice of boundary conditions?
For the case of the Fibonacci category, one can show the BCFT projectors do not depend on the choice of boundary condition when they are equal on both sides.
Suppose we choose both boundary conditions to be labelled by a;
the boundary fusion algebra is 

    ℒ_c^[a]×ℒ_d^[a] = ∑_e ∈ c × d^Ñ_cd^[a] eℒ_e^[a] ,

where N_cd^[a] e need not be integers, see <ref>.
The boundary fusion algebra identities eq:boundary-fusion-identities completely determine the boundary fusion rules for the Fibonacci category:

    ℒ_W^[a]×ℒ_W^[a] = 1/φℒ_1^[a] + φ - 1/φℒ_W^[a] .

The orthogonal projectors that are compatible with these fusion rules are

    Π_1^[a]   = 1/1+φ(ℒ_1 + φℒ_W)  , 
    Π_W^[a]   = φ/1+φ(ℒ_1 - ℒ_W) .


Finally, from the perspective of critical anyonic chains, it is somewhat natural to conclude that the boundary fusion rules are independent of the specific choice of boundaries.
We briefly discuss this for the case of Rep(S_3) symmetry.

There are numerous anyonic chains with Rep(S_3) symmetry with open boundary conditions.  For instance, the (anti) ferromagnetic SU(2)_4 spin-1/2 chain describes the (Tetracritical Ising) Z_4 parafermion CFT <cit.>.
Similarly, the SU(2)_4 spin-1 anyon chains can describe c = 1 compact boson CFTs at various compactification radii <cit.> (including non-rational points).
In the above cases, fixing a spin-1 anyon on the boundaries of the open chain will result in the Rep(S_3) symmetry being present.
The boundary fusion rules for Rep(S_3) will be those stated in <ref>.
Therefore, from the lattice perspective, it is obvious that with spin-1 anyons at the boundaries the Rep(S_3) projectors (<ref>) will project onto the irreps of Rep(S_3).

In contrast, from the continuum BCFT perspective, it is non-trivial to determine whether these projectors are also the appropriate projectors for Rep(S_3).
Firstly, this is because the BCFTs associated with the anyonic chains will generally admit non-Cardy boundary states.
Secondly, as shown for the golden chain, even if both boundary anyons are identical, the corresponding boundary states in the BCFT can be different.
For these reasons, it is not obvious that the boundary fusion rules in the presence of these boundaries are given by <ref> in the BCFT describing the continuum limit.
However, viewing these BCFTs as the continuum limit of the associated anyonic chains, the boundary fusion rules of the Rep(S_3) symmetry in the continuum must also satisfy <ref>.
This would suggest that as long as the boundary conditions in the CFT are weakly symmetric under Rep(S_3), then fusion of Rep(S_3) in the presence of these boundaries is identical to <ref>.
Therefore, we tentatively suggest the following picture. Suppose that the simple boundary conditions a and b are weakly symmetric under a symmetry category C that acts faithfully (see <cit.>) on the Hilbert space of interest.
Then the fusion of objects in C in the presence of the boundaries a and b are independent of the specific choice of a and b.






§ CONCLUSIONS
 
In this work we have analysed the symmetry-resolved entanglement entropy for non-invertible symmetries.
Specifically, we focused on non-invertible symmetries generated by Verlinde lines in 1+1d conformal field theories with diagonal partition function.
We constructed projectors onto irreducible  sectors for these non-invertible symmetries in the presence of boundaries and utilised them to study the equipartition of the symmetry-resolved entanglement entropy. A key element was the study of the defect fusion algebra in the presence of boundaries which permits non-integer coefficients.
Explicit examples of orthogonal projectors for the cases of the Fibonacci category and Rep(S_3) were provided.
Importantly, the Fibonacci projectors and the associated expressions for the symmetry-resolved entanglement entropy differ from those in the recent literature <cit.>.

An obvious direction for future work is extending the analysis to general fusion categories whose fusion rules may be non-commutative. This is likely best achieved by utilising the symmetry topological field theory <cit.>.
Indeed, the symmetry topological field theory proved to be a powerful tool when studying the symmetry resolution of the torus partition function <cit.>.
This is because the symmetry topological field theory can be used to construct the generalised charges associated with a symmetry <cit.>.
Work in this direction requires further understanding of the representation theory of non-invertible symmetries on manifolds with boundaries, which has been recently studied in the context of the representation theory of solitons <cit.>.

