Bipartite fidelity for models with periodic boundary conditions
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Bipartite fidelity for modelswith periodic boundary conditions
Alexi Morin-Duchesne , Gilles Parez , Jean Li´enardy
Universit´e catholique de LouvainInstitut de Recherche en Math´ematique et PhysiqueChemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium alexi.morin.duchesne @ gmail.com gilles.parez @ uclouvain.bejean.lienardy @ uclouvain.be
Abstract
For a given statistical model, the bipartite fidelity F is computed from the overlap between thegroundstate of a system of size N and the tensor product of the groundstates of the same modeldefined on two subsystems A and B , of respective sizes N A and N B with N = N A + N B . In thispaper, we study F for critical lattice models in the case where the full system has periodic boundaryconditions. We consider two possible choices of boundary conditions for the subsystems A and B ,namely periodic and open. For these two cases, we derive the conformal field theory predictionfor the leading terms in the 1 /N expansion of F , in a most general case that corresponds to theinsertion of four and five fields, respectively. We provide lattice calculations of F , both exact andnumerical, for two free-fermionic lattice models: the XX spin chain and the model of critical densepolymers. We study the asymptotic behaviour of the lattice results for these two models and findan agreement with the predictions of conformal field theory. Keywords:
Entanglement, bipartite fidelity, quantum spin chains, loop models, conformal field theory.1 ontents
A.1 Perturbation of the stress-energy tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 40A.2 Contribution to the periodic pants geometry . . . . . . . . . . . . . . . . . . . . . . . . . 42A.3 Contribution to the skirt geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.4 The constant C for the periodic pants geometry . . . . . . . . . . . . . . . . . . . . . . . 44A.5 The constant ˜ C for the skirt geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.6 Sub-leading term for the periodic pants geometry . . . . . . . . . . . . . . . . . . . . . . 46A.7 Sub-leading term for the skirt geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 B Asymptotics 47
B.1 Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47B.2 Asymptotics for P ( N ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50B.3 Asymptotics for P ( N , N , φ , φ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50B.4 Combinations of non-trivial terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50B.5 Asymptotics for Q ( N, φ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
Introduction
Understanding entanglement and correlations in quantum many-body systems is an important chal-lenge in modern theoretical physics. The interest for these questions stems from the understandingthat quantifying entanglement, an idea that originated in the information theory community [1, 2], isuseful to diagnose phase transitions and describe the critical behaviour of many-body systems [3–5].Entanglement now plays a prominent role in seemingly unrelated research areas, such as informationtheory, condensed matter, high energy physics and black holes physics.Among the numerous existing entanglement measures, the so-called entanglement entropy is themost broadly studied one, both in equilibrium situations [6–11] and out of equilibrium [12–15]. It is anefficient tool to measure bipartite entanglement in pure states. Consider a system in the pure state | ψ i ,composed of two complementary subsystems A and B . The entanglement entropy S A is defined as the von Neumann entropy [16] of the reduced density matrix of subsystem A , S A = − tr( ρ A log ρ A ) , ρ A = tr B ρ, ρ = | ψ ih ψ | , (1.1)where tr B indicates a trace over the degrees of freedom in B . The entanglement entropy does not dependon the subsystem that is traced over: S A = S B . For systems that are not critical, the entanglemententropy satisfies an area law [9, 17]: it is proportional to the area of the boundary between the twosubsystems. In particular, for non-critical one-dimensional quantum systems, it saturates to a constantvalue in the limit of large system size N → ∞ . In contrast, for critical one-dimensional quantumsystems, the entanglement entropy diverges logarithmically with the system size in the scaling limit: S A ∝ log N . The prefactor is predicted by conformal field theory (CFT) [5–7] to be proportional tothe central charge c , S A = ac N + O (1) , (1.2)where a is the number of contact points between the two subsystems. If the whole system is definedon a periodic lattice, we have a = 2. On the contrary, if one end of the subsystem A is attached to aboundary, then a = 1.Another observable that shares many features with the entanglement entropy is the fidelity [18–22].It is defined as the overlap between the groundstates of two Hamiltonians that differ by a smallperturbation. The Hamiltonian of the system H ( λ ), where λ parameterises the perturbation, hasthe groundstate | λ i . The fidelity is f ( λ , λ ) = (cid:12)(cid:12)(cid:12)(cid:12) h λ | λ i h λ | λ i h λ | λ i (cid:12)(cid:12)(cid:12)(cid:12) . (1.3)As a particular example, we consider the situation where λ parameterises the interaction between twocomplementary subsystems, H ( λ ) = H A + H B + λH int , (1.4)where H A and H B are the Hamiltonians of the subsystems A and B , respectively. The term H int contains the interaction between the two subsystems. We denote by | ψ A i , | ψ B i and | ψ AB i the ground-states of H A , H B and H AB = H (1), respectively. The logarithmic bipartite fidelity [23, 24] is thendefined as F A,B = − log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h ψ A ⊗ ψ B | ψ AB i h ψ A ⊗ ψ B | ψ A ⊗ ψ B i h ψ AB | ψ AB i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (1.5)with the notation | ψ A ⊗ ψ B i ≡ | ψ A i ⊗ | ψ B i for the groundstate of H (0). Similarly to the entanglemententropy, this quantity vanishes when the involved groundstates are of the form | ψ i = N Nj =1 | ψ j i with | ψ j i independent of j and N . Otherwise it is a positive real number.An integrable quantum model in one dimension often underlies a statistical model in two di-mensions [25]. The transfer matrices T ( u ) for the latter commute at different values of the spectral3 N A N B NN A N B NN A N B Figure 1: The periodic pants geometry, the skirt geometry and the flat pants geometry.parameter u , and the Hamiltonian of the former is obtained as a leading term in the expansion of T ( u )around u = 0. From its definition in terms of scalar products of groundstates, the bipartite fidelitythen has an interpretation in terms of partition functions in the two-dimensional model. If in theone-dimensional model, the boundary conditions for the system and the two subsystems are open, thecorresponding model in two dimensions is defined on a domain that resembles a pair of flat pants. Wecall it the flat pants domain . It is illustrated in the right panel of Figure 1.Similarly to the entanglement entropy, the logarithmic bipartite fidelity is an efficient tool to de-tect quantum phase transitions. For non-critical models it satisfies an area law, whereas it divergeslogarithmically with the system size for one-dimensional quantum critical systems. On the flat pantsgeometry, Dubail and St´ephan [23, 24] used CFT arguments to derive the 1 /N expansion of the log-arithmic bipartite fidelity up to order N − log N . In particular, they found that the leading term isproportional to log N , with a prefactor that depends on the central charge c of the theory. In thesimple case where there is no change in boundary conditions between the two subsystems, this term is F flat pants = c N + O (1) . (1.6)In [24], the authors in fact considered a more general case where the Hamiltonians H A , H B and H AB have different boundary conditions applied to the endpoints of the chains. In the conformal field theory,this corresponds to a situation where four primary fields are inserted on the flat pants domain. Thethree leading terms in the large- N expansion of F are proportional to log N , 1 and N − log N and theexplicit expressions for their coefficients are found to depend on the conformal data, namely the centralcharge, the conformal dimensions of the fields and the four-point function of these fields. In previouswork, we checked these conformal predictions with analytical lattice computations for the XXZ spinchain at ∆ = − [26], and for the model of critical dense polymers [27], a lattice model known tobe described by a logarithmic CFT of central charge c = − skirt domain . For the quantumchain, the state | ψ AB i used to compute (1.5) in this case is the groundstate of the Hamiltonian withperiodic boundary conditions, whereas | ψ A i and | ψ B i are the groundstates for the open chains of thesubsystems A and B . The results of [23, 24] cover the simplest case where no boundary conditionchanging fields are inserted. The resulting 1 /N expansion that they obtain for the bipartite fidelityreads F skirt = c N + c
24 ( ˜ G ( x ) + ˜ G (1 − x )) + ˜ C + O ( N − log N ) , (1.7)where x = N A /N is the aspect ratio, ˜ C is a non-universal constant, and˜ G ( x ) = 3 − x + 4 x − x log x. (1.8)4n this paper, we provide new CFT predictions and lattice computations of the bipartite fidelityon two domains with periodic boundary conditions. The first is the periodic pants domain , for whichthe full system and the two subsystems all have periodic boundary conditions. It is depicted in the leftpanel of Figure 1. To our knowledge, there are no previously known results for the bipartite fidelity onthis geometry. The second is the skirt domain, where we will push further the investigation initiatedin [23]. We will consider the more complicated case where boundary condition changing fields arepresent in the CFT context, and will derive the leading terms in (1.7) up to order N − log N .Accordingly, the bulk of the paper is divided into two large sections: Section 2 covers the case ofthe periodic pants domain, and Section 3 focuses on the skirt domain. Each of these two sections isdivided into four subsections. The first subsection gives the conformal predictions for the leading termsof F in its large- N expansion, with the details of the calculations presented in Appendix A. The secondand third subsections give the exact calculations of the bipartite fidelity for the XX spin chain and themodel of critical dense polymers, respectively. These are two free-fermionic models for which one candiagonalise the Hamiltonian explicitly and write F as a determinant, which can be evaluated in productform in certain favourable cases. The fourth subsection uses both exact asymptotic calculations andnumerical evaluations of the determinants to compare the lattice results with the CFT prediction.Some of the technical details of the asymptotic calculations are relegated to Appendix B. We presentfinal remarks in Section 4. In this section, we consider the bipartite fidelity for physical systems A , B and AB , of respectivelengths N A , N B and N = N A + N B , which are all endowed with periodic boundary conditions. Forone-dimensional chains, the fidelity is defined as F p = − log (cid:12)(cid:12) h X A ⊗ X B | X AB i (cid:12)(cid:12) (2.1)where h v ⊗ w | is a short-hand notation for h v | ⊗ h w | . The states h X S | and | X S i are respectively the leftand right groundstates of the Hamiltonian of the chain of the system S . These states are normalisedin such a way that h X S | X S i = 1.For two-dimensional lattice models, the fidelity is defined as F p = lim M →∞ − log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) Z ABp (cid:1) Z Ac Z Bc Z A ∪ Bc (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.2)Here Z ABp is the partition function defined on the periodic pants geometry depicted in the left panelof Figure 1. The perimeter at the top is N , the perimeters of the legs A and B are N A and N B , andthe height is 2 M . Likewise, Z Ac , Z Bc and Z A ∪ Bc are partitions functions on cylinders of height 2 M and perimeters N A , N B and N , respectively. Clearly, the bipartite fidelity depends on the choices ofboundary conditions assigned to the top and bottom of these lattices. For suitable choices of theseboundary conditions, the partition functions are all non-zero and the limit M → ∞ in (2.2) is well-defined.As is well-known, certain families of integrable models in two dimensions are related to one-dimensional quantum spin chains [25]. In these cases, the two definitions (2.1) and (2.2) coincide. InSection 2.1, we give the conformal prediction for the leading terms in the large- N expansion of thebipartite fidelity. In Sections 2.2 and 2.3, we compute F p for the XX spin chain and for the model ofcritical dense polymers. Finally, in Section 2.4, we compare the asymptotical behaviour of the latticeresults with the predictions of conformal field theory.5 − x z z z z = ∞ w p w w p (1 − x +0i + ) w p (1 − x +0i − ) NxN/ − x ) NxN/ w p (1 + +0i + ) w p (1 − +0i + ) w p (0 + +0i + ) w p (0 − ) w p (1 + +0i − ) w p (1 − +0i − ) w p (0 + +0i − ) w p ( ∞ ) Figure 2: The function w p ( z ) maps the complex plane onto the periodic pants geometry. On the rightpart of the figure, the dashed and dotted lines are identified pairwise. In this section, we give the conformal predictions for the bipartite fidelity on the periodic pants ge-ometry. The details of the calculations are given in Appendix A. Let us consider a one-dimensionalquantum critical system of characteristic size N . For large N , its free energy f behaves as f = f bulk N + f surface N + f shape log N + f cst + . . . (2.3)where the dots indicate that lower-order terms are omitted. In this expression, the first two terms areproportional to the area and the surface of the domain. They are non-universal in the sense that theydepend on the details of the theory’s short-range interactions. In contrast, the term f shape is universal.It depends on the geometry considered and on the data of the underlying CFT, namely the centralcharge c and the dimensions ∆ i , ¯∆ i of the conformal fields. Finally, the term f cst also depends onthe CFT data. Because the free energy can always be shifted by an overall non-universal constant,arguments of conformal field theory can only predict f cst up to a non-universal additive constant.From the definition (2.2), the bipartite fidelity on the periodic pants domain corresponds to thefollowing difference of free energies: F p = 2 f p − f c ( N ) − f c ( N A ) − f c ( N B ) , (2.4)where f p is the free energy on the periodic pants domain and f c ( P ) is the free energy on the cylinderof perimeter P . The bipartite fidelity can then be seen as a renormalised free energy. It dependson two independent characteristic lengths, N and N A , with the third characteristic length given by N B = N − N A . We consider the asymptotic behaviour of F p in the limit where N and N A are sent toinfinity with the aspect ratio x = N A /N kept fixed. It has the large- N expansion F p = g log N + g ( x ) + g ( x ) N − log N + . . . . (2.5)The coefficients g , g ( x ) and g ( x ) can be computed by the methods of conformal field theory. Inthis framework, the periodic pants domain is described as an infinite horizontal strip of width N drawnin the complex plane. The strip is decorated with two slits that divide its left half into two strips ofwidth N x and N (1 − x ). The boundary conditions are periodic along the strip’s edges and along theslits so as to reproduce the geometry of the left panel of Figure 1. In particular, due to the periodicity,the two endpoints of the slits are identified as a unique point in the domain. We refer to this point asthe crotch point . The mapping from the complex plane to the periodic pants geometry is w p ( z ) = N π (cid:0) x log( z −
