A tale of two Bethe ansätze

Transcription

1 UMTG 93 A tale of two Bethe ansätze arxiv: v [math-ph] 6 Apr 018 Antonio Lima-Santos 1, Rafael I. Nepomechie Rodrigo A. Pimenta 3 Abstract We revisit the construction of the eigenvectors of the single double-row transfer matrices associated with the Zamolodchikov-Fateev model, within the algebraic Bethe ansatz method. The left right eigenvectors are constructed using two different methods: the fusion technique Tarasov s construction. A simple explicit relation between the eigenvectors from the two Bethe ansätze is obtained. As a consequence, we obtain the Slavnov formula for the scalar product between on-shell off-shell Tarasov-Bethe vectors. 1 Departamento de Física, Universidade Federal de São Carlos, Caixa Postal 676, CEP , São Carlos, Brazil, dals@df.ufscar.br Physics Department, P.O. Box 48046, University of Miami, Coral Gables, FL 3314 USA, nepomechie@miami.edu 3 Instituto de Física de São Carlos, Universidade de São Paulo, Caixa Postal 369, , São Carlos, SP, Brazil, pimenta@ifsc.usp.br

2 1 Introduction The Zamolodchikov-Fateev ZF) model [1] is a nineteen-vertex model that can be seen as a generalization of the six-vertex model. Indeed, while the latter is associated with the spin- 1 XXZ chain, the former describes an integrable spin-1 XXZ chain. The ZF R-matrix is a solution of the Yang-Baxter equation, which is given by a 9 9 matrix with nineteen non-null entries. Starting from this R-matrix, one can use the quantum inverse scattering method [, 3, 4] to construct single double-row commuting transfer matrices, which are associated with closed open spin chains, respectively. The spectral problem for the transfer matrix can be solved by means of the algebraic Bethe ansatz. For the ZF model, the construction of the Bethe vectors of the single double-row transfer matrices can be performed in two different ways. The first one is by means of the fusion technique [5, 6, 7, 8, 9, 10, 11, 1, 13, 14, 15, 16, 17], the auxiliary space is still -dimensional, therefore the Bethe vectors are obtained similarly to the six-vertex model case. We shall refer to these as fusion-bethe vectors. The second way is by means of Tarasov s construction [18], which entails working with a 3-dimensional auxiliary space. This approach was originally developed for the Izergin-Korepin model [19] 1, see also [1] for the double-row case, was applied to the ZF model in [, 3]. We shall refer to these as Tarasov-Bethe vectors. The scalar product between on-shell off-shell fusion-bethe vectors, i.e., a formula of the type found in the celebrated paper by Slavnov [4], has been obtained for closed chains, see [5] for the rational case [6] for the trigonometric case. For open chains, the Slavnov formula has been obtained only for spin 1, see [7, 8] for the rational case [9] for the trigonometric case. On the other h, Slavnov-type formulas for Tarasov-Bethe vectors have not yet been found, to the best of our knowledge. This is probably due to the intricacy of the exchange relations arising from the Yang-Baxter reflection algebras when the auxiliary space has dimension greater than two, as is the case in Tarasov s construction. The purpose of this note is to initiate the investigation of the Slavnov scalar products for Tarasov-Bethe vectors. Here we hle this problem by showing that the fusion-bethe vectors the Tarasov-Bethe vectors are simply related: the result for the closed chain is given by 4.40,4.41) for the open chain it is given by 5.39,5.40). Thanks to these simple relations, we can easily obtain the Slavnov formula for the Tarasov-Bethe vectors, for both the closed 4.58) open chains 5.54), without going through Yang-Baxter reflection algebras in Tarasov s construction. Thispaperisorganizedasfollows. InSectionwerecallthebasicsofthefusiontechnique. The fundamental fused monodromy transfer matrices are given in Section 3. In Section 4 the Bethe ansatz is implemented the Slavnov formula is obtained for the single-row transfer matrix, while in Section 5 these results are generalized to the double-row transfer matrix. Our concluding remarks are given is Section 6. 1 This model is also a nineteen-vertex model, but it cannot be obtained by fusion from the fundamental six-vertex model. See, however, [0]. For simplicity, we focus here on the case of diagonal boundary conditions. 1

3 Fusion for R-matrices K-matrices In this section we recall some basic ingredients of the fusion technique, by means of which we can construct integrable generalizations of the fundamental six-vertex model, for both single-row [5, 6, 7, 8, 9, 10, 11, 1, 13, 14] double-row [15, 16, 17] transfer matrices..1 R-matrices We start with the fundamental R-matrix R 1,1 ) u) = sinhu + η) sinhu) sinhη) 0 0 sinhη) sinhu) sinhu + η),.1) which acts on C C η is the free anisotropy parameter. This R-matrix has the parity symmetry P is the permutation matrix, the unitarity property Ru) = PRu)P,.) Ru)R u) = ξu)ξ u)i I,.3) with ξu) = ξ 1,1 ) u) = sinhu+η). It also has crossing symmetry R 1 u) = V 1 R t 1 u ρ)v 1 = V t Rt 1 1 u ρ)v t,.4) with V = V 1,1 ) = ), ρ = ρ 1,1 ) = η+iπ..5) Hence, the matrix M defined by M = V t V.6) is given by M = M 1,1 ) = I..7) By means of the fusion procedure, we can construct R-matrices R i,j) u) associated with su) representations of spins i j, with i,j { 1,1}. These R-matrices, which map C i+1 C j+1 C i+1 C j+1, satisfy the various Yang-Baxter equations R i,j) 1 u v)r i,k) 13 u)r j,k) 3 v) = R j,k) 3 v)r i,k) 13 u)r i,j) 1 u v)..8)

4 Using the fusion procedure following [13, 14]), we define the fused R-matrix R 1,1) 1 3 u) = 1 sinhu+ η )F 3 R 1,1 ) 13 u+ η )R,1 ) 1 u η )E 3 = 1 sinhu+ 3η ) sinhu+ η ) 0 1 sinhη) sinhu η ) 0 sinhη) 0 0 sinhη) 0 sinhu η ) 0 0, sinhη) 0 sinhu+ η ) sinhu+ 3η ) which acts on C C 3. Here E = 1 0 0, F = Et = ) This R-matrix has the unitarity property.3) with ξu) = ξ 1,1) u) = sinhu+ 3η ). We also define the fused R-matrix R 1,1 ) 1 3 u) = 1 sinhu+ η )F 1 R 1,1 ) 13 u+ η )R1,1 ) which acts on C 3 C. The R-matrices.9).11) are related by 3 u η )E 1,.11) R 1,1 ) u) = P 1,1) R 1,1) u)p 1,1 ),.1).9) P 1,1) = ) is a permutation matrix that maps C C 3 to C 3 C, P 1,1 ) = P 1,1)) 1. Finally, we define the fused R-matrix 1 34 u) = F 1 R 1,1) 1 34 u+ η 1 )R,1) 34 u η )E 1 R 1,1) 3