Throughout this work we have emphasised that the fusion algebra of non-invertible symmetries in the presence of boundaries is different than the bulk fusion algebra. This leads to a natural question: does the fusion algebra explicitly depend on the choice of boundary conditions?
We commented on this briefly in <ref> and tentatively suggest that the boundary fusion relations are independent of the choice of boundary conditions, as long as the boundaries are weakly symmetric under the fusion category of interest.
While one may have physical reasons to believe this picture to be true, a mathematical proof is far from obvious as the boundary fusion algebra makes explicit reference to the choice of boundary conditions.

Additionally, it would be beneficial to further analyse the interplay between entanglement measures and non-invertible symmetries.
Entanglement measures require some choice of bipartition and consequently a choice of boundary conditions at the entangling surface.
Therefore, we believe our results will prove useful in analysing the relation between other entanglement measures and non-invertible symmetries, such as the entanglement asymmetry <cit.>.
On a similar note, to study these entanglement measures numerically requires realising the non-invertible symmetries on a lattice with open boundary conditions.
While recent works have constructed Rep(S_3) symmetries for periodic chains <cit.>, they have, to our knowledge, not yet been constructed on spin chains with open boundary conditions. 
We leave the discussion of spin chain realisations of non-invertible symmetries in the presence of boundaries to future work.




  
Note added. While in the process of completing this work we became aware of related work on symmetry-resolved entanglement measures for non-invertible symmetries by Yichul Choi, Brandon Rayhaun, and Yunqin Zheng <cit.> as well as Arpit Das, Javier Molina-Vilaplana and Pablo Saura-Bastida <cit.>. We also expect parts of other upcoming work on the SymTFT picture for physical theories with boundaries by Yichul Choi, Brandon Rayhaun, and Yunqin Zheng <cit.>, Lakshya Bhardwaj, Christian Copetti, Daniel Pajer, and Sakura Schäfer-Nameki <cit.> as well as Iñaki García Etxebarria, Jesús Huertas, and Angel Uranga <cit.> to be relevant in our context.




 §.§ Acknowledgement

  The research of JH was supported by an Australian Government Research Training Program (RTP) Scholarship.
  The authors thank Tyler Franke for helpful discussions and Hong-Hao Tu for bringing the concept of a ladder algebra to their attention. 
  We would like to thank everyone listed under “Note added” for willing to coordinate our respective submissions.




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§ PROJECTOR CALCULATIONS
  





 §.§ Tetrahedron symbols


The tetrahedron symbols are obtained by considering a defect network in the shape of a tetrahedron.
Following the conventions in Ref. <cit.> and using <ref>, they are related to the F-symbols by

    [baseline=(0, 0cm- pt)] 
      [black,thick] (0,0) circle (1);
      [black,thick] (0,0) – node[midway, anchor =east] i (0,1);
      [black,thick] (0,0) – node[midway, anchor =south west, xshift=-0.1cm] j(0.866,-0.5);
      [black,thick] (0,0) – node[midway, anchor =north west , xshift=-0.1cm, yshift=0.1cm] k(-0.866,-0.5);
      (45: 1) node[anchor=south west]c;
      (-90: 1) node[anchor=north]a;
      (135: 1) node[anchor=south east]b;
       =
      [ i j k; a b c ]^TET = √(d_a d_b d_i d_j) F_ck[ j i; a b ] .

There are numerous ways to shrink this tetrahedron to a point, which generates numerous identities between the F-symbols.
<ref> becomes more useful when used in conjunction with the tetrahedron symbols,
Due to the numerous way to shrink a tetrahedron to a point, <ref> has many symmetries.
For instance, <ref> is invariant under the cyclic permutation of its columns

    [ i j k; a b c ]^TET 
      = [ k i j; c a b ]^TET
      = [ j k i; b c a ]^TET ,

or switching the upper and lower labels of two different columns simultaneously.





 §.§ Pentagon identity
 

The pentagon identity (<ref>) reads 

    ∑_s ^ F_ps[b     c 
     a     q] F_qt[s     d 
     a     e] F_sr[c     d 
     b     t]    = F_qr[c     d 
    p     e] F_pt[b     r 
     a     e] .

Referring to <ref>, if p = 1 then we require a = b, c = q and e = r.
Then,  by the assumption of parity invariance of defects, F_pq[j k; il] = F_pq[i l; j k],[This is obtained by reflecting <ref> about the horizontal.] we find

    ∑_s ∈ C^ F_1s[b     c 
     b     c] F_ct[b     e 
     s     d] F_sr[c     d 
     b     t]    =  F_1t[b     r 
     b     r] δ_er .