1) + (1 − x ) log z (cid:1) + K x , K x = − N π (cid:0) x log x + (1 − x ) log(1 − x ) (cid:1) . (2.6)This map is illustrated in Figure 2.In a general setting, we consider the situation where four fields are inserted on the periodic pantsgeometry. A field φ is inserted at −∞ in the leg A , a field φ is inserted on the crotch point, a field φ
6s inserted at −∞ in the leg B , and a field φ is inserted at + ∞ . We denote by w i , with i = 1 , . . . , z i the corresponding positions in the complexplane obtained from the inverse map w − p ( z ). For simplicity, we assume that each field φ i is spinlesswith conformal dimensions ∆ i = ¯∆ i .The first term g in the expansion (2.5) is obtained as a direct application of the Cardy-Peschelformula [30] for conical singularities. Indeed, the slit’s endpoint in the pants domain corresponds toa single conical singularity of angle 4 π , as can be seen from the expansion of w p ( z ) about the point z = z . The resulting expression for g depends on the dimension of the field φ and on the centralcharge: g = c . (2.7)The next term g ( x ) depends on the aspect ratio x and on the dimensions ∆ i of the four fields. Inthe general setting where the four fields are non-trivial primary fields, g ( x ) also depends on the four-point function of the fields φ , φ , φ and φ . A similar dependence was found in [24] for the flat pantsdomain. Here, we only state the resulting expressions, with their derivations given in Appendix A. Wegive the result in two cases: (i) no field is inserted on the crotch point and the fields φ , φ and φ areprimary, and (ii) the four fields φ i are vertex operators with charges q i = ¯ q i and conformal dimensions∆ i = ¯∆ i = q i /
2. For case (i), the known form of the three-point functions allows us to find g ( x ) = 4 (cid:16) ∆ − ∆ x − ∆ − x (cid:17)(cid:0) x log x + (1 − x ) log(1 − x ) (cid:1) + c (cid:0) G ( x ) + G (1 − x ) (cid:1) + C (2.8)where G ( x ) = 3 − x + 2 x − x log x (2.9)and C is a constant with respect to x . This constant is non-universal. For case (ii), the n -point functionof the vertex fields is * n Y i =1 φ ( z i , ¯ z i ) + = Y i 2. For generic twist angles, we do not know how toevaluate this determinant in closed form. This is due to the numerators in (2.26) which are differentin the two parts of the matrix A . We can however evaluate the determinant if the twist parameters φ , φ A and φ B are such that1 + e − i πx (2 k ′ − e − i( φ A − xφ ) = 1 + e − i( φ − φ B ) e i π (1 − x )(2 k ′ − e i xφ . (2.27)The solutions are φ = φ A + φ B − π + 2 πℓ, ℓ ∈ Z . (2.28)In the following, we focus on the case ℓ = 0. Under this specialisation, we are able to simplify thedeterminant and obtain a closed-form formula. Indeed, we find |h X A ⊗ X B | X AB i| = N − N/ x − Nx/ (1 − x ) − N (1 − x ) / (cid:12)(cid:12)(cid:12)(cid:12) N/ Y k ′ =1 cos h πxk ′ − φ A ( x − xφ B i(cid:12)(cid:12)(cid:12)(cid:12) | det M| (2.29)where M k,k ′ = sin h π (2 k − − φ A Nx − π (2 k ′ − − φ N i − k = 1 , . . . , Nx , sin h π (2 k − Nx − − φ B N (1 − x ) − π (2 k ′ − − φ N i − k = Nx + 1 , . . . , N , k ′ = 1 . . . , N . (2.30)We use the Cauchy identitydet i,j x i − y j ) = Q i 1. The identity I for this algebra is the element I = Ω − Ω = ΩΩ − = ... N . (2.41)The diagrammatic rule describing the product in this algebra can be translated into relations satisfiedby the generators. These are given for instance in [29]. The value of β pertaining to the model ofcritical dense polymers is β = 0. In our calculations below, α is kept as a free parameter and N is setto an even integer. The transfer tangle and the Hamiltonian. The transfer tangle for the model of critical densepolymers with periodic boundary conditions is an element of E PTL N ( α, 0) defined as T ( u ) = . . . . . .u u u | {z } N , u = cos u + sin u . (2.42)It is a linear combination of 2 N connectivity diagrams. The isotropic value is u = π , and we usethe short-hand notation T = T ( π ). This transfer tangle commutes for different values of the spectralparameter: [ T ( u ) , T ( v )] = 0. Expanding T ( u ) in a Taylor series in u , we obtain T ( u ) = Ω( I − u H ) + O ( u ) , H = − N X j =1 e j , (2.43)where H is the Hamiltonian. It commutes with the transfer tangle: [ H , T ( u )] = 0.13 he standard modules W N, and W N, . The algebra E PTL N ( α, β ) possesses a family of standardmodules W N,d labelled by an integer number d of defects. Our calculation of F αp requires two standardmodules: W N, and W N, . These are respectively defined on the vector spaces generated by link stateswith zero and two defects. Link states are diagrams drawn over a segment where N marked nodes areconnected pairwise by non-intersecting loop segments or occupied by vertical defects that cannot beoverarched.The boundary conditions are periodic in the horizontal direction. For example, here are the linkstates for N = 4: W , : , , , , , , (2.44a) W , : , , , . (2.44b)The action of a connectivity in E PTL N ( α, β ) on a link state w in W N, is similar to the action of E PTL N ( α, β ) on itself. The link state w is drawn on top of the connectivity, the resulting new linkstate is read from the bottom nodes, and multiplicative factors of α and β are included for each non-contractible and contractible loop, respectively. Moreover, for W N, , the result is set to zero if the twodefects are connected. Here are examples of this standard action:= α , = β , = 0 . (2.45)We note that the standard module W N, can be defined more generally with a twist parameter thatkeeps track of how much the defects wrap around the cylinder. A link state module with identified connectivities. In the special case where α = β , thereexists another representation defined on link states with zero defects, with so-called identified connec-tivities [29]. We denote it by b W N, . Its vector space is spanned by the link states with zero defects in W N, that have no arcs travelling via the back of the cylinder. These are in fact the same link statesthat span the standard module with zero defects of the usual Temperley-Lieb algebra TL N ( β ). For N = 4, these link states are: b W , : , . (2.46)The action of connectivity diagrams on link states in b W N, is defined using the same construction asfor standard modules. One draws the link state above the connectivity diagram, reads the new link statefrom the bottom edge of the diagram, and includes factors of β for each closed loop, both contractibleand non-contractible. Crucially, in reading off the resulting link state, there is no distinction betweenarcs travelling via the front or back of the cylinder. Any arc travelling around the back of the cylinderis transformed into an arc travelling in the front of the cylinder. Here are two examples of this actionfor N = 4: = β , = β . (2.47) The XX representation. The XX representation of E PTL N ( α, X N , is definedon the vector space ( C ) ⊗ N . We use the canonical basis defined in Section 2.2.1. In this basis, thegenerators e j with j = 1 , . . . , N − X N ( e j ) = I ⊗ · · · ⊗ I | {z } j − ⊗ − 00 0 0 0 ⊗ I ⊗ · · · ⊗ I | {z } N − j − , j = 1 , . . . , N − . (2.48)14ikewise, the generators Ω ± and e N are represented by X N (Ω ± ) = t ± e ± i φσ z / , X N ( e N ) = X N (Ω) X N ( e ) X N (Ω − ) , (2.49)where φ is the twist angle and t is the translation operator: t (cid:0) | v i ⊗ | v i ⊗ · · · ⊗ | v N i (cid:1) = | v i ⊗ · · · ⊗ | v N i ⊗ | v i . (2.50)The matrices X N ( e j ) and X N (Ω ± ) realise a representation of E PTL N ( α, 0) for α = 2 cos (cid:16) φ (cid:17) . (2.51)They also commute with the total magnetisation S z = P Ni =1 σ zi . As a consequence, the representation X N splits as a direct sum of smaller representations labelled by the eigenvalues m of S z , which takethe values m = − N , − N − , . . . , N .In this representation, the Hamiltonian H is the twisted XX Hamiltonian (2.15): H = X N ( H ) = − N − X j =1 (cid:16) σ + j σ − j +1 + σ − j σ + j +1 (cid:17) − e i φ σ + N σ − − e − i φ σ − N σ +1 . (2.52)Likewise, the transfer matrix is defined as T ( u ) = X N ( T ( u )). It is the transfer matrix of the six-vertexmodel at the anisotropy ∆ = 0, with periodic boundary conditions and a diagonal twist. We use thenotation T = T ( π ) for this transfer matrix at the isotropic point. Homomorphisms. There exists a map from link states to spin states that intertwines the link stateand XX representations. Indeed, for a given link state w in W N, , W N, or b W N, , we write its image in( C ) ⊗ N under this map as | w i . It is defined from the following local maps: | i = ω |↑↓i + ω − |↓↑i , | i = |↓i , | i = ω − e − i φ |↑↓i + ω e i φ |↓↑i , ω = e i π/ . (2.53)For a given link state, these local rules are applied to each arc and each defect. For w ∈ W N, , W N, and b W N, , the resulting state | w i has the magnetisation m = 0 , − X N ( e j ) | w i = | e j w i , X N (Ω ± ) | w i = | Ω ± w i , j = 1 , . . . , N, (2.54)and is satisfied for each link state w . This holds for W N, , b W N, and W N, , with the action of thegenerators on the right sides of the equations adapted accordingly, for each case. For W N, , thehomomorphism with the spin-chain representation holds for α fixed as a function of φ as in (2.51).For b W N, , non-contractible loops have the weight β and the homomorphism holds provided that φ isfixed to ± π . In that case, | i and | i are equal up to a sign. For W N, , the map (2.53) is ahomomorphism for the special value φ = 0. This is consistent with the fact that our definition of thismodule was given without including a parameter that keeps track of the winding of the defects. The Gram bilinear form is an invariant form on the standardmodules. For critical dense polymers, it is defined as follows. Let w, w ′ be two link states in W N, or W N, . Performing a vertical flip of w and connecting its nodes to those of w ′ , we obtain a diagramwhere the loop segments form loops, that can be contractible or non-contractible loops. For W N, ,the Gram product of w and w ′ , denoted w · w ′ , is defined as α n α δ n β , , where n α and n β count the15on-contractible and the contractible loops. It is therefore non-zero only if the number of contractibleloops is zero.For W N, , the same diagram constructed from w and w ′ involves loops as before, but also findsthe defects connected pairwise. If both defects of w are connected to defects of w ′ and there are noloops, then w · w ′ = 1. Otherwise, w · w ′ = 0. To illustrate, for N = 4, the matrices encoding the Gramproducts between the link states in the bases (2.44) are α α α α α α αα α α α α α α , . (2.55)These bilinear forms allow us to express the partition functions Z A ∪ Bc , Z Ac and Z Bc in terms ofGram products: Z A ∪ Bc = 2 MN v AB · ( T AB ) M v AB (cid:12)(cid:12) W N, , (2.56a) Z Ac = 2 MN A v A · ( T A ) M v A (cid:12)(cid:12) W NA, , (2.56b) Z Bc = 2 MN B v B · ( T B ) M v B (cid:12)(cid:12) W NB, , (2.56c)where the boundary states are v = ... , v = ... . (2.57)The superscripts AB , A and B for T in (2.56) serve as a reminder that the corresponding objectsare elements of the enlarged periodic Temperley-Lieb algebra with N , N A and N B nodes, respectively.We have also indicated by subscripts on the right side which action is used. Finally, we note that thepowers of 2 ensure that each tile has a weight 1 instead of √ as it does in (2.42) for u = π . Gram products on the pants geometry. We define a new Gram product, denoted ( w × w ) · w ,that is needed to compute Z ABp . In this product, the states w , w and w belong to W N A , , b W N B , and W N, respectively, with N = N A + N B . The result of this product is obtained as follows. One flips w and w vertically, draws them on the legs A and B of the pants lattice, and connects its nodes to thoseof w drawn on the top part of the pants lattice. Then ( w × w ) · w equals α n A + n AB δ n β , δ n B , , where n β counts the number of contractible loops, and the numbers n A , n B and n AB count the non-contractibleloops in the three families, as explained in Section 2.3.1. Here are two examples to illustrate: (cid:16) × (cid:17) · = α , (2.58a) (cid:16) × (cid:17) · = 0 . (2.58b)In terms of this Gram product, the partition function Z ABp reads Z ABp = 2 MN (cid:16) ( T A ) M v A × ( T B ) M v B (cid:17) · ( T AB ) M v AB . (2.59)In this expression, the transfer tangles T A and T AB act on the boundary states v A and v AB under thestandard actions of W N A , and W N, respectively (where α is a free parameter). In contrast, ( T B ) M acts on v B under the module action of c W N B , (where α = 0).16e note that one can define more generally a Gram product on the pants geometry where theresult of the product is α n A α n B α n AB δ n β , and thus depends on three free parameters, one for eachfamily of non-contractible loops. Our choice to set α = 0 and α = α = α however allows for animportant simplification. Indeed, in this case, the connectivity of link states in leg B can be describedusing the module b W N B , , and the Gram products on the pants geometry can be computed using theGram product on the cylinder, for W N, . Our construction above achieves this by using the naturalembedding from W N A , × b W N B , into W N, , where the two link states w and w are simply drawn sideby side, and the loop segments of w that travel via the back of the cylinder (if any) are extended totravel via the back of the larger cylinder. For instance, × 7→ , × 7→ . (2.60)The same embedding in W N, does not naturally extended to W N A , × W N B , . For example, we cannotembed × in W N, in the same way as in (2.60).Thus, for the case α = 0, any Gram product on the pants geometry can be computed using thesame product on the cylinder. In the spin-chain language, the resulting embedding translates simplyto | w × w i = | w i ⊗ | w i . (2.61) Overlaps in the XX chain. At the end of Section 2.3.2, we defined a map w 7→ | w i that intertwinesthe link state and spin-chain representations of E PTL N ( α, w , we define the dualstate hh w | as hh w | = | w i t (cid:12)(cid:12)(cid:12) φ →− φ (2.62)where the superscript t stands for real transposition. The Gram product between two link states w and w can then be computed from the spin-chain representation [38]: w · w = hh w | w i . (2.63)In the spin-chain language, the partition functions read Z A ∪ Bc = 2 MN hh v AB | ( T AB ) M | v AB i , (2.64a) Z Ac = 2 MN A hh v A | ( T A ) M | v A i , (2.64b) Z Bc = 2 MN B hh v B | ( T B ) M | v B i , (2.64c) Z ABp = 2 MN hh v A ⊗ v B | (cid:0) ( T A ) M ⊗ ( b T B ) M (cid:1) ( T AB ) M | v AB i . (2.64d)We recall that T AB