5 = au) bu) 0 cu) fu) 0 du) 0 hu) cu) 0 bu) du) 0 eu) 0 du) bu) 0 cu) hu) 0 du) 0 fu) cu) 0 bu) au),.14) with au) = sinhu+η)sinhu+η), bu) = sinhu)sinhu+η), du) = sinhu)sinhη), du) = sinhu)sinhη)cosh η), cu) = sinhη)sinhu+η), fu) = sinhu)sinhu η), eu) = sinhu)sinhu+η)+sinhη)sinhη), hu) = sinhη)sinhη),.15) which acts on C 3 C 3. This R-matrix has the parity symmetry.) the unitarity property.3) with ξu) = ξ 1,1) u) = sinhu+η)sinhu+η). Although it does not have the crossing symmetry.4), it is related to a symmetric R-matrix R [ R1,1) t1 1,1) t u) = u)] by a constant gauge transformation R 1,1) u) = C 1 1 C 1 R 1,1) u)c 1 C, C = diagcoshη),coshη),1) ;.16) this symmetric R-matrix does have crossing symmetry, with V = V 1,1) = 0 1 0, ρ = ρ 1,1) = η +iπ,.17) therefore M = M 1,1) = I..18) The R-matrix.16) was firstly obtained by Zamolodchikov Fateev [1].. K-matrices We now construct corresponding K-matrices K ±i) u), which map C i+1 C i+1. For K i) u), the boundary Yang-Baxter equations are R i,j) 1 u v)k i) 1 u) R j,i) 1 u+v)k j) v) = K j) v)r i,j) 1 u+v)k i) 1 u)r j,i) 1 u v),.19) 4

6 R j,i) 1 u) = P i,j) R j,i) 1 u)p j,i)..0) For the fundamental K -matrix, we restrict for simplicity to the diagonal solution K 1 ) u) = diag sinhu+ξ ), sinhu ξ ) ),.1) ξ is an arbitrary boundary parameter. For K +, we take K +1 ) u) = K 1 ) u ρ 1,1 ) ) ξ + is another arbitrary boundary parameter. ξ ξ + = diag sinhu+η ξ + ), sinhu+η +ξ + ) ),.) The fused K -matrix is given by following [17]) K 1) u) = 1 sinhu+ η )F 1 K 1 ) 1 u+ η )R1,1 ) 1 u)k 1 ) u η )P 1E 1 = diag k 1 u),k u),k 3 u) ),.3) k1 u) = coshu+ η ) sinhu+ η +ξ ) sinhu η +ξ ), k u) = coshu+ η ) sinhu η +ξ ) sinhu η ξ ), k3 u) = coshu+ η ) sinhu+ η ξ ) sinhu η ξ )..4) For K +, we similarly take K +1) u) = K 1) u ρ 1,1) )..5) ξ ξ + 3 Monodromy transfer matrices Using the R K matrices introduced in the previous section, we can obtain families of commuting operators, i.e., transfer matrices. 3.1 Single-row Indeed, the single-row monodromy matrix for the R-matrix.9) is defined as 1 T 1,1) a u) = R 1,1) an u)...r,1) a1 u), 3.1) 5

7 the associated transfer matrix by Similarly, for the R-matrix.14) The Yang-Baxter equations.8) imply t 1,1) u) = tr a T 1,1) a u). 3.) T 1,1) a u) = R 1,1) an u)...r1,1) a1 u), 3.3) t 1,1) u) = tr a T 1,1) a u). 3.4) R 1,1 ) 1 u v)t,1) 1 1 u)t,1) 1 v) = T,1) 1 v)t,1) 1 1 u)r 1,1 ) 1 u v) 3.5) R 1,1) 1 u v)t 1,1) 1 u)t 1,1) v) = T 1,1) v)t 1,1) 1 u)r 1,1) 1 u v), 3.6) as well as [ ] t 1,1) u),t 1,1) v) = 0, [ t 1,1) u),t 1,1) v) ] = 0, 3.7) [ ] t 1,1) u),t 1,1) v) = ) The latter relation implies that t 1,1) u) t 1,1) v) can be diagonalized simultaneously. In addition, one obtains with the help of.14) the important relation 1 u) = F 1 T 1 T 1,1) which will be used in Section 4.3.,1) 1 u+ η )T1,1) u η )E 1, 3.9) 3. Double-row In order to construct double-row objects, one needs to introduce reflected single-row monodromy matrices similarly T 1,1) a u) = R 1,1) T 1,1) a u) = R 1,1) a1 u)...r 1,1) an a1 u)...r 1,1) an 6 u), 3.10) u). 3.11)

8 The corresponding double-row monodromy matrices are then defined as follows U 1,1) a u) = T 1,1) a u)k 1 ) a u) T 1,1) a u), a u) = T a 1,1) u)ka 1) 1,1) u) T a u). 3.1) U 1,1) They obey the boundary Yang-Baxter equations.19), in particular Moreover, they are related by R 1,1 ) 1 u v)u 1,1) 1 u) R 1,1 ) 1 u+v)u 1,1) v) = U 1,1) v)r 1,1 ) 1 u+v)u 1,1) 1 u)r 1,1 ) 1 u v), 3.13) R 1,1) 1 u v)u 1,1) 1 u) R 1,1) 1 u+v)u 1,1) v) U 1,1) 1 u) = 1 sinhu+ η )F 1 U 1 = U 1,1) v)r 1,1) 1 u+v)u 1,1) 1 u)r 1,1) 1 u v). 3.14),1) 1 u+ η )R1,1 ) 1 u)u 1,1) u η )P 1E 1, 3.15) which can be obtained with the help of.3). This relation, which is the double-row version of 3.9), will be exploited in Section 5.3. Finally, the corresponding transfer matrices are given by τ 1,1) u) = tr a K +1 ) a u)u 1,1) a u), 3.16) τ 1,1) u) = tr a K +1) a u)u 1,1) a u). 3.17) One can show that these transfer matrices obey [ ] [ τ 1,1) u),τ 1,1) v) = 0, τ 1,1) u),τ 1,1) v) ] = 0, 3.18) as well as [ ] τ 1,1) u),τ 1,1) v) = ) Similarly to the single-row case, the latter relation implies that τ 1,1) u) τ 1,1) v) can be diagonalized simultaneously. 7