Now we set b =c=X and t = d = Y

    ∑_s∈ C^ F_1s[X     X 
     X     X] F_XY[X     e 
     s     Y] F_sr[X     Y 
     X     Y]    =  F_1Y[X     r 
     X     r] δ_er
    1/F_1Y[X     r 
     X    r]∑_s^ F_1s[X     X 
     X     X]  F_sr[X     Y 
     X     Y] F_XY[X     e 
     s     Y]    = δ_re .

We identify the final term on the left side as the action of ℒ_s on the irreducible representation e with boundaries X and Y;
this is only non-zero if the boundaries X and Y are weakly symmetric with respect to ℒ_s.
Thus we arrive at 

    P_r^[a,b]   = 1/F_1b[a     r 
    a    r ]∑_s∈WS_[a,b]^ F_1s[a     a 
     a     a] F_sr[a     b 
     a     b] ℒ_s ,

as required.





 §.§ Orthogonal projectors
  

Let C ⊆ℳ where ℳ is a modular fusion category.
Consider a ∈ℳ such that a× a = ∑_c∈ C^ C.
Then using <ref> we obtain the projector onto each representation r ∈ C:

    P_r^[a]   = 1/F_1a[ a r; a r ]∑_s ∈ C ^ F_1s[ a a; a a ]
      F_sr[ a a; a a ]ℒ_s 
       = √(d_r)/d_a∑_s∈ C^√(d_s)  F_sr[ a a; a a ]ℒ_s .

Next, using the F-symbol identity (which is easily seen from the tetrahedron symbols)

    d_a   F_sr[ a a; a a ]   =  √(d_s d_r)  F_aa[ a r; s a ]

we then have

    P_r^[a]   = d_r/√(C)∑_s ∈ C^ d_s F_aa[ a r; s a ]ℒ_s

where we have used d_a^2 = √(ℳ). 
This recovers <ref>.
Next we show that these are orthogonal projectors.
We then compute

    P_r^[a] P_h^[a]   = ∑_s,d,λ^ N_sd^λÑ_sd^[a] λF_1s[ a a; a a ]
      F_sr[ a a; a a ]
    F_1d[ a a; a a ]
      F_dh[ a a; a a ]/ F_1a[ a r; a r ]
    F_1a[ a h; a h ]ℒ_λ
       = √(d_r d_h)/d_a^2∑_s,d,λ^√(d_s d_d)  N_sd^λ
      F_sr[ a a; a a ]
      F_dh[ a a; a a ]
     F_λ a[ s d; a a ]
     F_λ a[ s d; a a ]ℒ_λ

We use the following identity, which can be obtained from the tetrahedron symbols, 

    F_ea[ d c; a a ]   = √(d_e)/√(d_d) F_da[ c e; a a ]

to then write

    P_r^[a]P_h^[a]   = √(d_r d_h)/d_a^2∑_s,d,λ^√(d_d d_λ)   N_sd^λ
      F_sr[ a a; a a ]
      F_dh[ a a; a a ]
     F_λ a[ s d; a a ]
     F_s a[ d λ; a a ]ℒ_λ

Then, consider the following form of the pentagon identity <cit.> 

    F_rq[ e p; d c ]
      F_pt[ e r; a b ]   = ∑_s^ F_sp[ b c; a q ]
      F_tq[ s d; a e ]
      F_sr[ t d; b c ] .

Next, by using the tetrahedron symbols one finds 

    F_λ a[ s d; a a ] = F_a λ[ d s; a a ] .

The right hand side is only non-zero when N_sd^λ≠ 0.
Thus, summing over s in <ref> and utilising the pentagon identity, <ref> we find

    P_r^[a]P_h^[a]   = √(d_r d_h)/d_a^2∑_d,λ^√(d_d d_λ)  F_dh[ a a; a a ]
      F_aa[ a r; d a ]
      F_rλ[ a a; a a ]ℒ_λ .

From the tetrahedron symbol identity

    [ r a a; d a a ] ^TET   = [ a a r; a a d ]^TET         
      F_aa[ a r; d a ]   = d_a/√(d_r d_d)F_dr[ a a; a a ]

we then have

    P_r^[a]P_h^[a]   = √( d_h)/d_a∑_d,λ^√(d_λ)F_dh[ a a; a a ]
      F_dr[ a a; a a ]_pentagon  identity
      F_rλ[ a a; a a ]ℒ_λ
       = δ_rh√(d_h)/d_a∑_λ^√(d_λ)  F_r λ[ a a; a a ]ℒ_λ
       = δ_rh P_h^[a] ,

as required.