and T A are the transfer matrices of the six-vertex model with a twist φ for systemsizes N and N A respectively, whereas T B and b T B are the transfer matrices for the system size N B withthe twists φ = 0 and φ = π , respectively. The diagonalisation of H is given in Section 2.2.2. We recallthat for φ ∈ ( − π, π ), the groundstate of H is unique, belongs to the magnetisation sectors S z = 0 andis given by | X i = ( µ † (4 − N ) / · · · µ † N/ | i N even, µ † (2 − N ) / · · · µ † ( N − / | i N odd, | i = |↓ · · · ↓i . (2.65)17ikewise, in the sector S z = − 1, the groundstate of H for φ = 0 is unique and given by | X − i = µ † (4 − N ) / · · · µ † ( N − / | i (cid:12)(cid:12) φ =0 N even, µ † (6 − N ) / · · · µ † ( N − / | i (cid:12)(cid:12) φ =0 N odd. (2.66)The transfer matrix and Hamiltonian commute, and thus we have T | X i = Λ | X i , ( T | φ =0 ) | X − i = Λ − | X − i . (2.67)The eigenvalues are [29, 39]Λ = cos( φ )2 N − N/ Y j =1 (1 + tan x j ) N Y j =( N +2) / (1 − tan x j ) , x j = π ( j − ) − φ N , (2.68a)Λ − = N N − N − / Y j =1 (cid:0) πjN ) (cid:1) . (2.68b)and are the largest eigenvalues in the sectors S z = 0 and S z = − 1, respectively. We note in particularthat Λ − = lim φ → π Λ .In the representations W N, and W N, , the transfer tangle T has the right eigenstates X and X − with respective eigenvalues Λ and Λ − , and | X i and | X − i are their images under the homomorphismmap. Their duals hh X | and hh X − | are obtained from the definition (2.62). Because µ t k (cid:12)(cid:12) φ →− φ = ( µ †− k N + S z odd, µ † − k N + S z even, (2.69)we have hh X | = ( − N ( N − / h X | , hh X − | = ( − ( N − N − / h X − | (2.70)where h X | = | X i † and h X − | = | X − i † . The state h X | is given in (2.23) and is the left groundstateof H in the sector S z = 0. Likewise, h X − | is the left groundstate in the sector S z = − 1. Because ofthe fermionic relations (2.21), the groundstates have unit norms, and therefore we have hh X | X i = ( − N ( N − / , hh X − | X − i = ( − ( N − N − / . (2.71) Ratios of overlaps. We now extract the leading behaviours of the partition functions (2.56) and(2.59) as M tends to infinity. We start with Z ABp and assign the extra labels AB , A and B tothe groundstates over ( C ) ⊗ N , ( C ) ⊗ N A and ( C ) ⊗ N B , respectively. From (2.71), we deduce that theidentity matrix over ( C ) ⊗ N in the sector of zero magnetisation has a contribution along the groundstateof the form I (cid:12)(cid:12) S z =0 = ( − N ( N − / | X AB ihh X AB | + . . . . (2.72)The next terms involve states that are not groundstates. Likewise, for ( C ) ⊗ N A and ( C ) ⊗ N B , we have I (cid:12)(cid:12) S z =0 = ( − N A ( N A − / | X A ihh X A | + . . . , (2.73a) I (cid:12)(cid:12) S z =0 = ( − N B ( N B − / | X B ihh X B | + . . . . (2.73b)We then have ( T AB ) M | v AB i ≃ ( − N ( N − / (Λ AB ) M | X AB ihh X AB | v AB i , (2.74a) hh v A | ( T A ) M ≃ ( − N A ( N A − / (Λ A ) M hh v A | X A ihh X A | , (2.74b) hh v B | ( b T B ) M ≃ ( − N B ( N B − / (Λ B ) M hh v B | X B ihh X B | , (2.74c)18here Λ AB , Λ A and Λ B are the eigenvalues of T AB , T A and b T B in the zero magnetization sector. Thesymbol ≃ indicates that the smaller contributions coming from excited states have been omitted. Wethus find Z ABp ≃ ǫ MN (Λ AB Λ A Λ B ) M hh v A | X A ihh v B | X B ihh X A ⊗ X B | X AB ihh X AB | v AB i , (2.75)where ǫ is a sign that will be irrelevant for our computation of F αp . We repeat the same argument forthe other partition functions and find Z A ∪ Bc ≃ MN (Λ AB ) M hh v AB | X AB ihh X AB | X AB ihh X AB | v AB i , (2.76a) Z Ac ≃ MN A (Λ A ) M hh v A | X A ihh X A | X A ihh X A | v A i , (2.76b) Z Bc ≃ MN B (Λ B − ) M hh v B | X B − ihh X B − | X B − ihh X B − | v B i , (2.76c)where Λ B − is the groundstate eigenvalue of T B in the sector S z = − 1. Using (2.71) and Λ B = Λ B − , wefind F αp = − log σ hh X A ⊗ X B | X AB i hh X AB | v AB ihh v AB | X AB i hh v A | X A ihh X A | v A i hh v B | X B i hh v B | X B − ihh X B − | v B i ! (2.77a)with σ = ( − N ( N − ( − NA ( NA − ( − ( NB − NB − . (2.77b) Product expressions for the overlaps involving boundary states. We use Wick’s theorem toevaluate the various overlaps in (2.77a). To start, we note that the boundary states hh v | and hh v | canbe written in terms of the fermionic operators c j as hh v | = h | a N − a N − · · · a a a , hh v | = h | a N − a N − · · · a a a , a j = ω c j + ω − c j +1 . (2.78)We have the anti-commutation relation { a ℓ − , µ † k } = e − ℓθ k √ N (cid:0) ω e i θ k + ω − (cid:1) . (2.79)Using Wick’s theorem, we write the overlaps hh X | v i and hh X − | v i as hh v | X i = det k ∈ K ℓ =1 ,...,N/ { a ℓ − , µ † k } , hh v | X − i = det k ∈ K − ℓ =1 ,..., ( N − / { a ℓ − , µ † k } (cid:12)(cid:12) φ =0 , (2.80)where K = ( (cid:8) − N , · · · , N (cid:9) N even, (cid:8) − N , · · · , N − (cid:9) N odd, K − = ( (cid:8) − N , · · · , N − (cid:9) N even, (cid:8) − N , · · · , N − (cid:9) N odd. (2.81)The factors that depend only on k in (2.79) can be factorised from the determinant, which can thenbe evaluated in terms of a product using the Vandermonde identity. We simplify the result using theidentities N/ Y j =1 sin (cid:0) πjN (cid:1) = N / ( N − / , N/ Y j =1 N/ Y k = j +1 sin (cid:0) π ( k − j ) N (cid:1) = N N/ N / , (2.82)and find hh v | X i = 2 N/ e i φ ( N +1) / e − i πN ( N − / N/ Y j =1 sin (cid:0) (2 j − π + φ N (cid:1) , (2.83a) hh v | X i (cid:12)(cid:12) φ = π = N / ( N − / e − i π ( N − N − / , (2.83b) hh v | X − i = e − i π ( N − N − / ( N − / . (2.83c)19t also follows from the definition (2.62) of dual states that hh X | v i = hh v | X i (cid:12)(cid:12) φ →− φ , hh X − | v i = hh v | X − i . (2.84)Putting these results together, we obtain hh X AB | v AB ihh v AB | X AB i = e − i φ ( N +1) / N/ Y j =1 sin (cid:0) (2 j − π − φ N (cid:1) sin (cid:0) (2 j − π + φ N (cid:1) , (2.85a) hh v A | X A ihh X A | v A i = e i φ ( N A +1) / N A / Y j =1 sin (cid:0) (2 j − π + φ N A (cid:1) sin (cid:0) (2 j − π − φ N A (cid:1) , (2.85b) hh v B | X B i hh v B | X B − ihh X B − | v B i = − i N B . (2.85c) Exact expression for the bipartite fidelity. It remains to compute the overlap hh X A ⊗ X B | X AB i .Up to unimportant signs that come from (2.71), the overlap is exactly the one computed in (2.34) withthe specialisations φ A = φ and φ B = π . We note that these specialisations are compatible with (2.28)for ℓ = 0, for which we were able to obtain product expressions from the determinant expressions.From (2.77a) and (2.85) we find F αp = F XX p (cid:12)(cid:12) φ A = φφ B = π − Q ( N, φ ) + Q ( N x, φ ) − log (cid:0) N (1 − x ) (cid:1) (2.86)where Q ( N, φ ) = N/ X j =1 log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin (cid:0) (2 j − π − φ N (cid:1) sin (cid:0) (2 j − π + φ N (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.87)The closed-form expression for F XX p (cid:12)(cid:12) φ A = φφ B = π is read from (2.34). In this section, we study the large- N asymptotics of the bipartite fidelity for the XX chain and themodel of critical dense polymers on the periodic pants lattice and compare the results with the CFTpredictions of Section 2.1. F XX p with φ = φ A + φ B − π . The calculation of the first terms in the large- N asymptotics of (2.34) is a long yet straightforward calculation. The details are given in Appendix B.We find F XX p (cid:12)(cid:12) φ = φ A + φ B − π = 12 log N − π ( − x )(1 + 2 x ) − πx (cid:16) ( − x ) φ A + xφ B (cid:17) + 3 (cid:16) ( − x ) φ A + xφ B (cid:17) π x log(1 − x )+ π (3 − x ) x + 6 π ( − x ) (cid:16) ( − x ) φ A + xφ B (cid:17) − (cid:16) ( − x ) φ A + xφ B (cid:17) π (1 − x ) log x − − 12 log π + 6 log A + O (cid:0) N − (cid:1) (2.88)20 XX p x F XX p N Figure 5: The standard bipartite fidelity F XX p as a function of x for N = 800 (left panel) and as afunction of N for x = (right panel). In both cases, the twists are set to φ = 1, φ A = 2 and φ B = 3.The data points are computed from the determinant expression (2.25), whereas the continuous line is(2.92) with (2.93), Ξ = 0 and C ′ = 0 . A ≃ . x → − x and φ A ↔ φ B . We see that this result is identical to theCFT prediction (2.5) for case (ii), where g and g ( x ) are given in (2.7) and (2.11), the conformal datais c = 1 , q = φ A π , q = − , q = φ B π , q = − φ A + φ B − π π , (2.89)and the non-universal constant is given by C ′ = − − 12 log π + 6 log A ≃ . . (2.90)We also remark that there is no term proportional to N − log N in (2.88). Comparing with (2.12),we observe that the content of the large parenthesis in this equation does not vanish in general. Wetherefore deduce that the extrapolation length Ξ vanishes in this case. Numerical results for F XX p ( n ). We now study the large- N behaviour of the bipartite fidelitiesin the case where the three twists φ , φ A and φ B are arbitrary. We consider the standard bipartitefidelity (2.17) as well as the modified instances F XX p ( n ) defined in (2.35). Since we do not have anexact product formula, we instead use the determinant expression (2.25), and very similar formulas forthe modified instances, to study the large- N behaviour of these quantities numerically. To get precisenumerical estimates, we follow the strategy of [23] and fit F XX p ( n ) as F XX p ( n ) = g log N + g ( x ) + g log NN + γ N + γ log NN + γ N + γ log NN + γ N . (2.91)From these numerical explorations, we find that the leading terms in the asymptotic expansion preciselyreproduce the conformal prediction of Section 2.1. Namely, we have F XX p ( n ) = (cid:18) 14 + 14 (cid:16) φ − φ A − φ B π (cid:17) (cid:19) log N + g ( x ) + g ( x ) N − log N + O (cid:0) N − (cid:1) , (2.92)where g ( x ) is given by the formula (2.11) for vertex operators, with c = 1 , q = φ A π , q = φ − φ A − φ B π , q = φ B π , q = − φ π . (2.93)Likewise, g ( x ) is given by (2.12) with the conformal dimensions ∆ i = q i / 2, the charges q i in (2.93),and the extrapolation length Ξ = − n. (2.94)21 ′ q C ′ q Figure 6: Numerical results for the constant C ′ as a function of q for the standard bipartite fidelity(left panel) and n = 4 (right panel). g ( x ) xn = 2 g ( x ) n = 4 x Figure 7: The function g ( x ) for the XX spin chain with φ = 1, φ A = 2 and φ B = 3 for n = 2 and n = 4. The blue dots are obtained by numerical evaluation of determinants similar to (2.25). The bluecurves plot the CFT predictions (2.12) with the conformal data (2.93) and Ξ given in (2.94).In particular, we have Ξ = 0 for the standard bipartite fidelity.The constant C ′ is unknown and obtained from a fit in our analysis. In Figure 5, we plot (2.92)and compare it with numerical values obtained from the determinant formula (2.25), as a function of x and N . The agreement is remarkable. The same plots for F XX p ( n ) with n = 2 , , C ′ depends on the twists andon the spin state inserted at the crotch point. For a given choice of this spin state, we find that C ′ depends on the twist parameters only through the difference φ − φ A − φ B , or equivalently on the valueof q . In Figure 6, we plot C ′ for various values of q in the range ( − , ), for n = 0 and n = 4.Finally, in Figure 7 we plot the function g ( x ) for n = 2 and n = 4. This function depends on the spinstate inserted at the crotch point only via the extrapolation length (2.94). The match between ournumerical results and the CFT prediction is clear. Conformal interpretation. Let us now discuss how our results fit with previously known resultsabout the conformal interpretation of the XX chain. It is well-known that the XX chain (and moregenerally the XXZ chain) is described by a CFT with central charge c = 1. In this context, thepresence of a twist line φ between two points z and z is accounted for by the insertion of two fields ϕ ( z ) and ϕ ( z ). The field ϕ is a so-called electric operator [40]. Its multi-point functions are knownto have the form (2.10), and our analysis in this subsection confirms this. It is also a spinless field. Itsconformal dimensions ∆ XX φ = ¯∆ XX φ can be computed from the N finite-size correction term of the largesteigenvalue of the periodic transfer matrix T with a diagonal twist. Indeed, this finite-size correction22an be obtained either via the Bethe ansatz [41] or with the method of functional relations [42], and itallows one to compute the difference c − 24∆ explicitly: c − 24∆ = 1 − φ π . (2.95)With c = 1, the resulting conformal dimension is∆ XX φ = φ π . (2.96)This is consistent with the known values q = ¯ q = φ π for the charges of electric operators, and the values∆ i = q i / g . We note that the field inserted on the crotch point is special, as it lies at theintersection of three twist lines. Accordingly, its conformal dimension is obtained from (2.96) with φ replaced by the difference φ − φ A − φ B of the twists at this intersection. The insertion of a N´eel stateof size n on the crotch point does not modify this conformal dimension. The large- N expansion for F αp follows directly from the same expansions for F XX p (cid:12)(cid:12) φ = φ A + φ B − π and thefunction Q ( N, φ ) defined in (2.87). These are given in (2.88) and (B.24), respectively. This yields F αp = F XX p (cid:12)(cid:12) φ A = φφ B = π − log N − π log(1 − x ) + φ log xπ = − 12 log N − 16 ( G ( x ) + G (1 − x )) − (cid:16) φ π − (cid:17)(cid:16) (1 − x ) log x + (1 − x ) x log(1 − x ) (cid:17) − − 12 log π + 6 log A + O (cid:0) N − (cid:1) (2.97)where G ( x ) is defined in (2.9). This exactly corresponds to the CFT prediction for case (i) with c = − , ∆ = ∆ = 18 (cid:16) φ π − (cid:17) , ∆ = 0 , C = − − 12 log π + 6 log A, Ξ = 0 . (2.98)This fits with previously known results for the conformal interpretation of the model of criticaldense polymers. The central charge of this model is known to be c = − α = β to the non-contractible loops amountsto inserting two copies of a conformal field ρ ( z ) at the top and bottom of the cylinder. The propertiesof this field were investigated in detail in [43]. The conformal dimensions ∆ CDP φ and ¯∆ CDP φ of ρ ( z ) areequal and can be obtained from the N finite-size correction term for the groundstate eigenvalue of theperiodic transfer matrix. The spectrum of the transfer matrix for the six-vertex model and for themodel of critical dense polymers are identical for α = 2 cos( φ ). As a result, the difference c − 24∆ forcritical dense polymers is also given by (2.95). Setting c = − CDP φ = 18 (cid:16) φ π − (cid:17) . (2.99)This is precisely the conformal dimension that we obtained in (2.98) for ∆ and ∆ . It is also the valuepreviously known from Coulomb gas calculations [44].23 Bipartite fidelity on the skirt geometry In this section, we consider the bipartite fidelity for physical systems A , B and AB , of respectivelengths N A , N B and N = N A + N B , where the system AB has periodic boundary conditions, whereasthe systems A and B have open boundary conditions. For one-dimensional chains, the fidelity is definedas F s = − log (cid:12)(cid:12) h x A ⊗ x B | X AB i (cid:12)(cid:12) . (3.1)The states h x S | and | x S i are respectively the left and right groundstates of the Hamiltonian of thesystem S endowed with open boundary conditions. Moreover, h X S | and | X S i are the groundstates ofthe Hamiltonian of the system S with periodic boundary