9 4 Bethe ansatz: single-row transfer matrix In this section we obtain the eigenvectors of the single-row transfer matrices 3.) 3.4). 4.1 Bethe vectors from fusion Westart by applying thebethe ansatzto 3.), we refer to [5, 6, 7, 8, 9, 10, 11, 1, 13, 14] for more details. For that, we use the -dimensional auxiliary space representation, i.e., we set T,1) ) 1 Au) Bu) a u) =, 4.1) Cu) Du) in which each entry is an operator acting on the quantum space C 3 ) N. These operators satisfy exchange relations dictated by 3.5). We also introduce the reference state vector 1 0 = 0 0 as well as its dual N a 4.) 0 = ) N, 4.3) such that 0 0 = 1. The action of the monodromy operators on 4.,4.3) is given by Au) 0 = λ 1 u) 0, Du) 0 = λ u) 0, Cu) 0 = 0, 4.4) 0 Au) = 0 λ 1 u), 0 Du) = 0 λ u), 0 Bu) = 0, 4.5) λ 1 u) = sinh N u+ 3η ), λ u) = sinh N u η ). 4.6) The transfer matrix 3.) is given by t 1,1) u) = Au)+Du). 4.7) The Bethe vectors are given by φ m u 1,...,u m ) = Bu 1 )...Bu m ) 0 4.8) φ m u 1,...,u m ) = 0 Cu 1 )...Cu m ). 4.9) 8

10 In fact, using the Yang-Baxter algebra 3.5) one can show that t,1) 1 u) φ m u 1,...,u m ) = λu,u 1,...u m ) φ m u 1,...,u m ) + sinhη)fu,u j ) E ju j ) Q j u j ) Bu) φ m 1u 1,...,û j,...,u m ) j=1 φ m u 1,...,u m ) t,1) 1 u) = φ m u 1,...,u m ) λu,u 1,...u m ) + φ m u 1,...,û j,...,u m ) Cu) E ju j ) Q j u j ) Fu,u j)sinhη), j=1 4.10) 4.11) λu,u 1,...u m ) = λ 1 u) Qu η) Qu) +λ u) Qu+η) Qu), 4.1) E j u) = λ 1 u)q j u η) λ u)q j u+η), 4.13) Fu,v) = 1 sinhu v). 4.14) In the above formulae we have introduced the Baxter Q-polynomial Qu) = sinhu u i ), 4.15) i=1 which is indeed a Laurent polynomial in z e u, as well as the indexed Baxter Q-polynomial Q j u) = sinhu u i ). 4.16) i j Let us also introduce the two-index polynomial which we will use later Q j,k u) = sinhu u i ). 4.17) i j,k The equations E j u j ) = 0 for j = 1,...,m, 4.18) 9

11 are the conditions for the unwanted terms in the off-shell equations 4.10) 4.11) to vanish; therefore, for the Bethe vectors 4.8) 4.9) to be eigenvectors of the transfer matrix t 1,1) u), with corresponding eigenvalue λu,u 1,...u m ) given by 4.1). These polynomial equations are known as the Bethe ansatz equations, we shall refer to E j u) as the Bethe ansatz polynomials. As usual, the notation û j means that the rapidity u j is absent from a given function or operator argument. Finally, let us suppose that {u 1,...,u m } satisfy the Bethe equations 4.18) i.e., they are on-shell rapidities) that there is no restriction on {v 1,...,v m } i.e., they are offshell rapidities). Then, the scalar product φ m u 1,...,u m ) φ m v 1,...,v m ) is given by the determinant formula [4, 6] ) det m u i λv j,u 1,...,u m ) φ m u 1,...,u m ) φ m v 1,...,v m ) = λ u i ), det m Fv i,u j )) i=1 which is known as the Slavnov formula. Using 4.19) λv j,u 1,...,u m ) = sinhη)fu i,v j )Fv j,u i ) E iv j ) u i Q i v j ), 4.0) we can rewrite the Slavnov formula as ) det m sinhη)fu i,v j )Fv j,u i ) E iv j ) Q i v j ) φ m u 1,...,u m ) φ m v 1,...,v m ) = λ u i ). det m Fv i,u j )) i=1 4.1) Although the Slavnov formula is frequently written in the form 4.19), which was introduced in [30], it seems more suitable for generalization to rewrite the Slavnov formula as in 4.1) in terms of the Bethe ansatz Baxter polynomials instead of the derivative of the transfer matrix eigenvalue, as it will become clear in Section 4.4. The form 4.1) is closer to the expression presented by Slavnov in his pioneering paper [4]. 4. Bethe vectors from Tarasov s construction WenowapplytheBetheansatzto3.4), see[18,]. Here, weusethe3-dimensionalauxiliary space representation, i.e., we set A 1 u) B 1 u) B u) T a 1,1) u) = C 1 u) A u) B 3 u), 4.) C u) C 3 u) A 3 u) each entry acts on the quantum space C 3 ) N. These operators satisfy exchange relations dictated by 3.6). The action of the monodromy operators 4.) on the reference state 4.,4.3) is given by A j u) 0 = Λ j u) 0, C j u) 0 = 0, 4.3) 10 a

12 0 A j u) = 0 Λ j u), 0 B j u) = 0, 4.4) for j = 1,,3 Λ 1 u) = sinhu+η)sinhu+η)) N, Λ u) = sinhu)sinhu+η)) N, Λ 3 u) = sinhu)sinhu η)) N. 4.5) The transfer matrix 3.4) is given by t 1,1) u) = A 1 u)+a u)+a 3 u). 4.6) The Bethe vector is constructed by means of the recursion relation ψ m u 1,...,u m ) = B 1 u 1 ) ψ m 1 u,...,u m ) B u 1 ) γ m) i u 1,...,u m ) ψ m u,...,û i,...,u m ), 4.7) γ m) i u 1,...,u m ) = sinhη)λ 1 u i )Fu 1,u i +η) Q 1,iu i η) Q 1,i u i ) i= j=,j<i Ωu i,u j ), 4.8) Ωu,v) = sinhu v η)sinhu v +η) sinhu v η)sinhu v+η). 4.9) The dual Bethe vector is given by ψ m u 1,...,u m ) = ψ m 1 u,...,u m ) C 1 u 1 ) γ m) i u 1,...,u m ) ψ m u,...,û i,...,u m ) C u 1 ), 4.30) i= γ m) i u 1,...,u m ) = sinhη)coshη)λ 1 u i )Fu 1,u i +η) Q 1,iu i η) Q 1,i u i ) Note that the the initial conditions are assumed for 4.7) 4.30). j=,j<i Ωu i,u j ).4.31) ψ 0 = 0, ψ 0 = 0, 4.3) Using the Yang-Baxter algebra 3.6) one can show that the Tarasov-Bethe vectors 4.7) 4.30) satisfy the off-shell equations t 1,1) u) ψ m u 1,...,u m ) = Λu,u 1,...,u m ) ψ m u 1,...,u m ) 11

13 + + + j=1 j=1 j<k sinhη)fu,u j )Ēju j )Q j u j η) Q j u j )Q j u j η) Ωu j,u p )B 1 u) ψ m 1 u 1,...,û j,...,u m ) p<j sinhη)fu,u j +η)ēju j )Q j u j η) Q j u j )Q j u j η) Ωu j,u p )B 3 u) ψ m 1 u 1,...,û j,...,u m ) p<j H m) jk u,u 1,...,u m )B u) ψ m u 1,...,û j,...,û k,...,u m ), 4.33) ψ m u 1,...,u m ) t 1,1) u) = ψ m u 1,...,u m ) Λu,u 1,...,u m ) + ψ m 1 u 1,...,û j,...,u m ) C 1 u) Ωu j,u p )Ēju j )Q j u j η) Q j u j )Q j u j η) Fu,u j)sinhη) + + j=1 j=1 p<j ψ m 1 u 1,...,û j,...,u m ) C 3 u) p<j Ωu j,u p )Ēju j )Q j u j η) Q j u j )Q j u j η) sinhη)coshη)fu,u j +η) ψ m u 1,...,û j,...,û k,...,u m ) C u)h m) jk u,u 1,...,u m )cosh η), 4.34) j<k Λu,u 1,...,u m ) = Λ 1 u) Qu η) Qu) +Λ u) Qu η)qu+η) Qu)Qu η) +Λ 3 u) Qu+η) Qu η), 4.35) Ē j u) = Λ 1 u)q j u η) Λ u)q j u+η), 4.36) H m) jk u,u 1,...,u m ) = 4coshη)sinh Q j u j η)q j,k u k η) η) Q j u j η)q j,k u j )Q j,k u k )Q j,k u k η) { Fu,u j )Fu,u k )Fu j,u k +η)fu j,u k η)ēju j )Ēku k ) Ωu j,u p ) p<j q<k,q j + sinhη)fu,u k )Fu,u k +η)fu j,u k )Fu j,u k +η)q j,k u j +η)λ u j )Ēku k ) Ωu k,u q ) + sinhη)fu,u j )Fu,u j +η)fu j,u k )Fu j,u k η)q j,k u k +η)λ u k )Ēju j ) }.4.37) The set of polynomial equations Ē j u j ) = 0 for j = 1,...,m, 4.38) are the Bethe ansatz equations from Tarasov s construction. We remark that the unwanted terms have been written explicitly in terms of the Bethe polynomials; in this way, it is clear that all unwanted terms disappear on-shell. 1

14 4.3 Relating the two closed-chain Bethe ansätze The Bethe equations from fusion4.18) from Tarasov s construction4.38) are equivalent. Indeed, since the factors in the Bethe polynomials E j u) 4.13) Ēju) 4.36) are related by Λ 1 u) Λ u) = λ 1u+ η ) λ u+ η ), 4.39) the solutions of the two sets of Bethe equations are related by η shifts. We claim that the relation between the off-shell fusion-bethe vectors 4.8), 4.9), the off-shell Tarasov-Bethe vectors 4.7), 4.30) is given by ψ m u 1,...,u m ) = su 1,...,u m ) φ m u 1 + η,...,u m + η ), 4.40) ψ m u 1,...,u m ) = φ m u 1 + η,...,u m + η ) su1,...,u m )cosh m η), 4.41) su p,...,u m ) = m p 1) i=p λ 1 u i η ) m sinhu j u k η) sinhu j u k η). 4.4) j,k=p,j<k We prove 4.40) by induction; the proof of 4.41) is similar. The cases m = 1 m = follow from brute force computation. We first note that the representations 4.1) 4.) are connected by means of the relation 3.9), which gives us in particular see e.g. [5]) B 1 u) = 1 A u+ η ) B u η ) +B u+ η ) A u η )) = B u+ η ) A u η ), B u) = B u+ η ) B u η ), 4.43) then use commutation relations from 3.5), the action 4.4) the identity Λ 1 u) = λ 1 u+ η )λ 1 u η ). 4.44) Next, let us compute ψ m+1 u 1,...,u m+1 ) supposing 4.40) is valid. By 4.7) we have ψ m+1 u 1,...,u m+1 ) = B 1 u 1 ) ψ m u,...,u m+1 ) m+1 B u 1 ) γ m+1) i u 1,...,u m+1 ) ψ m 1 u,...,û i,...,u m+1 ), 4.45) i=, by means of 4.40), we obtain ψ m+1 u 1,...,u m+1 ) = su,...,u m+1 )B 1 u 1 ) φ m ũ,...,ũ m+1 ) 13

15 m+1 B u 1 ) γ m+1) i u 1,...,u m+1 )su,...,û i,...,u m+1 ) φ m 1 ũ,...,ˆũ i,...,ũ m+1 ), i= 4.46) we have introduced su p,...,û l,...,u m ) = m p i=p,i l λ 1 u i η ) m sinhu j u k η) sinhu j u k η) j,k=p,j<k,j l,k l 4.47) the notation ũ = u+ η. 4.48) Using 4.43) we express B 1 u 1 ) B u 1 ) in terms of Au 1 ) Bu 1 ), compute B 1 u 1 ) φ m ũ,...,ũ m+1 ) = Bũ 1 )A u 1 η ) φ m ũ,...,ũ m+1 ), 4.49) B u 1 ) φ m 1 ũ,...,ˆũ i,...,ũ m+1 ) = Bũ 1 )B u 1 η ) φ m 1 ũ,...,ˆũ i,...,ũ m+1 ) = B u 1 η ) φ m ũ 1,ũ,...,ˆũ i,...,ũ m+1 ). 4.50) By means of 3.5) the action 4.4), we obtain Au) φ m u 1,...,u m ) = λ 1 u) Qu η) φ m u 1,...,u m ) Qu) sinhη) + sinhu u j ) λ 1u j ) Q ju j η) Bu) φ m 1 u 1,...,û j,...,u m ), 4.51) Q j u j ) which leads to j=1 B 1 u 1 ) φ m ũ,...,ũ m+1 ) + m+1 j= sinhη)λ 1 ũ j ) sinhu 1 u j η) Using 4.50) 4.5) we obtain = λ 1 u 1 η )m+1 sinhu 1 u j η) sinhu j= 1 u j η) φ m+1ũ 1,ũ,...,ũ m+1 ) m+1 k=,k j sinhu k u j +η) B u 1 η ) φ m ũ 1,ũ,...,ˆũ j,...,ũ m+1 ). sinhu k u j ) ψ m+1 u 1,...,u m+1 ) = su,...,u m+1 ) λ 1 u 1 η )m+1 sinhu 1 u j η) sinhu j= 1 u j η) φ m+1ũ 1,ũ,...,ũ m+1 ) )