conditions. These states are assumed to benormalised in such a way that h X S | X S i = 1 and h x S | x S i = 1.For two-dimensional lattice models, the fidelity is defined as F s = lim M →∞ − log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) Z ABs (cid:1) Z Ar Z Br Z A ∪ Bc (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.2)Here Z ABs is the partition function defined on the skirt geometry depicted in the central panel ofFigure 1. The height is 2 M , the perimeter at the top is N , the width of the legs A and B are N A and N B . Likewise, Z Ar and Z Br are partitions functions on rectangles of sizes 2 M × N A and 2 M × N B ,respectively. Finally, Z A ∪ Bc is the model’s partition function on the cylinder of perimeter N andheight 2 M . For suitable choices of these boundary conditions, the partition functions are all non-zeroand the limit M → ∞ in (3.2) is well-defined.For two-dimensional models that have one-dimensional quantum analogues [25], the definitions(3.1) and (3.2) coincide. In Section 3.1, we give the conformal prediction for the leading terms in thelarge- N expansion of the bipartite fidelity. In Sections 3.2 and 3.3, we compute F s for the XX spinchain and for the model of critical dense polymers. Finally, in Section 3.4, we compare the asymptoticalbehaviour of the lattice results with the predictions of conformal field theory. This section gives the conformal predictions for the bipartite fidelity on the skirt geometry. The detailsof the derivations are given in Appendix A. From the definition (3.2), the bipartite fidelity on the skirtgeometry corresponds to the following difference of free energies: F s = 2 f s − f c ( N ) − f r ( N A ) − f r ( N B ) , (3.3)where f s is the free energy on the skirt geometry, f c ( N ) is the free energy on the cylinder of perimeter N and f r ( N ) is the free energy on the rectangle of width N . The bipartite fidelity depends on twoindependent characteristic lengths, N and N A , with the third characteristic length given by N B = N − N A . We consider the asymptotic behaviour of F s in the limit where N and N A are sent to infinitywith the aspect ratio x = N A /N kept fixed. It has the large- N expansion F s = ˜ g log N + ˜ g ( x ) + ˜ g ( x ) N − log N + . . . . (3.4)The coefficients ˜ g , ˜ g ( x ) and ˜ g ( x ) can be computed by the methods of conformal field theory. Inthis framework, the skirt domain is described as an infinite horizontal strip of width N drawn in thecomplex plane, and decorated with two slits that divide the left half into two strips of width N x and N (1 − x ). The boundary conditions are periodic along the strip’s edges for Re( w ) > 0, but are openalong the strips’ edges for Re( w ) < 0. This indeed reproduces the geometry of the central panel ofFigure 1. In contrast with the periodic pants domain, the skirt domain really has two distinct slitendpoints. The map from the upper half-plane to the skirt geometry is w s ( z ) = N π (cid:0) x log (cid:0) − ( z − (cid:1) + (1 − x ) log z − log( z + ( z − ) (cid:1) + K x , (3.5)24 z z z z z = ∞ w s w w s ( z ) w s ( z ) NxN/ − x ) NxN/ w s ( z ) w s ( z ) w s ( z ) Figure 8: The function w s ( z ) maps the upper half-plane onto the skirt geometry. The positions z and z that are the preimages of the slits’ endpoints are z = ( √ x − x + x − / (2 x − 1) and z = ( −√ x − x + x − / (2 x − K x is given in (2.6). This map is illustrated in Figure 8. It can be understood as the composition w s = w p ◦ v of the function w p ( z ) given in (2.6) and the function v ( z ) = z / (2 z − z + 1). The lattermaps the upper half-plane into the complex plane with a cut along the segment [0 , φ is inserted at −∞ in the leg A , a field φ is inserted on the first slit’s endpoint,a field φ is inserted at −∞ in the leg B , a field φ is inserted on the second slit’s endpoint, and afield φ is inserted at + ∞ . The fields φ i with i = 1 , , , i . We denote by w i these positions in the skirt domain,and by z i the corresponding positions in the complex plane obtained from the inverse map w − s ( z ). Incontrast, φ is a bulk field that depends on two variables, which we write as w , ¯ w in the skirt domainand as z , ¯ z in the complex plane. For simplicity, we assume that φ is spinless. It has the conformaldimensions ∆ = ¯∆ .The first function ˜ g in the expansion (3.4) is obtained as a direct application of the Cardy-Peschelformula [30] for domains with corners. Indeed, the skirt domain has two corners of angles 2 π . Theresulting expression for ˜ g depends on the dimensions of the field φ and φ and on the central charge:˜ g = c + ∆ . (3.6)The next-leading terms ˜ g ( x ) and ˜ g ( x ) depend non-trivially on the aspect ratio x . In this section,we write down the resulting expressions, with their derivations given in Appendix A. In a general settingwhere the five fields are primary fields, the function ˜ g ( x ) also depends on the non-trivial functionsof the cross-ratios that arise in the five-point function of these fields in the upper half-plane. Because φ is a bulk field, from Cardy’s method of images, we know that this correlator is in fact a six-pointfunction in the full complex plane. Here we restrict our focus to the following special cases: (i) onlyone non-trivial primary field is present, namely the bulk field φ , and (ii) each of the fields φ i is avertex operator. In this case, the fields φ i with i = 1 , . . . , q i anddimension ∆ i = q i / 2. In contrast, the field φ is a bulk field with the charges q = ¯ q and dimensions∆ = ¯∆ = q / g ( x ) = c (cid:16) ˜ G ( x ) + ˜ G (1 − x ) (cid:17) + 4∆ ( x log x + (1 − x ) log(1 − x ) + 2 log 2) + ˜ C (3.7)where ˜ G ( x ) = 3 − x + 4 x − x log x (3.8)25nd ˜ C is a constant. For case (ii), we have˜ g ( x ) = c (cid:16) ˜ G ( x ) + ˜ G (1 − x ) (cid:17) + 4∆ ( x log x + (1 − x ) log(1 − x ) + 2 log 2) − ∆ x log(1 − x ) − ∆ − x log x + (∆ + ∆ )(log x + log(1 − x )) + ˜ C ′ (3.9)where ˜ C ′ is a constant. We note that the functions ˜ g and ˜ g ( x ) were obtained in [23] for the specialcase where all the conformal dimensions are zero.Similarly, we compute the function ˜ g ( x ) using the arguments of St´ephan and Dubail. The detailsare given in Appendix A. The authors argue that this function depends on a non-universal constantcalled the extrapolation length. As we shall see in Section 3.4, a correct conformal interpretation ofour lattice results requires a generalisation of their derivation to cases where each slit is assigned itsown extrapolation length. We denote them by Ξ and Ξ , and they correspond to the corners situatedat w and w , respectively. For case (i), we obtain˜ g ( x ) = (Ξ + Ξ ) × (cid:18) c (1 − x ) x (1 − x ) + 2∆ (cid:19) . (3.10)For the case (ii), the expression for ˜ g ( x ) instead reads˜ g ( x ) = Ξ × (cid:20) c (1 − x ) − x ) x + q − q x (1 − x ) − q ( q − q + q )4 x − q ( q − q + q )4(1 − x ) + q (cid:21) + { Ξ → Ξ , q ↔ q } . (3.11)We note that this expression greatly simplifies for Ξ = Ξ , and becomes independent of thecharges q and q of the fields inserted at the endpoints of the slits. The function ˜ g ( x ) also simplifiesfor q = q . In that case, it only depends on the sum Ξ + Ξ , as in (3.10), and can be written in termsof the dimensions ∆ i = q i / g ( x ) (cid:12)(cid:12) q = q = (Ξ + Ξ ) × (cid:18) c (1 − x ) x (1 − x ) + 2∆ − ∆ x − ∆ − x ) (cid:19) . (3.12) The first model for which we compute the bipartite fidelity on the skirt geometry is the XX chain. Inthis case, the system AB is a periodic chain of length N . Its Hamiltonian is (2.15) and depends on thetwist φ . The subsystems A and B are instead open chains of lengths N A and N B . The Hamiltonianfor the open chain of length N with free boundary conditions, namely without fields applied to theendpoints, is H f = − N − X j =1 (cid:16) σ xj σ xj +1 + σ yj σ yj +1 (cid:17) . (3.13)We denote by h x A | and h x B | the left groundstates of H f with N A and N B sites, respectively, in theirzero-magnetisation sectors. As before, N A + N B = N and all three lengths are even numbers. Thelogarithmic bipartite is F XX s = − log (cid:12)(cid:12) h x A ⊗ x B | X AB i (cid:12)(cid:12) (3.14)where | X AB i is the groundstate of H given in (2.22). The states | x A i and | x B i , like | X AB i , are assumedto have unit norms. 26 .2.2 Bipartite fidelity The diagonalisation of H f is standard. The tensor product of the left groundstates is h x A ⊗ x B | = h | d BN B / · · · d B d AN A / · · · d A (3.15)where d Ak = (cid:16) N A + 1 (cid:17) / N A X j =1 sin (cid:16) πkjN A + 1 (cid:17) c j , (3.16a) d Bk = (cid:16) N B + 1 (cid:17) / N X j = N A +1 sin (cid:16) πk ( j − N A ) N B + 1 (cid:17) c j . (3.16b)The operators c j are the fermionic operators in (2.20). With the choice of normalisation in (3.16), both h x A | and h x B | have unit norms. To compute (3.14), we use Wick’s theorem and find (cid:12)(cid:12) h x A ⊗ x B | X AB i (cid:12)(cid:12) = 2 − N/ N − N/ ( N A + 1) − ( N A − / ( N B + 1) − ( N B − / | det D| (3.17a)where D k,k ′ = − ( − k e − i( N A +1) ϑ k ′ cos (cid:0) πkN A +1 (cid:1) − cos ϑ k ′ k = 1 , . . . , N A , e − i N A ϑ k ′ − ( − k − N A / e − i( N B +1) ϑ k ′ cos (cid:0) π ( k − N A / N B +1 (cid:1) − cos ϑ k ′ k = N A + 1 , . . . , N , k ′ = 1 , . . . , N , (3.17b)and ϑ k = 2 π ( k − − N ) − φN . (3.17c)This holds for both parities of N/ 2. Sadly, we have been unable to push the calculation further andevaluate this determinant in product form using the methods employed in [24, 27], even for specialvalues of x and φ . Our analysis in Section 3.4.1 of the asymptotics of F XX s will thus rely on thenumerical evaluations of determinants. The CFT predictions of Section 3.1 covers cases where fields are inserted in the legs and on the slits’endpoints. In order to investigate these cases on the lattice, we consider various modified instancesof the bipartite fidelity on the skirt domain. In each case, the state | X AB i is kept unchanged in theoverlap, whereas the state h x A ⊗ x B | is replaced by a state of the form h x Am A ⊗ s ⊗ x Bm B ⊗ s | . (3.18)Here h x Sm | is the left groundstate of the open XX chain of the system S in the magnetisation sector m ,and s , s are selected from the set {∅ , ↑ , ↓ , ↑↓ , ↑↑ , ↑↑↑} . The condition on the even parity of N A and N B is relaxed and the relation tying them to N is N = N A + n + N B + n , where n and n are thelengths of the states | s i and | s i . Table 1 gives an overview of the thirteen cases that we consider. Inparticular, case 1 corresponds to the standard bipartite fidelity, as defined in (3.14) with the overlap(3.17). In the other cases, N A is chosen even if m A is an integer and odd if m A is a half-integer, andlikewise for N B in terms of m B . In each case, the magnetisation of the state (3.18) vanishes, and thecorresponding overlap (cid:12)(cid:12) h x Am A ⊗ s ⊗ x Bm B ⊗ s | X AB i (cid:12)(cid:12) is non-zero. We obtain a determinant expressionfor each of these overlaps similar to (3.17), which we do not reproduce here. We are unable to evaluatethese determinants in product form. Our analysis in Section 3.4 will instead rely on the numericalevaluation of these determinants. 27ase m A s m B s ∅ ∅ − / ∅ +1 / ∅ − ∅ ∅ − / ∅ +3 / ∅ − ∅ ∅ ↑ ↓ − / ↑ +1 / ↓ / ↓ +1 / ↓ ∅ ↑↓ − / ∅ − / ↑↑ − / ∅ ↑ − ∅ ↑↑ − / ∅ ↑↑↑ Table 1: The thirteen cases considered for the XX chain on the skirt geometry. We study the model of dense polymers on the skirt geometry. The lattice is a cylinder of height 4 M and width N , which we choose to be even. There are two vertical slits that extend halfway across thecylinder. They divide the lower edge into two open segments of lengths N A and N B , which are evennumbers too. A configuration of the model of critical dense polymers on the skirt lattice is a tiling ofthe faces of this lattice by one of the two elementary tiles, and , with equal probability.The top circular edge, the bottom segments as well as the interior of the two slits are decoratedwith simple half-arcs. The situation is then similar to the example of Figure 3, with the followingdifferences: (i) the total height of the lattice is 4 M instead of 2 M , and (ii) the four diagonal edges inthe diagram’s lower-half are decorated with simple half-arcs. The contractible loops are given a weight β = 0, whereas the non-contractible loops have a fugacity α . The Boltzmann weight of a configuration σ is then given by W σ = α n α δ n β , , where n α and n β are the numbers of non-contractible and contractibleloops in the configuration. The partition function on the skirt geometry, denoted Z ABs , is defined as Z ABs = X σ W σ . (3.19)The definition of the bipartite fidelity given below involves three more partition functions. Thefirst, Z A ∪ Bc , is the partition function of the same model defined on a cylinder of height 4 M andperimeter N . The other two partition functions, Z Ar and Z Br , are defined on 4 M × N A and 4 M × N B rectangles, respectively. The boundary conditions consist of simple half-arcs on all four segments, andwe restrict to configurations that have a single (contractible) loop. The partition functions Z Ar and Z Br are then the numbers of these restricted configurations.Similarly to Z Bc discussed in Section 2.3.1, the partition functions Z Ar and Z Br can alternativelybe defined on a lattice where the rightmost arcs on both the top and bottom edges are removed andreplaced by a pair of defects. One then imposes that both defects from the top segment connect tothose of the bottom segment (with weight 1). This produces exactly the same set of configurations andtherefore the correct partition functions. 28he logarithmic bipartite fidelity is then defined as F αs = − lim M →∞ log (cid:0) Z ABs (cid:1) α Z A ∪ Bc Z Ar Z Br ! . (3.20)As we shall see, the limit M → ∞ of this ratio is well defined. The factor α in the denominator ensuresthat F sα is well-defined in the limit α → 0. Indeed, Z ABs and Z A ∪ Bc both vanish linearly in α as it tendsto zero. With this convention for F sα , the leading powers of α coincide between the numerator and thedenominator.The choice to include Z Ar and Z Br in the denominator is justified as follows. We note that forthe other dense loop models for which the fugacity of contractible loops is non-zero, the natural choicewould be to include in the denominator the partition functions with no defects. For the model of criticaldense polymers, these partition functions are zero. As argued in [31], the reference partition functionsin this case are instead those with pairs of defects on the top and bottom segments, as explained above. Our goal is first to express the partition functions in (3.20) in the language of the Temperley-Liebalgebra. The skirt geometry involves both periodic and open boundary conditions. For that reason,both the ordinary Temperley-Lieb algebra TL N ( β ) and the enlarged periodic Temperley-Lieb algebra E PTL N ( α, β ) are useful. Since TL N ( β ) is a subalgebra of E PTL N ( α, β ), all the tools needed to define TL N ( β ) are given in Section 2.3.2, and below we only give a brief reminder. Definition of TL N ( β ). The Temperley-Lieb algebra [45–50] is a unital, associative algebra generatedby the linear span of connectivities. It is the subalgebra of E PTL N ( α, β ) generated by the generators I and e j , with j = 1 , . . . , N − 1. With products of these generators, one can produce all the connectivitydiagrams in E PTL N ( α, β ) that have no loop segments travelling via the back of the cylinder. Thediagrammatic rules for computing products of connectivity diagrams are the same as those describedin the periodic case. Non-contractible loops are never created in such products, and the algebra thusdepends only on the weight β of the contractible loops. We recall that the value of β pertaining to themodel of critical dense polymers is β = 0. The transfer tangle D ( u ). The double-row transfer tangle for the model of dense polymers is anelement of TL N (0) defined as [28] D ( u ) = 1sin 2 u . . .. . . . . .. . .uu uu uu | {z } N , u = cos u + sin u , (3.21)where u is the spectral parameter. We refer to the value u = π , for which both tiles have equal weights,as the isotropic point , and use the short-hand notation D = D ( π ).Two copies of the transfer tangle evaluated at different values of the spectral parameter commute:[ D ( u ) , D ( v )] = 0. Furthermore, the Hamiltonian H o of the model is related to the transfer tangle viathe relation D ( u ) = I − u H o + O ( u ) , H o = − N − X j =1 e j , (3.22)and is also in the commuting family. 