16 + m+1 j= { su,...,u m+1 ) sinhη)λ 1ũ j ) sinhu 1 u j η) m+1 k=,k j sinhu k u j +η) sinhu k u j ) } γ m+1) j u 1,...,u m+1 )su,...,û j,...,u m+1 ) B u 1 η ) φ m ũ 1,ũ,...,ˆũ j,...,ũ m+1 ). 4.53) We can check that su 1,...,u m+1 ) = su,...,u m+1 ) λ 1 u 1 η )m+1 sinhu 1 u j η) sinhu j= 1 u j η), 4.54), thanks to 4.44), which implies su,...,u m+1 ) sinhη)λ 1u j + η ) sinhu 1 u j η) m+1 k=,k j sinhu k u j +η) sinhu k u j ) = γ m+1) j u 1,...,u m+1 )su,...,û j,...,u m+1 ), 4.55) ψ m+1 u 1,...,u m+1 ) = su 1,...,u m+1 ) φ m+1 ũ 1,ũ,...,ũ m+1 ), 4.56) therefore completes the inductive proof of 4.40). 4.4 Scalar products for Tarasov-Bethe vectors Combining the scalar product for the fusion-bethe vectors 4.19) with the relations between Tarasov-Bethe vectors fusion-bethe vectors 4.40), 4.41), it is now straightforward to compute the scalar product between the off-shell state ψ m u 1,...,u m ) 4.30) the onshell state ψ m v 1,...,v m ) 4.7) ψ m u 1,...,u m ) ψ m v 1,...,v m ) = cosh m η)su 1,...,u m )sv 1,...,v m ) λ u i + η ) det m u i λ v j + η,u 1 + η,...,u ) ) m + η ), 4.57) det m Fv i,u j )) i=1 which is in terms of the quantities λ λ arising from fusion. We can rewrite 4.57) in a similar way as in 4.1) ψ m u 1,...,u m ) ψ m v 1,...,v m ) = j<k sinhu j u k η) sinhv j v k η) sinhu j u k η) sinhv j v k η) 15

17 ) det m sinhη)fu i,v j )Fv j,u i )Ēiv j ) Q i v j ) Λ u i ), det m Fv i,u j )) i=1 4.58) which is instead in terms of quantities arising from Tarasov s construction, namely, the Bethe ansatz polynomial 4.38) the Baxter polynomial 4.15). The Slavnov formula written in 4.1) or 4.58) therefore seems more universal than the formula given in terms of the eigenvalue of the transfer matrix. 5 Bethe ansatz: double-row transfer matrix In this section we obtain the eigenvectors of the double-row transfer matrices 3.16) 3.17). 5.1 Bethe vectors from fusion We firstly apply the Bethe ansatz to 3.16), see [4, 15]. For that, we use the -dimensional auxiliary space representation, i.e., we set ) U 1,1) Au) Bu) a u) = Cu) Du)+ sinhη) Au) 5.1) sinhu+η) in which each entry is an operator acting on the quantum space C 3 ) N. These operators satisfy exchange relations dictated by 3.13). The action of the double-row monodromy operators on the reference states 4.,4.3) is given by Au) 0 = δ 1 u) 0, Du) 0 = δ u) 0, Cu) 0 = 0, 5.) a 0 Au) = 0 δ 1 u), 0 Du) = 0 δ u), 0 Bu) = 0, 5.3) δ 1 u) = sinhu+ξ )sinh N u+ 3η ), The transfer matrix 3.16) is given by δ u) = sinhu)sinhξ u η) sinhu+η) sinh N u η ). 5.4) τ 1,1) u) = sinhu+η))sinhu ξ+ ) Au) sinhu+η +ξ + )Du). 5.5) sinhu+η) 16

18 The fusion-bethe vectors are given by Φ m u 1,...,u m ) = Bu 1 )...Bu m ) 0, 5.6) Φ m u 1,...,u m ) = 0 Cu 1 )...Cu m ). 5.7) In fact, using the reflection algebra 3.13) we can show that τ,1) 1 u) Φ m u 1,...,u m ) = δu,u 1,...,u m ) Φ m u 1,...,u m ) F u,u j ) E j u j ) + sinhη) sinhu + η)) sinhu j +η) Q j u j ) Bu) Φ m 1u 1,...,û j,...,u m ), j=1 5.8) Φ m u 1,...,u m ) τ,1) 1 u) = Φ m u 1,...,u m ) δu,u 1,...u m ) + Φ m u 1,...,û j,...,u m ) Cu) E ju j ) F u,u j ) Q j u j ) sinhu j +η) sinhu+η))sinhη), j=1 5.9) δu,u 1,...,u m ) = sinhu+η))sinhu ξ+ ) δ 1 u) Qu η) sinhu+η) Qu) sinhu+η +ξ + )δ u) Qu+η) Qu) E j u) = sinhu)sinhu ξ + )δ 1 u)q j u η), 5.10) +sinhu+η)sinhu+η +ξ + )δ u)q j u+η) 5.11) Fu,v) = 1 sinhu v)sinhu+v +η). 5.1) In the above formulae we have introduced the double-row Baxter Q-polynomial, Qu) = sinhu u i )sinhu+u i +η), 5.13) i=1 the indexed double-row Baxter Q-polynomial, Q j u) = sinhu u i )sinhu+u i +η). 5.14) i j 17

19 The set of polynomial equations E j u j ) = 0 for j = 1,...,m, 5.15) are known as the Bethe ansatz equations for the double-row transfer matrix. Again, we will refer to E j u) as the Bethe ansatz polynomials for the double-row transfer matrix. Finally, let us suppose that {u 1,...,u m } satisfy the Bethe equations 5.15) i.e., they are on-shell rapidities) that there is no restriction on {v 1,...,v m } i.e., they are offshell rapidities). Then, the scalar product Φ m u 1,...,u m ) Φ m v 1,...,v m ) is given by the compact formula [9], Φ m u 1,...,u m ) Φ m v 1,...,v m ) = i=1 δ u i ) sinhv i +η))sinhu i ξ + ) det m u i δv j,u 1,...,u m ) det m Fv i,u j )) ) j<i sinhu i +u j +η) sinhu i +u j ), 5.16) which is the Slavnov formula for the double-row transfer matrix. Using δv j,u 1,...,u m ) = sinhη)sinhu i +η) sinhv j +η)) u i sinhv j +η) Fu i,v j )Fv j,u i ) E iv j ) Q i v j ) 5.17) we can rewrite the Slavnov formula as Φ m u 1,...,u m ) Φ m v 1,...,v m ) = i=1 sinhu i +η)δ u i ) sinhv i +η)sinhu i ξ + ) j<i ) det m sinhη)fu i,v j )Fv j,u i ) E iv j ) det m Fv i,u j )) Q i v j ) sinhu i +u j +η) sinhu i +u j ), 5.18) which is given in terms of the double-row Bethe ansatz Baxter polynomials, similarly to the single-row case. 5. Bethe vectors from Tarasov s construction We now apply the Bethe ansatz to 3.17), see [1, 3]. Here, we use the 3-dimensional auxiliary space representation, i.e., we set U a 1,1) u) = A 1 u) B 1 u) B u) C 1 u) A u)+ sinhη) A sinhu+η)) 1u) B 3 u) C u) C 3 u) A 3 u)+ sinhη) sinhη) A sinhu+η))sinhu+η) 1u)+ sinhη) A sinhu) u) a, 18

20 each entry acts on the quantum space C 3 ) N. These operators satisfy exchange relations dictated by 3.14). The action of the monodromy operators 5.19) on the reference state 4.,4.3) is given by A j u) 0 = j u) 0, C j u) 0 = 0, 5.0) 5.19) for j = 1,,3 1 u) = cosh u+ η u) = cosh u+ η ) sinh 0 A j u) = 0 j u), 0 B j u) = 0, 5.1) u η ) +ξ sinh u+ η ) +ξ sinhu+η)sinhu+η)) N, ) sinhu)sinh u+ 3η ξ ) sinh u η +ξ ) sinhu)sinhu+η)) N, sinhu+η)) 3 u) = sinhu η)sinh u+ η ξ ) sinh u+ 3η ξ ) sinh ) u+ η sinhu)sinhu η)) N. The transfer matrix 3.17) is given by 5.) τ 1,1) u) = sinhu+3η)sinh u η ξ+) sinh u+ η ξ+) sinh ) u+ η A 1 u) + cosh ) u+ η sinhu+η))sinh u η ξ+) sinh u+ 3η +ξ+) A u) sinhu) cosh u+ η ) sinh u+ η ) +ξ+ sinh u+ 3η ) +ξ+ A 3 u). 5.3) The Bethe vector is constructed by means of the recursion relation Ψ m u 1,...,u m ) = B 1 u 1 ) Ψ m 1 u,...,u m ) B u 1 ) i= Γ m) i u 1,...,u m ) = sinhη) Q 1,i u i η) Q 1,i u i ) Γ m) i u 1,...,u m ) Ψ m u,...,û i,...,u m ), 5.4) j=,j<i sinhu i ) sinhu 1 u i η)sinhu i +η)) 1u i ) 19 Ωu i,u j ) 1 sinhu 1 +u i +η) ) Q 1,i u i +η) Q 1,i u i η) u i ),

21 5.5) we have introduced a new Baxter double-row Q-polynomial Qu) = sinhu u i )sinhu+u i +η), 5.6) i=1 Q j u) = sinhu u i )sinhu+u i +η), 5.7) i j Q j,k u) = sinhu u i )sinhu+u i +η). 5.8) i j,k The dual Bethe vector is given by Ψ m u 1,...,u m ) = Ψ m 1 u,...,u m ) C 1 u 1 ) i= Γ m) i u 1,...,u m ) = sinhη)coshη) Q 1,i u i η) Q 1,i u i ) Γ m) i u 1,...,u m ) Ψ m u,...,û i,...,u m ) C u 1 ), 5.9) sinhu i ) sinhu 1 u i η)sinhu i +η)) 1u i ) Notice that the the initial conditions are assumed for 5.4) 5.9). j=,j<i Ωu i,u j ) 1 sinhu 1 +u i +η) ) Q 1,i u i +η) Q 1,i u i η) u i ). 5.30) Ψ 0 = 0, Ψ 0 = ) Using the reflection algebra 3.14) one can show that the vectors 5.4) 5.9) satisfy the off-shell equations τ 1,1) u) Ψ m u 1,...,u m ) = u,u 1,...,u m ) Ψ m u 1,...,u m ) + G 1 u,u j )Ēju j ) Q j u j η) Q j u j ) Q Ωu j,u p )B 1 u) Ψ m 1 u 1,...,û j,...,u m ) j u j η) + j=1 j=1 G u,u j )Ēju j ) Q j u j η) Q j u j ) Q j u j η) p<j Ωu j,u p )B 3 u) Ψ m 1 u 1,...,û j,...,u m ) p<j 0

22 + j<k H m) jk u,u 1,...,u m )B u) Ψ m u 1,...,û j,...,û k,...,u m ), 5.3) Ψ m u 1,...,u m ) τ 1,1) u) = Ψ m u 1,...,u m ) u,u 1,...,u m ) + Ψ m 1 u 1,...,û j,...,u m ) C 1 u) Ωu j,u p )Ēju j ) Q j u j η) Q j u j ) Q j u j η) G 1u,u j ) + + j=1 j=1 p<j Ψ m 1 u 1,...,û j,...,u m ) C 3 u) j<k p<j Ωu j,u p )Ēju j ) Q j u j η) Q j u j ) Q j u j η) G u,u j )cosh η) Ψ m u 1,...,û j,...,û k,...,u m ) C u)h m) jk u,u 1,...,u m )cosh η), 5.33) u,u 1,...,u m ) = sinhu+3η)sinh u η ξ+) sinh u+ η ξ+) sinh ) u+ η 1 u) Qu η) Qu) + cosh ) u+ η sinhu+η))sinh u η ξ+) sinh u+ 3η +ξ+) Qu η) Qu+η) u) sinhu) Qu) Qu η) cosh u+ η ) sinh u+ η ) +ξ+ sinh u+ 3η ) +ξ+ 3 u) Qu+η) Qu η), 5.34) Ē j u) = sinhu)sinh u+ η ) ξ+ 1 u) Q j u η) +sinhu+η))sinh u+ 3η ) +ξ+ u) Q j u+η), 5.35) H m) jk u,u 1,...,u m ) = Q j u j η) Q j,k u k η) Q j u j η) Q j,k u j ) Q j,k u k ) Q j,k u k η) Ωu j,u p ) p<j q<k,q j Ωu k,u q ) { β 1 u,u j,u k )Ēju j )Ēku k )+β u,u k,u j ) Q j,k u j +η)λ u j )Ēku k ) +β u,u j,u k ) Q j,k u k +η)λ u k )Ēju j ) }, 5.36) the functions G 1 u,u j ), G u,u j ), β 1 u,u j,u k ) β u,u j,u k ) are given in the appendix A. The set of polynomial equations Ē j u j ) = 0 for j = 1,...,m, 5.37) are the Bethe ansatz equations for the double-row transfer matrix from Tarasov s construction. We remark that the unwanted terms have been written explicitly in terms of the Bethe polynomials; in this way, it is clear that all unwanted terms disappear on-shell. 1