29 he standard modules V N, and V N, . The standard modules V N, and V N, for TL N ( β ) are builton the vector space generated by link states on N nodes with zero and two defects, respectively. Thestandard module V N, is defined on the same vector space as the module b W N, over E PTL N ( β, β ). Itis in fact the restriction of this module to the action of elements of TL N ( β ) ⊂ E PTL N ( β, β ). Likewise,the module V N, is defined on the subspace of W N, spanned by link states with no arcs travelling viathe back of the cylinder. The diagrammatic action of the connectivities in TL N ( β ) on the link state of V N, and V N, is the same as in the periodic case, with the difference that non-contractible loops arenever formed. Partition functions. The Gram products for the standard modules of TL N ( β ) are defined in thesame way as those introduced in Section 2.3.3 for the periodic case. We express the partition functionsin (3.20) as Z ABs = 2 MN ( v A ⊗ v B ) · (cid:0) D A ⊗ D B (cid:1) M (cid:0) T AB (cid:1) M v AB (cid:12)(cid:12) W N, , (3.23a) Z A ∪ Bc = 2 MN v AB · ( T AB ) M v AB (cid:12)(cid:12) W N, , (3.23b) Z Ar = 2 MN A v A · ( D A ) M v A (cid:12)(cid:12) V NA, , (3.23c) Z Br = 2 MN B v B · ( D B ) M v B (cid:12)(cid:12) V NB, , (3.23d)where v and v are defined in (2.57). The conventions are the same as those used in Section 2.3.3. The XX representation and spin-chain overlaps. The XX representation of TL N ( β = 0) isgiven in (2.48). In this representation, the Hamiltonian with open boundary conditions H o is the XXHamiltonian with the U q ( sℓ )-invariant boundary magnetic fields of Pasquier and Saleur [37], H o = X N ( H o ) = − N − X j =1 (cid:0) σ xj σ xj +1 + σ yj σ yj +1 (cid:1) − i2 ( σ z − σ zN ) . (3.24)The representative of D ( u ) in the XX representation is the double-row transfer matrix D ( u ). Weuse the notation D = D ( π ) for the transfer matrix at the isotropic point. The map from link statesto spin states given in (2.53) is a homomorphism between link state and spin representations for thealgebra TL N ( β ) as well. The partition functions can then be written in terms of spin-chain overlaps as Z ABs = 2 MN hh v A ⊗ v B | (cid:0) D A ⊗ D B (cid:1) M (cid:0) T AB (cid:1) M | v AB i , (3.25a) Z A ∪ Bc = 2 MN hh v AB | ( T AB ) M | v AB i , (3.25b) Z Ar = 2 MN A hh v A | ( D A ) M | v A i , (3.25c) Z Br = 2 MN B hh v B | ( D B ) M | v B i . (3.25d)We note that all the states that appear as dual states do not depend on the twist parameter φ , so inthis case (2.62) simply becomes hh w | = | w i t . H o . The first step is to use the Jordan-Wigner transformation to write H o as H o = − N − X j =1 (cid:0) c † j +1 c j + c † j c j +1 (cid:1) − i (cid:16) c † c − c † N c N (cid:17) (3.26)30here the fermionic operators c j and c † j are defined in (2.20). Recalling that ω = e i π/ , the second stepis to perform a Fourier transform of these operators, by defining η k = 1 κ k N − X j =1 sin( πkjN ) a j , η t k = 1 κ k N − X j =1 sin( πkjN ) a t j , κ k = q N cos( πkN ) , (3.27)where a j = ω c j + ω − c j +1 , a t j = ω c † j + ω − c † j +1 . (3.28)These operators satisfy the fermionic relations { a j , a t k } = δ j,k − + δ j,k +1 , { η k , η t ℓ } = δ k,ℓ , { a j , a k } = { a t j , a t k } = { η k , η ℓ } = { η t k , η t ℓ } = 0 . (3.29)The Hamiltonian can be expressed in Jordan-normal form using these operators. For N even, the setof operators η k and η t k , with k ∈ { , . . . , N − } ∪ { N +22 , . . . , N − } is complemented with the operators χ = − ω r N N X j =1 i − ( j − (cid:0)(cid:4) j (cid:5) − N (cid:1) c j , χ t = − ω r N N X j =1 i − ( j − (cid:0)(cid:4) j (cid:5) − N (cid:1) c † j , (3.30)as well as with the operators ϕ and ϕ t , ϕ = ω − r N N X j =1 i − ( j − c j , ϕ t = ω − r N N X j =1 i − ( j − c † j . (3.31)The anti-commutation relations are { ϕ, χ t } = { ϕ t , χ } = 1 , { ϕ t , ϕ } = { χ t , χ } = { ϕ, χ } = { ϕ t , χ t } = 0 . (3.32)All the anti-commutators involving the operators η k and η t k and one of ϕ, ϕ t , χ and χ t also vanish. Interms of these operators, the Hamiltonian takes the form H o = ϕ t ϕ − N − X k =1 k = N/ cos( πkN ) η t k η k . (3.33)The groundstate eigenspace in the zero-magnetisation sector is two-dimensional and is spanned by thestates | w i = ϕ t η t1 η t2 . . . η t N/ − | i , | ˆ w i = χ t η t1 η t2 . . . η t N/ − | i . (3.34)These form a rank-two Jordan cell: H o | w i = h | w i , H o | ˆ w i = h | ˆ w i + | w i , h = 1 − cot( π N ) . (3.35)Likewise, in the sectors of magnetisation m = − 1, the state with the lowest energy is | w − i = η t1 η t2 . . . η t N/ − | i (3.36)and its eigenvalue is also h . Because D ( u ) and H o commute, we have D ( u ) | w i = λ ( u ) | w i , D ( u ) | ˆ w i = λ ( u ) | ˆ w i + f ( u ) | w i , D ( u ) | w − i = λ − ( u ) | w − i . (3.37)where λ ( u ) = λ − ( u ) are the eigenvalues of D ( u ) and f ( u ) is a non-zero function. At the isotropicpoint, these states generate the eigenspace of D of maximal eigenvalue in the sectors S z = 0 , − atios of partition functions in the limit M → ∞ . Following the same steps and logic as inSection 2.3.4 and in [27], we extract the leading behaviour of the overlaps (3.25) in the limit M → ∞ .The result is F αs = − log α − hh w A ⊗ w B | X AB i hh v A | ˆ w A i hh v A | w A − i hh v B | ˆ w B i hh v B | w B − i hh X AB | v AB ihh v AB | X AB i ! (3.38)where hh v | and hh v | are defined in (2.78). Determinant forms for the overlaps. We express the overlaps in (3.38) in determinant form withWick’s theorem. Two of the ratios involving boundary states were already computed in [27], hh v A | ˆ w A ihh v A | w A − i = ω r N A , hh v B | ˆ w B ihh v B | w B − i = ω r N B . (3.39)To compute the last ratio in (3.38), we need the explicit form of the anti-commutators (2.79). Aftersome algebra, we obtain hh X AB | v AB ihh v AB | X AB i = N/ Y k =1 cos( π − ϑ k )cos( π + ϑ k ) ! e − i φ ( N +1) ( − N +1+ ⌊ N − ⌋ (3.40)where ϑ k is defined in (3.17c).It only remains to compute the overlap hh w A ⊗ w B | X AB i . It involves the state hh w A ⊗ w B | = h | η B NB − . . . η B ϕ B η A NA − . . . η A ϕ A (3.41)where η Ak = 1 κ Ak N A − X j =1 sin( πkjN A ) a j , ϕ A = ω − r N A N A X j =1 i − ( j − c j , (3.42a) η Bk = 1 κ Bk N − X j = N A +1 sin( πk ( j − N A ) N B ) a j , ϕ B = ω − r N B N X j = N A +1 i − ( j − N A − c j , (3.42b) κ Ak = (cid:0) N A cos( πkN A ) (cid:1) / , κ Bk = (cid:0) N B cos( πkN B ) (cid:1) / . (3.42c)To compute the overlap, we introduce the rescaled operators˜ η k = N − X j =1 sin( πkjN ) a j , k = 1 , . . . , N − . (3.43)We similarly define rescaled operators for ˜ η Ak and ˜ η Bk from (3.42), by removing the prefactors κ Ak and κ Bk . These operators have the advantage of being well defined for k = N/ 2. All the fermionic operatorsappearing in hh w A ⊗ w B | can in fact be written in terms of the ˜ η k , as indeed we have ˜ η k = κ k η k and˜ η N = ω p N/ ϕ .The anti-commutators between the operators ˜ η Ak , ˜ η Bk and µ † j are { ˜ η Ak , µ † j } = 1 √ N ( ω + ω − e − i θ j ) sin (cid:16) πkN A (cid:17) − ( − k e − i N A θ j θ j − πkN A ) , (3.44a) { ˜ η Bk , µ † j } = e − i N A θ j √ N ( ω + ω − e − i θ j ) sin (cid:16) πkN B (cid:17) − ( − k e − i N B θ j θ j − πkN B ) . (3.44b)32he final overlap hh w A ⊗ w B | X AB i is then obtained as the determinant of these commutators. Usingthe identities N − Y k =1 sin (cid:16) πkN (cid:17) = N − Y k =1 cos (cid:16) πkN (cid:17) = N / − N − , (3.45a) N Y k =1 cos (cid:16) π θ k (cid:17) cos (cid:16) π − θ k (cid:17) = 2 − N +1 cos (cid:16) φ (cid:17) = 2 − N α, (3.45b)we find that the result simplifies to e −F αs = N − N ( N x ) − Nx + ( N (1 − x )) − N (1 − x )2 + − N +1 (det C ) (3.46)where C k,k ′ = sin (cid:0) Nxϑ k ′ − πk (cid:1)(cid:16) cos ϑ k ′ − cos (cid:0) πkNx (cid:1)(cid:17) − k = 1 , . . . , Nx , ( − k ′ sin (cid:0) N (1 − x ) ϑ k ′ − π ( k − Nx )2 (cid:1)(cid:16) cos ϑ k ′ − cos (cid:0) π ( k − Nx ) N (1 − x ) (cid:1)(cid:17) − k = Nx + 1 , . . . , N , (3.47)with k ′ = 1 , . . . , N . For arbitrary values of x and φ , we are unable to evaluate the determinant in(3.46) in closed form using the Cauchy determinant formula (2.31). We are however able to computethe determinant for two specialisations of x and φ : (i) x = 1 / φ arbitrary and (ii) φ = 0 with x arbitrary. For simplicity, our results for these two cases are given below for N ≡ Specialisation (i): x = 1 / φ . For x = 1 / N ≡ φ , after some simplifications, we find | det C| = cos( φ ) N/ (cid:12)(cid:12)(cid:12)(cid:12) N/ det k,k ′ =1 (cid:16) cos ϑ k ′ − cos (cid:0) πkN (cid:1)(cid:17) − N/ det k,k ′ =1 (cid:16) cos ϑ k ′ − − cos (cid:0) πkN (cid:1)(cid:17) − (cid:12)(cid:12)(cid:12)(cid:12) . (3.48)In this case, we can apply (2.31) to evaluate both determinants. Specialisation (ii): φ = 0 and arbitrary x . For φ = 0, N ≡ x ,after some simplifications, we find | det C| = 2 N/ (cid:12)(cid:12)(cid:12) det C det C N/ Y k =1 cos (cid:16) Nxϑ k (cid:17) N/ Y k = N/ sin (cid:16) Nxϑ k (cid:17)(cid:12)(cid:12)(cid:12) (3.49a)with C k,k ′ = (cid:16) cos ϑ k ′ − cos (cid:0) π (2 k − Nx (cid:1)(cid:17) − k = 1 , . . . , Nx , (cid:16) cos ϑ k ′ − cos (cid:0) π (2 k − Nx ) N (1 − x ) (cid:1)(cid:17) − k = Nx + 1 , . . . , N , k ′ = 1 , . . . N , (3.49b) C k,k ′ = (cid:16) cos ϑ k ′ − cos (cid:0) π kNx (cid:1)(cid:17) − k = 1 , . . . , Nx , (cid:16) cos ϑ k ′ − cos (cid:0) π (2 k − Nx − N (1 − x ) (cid:1)(cid:17) − k = Nx + 1 , . . . , N , k ′ = N + 1 , . . . N . (3.49c)In this case as well, both determinants can be evaluated with (2.31).33ase ∆ ∆ ∆ ∆ ∆ ˜ C ′ Ξ Ξ XX φ . / / XX φ . / / XX φ . / / XX φ . XX φ . / / XX φ . / / / / XX φ . / / / / XX φ . XX φ . − 110 1 / / / XX φ . − 111 1 / / XX φ . / / XX φ . − 113 9 / / XX φ . − C ′ and the extrapolation lengths for each of thethirteen cases, with ∆ XX φ defined in (2.96). In this subsection, we study the large- N asymptotics of the bipartite fidelity for the XX chain and themodel of critical dense polymers on the skirt geometry. We compare these results with the conformalpredictions of Section 3.1. We study the asymptotic behaviour of the bipartite fidelity for the XX chain numerically, for each casedefined in Table 1. The data points are obtained by evaluating the determinants numerically, namelythe expression (3.17) for case 1 and similar determinant expressions for the other cases. We comparethese numerical values with a fit of the form F XX s = ˜ g log N + ˜ g ( x ) + ˜ g log NN + γ N + γ log NN + γ N + γ log NN + γ N . (3.50)From this analysis, we find that the asymptotic behaviour of F XX s is correctly predicted by theCFT formula (3.4) for case (ii), with the functions ˜ g , ˜ g ( x ) and ˜ g ( x ) given in (3.6), (3.9) and (3.11).These are specialised to the value c = 1 of the central charge and to values of the conformal dimensions∆ and of the constant ˜ C ′ that depend on the case considered. These values are given in Table 2. InFigure 9, we plot the numerical data and the curve ˜ g log N + ˜ g ( x ), for the cases 1, 6, 8 and 12. Wefind a perfect agreement.We observe that the constant ˜ C ′ is independent of φ . In fact, one should note the presence ofa factor of 8∆ log 2 in (3.9) which could a priori be included in the constant C ′ . In general, C ′ is aconstant with respect to x , but can depend on the dimensions of the fields. In Appendix A.5, we arguethat the overall additive term in (3.9) should indeed include a factor 8∆ log 2, which we have chosento write separately from ˜ C ′ . With this choice, we find that the values of ˜ C ′ depend only on the fieldsinserted on the endpoints of the slits, corresponding to the states s and s in (3.18).The conformal dimensions in Table 2 are consistent with known results for the CFT description ofthe XX spin chain and the six-vertex model. The field ϕ inserted at z = z accounts for the presenceof the twist φ in the model. As discussed in Section 2.4.1, this electric operator has the dimensions∆ = ¯∆ = ∆ XX φ . In the current setting, the twist line connects the point z with the endpoint of one of34 g log N + ˜ g ( x ) x case 1 ˜ g log N + ˜ g ( x ) x case 6˜ g log N + ˜ g ( x ) x case 8 ˜ g log N + ˜ g ( x ) x case 12Figure 9: The function ˜ g log N + ˜ g ( x ) as a function of x for N = 800 and φ = 2 for the cases 1, 6, 8and 12. For the data points, ˜ g and ˜ g ( x ) are obtained by fitting (3.50), whereas the solid curve is theCFT prediction for these two functions with the data of Table 2.the two slits, which corresponds to the boundary of the domain. In the six-vertex model, the point onthe boundary where this twist line is chosen to terminate can be chosen arbitrary, and the resultingpartition function is independent of this choice. In the conformal interpretation, the method of imagestells us that we can replace the field ϕ ( z , ¯ z ) by the product ϕ ( z ) ϕ (¯ z ) of two chiral fields. These twofields have charges q and − q , respectively, and the conformal dimensions are ∆ XX φ = ¯∆ XX φ = q / 2. Thisis precisely the assumption we make in Appendix A to derive the CFT prediction for the asymptoticexpansion of F s .The other fields in the positions z i with i = 1 , . . . , magnetic operators . These areprimary fields that account for the presence of spin states of fixed magnetisation m . Here they liveon the boundary and thus depend on a single variable. Their conformal dimension ∆ m depends onlyon the magnetisation m of the state inserted at z = z i . This dimension can be computed from thefinite-size correction of the groundstate of the Hamiltonian H f with free boundary conditions, definedin (3.13). The scaling limit of this Hamiltonian was for instance studied in [51]. In this case, from thecorrection term proportional to 1 /N , one obtains the difference c − m as c − m = 1 − m . (3.51)Setting c = 1 and solving for ∆ m yields ∆ m = m . (3.52)The values of ∆ , ∆ , ∆ and ∆ in Table 2 are precisely given by (3.52) with the corresponding