23 5.3 Relating the two open-chain Bethe ansätze As in the periodic case, the open-chain Bethe equations from fusion5.15) from Tarasov s construction 5.37) are equivalent. Indeed, since the factors in the Bethe polynomials E j u) 5.11) Ēju) 5.35) are related by sinhu)sinh u+ η ξ+) ) 1 u) sinhu+η))sinh u+ 3η +ξ+) u) = sinhu)sinhu ξ + )δ 1 u) sinhu+η)sinhu+η +ξ + )δ u) the solutions of the two sets of Bethe equations are related by η shifts. u u+ η, 5.38) We claim that the off-shell Tarasov-Bethe vectors 5.4), 5.9) are related to the off-shell fusion-bethe vectors 5.6), 5.7) by Ψ m u 1,...,u m ) = su 1,...,u m ) Φ m u 1 + η,...,u m + η ), 5.39) Ψ m u 1,...,u m ) = Φ m u 1 + η,...,u m + η ) su1,...,u m )cosh m η), 5.40) su p,...,u m ) = m p+1 ) sinhu i )δ 1 ui η sinh ) u i + η i=p j=k=p,j<k sinhu j +u k )sinhu j u k η) sinhu j +u k +η)sinhu j u k η). 5.41) We now prove 5.39) by induction; the proof of 5.40) is similar. The cases m = 1 m = can be proved by brute-force computation. We begin by using 5.4) to obtain Ψ m+1 u 1,...,u m+1 ) = B 1 u 1 ) Ψ m u,...,u m+1 ) m+1 B u 1 ) Γ m+1) i u 1,...,u m+1 ) Ψ m 1 u,...,û i,...,u m+1 ). 5.4) i= The induction hypothesis 5.39) implies Ψ m+1 u 1,...,u m+1 ) = su,...,u m+1 )B 1 u 1 ) Φ m ũ,...,ũ m+1 ) 5.43) m+1 B u 1 ) Γ m+1) i u 1,...,u m+1 ) su,...,û i,...,u m+1 ) Φ m 1 ũ,...,ˆũ i,...,ũ m+1 ), i= we have introduced su p,...,û l,...,u m ) 5.44)

24 = m p i=p,i l sinhu i )δ 1 ui η sinh ) u i + η sinhu) = )B sinh u+ η ) j=k=p,j<k,j l,k l sinhu j +u k )sinhu j u k η) sinhu j +u k +η)sinhu j u k η), we have again employed the notation 4.48). We now use the relations ) cosh u+ η B 1 u) = A u+ η ) B u η ) sinhη) + 4sinh )B u+ η u+ η ) D u η ) sinh u)+sinh η) ) + 4sinhu)sinh ) u+ η B u+ η ) A u η ) u+ η B u) = sinhu) sinh )B u+ η ) A u η ), 5.45) u+ η ) B u η ), 5.46) which follow from 3.15) the reflection algebra 3.13). In this way, we obtain { Ψ m+1 u 1,...,u m+1 ) = sinhu 1) sinh ) su u 1 + η,...,u m+1)bũ 1 )Au 1 η ) Φ mũ,...,ũ m+1 ) m+1 i= Γ m+1) i u 1,...,u m+1 ) su,...,û i,...,u m+1 )Bũ 1 )Bu 1 η ) Φ m 1ũ,...,ˆũ i,...,ũ m+1 ) Using the result Au) Φ m u 1,...,u m ) = δ 1 u) Qu η) Φ m u 1,...,u m ) Qu) [ sinhη) sinhu j ) + sinhu u j ) sinhu j +η) δ 1u j ) Q ju j η) Q j u j ) j=1 5.47) sinhη) sinhu+u j +η) δ u j ) Q ju j +η) ] Bu) Φ m 1 u 1,...,û j,...,u m ) 5.48) Q j u j ) that can be obtained using the reflection algebra 3.13), we find that Ψ m+1 u 1,...,u m+1 ) = sinhu 1) sinh ) u 1 + η su,...,u m+1 )δ 1 u 1 η ) m+1 k= sinhu 1 u k η) sinhu 1 +u k ) sinhu 1 u k η) sinhu 1 +u k +η) Φ m+1ũ 1,...,ũ m+1 ) + sinhu 1) sinh ) u 1 + η m+1 j= { 3 α j u 1,...,u m+1 ) }.

25 Γ m+1) j u 1,...,u m+1 ) su,...,û j,...,u m+1 ) } Bu 1 η ) Φ mũ 1,...,ˆũ j,...,ũ m+1 ), α j u 1,...,u m+1 ) = su,...,u m+1 ) sinhη) [ sinhũ j )δ 1 ũ j ) m+1 sinhu j u k η)sinhu j +u k +η) sinhu 1 ũ j η ) sinhũ j +η) sinhu j u k )sinhu j +u k +η) m+1 δ ũ j ) sinhu 1 +ũ j + η ) Finally, using the identities k=,k j k=,k j sinhu j u k +η)sinhu j +u k +3η) sinhu j u k )sinhu j +u k +η) su 1,...,u m+1 ) = sinhu 1) sinh ) u 1 + η su,...,u m+1 )δ 1 u 1 η ) ] 5.49). 5.50) m+1 k= sinhu 1 u k η) sinhu 1 +u k ) sinhu 1 u k η) sinhu 1 +u k +η) α j u 1,...,u m+1 ) = Γ m+1) j u 1,...,u m+1 ) su,...,û j,...,u m+1 ), 5.51) we conclude that Ψ m+1 u 1,...,u m+1 ) = su 1,...,u m+1 ) Φ m+1 ũ 1,...,ũ m+1 ), 5.5) which completes the inductive proof of 5.39). 5.4 Scalar products for Tarasov-Bethe vectors Combining the scalar product for the fusion-bethe vectors 5.16) with the relations between Tarasov-Bethe vectors fusion-bethe vectors 5.39), 5.40), it is now straightforward to compute the scalar product between the off-shell state Ψ m u 1,...,u m ) 5.9) the onshell state Ψ m v 1,...,v m ) 5.4) Ψ m u 1,...,u m ) Ψ m v 1,...,v m ) = cosh m η) su 1,...,u m ) sv 1,...,v m ) ) δ ui + η sinh m sinhu )) i +u j +3η) v i=1 i + 3η sinh ui + η ξ+) sinhu j<i i +u j +η) det m u i δ v j + η,u 1 + η,...,u ) ) m + η ) det m Fvi,u j ) ). 5.53) 4