valuesfor m ∈ { , ± , ± , ± , ± } .We plot our numerical data for ˜ g ( x ) in Figure 10 for the cases 2, 3, 10 and 13. This data appears35 g ( x ) x case 2 ˜ g ( x ) x case 3˜ g ( x ) x case 10 ˜ g ( x ) x case 13Figure 10: The function ˜ g ( x ) for the XX spin chain for φ = 2 for the cases 2, 3, 10 and 13. Thedata points are obtained by fitting (3.50). The blue lines are the CFT predictions with the conformaldimensions and extrapolation lengths fixed as in Table 2.alongside the CFT prediction for ˜ g ( x ), with the suitably chosen values of the extrapolation lengthsΞ and Ξ , given in Table 2.The original derivation of the N − log N term in [24] was claimed to be valid for domains with anarbitrary number of corners of interior angle 2 π . It is based on a perturbative calculation. By varyingthe position of a boundary near a corner by a distance Ξ, one obtains the variation of the free energy asa function of the perturbation, and then integrates it to obtain ˜ g ( x ). On the skirt geometry, there aretwo corners, each of internal angle 2 π , and in this case, one can perform two separate perturbations nearthose corners. One can then assign an extrapolation length to each of the two slits. In Appendix A,we repeat the derivation of [24] while allowing for two such lengths, one for each slit. Because theyare assigned to the corners situated at w = w and w = w , we name them Ξ and Ξ . We obtain themore general expression (3.11). This generalised result is necessary to correctly reproduce the data forthe cases 9 to 13, for which Ξ = Ξ . In the general case, our numerics reveal that the extrapolationlengths are given by the formula Ξ = 1 − n , Ξ = 1 − n , (3.53)where n i , with i = 1 , 2, is the length of s i defined in Section 3.2.2. We now study the large- N expansion for the bipartite fidelity for the model of dense polymers onthe skirt lattice. In Section 3.3.3, we obtained product expressions for F αs for two specialisations:(i) x = 1 / φ , and (ii) φ = 0 with arbitrary x . We extract the asymptotic behaviour in36hese two cases using arguments that are very similar to those presented in Appendix B for the caseof the periodic pants lattice. We obtain F αs (cid:12)(cid:12) x = = − 12 log N + log 23 + 4∆ CDP φ log 2 + 12 ( − A − log π + 4 log 2) + O ( N − ) , (3.54a) F αs (cid:12)(cid:12) φ =0 = − 12 log N − 112 ( ˜ G ( x ) + ˜ G (1 − x )) − 12 ( x log x + (1 − x ) log(1 − x ) + 2 log 2)+ 12 ( − A − log π + 4 log 2) + O ( N − ) . (3.54b)We recall that ˜ G ( x ) and ∆ CDP φ are defined in (3.8) and (2.99), respectively.We performed numerical evaluations of the determinant expression (3.46) for values of x and φ that do not enter the specialisations (i) and (ii) defined in Section 3.3.3. In Figure 11, we plot theresults for the value φ = 2. From the exact results (3.54) and our numerical experiments, we conjecturethat the general formula is F αs = − 12 log N − 112 ( ˜ G ( x ) + ˜ G (1 − x )) + 4∆ CDP φ (cid:0) x log x + (1 − x ) log(1 − x ) + 2 log 2 (cid:1) + 12 ( − A − log π + 4 log 2) + O ( N − ) . (3.55)As illustrated in the example of Figure 11, this conjecture reproduces the numerical results with greatprecision. It also coincides with the CFT prediction (3.4) for case (i), with ˜ g given in (3.6), ˜ g ( x ) in(3.7), ˜ g ( x ) in (3.10), the conformal data specified to c = − , ∆ = ∆ CDP φ , (3.56)the non-universal constant taking the value˜ C ′ = 12 ( − A − log π + 4 log 2) , (3.57)and with vanishing extrapolation lengths: Ξ = Ξ = 0. Indeed, since the conditions at the twoslits’ endpoints are identical, we must have Ξ = Ξ . From the exact results (3.54) and the numericalevaluations, we find that the term proportional to N − log N vanishes, namely ˜ g ( x ) = 0. A glance at(3.11) indicates that this is only possible if both extrapolation lengths vanish.The values (3.56) are precisely those expected in the conformal description of the model of densepolymers. Indeed, there is no change of boundary condition at the endpoints of the slits. Moreover,the boundary conditions in the legs A and B are known to correspond to the groundstate of the model.Both of these features correspond to identity fields with the dimension ∆ = 0. Similarly to what isdiscussed in Section 2.4.2, the field ρ ( z ) at z = z assigns weights α = 0 to the non-contractible loopsand has the conformal dimension ∆ CDP φ . In this paper, we investigated the bipartite fidelity for critical lattice models on two geometries: theperiodic pants domain and the skirt domain. Using arguments of conformal field theory, we obtainedpredictions for the leading terms in the asymptotic large- N expansions of F . We compared these withexact and numerical lattice calculations and found a precise match for two lattice models: the XX spinchain and the model of critical dense polymers. These models are known to be described by CFTswith central charges c = 1 and c = − 2, respectively.For the XX spin chain, we considered different instances of the bipartite fidelity with certain spinstates of magnetisation m fixed in the domain, or in the presence of twist lines φ connecting certain37 αs x F αs N Figure 11: The bipartite fidelity F αs as a function of x for N = 800 (left panel) and as a function of N for x = (right panel). In both cases, the fugacity is set to α = 2 cos( φ ) with φ = 2. The data pointsare computed from the determinant (3.46), whereas the continuous line is (3.55).points of the domain. In the CFT, these correspond to the insertions of magnetic and electric operators,respectively. The conformal dimensions of these operators are known from the CFT description of theXXZ spin chain in terms of a compact boson [40, 44, 52]. The electric operators are also known to bevertex operators whose n -point functions are given by the simple formula (2.10). Our analysis of F for the XX chain on the periodic pants domain confirms this. The charges of the electric operator are q = ¯ q = φ π , and the conformal dimensions are ∆ = ¯∆ = q . If a field marks a transition between twotwist lines φ and φ A , then its charge is q = φ − φ A π . We also investigated the case of an intersectionpoint between three twist lines of twists φ , φ A and φ B , and found that the corresponding charge is q = φ − φ A − φ B π .Our analysis of F on the skirt domain goes further, by considering six-point correlators thatmix electric and magnetic operators. Surprisingly, our numerical analysis reveals that these mixedcorrelators also take the simple form (2.10) for vertex operators, with the charge of the magneticoperators given by q = m . This is quite intriguing. In the general case, one expects that mixedcorrelators will involve non-trivial functions of the cross-ratios, sums over conformal blocks, etc, seefor instance [53]. We are currently unaware of other results in the literature that confirm or infirm ourfindings. Is this feature unique to free-fermionic models? To push our understanding further, we hopeto return to the problem of computing F for the XXZ spin chain with ∆ = 0.For the model of critical dense polymers, we studied the model with non-contractible loops thatare assigned a fugacity α = 2 cos( φ ), different from the fugacity β = 0 of the contractible loops. In theconformal field theory description, this amounts to inserting a bulk operator with conformal dimension∆ CDP φ = (cid:0) φ π − (cid:1) . Our investigation uses the known map from the loop model to the spin chain, andonly covers the case where at most two such operators are inserted in the domain. It is then natural tosearch for an extension of these results to the case where loops are assigned different weights accordingto how they encircle the various marked points. However, this appears not to be feasible using the XXrepresentation of the periodic Temperley-Lieb algebra. This hints at the fact that the correspondingconformal fields are not vertex operators.A most interesting result that we found regards the extrapolation lengths for the terms proportionalto N − log N . In our investigation of the XX spin chain on the skirt domain, we studied special instancesof F s where certain spin states s and s are inserted on the endpoints of the two slits, see (3.18). As aresult, we found that the CFT prediction of Dubail and St´ephan [23, 24] had to be generalised to allowfor two extrapolation lengths Ξ and Ξ , one for each slit. Moreover, contrary to what these authorsclaimed, these lengths sometimes take negative values. In fact, for the XX chain on the skirt, we founda linear relation between the extrapolation lengths and the lengths of the inserted states, with thenegative values arising when s and s are non-trivial states. We speculate that this is a consequence38f the unusual finite shape of the corner in the corresponding two-dimensional lattice model. Indeed,in the underlying six-vertex model, the endpoints of the slits take the form of two subsequent corners,distant by n i lattice spacings. In the scaling limit, this shape becomes a corner of internal angle 2 π ,and the extrapolation length is modified accordingly.On the periodic pants domain, the asymptotic expansion of the bipartite fidelity also has an N − log N term. This is an important new result of this paper: for the bipartite fidelity, conical singu-larities with internal angles of 4 π act similarly to corners with internal angles 2 π . Both these featuresresult in N − log N contributions to F . On the pants domain, there is a single conical singularity andtherefore a unique extrapolation length. For the XX spin chain, we find that the extrapolation lengthis Ξ = − n , where n is the length of the spin state on the crotch point. Moreover, we remark that theconformal predictions for g ( x ), g ( x ) and g ( x ) are precisely equal to twice the same functions, previ-ously obtained by Dubail and St´ephan, for the asymptotic expansion of F on the flat pants geometry.This can be traced back to the fact that the map w ( z ) for the flat pants domain is equal to two times w p ( z ), with z restricted to the upper half-plane.While this has not been fully exploited in this paper, we believe that the bipartite fidelityis a useful tool to compute the structure constants C that arise in the three-point functions h φ ( z , ¯ z ) φ ( z , ¯ z ) φ ( z , ¯ z ) i C . Computing a correlation function of a lattice model for a numberof arbitrarily located positions in the complex plane is in general a notoriously difficult task, even fortwo-point functions. In this case, a simple way to measure the conformal dimension ∆ that appearsin a two-point correlator is to consider the problem defined on an infinite cylinder and insert the twofields at infinity at the endpoints of this cylinder. The dimension ∆ then appears in the 1 /N finite-sizecorrection term of the largest eigenvalue of the transfer matrix, and computing it is possible with theusual methods of Yang-Baxter integrability. Similarly for the three-point functions, the pants and skirtdomains are useful as they have three points at infinity where the three fields can be inserted. (Nofield should then be inserted on the crotch point or on the endpoints of the slits.) This idea has thepotential to make the lattice computation of C a manageable problem. In the present paper, theonly case where we hoped to get a glimpse of the structure constants was in our investigation of F XX p .Indeed, in Appendix A.4, we set q = 0, calculate the difference F − F ′ of two bipartite fidelities fortwo different sets of fields, and find that the difference of constants C in (2.8) is simply the logarithmof the ratio of the two structure constants, see (A.31). For F XX p , this analysis applies for the twist φ specified to φ = φ A + φ B , so that q = 0. In Section 2.4.1, we however found that varying φ , φ A and φ B while keeping φ − φ A + φ B fixed does not change C . This constant instead depends only on q ,see Figure 6. We conclude that the structure constant in this case has no non-trivial dependence onthe three fields inserted at infinity. This is consistent with the interpretations of these fields as vertexoperators, where the structure constant are known to be equal to one. Acknowledgments AMD and GP acknowledge the support from the Fonds de la Recherche Scientifique – FNRS: AMD wassupported by the Postdoctoral Research Project CR28075116 and GP was supported by the AspirantFellowship FC23367. The authors also acknowledge support from the EOS contract O013018F. AMDthanks Y. Ikhlef and J.L. Jacobsen for useful exchanges. A The bipartite fidelity from conformal field theory calculations In this appendix, we derive the CFT prediction for the leading terms in the N expansion of the bipartitefidelity, for the periodic pants and the skirt geometry. It closely follows the arguments of St´ephan andDubail [23, 24]. 39 N Λ − Λ ε = i δxN ε = 0 d Figure 12: The perturbation ε . The darkened area is the support A of ε , and the slit is moved upwardin A .The outline of this appendix is as follows. In Appendix A.1, we derive a general formula forthe constant term of the large- N expansion of F . We find g ( x ) and ˜ g ( x ), namely the constantcontributions for the periodic pants and the skirt geometry, in Appendices A.2 and A.3, respectively.We evaluate the constants C and ˜ C in Appendices A.4 and A.5. Finally, we compute the sub-leadingcontributions, g ( x ) and ˜ g ( x ) in Appendices A.6 and A.7. A.1 Perturbation of the stress-energy tensor To compute the constant term in the 1 /N expansion of F , we follow the strategy of [23,24] that consistsin varying the aspect ratio x to find δ F /δx . The general formula derived below applies to both the skirtand the pants geometry and will allow us to compute g ( x ) and ˜ g ( x ). For this reason, we temporarilydrop the subscripts of the maps w and the free energies f .We recall the relation between the action and the stress-energy tensor. Under a transformation w µ w µ + ε µ whose support is A , the action varies according to δS = 12 π Z A ∂ µ ε ν T µν d w d w . (A.1)The free energy f is defined as minus the logarithm of the partition function: f = − log Z with Z = h φ · · · φ n i . Here, φ i is a short-hand notation for φ i ( w i, , w i, ). Under the perturbation w µ w µ + ε µ ,the free energy varies as δf = 12 π Z A ∂ µ ε ν h T µν φ · · · φ n ih φ · · · φ n i d w d w . (A.2)On the periodic pants and the skirt domain, we perform the perturbation w = w +i w w + ε ( w )given by ε ( w ) = ( i δxN if | w − xN | < d and | w | < Λ , A of ε consists of the rectangle where the perturbation takes the value i δxN . It has awidth 2 d and a length 2Λ. We will take the limit where d tends to zero. In contrast, Λ will play therole of a cut-off for certain integrals and will be sent to infinity at the end of the calculation. The effectof the perturbation ε is to slightly shift the slit in position i xN/ δxN , as depicted inFigure 12. This is equivalent to changing x to x + δx while keeping N unchanged.The derivative of the perturbation yields linear integrals on the boundaries of the support A .Because ∂ ε = ∂ ε = 0, the only contributions to the integral (A.2) come from ∂ ε on the widthof A , and from ∂ ε on its length. We take the limit d → δf = 12 π δxN (cid:18)Z U + h T φ · · · φ n ih φ · · · φ n i d w − Z U − h T φ · · · φ n ih φ · · · φ n i d w (cid:19) . (A.4)40ere U ± = [ − Λ , 0] + i xN − i0 ± are segments on the boundary of A , on each side of the slit. Theintegrals along [0 , Λ] + i xN ± d cancel each other in the limit d → w = w + i w , ¯ w = w − i w and rewrite the integrand in termsof the non-vanishing components of the stress-energy tensor T ( w ), ¯ T ( ¯ w ) using T = − ( T ( w ) + ¯ T ( ¯ w )).We denote the fields after the transformation by φ i ( w i , ¯ w i ), i = 1 , . . . , n . From here onwards, weonly consider the holomorphic part of the expression and write + c.c. for the addition of its complexconjugate: δf = − π δxN (cid:16) Z U + h T ( w ) φ ( w , ¯ w ) · · · φ n ( w n , ¯ w n ) ih φ ( w , ¯ w ) · · · φ n ( w n , ¯ w n ) i d w − Z U − h T ( w ) φ ( w , ¯ w ) · · · φ n ( w n , ¯ w n ) ih φ ( w , ¯ w ) · · · φ n ( w n , ¯ w n ) i d w (cid:17) + c.c. (A.5) Integral over the real line. In order to compute these integrals, we pull them back via the inverseof the transformation w . As illustrated in Figures 2 and 8, the pre-image of the integration curves are w − ( U + ) = [ w − ( − Λ + i xN − i0 + ) , w − (i xN )] ≡ [1 − Λ , w − (i xN )] , (A.6a) w − ( U − ) = [ w − ( − Λ + i xN + i0 + ) , w − (i xN )] ≡ [0 +Λ , w − (i xN )] , (A.6b)where we introduced the short-hand notations 0 +Λ and 1 − Λ , highlighting the fact that those points tendto 0 and 1 for large Λ, respectively. These values for 0 +Λ and 1 − Λ are different for w p ( z ) and w s ( z ), andtheir leading behaviour for large Λ can be computed directly from (2.6) and (3.5).We change the variable from w to z in the integrals in (A.5) and use the transformation law of thestress-energy tensor T ( w )( w ′ ( z )) = T ( z ) − c { w ( z ) , z } , where { w ( z ) , z } is the Schwarzian derivative.We obtain the following integral: δf = δxN π Z − Λ +Λ (cid:18) d w d z (cid:19) − (cid:20) h T ( z ) φ ( z , ¯ z ) · · · φ n ( z n , ¯ z n ) ih φ ( z , ¯ z ) · · · φ n ( z n , ¯ z n ) i − c { w ( z ) , z } (cid:21) d z + c.c. (A.7)Finally, we use the conformal Ward identity on the complex plane: h T ( z ) φ ( z , ¯ z ) · · · φ n ( z n , ¯ z n ) i = n X i =1 (cid:20) ∆ i ( z − z i ) + ∂ i z − z i (cid:21) h φ ( z , ¯ z ) · · · φ n ( z n , ¯ z n ) i (A.8)to write the first part of the integrand in terms of an n -point correlator. Free energy variation of the cylinder and the strip. The expressions (2.4) and (3.3) for thebipartite fidelity on the periodic pants domain and the skirt domain involve the free energy of thecylinder and rectangle strip geometry. These expressions are standard [54]. For a cylinder and arectangle of length 2Λ with inserted fields of dimension ∆, they read f c ( N ) = − (2Λ) πN (cid:16) c − (cid:17) , f r ( N ) = − (2Λ) πN (cid:16) c − ∆ (cid:17) . (A.9)We use these expressions with N replaced by N A = xN , N B = N − N A = (1 − x ) N . On the cylinder,the variation of these free energies with respect to x is δf c ( N ) = 0 , δf c ( N A ) = (2Λ) πδxx N (cid:16) c − (cid:17) , δf c ( N B ) = − (2Λ) πδx (1 − x ) N (cid:16) c − (cid:17) . (A.10)These free energies tend to infinity with Λ. However, the differences (2.4) and (3.3) will turn out to befinite. 41 .2 Contribution to the periodic pants geometry We consider the periodic pants domain and compute the variation δf p from (A.7). The term involvingthe Schwartzian derivative in the integrand is readily computed. One finds Z − Λ +Λ (cid:18) d w p d z (cid:19) − { w p ( z ) , z } d z + c.c. = 4 π N (cid:18) Λ(1 − x ) − Λ x (cid:19) + 2 πN (cid:0) b ( x ) − b (1 − x ) (cid:1) , (A.11)with b ( x ) = ( x − x (1 − x ) log x − x . We now compute the second term involving the fields, for the two specialcases (i) and (ii) discussed in Section 2.1. Case (i): Three primary fields. There are three primary fields φ , φ and φ in the legs but nofield at the crotch point. Following the notation of Section 2.1, we denote the insertion points by w , w and w . These points correspond to z = 1, z = 0 and z = ∞ in the complex plane via the map w p ( z ).The three-point function is given by D Y i =1 , , φ i ( z i , ¯ z i ) E C = C | z − z | +∆ − ∆ ) | z − z | +∆ − ∆ ) | z − z | +∆ − ∆ ) , (A.12)where C is the three-point structure constant. Inserting this in (A.7) and using the Ward identity(A.8), we find after some algebra Z − Λ +Λ (cid:18) d w p d z (cid:19) − h T ( z ) φ φ φ ih φ φ φ i d z + c.c. = 8 π N (cid:18) ∆ Λ(1 − x ) − ∆ Λ x (cid:19) + 4 πN (cid:18) ∆ − ∆ (1 − x ) (cid:19) log x − πN (cid:18) ∆ − ∆ x (cid:19) log(1 − x ) . (A.13)We can now take the linear combination (2.4) of variations of free energies. The terms proportionalto the cut-off vanish and we obtain δ F p δx = 4 (cid:18) ∆ − ∆ (1 − x ) (cid:19) log x − (cid:18) ∆ − ∆ x (cid:19) log(1 − x ) − c b ( x ) − b (1 − x )) . (A.14)Integrating with respect to x , we find (2.8). Case (ii): Four vertex operator fields. We now consider the case (ii) where the four fields φ i , i = 1 , . . . , 4, are vertex operators with charges q i = ¯ q i . Their positions in the complex plane are z = 1, z = 1 − x , z = 0 and z = ∞ . The charges are constrained to satisfy the neutrality condition P i q i = 0. In this case, the n -point function has the simple form (2.10). The same correlator vanishesif the sum of the charges is non-zero. Using the Ward identity, we find the simple formula h T ( z ) Q ni =1 φ i ( z i , ¯ z i ) ih Q ni =1 φ i ( z i , ¯ z i ) i = 12 n X i =1 q i z − z i ! . (A.15)This expression allows us to compute the first term of the integral (A.7). As in the case (i), we obtaintwo terms that depend on the cut-off as well as an expression that is regular as Λ tends to infinity. Wetake the linear combination (2.4) so that the divergences vanish, integrate with respect to x and find(2.11).We note that in both cases (i) and (ii), the term g ( x ) of the fidelity is calculated up to an additiveconstant that cannot be obtained from this perturbative argument presented here and involving δ F /δx .These overall constants will be discussed further in Appendices A.4 and A.5 for the periodic pantsdomain and the skirt domain, respectively. 42 .3 Contribution to the skirt geometry Let us focus on the skirt geometry. First, we compute in (A.7) the term proportional to c in δf p . Wehave Z − Λ +Λ (cid:18) d w s d z (cid:19) − { w s ( z ) , z } d z + c.c. = π N (cid:18) Λ(1 − x ) − Λ x (cid:19) (A.16)+ π (2 x − N (cid:18) x − − x ) log x − x + 1 x log(1 − x ) + 2 x (1 − x ) (cid:19) . The other term in (A.7) depends on the fields that are inserted. We compute it for the cases (i) and (ii)defined in Section 3.1. Case (i): One bulk field. We apply (A.7) in the case where one primary field φ is inserted at w = + ∞ . The corresponding position on the upper half-plane is z = (1 + i) / 2. To compute thevariation of free energy, we apply the method of images to express the one-point function on the upperhalf-plane as a two-point function on the complex plane. We introduce the image field φ of dimension∆ = ∆ at the position z = (1 − i) / h φ ( z , ¯ z ) i H = h φ ( z ) φ ( z ) i C . After some algebra, we obtain Z − Λ +Λ (cid:18) d w s d z (cid:19) − h T ( z ) φ φ ih φ φ i d z + c.c. = 4 πN (log x − log(1 − x ))∆ . (A.17)The absence of divergences in this case can be traced back to the fact that there are no fields presentin the legs A and B . We take the linear combination of terms (3.3) and integrate over x to find (3.7). Case (ii): Five vertex operator fields. We investigate the case where fields φ , φ are insertedon the endpoints of the slits, φ , φ in the legs A and B , respectively, and φ is inserted at + ∞ . Thecorresponding points on the upper half-plane are z = 1 , z = √ x − x + x − x − , z = 0 , z = −√ x − x + x − x − , z = 1 + i2 . (A.18)As in the previous case, we introduce the image field φ at the position z = (1 − i) / φ i is a vertex operator with charge q i .In the case of boundary fields, the ratio of n -point functions that appears in (A.7) is similar to(A.15) and reads h T ( z ) Q ni =1 φ i ( z i ) ih Q ni =1 φ i ( z i ) i = 12 n X i =1 q i z − z i ! . (A.19)We insert the right-hand side of this equality in the integral (A.7). We evaluate this integral andobtain the real part of the contributions to δf s . It is invariant under the simultaneous transformation x → − x, q ↔ q . It also contains terms that diverge for Λ → ∞ . These divergences however cancelout in the linear combination (3.3). We integrate over x and obtain˜ g ( x ) = c 24 ˜ G ( x ) + (cid:18) x ( q + q ) + 12 ( q + q ) − x − x ) q − ( q + q + q ) (cid:19) log x + { x → − x, q ↔ q } . (A.20)Applying the method of images to the case where φ is an electric operator, the charge of the imagefield φ is q = − q . In this particular case, the expression (A.20) is further simplified and can beexpressed solely in terms of the dimensions ∆ i = q i / 2, yielding (3.9).43 .4 The constant C for the periodic pants geometry In the two previous sections, we computed the correction to the free energy using a perturbation δx of the aspect ratio x . These involve additive constants C and C ′ that do not depend on x and are notfixed by the perturbative argument. Let us investigate this further for case (i), defined in Section 2.1.In this case, the bipartite fidelity F p can be understood in terms of conformal correlation functions as F p = − log (cid:12)(cid:12)(cid:12)(cid:12) (cid:10) Q i =1 , , φ i ( w i , ¯ w i ) (cid:11) p h φ − φ +1 i c ( Nx ) h φ − φ +3 i c ( N − Nx ) h φ − φ +4 i c ( N ) (cid:12)(cid:12)(cid:12)(cid:12) (A.21)where h φ − i φ + i i c ( P ) = h φ i ( w − , ¯ w − ) φ i ( w + , ¯ w + ) i c ( P ) . (A.22)The correlators h φ − i φ + i i c ( P ) are evaluated on the cylinder of perimeter P , and the fields φ i are insertedat the positions w ± = ± Λ where Λ is a cut-off. The three-point function in the numerator of (A.21)is evaluated on the periodic pants geometry. We set the positions of the three fields to be functions ofthe cut-off Λ: w = ( i xN ) − − Λ , w = ( i xN ) + − Λ , w = Λ . (A.23)For large Λ, the corresponding positions in the complex plane are z ≃ − e − π (Λ+ Kx ) Nx , z ≃ e − π (Λ+ Kx ) N (1 − x ) , z ≃ e π (Λ − Kx ) N . (A.24)Let us also consider the bipartite fidelity F ′ p where the fields φ ′ , φ ′ and φ ′ are different fields.The conformal dimensions of these new fields are ∆ ′ i = ¯∆ ′ i , i = 1 , , 4. We assume in the following thatall these fields are primary. The difference F p − F ′ p is F p − F ′ p = − log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:10) Q i =1 , , φ i ( w i , ¯ w i ) (cid:11) p (cid:10) Q i =1 , , φ ′ i ( w i , ¯ w i ) (cid:11) p × h φ ′ − φ ′ i c ( Nx ) h φ ′ − φ ′ i c ( N − Nx ) h φ ′ − φ ′ i c ( N ) h φ − φ +1 i c ( Nx ) h φ − φ +3 i c ( N − Nx ) h φ − φ +4 i c ( N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A.25)We recall that under a map w ( z ), correlation functions of primary fields φ i of dimensions ∆ i = ¯∆ i transform as D n Y i =1 φ i ( w i , ¯ w i ) E = n Y i =1 (cid:12)(cid:12)(cid:12)(cid:12) d w d z (cid:12)(cid:12)(cid:12)(cid:12) − i w = w i D n Y i =1 φ i ( z i , ¯ z i ) E . (A.26)For the two-point functions, we use the map w c ( z ) = P π log z from the complex plane to the cylinderof perimeter P , and find h φ − i φ + i i c ( P ) = (cid:12)(cid:12)(cid:12)(cid:12) d w c d z (cid:12)(cid:12)(cid:12)(cid:12) − i w = w − (cid:12)(cid:12)(cid:12)(cid:12) d w c d z (cid:12)(cid:12)(cid:12)(cid:12) − i w = w + h φ i ( z − , ¯ z − ) φ i ( z + , ¯ z + ) i C ≃ (cid:12)(cid:12)(cid:12)(cid:12) P π (cid:12)(cid:12)(cid:12)(cid:12) − i e − π Λ∆ i /P , (A.27)where z ± = e ± π Λ /P is such that w c ( z ± ) = w ± . The symbol ≃ means that the equality holds at theleading order in Λ. Furthermore, in the last equality, we used the known value of the two-point function h φ i ( z − , ¯ z − ) φ i ( z + , ¯ z + ) i C = | z − − z + | − i .The three-point function is given in (A.12). For the values (A.24), we have D Y i =1 , , φ i ( z i , ¯ z i ) E C ≃ C e − π ∆ (Λ − K x ) /N . (A.28)Using (A.26), we find D Y i =1 , , φ i ( w i , ¯ w i ) E p ≃ C (cid:0) πN (cid:1) +∆ +∆ ) x (1 − x ) e − π ∆1(Λ+ Kx ) Nx e − π ∆3(Λ+ Kx ) N (1 − x ) e − π ∆4(Λ − Kx ) N . (A.29)44fter some simplifications, we find that the difference of bipartite fidelities is finite for Λ → ∞ . Theresult is F p − F ′ p = 4 (cid:16) δ − δ − x − δ − x (cid:17) ( x log x + (1 − x ) log x ) − (cid:18) C C ′ (cid:19) (A.30)where δ i = ∆ i − ∆ ′ i and C ′ is the structure constant for the correlator h φ ′ φ ′ φ ′ i . Comparing with(2.8), we see that the difference of constants C between F p and F ′ p is simply C − C ′ = − C / C ′ ) . (A.31) A.5 The constant ˜ C for the skirt geometry In Appendix A.3, we found an explicit expression for the function ˜ g ( x ) for the skirt domain. Theresult is given in (3.7) and (3.9) up to a non-universal constant, which a priori can depend on thedimension of the inserted fields. In this section, we extract from this additive constant the dependenceon ∆ .First, we investigate this constant in greater detail for F s in the special case (i) where the onlynon-trivial field is the bulk field φ ( z , ¯ z ). We consider a second realisation F ′ s where a primary field φ ′ ( z , ¯ z ) replaces the field φ ( z , ¯ z ) and is of conformal dimensions ∆ ′ = ¯∆ ′ . In the conformalinterpretation, the difference of bipartite fidelities is F s − F ′ s = − log (cid:12)(cid:12)(cid:12)(cid:12) Z s Z c Z ′ c ( Z ′ s ) (cid:12)(cid:12)(cid:12)(cid:12) = − log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h φ ( w , ¯ w ) i s h φ − φ +5 i c ( N ) · h φ ′ − φ ′ i c ( N ) h φ ′ ( w , ¯ w ) i s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A.32)Here, the two-point functions are evaluated on the cylinder of perimeter N and height Λ. Theirexpression is given in the previous section in the large-Λ limit. Moreover, we calculate the one-pointfunction h φ ′ ( w , ¯ w ) i s for Re( w ) = Λ and Λ large. We compute it using the map to the upperhalf-plane and the transformation law (A.26): h φ ( w , ¯ w ) i s = (cid:12)(cid:12)(cid:12)(cid:12) d w s d z (cid:12)(cid:12)(cid:12)(cid:12) − w = w h φ ( z , ¯ z ) i H . (A.33)In the large-Λ limit, z tends to (1 + i) / 2. The one-point correlator is calculated using the method ofimages and equals 1 in this limit. As for the partial derivatives, they diverge for w → ∞ . Hence weparameterise w ( m ) = w s ( z ( m )), with z ( m ) = (1 − e − mπ/N ), to obtain for large m h φ ( w ( m ) , ¯ w ( m )) i s ≃ (cid:12)(cid:12)(cid:12)(cid:12) √ πN (cid:12)(cid:12)(cid:12)(cid:12) ( e mπ/N ) − . (A.34)With this parameterisation, the position of w ( m ) along the skirt is obtained using the explicit formof w s ( z ), and is of order m : Re( w ( m )) = m + K x − N π log 2 + O (1 /m ) . (A.35)Hence we set m = Λ − K x + N π log 2 to compare the two partition functions. The evaluation for largeΛ yields F s = 4 − ( e − πK x /N ) − . (A.36)Inserting in (A.32), we find F s − F ′ s = 4(∆ − ∆ ′ ) [ x log x + (1 − x ) log(1 − x ) + 2 log 2] . (A.37)We deduce that the additive constant in the function ˜ g ( x ) has a term proportional to 8∆ log 2, asgiven in (3.7). The remaining constant ˜ C in this equation is then independent of ∆ .45econd, we focus on the case (ii), where five vertex operators are present. We consider a secondinstance of F ′ s where the field φ is replaced by φ ′ with conformal dimensions ∆ ′ = ¯∆ ′ . The four otherfields remain unchanged. The difference of bipartite fidelities reads F s − F ′ s = − log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) h φ ( w ) φ ( w ) φ ( w ) φ ( w ) φ ( w , ¯ w ) i s h φ ( w ) φ ( w ) φ ( w ) φ ( w ) φ ′ ( w , ¯ w ) i s · h φ ′ − φ ′ i c ( N ) h φ − φ +5 i c ( N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (A.38)We map the ratio of five-point functions into the upper half-plane and obtain h φ ( w ) φ ( w ) φ ( w ) φ ( w ) φ ( w , ¯ w ) i s h φ ( w ) φ ( w ) φ ( w ) φ ( w ) φ ′ ( w , ¯ w ) i s = (cid:12)(cid:12)(cid:12)(cid:12) d w s d z (cid:12)(cid:12)(cid:12)(cid:12) − − ∆ ′ ) w = w h φ ( z ) φ ( z ) φ ( z ) φ ( z ) φ ( z , ¯ z ) i H h φ ( z ) φ ( z ) φ ( z ) φ ( z ) φ ′ ( z , ¯ z ) i H . (A.39)We apply the method of images to express this ratio in terms of a six-point function, where we introducethe new coordinate z = ¯ z . We then use the generic expression for n -point functions of products ofvertex operators (2.10) and find h φ ( z ) φ ( z ) φ ( z ) φ ( z ) φ ( z , ¯ z ) i H h φ ( z ) φ ( z ) φ ( z ) φ ( z ) φ ′ ( z , ¯ z ) i H = h φ ( z ) φ ( z ) φ ( z ) φ ( z ) φ ( z ) φ ( z ) i C h φ ( z ) φ ( z ) φ ( z ) φ ( z ) φ ′ ( z ) φ ′ ( z ) i C = Q i =1 | z i − z | q i q | z i − z | q i q Q i =1 | z i − z | q i q ′ | z i − z | q i q ′ . (A.40)We exploit the fact that z , z , z and z lie on the real line while z is the complex conjugate of z .Hence, | z i − z | = | z i − z | for each i = 1 , . . . , 4. We finally obtain h φ ( z ) φ ( z ) φ ( z ) φ ( z ) φ ( z , ¯ z ) i H h φ ( z ) φ ( z ) φ ( z ) φ ( z ) φ ′ ( z , ¯ z ) i H = Q i =1 | z i − z | q i ( q + q ) Q i =1 | z i − z | q i ( q ′ + q ′ ) = Q i =1 | z i − z | − q i ( q + q + q + q ) Q i =1 | z i − z | − q i ( q + q + q + q ) = 1 , (A.41)where we used the neutrality condition P i q i = 0.The rest of the computation of the difference F s − F ′ s is similar to the case (i) and yields a termproportional to 8∆ log 2 in the additive constant in ˜ g ( x ), as stated in (3.9). A.6 Sub-leading term for the periodic pants geometry The computation of the sub-leading correction g ( x ) for the periodic pants domain follows closely thesame computation by St´ephan and Dubail [24] for the flat pants domain. Here we only state the maindifferences with their proof.For a conical singularity of internal angle θ , the geometry is that of a simply connected domainwith a conical corner of angle θ . The two edges leaving this corner are endowed with periodic boundaryconditions. If the corner is at the origin, the map w ( z ) from the complex plane to this domain behavesas w ( z ) = z θ/ π (1 + κ z + κ z + . . . ) . (A.42)The preimages in the z -plane of the edges leaving the corner are two lines leaving the origin at angles 0and 2 π . In comparison, for the case considered in [24], the domain of the transformation is the upperhalf-plane, the origin is mapped to a corner that lies on the boundary of the domain, and the prefactorin (A.42) is instead z θ/π .The rest of the computation is similar to [24] and we have g ( x ) = Ξ × N π Res " e i ϕ c (cid:18) d w d z (cid:19) − (cid:18) c { w ( z ) , z } − h T ( z ) φ · · · φ n ih φ · · · φ n i (cid:19) , z = z c + c.c. (A.43)where ϕ c = arg ( w ′′ ( z c )). Computing the residue with w ( z ) = w p ( z ), four arbitrary primary fields φ , . . . , φ inserted in the positions z = 0, z = 1 − x , z = 1 and z = ∞ , and with z c = z , we obtain(2.12). This is exactly twice the result obtained in [24]. This factor of 2 can be traced back to w p ( z )being exactly half of the map for the flat pants geometry considered in [27].46 .7 Sub-leading term for the skirt geometry In this subsection, we compute the sub-leading corrections to the free-energy ˜ g ( x ) for the skirt domain,see (3.4). For a geometry with a boundary and a given number of corners of internal angle 2 π , thisfunction is˜ g ( x ) = N π X π corners Ξ c × Res " e i ϕ c (cid:18) d w d z (cid:19) − (cid:18) c { w ( z ) , z } − h T ( z ) φ · · · φ n ih φ · · · φ n i (cid:19) , z = z c + c.c. (A.44)Here, the sum runs over all the corners of internal angle 2 π . The phase ϕ c is arg ( w ′′ ( z c )). The Ξ c is theextrapolation length corresponding to the corner situated at z = z c . This formula is a generalisationof the one given in [24], where the authors take the same extrapolation length for all corners, Ξ c = Ξ.However, our lattice analysis of F XX s in Section 3.4.1 includes cases where spin states of different lengthsare assigned to the endpoints of the slits, and this requires a CFT analysis with non-equal extrapolationlengths.We now apply the formula to the skirt geometry. The phase verifies e i ϕ c = − z = z and z = z . We write Ξ and Ξ for the corresponding extrapolation lengths.For case (i), we use the method of the images to express the one-point correlator as a two-pointfunction and readily find (3.10). In the case (ii), there is a vertex operator φ i at the position z i on theupper half-plane, with i = 1 , . . . , 5. We use the method of the images as well as (A.15) to find the realpart of the sub-leading contribution to the free-energy:˜ g ( x ) = Ξ × (cid:20) c (1 − x ) − x ) x + q − q x (1 − x ) − q ( q − q + q )4 x − q ( q − q + q )4(1 − x ) + q + q (cid:21) (A.45)+ { Ξ → Ξ , q ↔ q } . We note that this formula greatly simplifies if the two extrapolation lengths are identical. In that case,the result does not depend on q or q . We particularise our result to the case q = − q and obtain(3.11). B Asymptotics In this appendix, we derive the asymptotic expansion of F on the periodic pants domain, for theXX chain (2.88) and for the model of dense polymers (2.97). The starting point is the closed-formexpressions (2.34) and (2.86) for finite N . B.1 Toolbox We define the mathematical tools and functions used in the computation. Exact expressions. The building blocks of the calculations are the functions s [ x ] = sin xx , (B.1a) X ( a, b, N ) = N/ X k =0 log s h πaN ( k + b ) i , (B.1b) Y ( a, b, N, x ) = Nx/ X k =0 N/ X k ′ =0 log s h πaN (cid:0) kx − k ′ + b (cid:1)i , (B.1c) Z ( a, b, N, x ) = Nx/ X k =1 N/ X k ′ =1 log (cid:12)(cid:12)(cid:12)(cid:12) k ′ − a − k − bx (cid:12)(cid:12)(cid:12)(cid:12) . (B.1d)47 symptotics for X and Y . To compute the asymptotic expansion of the X and Y functions, weuse the Euler-Maclaurin formula b X i = a f ( i ) = Z ba f ( x ) dx + f ( a ) + f ( b )2 + ∞ X k =1 B k (2 k )! ( f k − ( b ) − f k − ( a )) (B.2)where B k is the k -th Bernoulli number. The results read X ( a, b, N ) = N I ( πa, 0) + (cid:16) b + 12 (cid:17) log s [ πa ] + O (cid:0) N − (cid:1) , (B.3a) Y ( a, b, N, x ) = N x I ( πa, − πa ) + N x (cid:16) − b + 12 (cid:17) I ( − πa, πa ) + N x (cid:16) b + 12 (cid:17) I ( − πa, N I ( πa, − πa ) + I ( πa, s [ πa ] n 12 + xb + x x o + O (cid:0) N − (cid:1) , (B.3b)with I ( a, b ) = Z d y log s [ ya + b ] , I ( a, b ) = Z Z d w d z log s [ wa + bz ] . (B.4) Asymptotics for Z . The function Z ( a, b, N, x ) is written in terms of Gamma functions as Z ( a, b, N, x ) = − N x π + Nx/ X k =1 log (cid:12)(cid:12)(cid:12) sin h π (cid:0) − k + bx − a (cid:1)i(cid:12)(cid:12)(cid:12)| {z } K ( a,b,N,x ) + Nx/ X k =1 (cid:26) log Γ (cid:16) kx + b − x + 1 − a (cid:17) + log Γ (cid:16) kx − bx + a (cid:17)(cid:27)| {z } ˜ Z ( a,b,N,x ) . (B.5)Let us focus on the function ˜ Z ( a, b, N, x ). We use the integral representation for the logarithm of theGamma function: log Γ( z ) = Z ∞ dtt n ( z − e − t − e − t − e − zt − e − t o , Re z > , (B.6)and find˜ Z ( a, b, N, x ) = Nx/ X k =1 Z ∞ dtt (cid:16) kx − x − (cid:17) e − t − e − t − e − ktx (cid:16) e − (cid:0) b − x +1 − a (cid:1) t + e − (cid:0) − bx + a (cid:1) t (cid:17) − e − t = Z ∞ dtt N x (cid:16) N − (cid:17) e − t − N x e − t − e − t + e − Nt − − e tx · e − (cid:0) b − x +1 − a (cid:1) t + e − (cid:0) − bx + a (cid:1) t − e − t . (B.7)To proceed further, we use the relations [27] i ( n ) ≡ Z ∞ ǫ dtt e − nt = − log ǫ − log n − γ + O ( ǫ ) , (B.8a) i ( n ) ≡ Z ∞ ǫ dtt e − nt = 1 ǫ + n log ǫ + n log n + n ( γ − 1) + O ( ǫ ) , (B.8b) i ( n ) ≡ Z ∞ ǫ dtt e − nt = 12 ǫ − nǫ − n ǫ − n (cid:16) log n − 32 + γ (cid:17) + O ( ǫ ) , (B.8c) j ( α ) ≡ Z ∞ ǫ dtt e αt − αǫ + 12 log( αǫ ) + γ − 12 log(2 π ) + O ( ǫ ) , (B.8d)48alid for n > α > 0. Here γ ≃ . ǫ and take the limit ǫ → Z ∞ ǫ dtt (cid:20) N x (cid:16) N − (cid:17) e − t − N x e − t − e − t (cid:21) = N x (cid:16) N − (cid:17) i (1) − N x j (1) . (B.9)For the second part, we first consider the term with a factor of e − Nt/ . We keep the exponentials inthe numerator unchanged and expand the denominator around t = 0. We only need to keep term upto order t − . Indeed, for k > 0, the integral R ∞ t k e − Nt vanishes as N − ( k +1) for N → ∞ . Hence, wehave Z ∞ ǫ dtt (cid:20) e − Nt − e tx · e − (cid:0) b − x +1 − a (cid:1) t + e − (cid:0) − bx + a (cid:1) t − e − t (cid:21) = 3 − x − x (cid:20) i (cid:16) N b − x + 1 − a (cid:17) + i (cid:16) N − bx + a (cid:17)(cid:21) + 1 − x (cid:20) i (cid:16) N b − x + 1 − a (cid:17) + i (cid:16) N − bx + a (cid:17)(cid:21) − x (cid:20) i (cid:16) N b − x + 1 − a (cid:17) + i (cid:16) N − bx + a (cid:17)(cid:21) + O ( N − ) . (B.10)The last integral in (B.7) is also divergent. Changing the lower bound to ǫ , we simplify the integralby adding and subtracting a function m ( a, b, x, t ) e − t from the integrand. We choose m ( a, b, x, t ) so that(i) the leading terms in its expansion around t = 0 are identical to those of the original integrand, and(ii) it can be easily integrated in terms of the integrals i ( n ), i ( n ) and i ( n ) introduced in (B.8). Wefind Z ∞ ǫ dtt − − e tx · e − (cid:0) b − x +1 − a (cid:1) t + e − (cid:0) − bx + a (cid:1) t − e − t = Z ∞ ǫ dtt (cid:2) J ( a, b, x, t ) − m ( a, b, x, t ) e − t (cid:3) (B.11)+ 2 x ( i (1) + i (1)) + (cid:16) b + 16 (cid:16) − x (cid:17) + ( − b ) bx + 7 x a x − a ( − b + x ) (cid:17) i (1)with J ( a, b, x, t ) = − − e tx · e − (cid:0) b − x +1 − a (cid:1) t + e − (cid:0) − bx + a (cid:1) t − e − t , (B.12a) m ( a, b, x, t ) = 2 x (cid:16) t + 1 t (cid:17) + (cid:16) b + 16 (cid:16) − x (cid:17) + ( − b ) bx + 7 x a x − a ( − b + x ) (cid:17) . (B.12b)We obtain the leading terms in the expansion of ˜ Z by combining (B.9), (B.10), (B.11) and (B.12).Using (B.8), we find that the divergent contributions cancel as expected. The result has a well-defined ǫ → Z ( a, b, N, x ) = Z ∞ dtt (cid:2) J ( a, b, x, t ) − m ( a, b, x, t ) e − t (cid:3) − N x − N ) + N x π )+ log N b − b (1 − x + 2 ax ) + x ( − x + 6 a (1 − x + ax ))6 x − x + (1 + x − x + 6( b − ax )( − b + x − ax )) log 26 x + O ( N − ) . (B.13)We note that the integral R ∞ dtt (cid:2) J ( a, b, x, t ) − m ( a, b, x, t ) e − t (cid:3) is convergent.49 .2 Asymptotics for P ( N ) To derive the large- N expansion of P ( N ), we start from the definition (2.33) and rewrite each sinefactor using sin x = x s [ x ]. Each product then splits into two products, the first involving the function s [ x ] and the second involving the arguments x . The former products are expressed in terms of thefunctions X and Y in (B.1). The latter are written in terms of Barnes’ G -function. The result islog P ( N ) = − N ( N − N − log π ) + log G (cid:16) N (cid:17) + 12 Y (cid:16) , , N, (cid:17) − X (cid:16) , , N (cid:17) . (B.14)We recall the asymptotics of the Barnes’ G-function:log G ( z ) = (cid:18) ( z − − (cid:19) log( z − − z − z − 12 log(2 π ) + 112 − log A + O ( z − ) , (B.15)where A ≃ . X and Y , we obtain the large- N expansion of P up to order O ( N − ). B.3 Asymptotics for P ( N , N , φ , φ ) We follow the same strategy for P and findlog P ( N , N , φ , φ ) = − N N N π + log s h πN (cid:16) − z + 12 − φ πz + φ π (cid:17)i + Z (cid:16) 12 + φ π , 12 + φ π , N , z (cid:17) + Y (cid:16) , − z − φ πz + 12 + φ π , N , z (cid:17) (B.16) − X (cid:16) , z (cid:0) − z − φ πz + 12 + φ π (cid:1) , N (cid:17) − X (cid:16) , z + φ πz − − φ π , N (cid:17) where z ≡ N N . The term log s h πN (cid:16) − z + − φ πz + φ π (cid:17)i does not contribute to the leading orders inthe large- N expansion. Using (B.3), we obtain the large- N expansion of P up to order O ( N − ). B.4 Combinations of non-trivial terms We consider the combination of log P and log P corresponding to (2.34). In doing so, we find thatall the terms involving the integrals I and I that appear in the large- N expansion (B.3) of X and Y cancel out. Let us now investigate the remaining terms. Combination of ˜ Z . We are interested in the combination˜ Z (cid:0) r B , r A , N (1 − x ) , x − x (cid:1) − ˜ Z ( r, r A , N, x ) − ˜ Z ( r, r B , N, − x ) (B.17)with r A,B = + φ A,B π and r = + φ π = r A + r B − 1. This last constraint followsfrom (2.28) with ℓ = 0. The only non-trivial combination involves the (convergent) integral R ∞ dtt (cid:2) J ( a, b, x, t ) − m ( a, b, x, t ) e − t (cid:3) . From (B.12a), we observe that J (cid:16) r B , r A , x − x , t (cid:17) − J (cid:16) r A + r B − , r A , x, t (cid:17) − J (cid:16) r A + r B − , r B , − x, t − xx (cid:17) = 0 . (B.18)Hence, in computing (B.17), we make the change of variable t → t − xx in the integral of the third term.The integrals involving the functions J cancel and the resulting combination Z ∞ dtt (cid:20) − m (cid:16) r B , r A , x − x , t (cid:17) e − t + m (cid:16) r A + r B − , r A , x, t (cid:17) e − t + m (cid:16) r A + r B − , r B , − x, t − xx (cid:17) e − t − xx (cid:21) (B.19)50an be written in terms of the functions i ( n ), i ( n ) and i ( n ) defined in (B.8). Including the remainingterms in (B.13), we find that the result simplifies to˜ Z (cid:0) r B , r A , N (1 − x ) , x − x (cid:1) − ˜ Z ( r A + r B − , r A , N, x ) − ˜ Z ( r A + r B − , r B , N, − x ) =18 N n − x ) x (cid:16) log N + log(1 − x ) (cid:17) − N + (cid:16) ( x − x + 1 (cid:17) (3 + log 4) o + 12 N ( x − 1) log(2 π )+ 6 r A ( x − + 6 r A ( x − r B x − x + 1) + x (cid:16) r B ( r B x − x + 1) + 13 x − (cid:17) + 16 x log(1 − x )+ 6 r A ( x − + 6 r A ( x − r B x − x + 2) + x (cid:16) r B ( r B x − x + 2) + 13 x − (cid:17) + 56(1 − x ) log x + O ( N − ) . (B.20) Combination of trigonometric functions. The logarithm of the product of cosines in (2.34) andthe functions K appearing in the definition (B.5) of Z combine to give the identity K (cid:16) r B , r A , N (1 − x ) , x − x (cid:17) − K (cid:16) r A + r B − , r A , N, x (cid:17) − K (cid:16) r A + r B − , r B , N, − x (cid:17) + N/ X k ′ =1 log (cid:12)(cid:12)(cid:12)(cid:12) cos h πxk ′ − φ A ( x − 1) + xφ B i(cid:12)(cid:12)(cid:12)(cid:12) = − N x log 2 , (B.21)whose proof is straightforward.We now have all the ingredients needed to compute the large- N expansion of F XX p | φ = φ A + φ B − π ,defined in (2.34). Simplifying the result, we find (2.88). B.5 Asymptotics for Q ( N, φ ) We obtain the large- N expansion (2.97) of F αp from (2.86) and (2.88) by computing the asymptoticexpansion of the function Q , defined in (2.87). Using the same strategy as for the functions P and P ,we find Q ( N, φ ) = log Γ (cid:16) N − φ π (cid:17) − log Γ (cid:16) − φ π (cid:17) − X (cid:16) , − 12 + φ π , N (cid:17) − n φ → − φ o . 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