26 We can rewrite 5.53) like 5.18) Ψ m u 1,...,u m ) Ψ m v 1,...,v m ) sinhu j +u k )sinhu j u k η) sinhv j +v k )sinhv j v k η) = sinhu j +u k +η)sinhu j u k η) sinhv j +v k +η)sinhv j v k η) j<k i=1 sinhu i +η) u i ) sinhv i +η)sinh m u i + η ξ+) j<i sinhu i +u j +3η) sinhu i +u j +η) det m sinhη) Fu i,v j ) Fv ) j,u i ) Ēiv j ) Q i v j ) det m Fvi,u j ) ), 5.54) Fu,v) = 1 sinhu v)sinhu+v +η), 5.55) i.e., in terms of the Bethe ansatz 5.35) the Baxter polynomial 5.6). 6 Conclusion We have considered the Bethe vectors of the ZF model from two different perspectives, namely, the fusion technique Tarasov s construction. For the single-row transfer matrix, the relations between the two sets of Bethe vectors are given by 4.40,4.41). For the doublerow transfer matrix, the associated formulas are given by 5.39,5.40). By means of these simple relations, we have computed the Slavnov scalar products for the Bethe vectors within Tarasov s construction, see 4.58) for the single-row case 5.54) for the double-row case. The Slavnov formulas have been written in terms of Bethe ansatz Baxter polynomials, which seems to have more universal structure. WehopethatourresultscanbeusefulinthestudyoftheSlavnovscalarproductsforother 19-vertex models, since they have similar commutation relations in the 3-dimensional auxiliary space representation. One first step would be to derive the formulas 4.58) 5.54) for the ZF model without resorting to the fusion-bethe vectors. A particularly important 19-vertex model is the Izergin-Korepin IK) model. We hope that there exist scalar product formulas for IK analogous to 4.58) or 5.54), at least for the quantum-group-invariant [31] or root of unity cases [3], even though IK unlike ZF) cannot be obtained from some more elementary model by fusion. Acknowledgments RP thanks Alexer Garbali for insightful discussions about the scalar product of 19-vertex models. This work was supported by the São Paulo Research Foundation FAPESP) 5

27 the University of Miami under the SPRINT grant #016/ Additional support was provided by the Brazilian Research Council CNPq) grant # 31065/013-0 FAPESP #011/1879-1ALS), by a Cooper fellowship a Fulbright Specialist grant RN), by FAPESP/CAPES grant # 017/0987-8RP). RN acknowledges the hospitality at UFSCar, ICTP-SAIFR AIMS-SA. RP thanks the University of Miami for its warm hospitality. A Auxiliary formulas The auxiliary functions entering the off-shell equations 5.3) 5.33) are given by cosh u+ η G 1 u,u j ) = ) sinhη)sinhu+η)) 4sinhu u j )sinhu η u j )sinhη +u j ))sinhu+η +u j )sinhu+η+u j ) sinh u η ) u j ξ + +sinh u+ η ) +ξ+ sinh 3u+ 3η ) ξ+ +sinh u+ 5η ) +ξ+ +sinh u+ 7η ) +u j ξ + sinh u+ 9η )) +ξ+, A.1) G u,u j ) = cosh ) u+ η sinhη)sinhu+η))sinh u+ 3η +ξ+), A.) sinhu η u j )sinhη +u j ))sinhu+η +u j ) β 1 u,u j,u k ) = sinh η)coshη) cosh u η ) cosh u+ 3η ) +cosh 3u+ 9η ) + cosh 5u+ 11η )) W1 u,u j,u k ) A.3) W u,u j,u k ) W 1 u,u j,u k ) = cosh u+u j u k ξ + 3η ) +cosh 3u j u k ξ + +η ) cosh u u j +u k ξ + 3η ) +cosh u j +u k ξ + +η ) +cosh 4u+u j +u k ξ + +η ) cosh u j +u k ξ + +3η ) cosh u+3u j +u k ξ + +3η ) +cosh 3u k u j ξ + +η ) cosh u+u j +3u k ξ + +3η ) +cosh 3u j +3u k ξ + +5η ) cosh u u j 3u k +ξ + η ) cosh u 3u j u k +ξ + η ) +cosh u j u k +ξ + +η ) +cosh u j u k +ξ + +3η ) +cosh 4u u j u k +ξ + +3η ) cosh u+u j u k +ξ + +5η ) cosh 3u j u k +ξ + +3η ) +cosh 3u j u k +ξ + +5η ) cosh u u j +u k +ξ + +5η ) +cosh u j +u k +ξ + +η ) 6

28 cosh u j +u k +ξ + +3η ) +cosh u j +u k +ξ + +5η ) cosh u j +3u k +ξ + +3η ) +cosh u j +3u k +ξ + +5η ) +coshu 4η u j u k )+coshu u j u k η) coshu+η) u j u k ) cosh4u+u j u k +η) +coshu+η)+3u j u k ) cosh4u u j +u k +η) +coshu+u j +u k +4η)+coshu+u j +u k +6η) +coshu+η)+u j +u k ) cosh3u j +u k +6η) +coshu+η) u j +3u k ) coshu j +3u k +6η)+coshu+u j 3u k ) +coshu u j u k ) 3coshu j u k ) cosh3u j u k )) +coshu 3u j +u k ) coshu+u j +u k ), A.4) W u,u j,u k ) = 3sinhu u j )sinhu u k )sinhu u j η)sinhu j +η)) sinhu+u j +η)sinhu+u j +η)sinhu u k η)sinhu k +η)) sinhu+u k +η)sinhu+u k +η)sinhu j u k +η)sinhu k u j +η)sinhu j +u k +η) sinhu j +u k +η)sinh u j ξ + + η ) sinh u k ξ + + η ), A.5) β u,u j,u k ) = cosh u+ η ) sinh η)sinhη)sinhu+η))sinhu+3η) sinhu u j )sinhu u j η)sinhu+u j +η)sinhu+u j +η) sinhu k +η)) sinhu j +η))sinhu j u k )sinhu j u k +η)sinhu j +u k +η)sinhu j +u k +η) sinh u j η ξ+) sinh u j + 5η +ξ+) sinh u k + η. A.6) ξ+) References [1] A. B. Zamolodchikov V. A. Fateev, Model Factorized S Matrix And An Integrable Heisenberg Chain With Spin 1., Sov. J. Nucl. Phys ) [Yad. Fiz.3, )]. [] L. D. Faddeev, E. K. Sklyanin, L. A. Takhtajan, The Quantum Inverse Problem Method. 1, Theor. Math. Phys ) [Teor. Mat. Fiz.40, )]. [3] L. D. Faddeev, How algebraic Bethe ansatz works for integrable models, in Symétries Quantiques Les Houches Summer School Proceedings vol 64), A. Connes, K. Gawedzki, J. Zinn-Justin, eds., pp North Holl, arxiv:hep-th/ [hep-th]. 7

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