Quantum Group U_q(sl(2)) Symmetry and Explicit Evaluation of the One-Point Functions of the Integrable Spin-1 XXZ Chain

We show some symmetry relations among the correlation functions of the integrable higher-spin XXX and XXZ spin chains, where we explicitly evaluate the multiple integrals representing the one-point functions in the spin-1 case. We review the multiple-integral representations of correlation functions for the integrable higher-spin XXZ chains derived in a region of the massless regime including the anti-ferromagnetic point. Here we make use of the gauge transformations between the symmetric and asymmetric R-matrices, which correspond to the principal and homogeneous gradings, respectively, and we send the inhomogeneous parameters to the set of complete 2s-strings. We also give a numerical support for the analytical expression of the one-point functions in the spin-1 case.


Introduction
The correlation functions of the spin-1/2 XXZ spin chain has attracted much interest during the last decades in mathematical physics, and several nontrivial results such as their multipleintegral representations have been obtained explicitly [1,2,3]. The Hamiltonian of the XXZ spin chain under the periodic boundary conditions (P.B.C.) is given by Here σ a j (a = X, Y, Z) are the Pauli matrices defined on the jth site and ∆ denotes the anisotropy of the exchange coupling. The P.B.C. are given by σ a L+1 = σ a 1 for a = X, Y, Z. The XXZ Hamiltonian shows the quantum phase transition: the ground state of the XXZ spin chain depends on ∆. For |∆| > 1 the low-lying excitation spectrum at the ground state has a gap, while for |∆| ≤ 1 it has no gap. Here we remark that the quantum phase transition that we have discussed is associated with the behavior of the XXZ spin chain in the thermodynamic limit: L → ∞. In terms of the q parameter of the quantum group U q (sl 2 ), we express ∆ as follows It is often convenient to define parameters η and ζ by q = exp η with η = iζ. Here we have ∆ = cosh η = cos ζ. In the massive regime ∆ > 1, we set η > 0. In the massless regime −1 < ∆ ≤ 1, we set η = iζ where ζ satisfies 0 ≤ ζ < π. Here, the XXX limit is given by η → +0 or ζ → +0. Here we remark that the XXZ Hamiltonian can be derived from the R-matrix of the affine quantum group with q parameter, U q ( sl 2 ): we derive the R-matrix by solving the intertwining relations, construct the XXZ transfer matrix from the product of the R matrices, and then we derive the XXZ Hamiltonian by taking the logarithmic derivative of the XXZ transfer matrix. Thus, the q parameter of the affine quantum group is related to the ground state of the XXZ spin chain through ∆. The multiple-integral representations of the XXZ correlation functions were derived for the first time by making use of the q-vertex operators through the affine quantum-group symmetry in the massive regime for the infinite lattice at zero temperature [4,2]. In the massless regime they were derived by solving the q-KZ equations [5,6]. Making use of the algebraic Betheansatz techniques [7,1,8,9,10], the multiple-integral representations were derived for the spin-1/2 XXZ correlation functions under a non-zero magnetic field [11]. Here, they are derived through the thermodynamic limit after calculating the scalar product for a finite chain. The multiple-integral representations were extended into those at finite temperatures [12], and even for a large finite chain [13]. Interestingly, they are factorized in terms of single integrals [14]. We should remark that the multiple-integral representations of the dynamical correlation functions were also obtained under finite-temperatures [15]. Furthermore, the asymptotic expansion of a correlation function of the XXZ model has been systematically discussed [16]. Thus, the exact study of the XXZ correlation functions should play an important role not only in the mathematical physics of integrable models but also in many areas of theoretical physics.
Recently, the form factors of the integrable higher-spin XXX spin chains and the multipleintegral representations of correlation functions for the integrable higher-spin XXX and XXZ chains have been derived by the algebraic Bethe-ansatz method [17,18,19,20,21] (see also [22]). The spin-1 XXZ Hamiltonian under the P.B.C. is given by the following [23]: − (q + q −1 − 2)[(S x j S x j+1 + S y j S y j+1 )S z j S z j+1 + S z j S z j+1 (S x j S x j+1 + S y j S y j+1 ) .
(1.1) Furthermore, the multiple-integral representations have been obtained for the correlation functions at finite temperature of the integrable spin-1 XXX chain [24]. The solvable higher-spin generalizations of the XXX and XXZ spin chains have been derived by the fusion method in several references [25,26,27,28,29,30,31,32]. In the region: 0 ≤ ζ < π/2s, the spin-s groundstate should be given by a set of string solutions [33,34]. Furthermore, the critical behavior should be given by the SU(2) WZWN model of level k = 2s with central charge c = 3s/(s + 1) [35,36,37,38,39,40,41,42,43,44,31,45,46,47]. For the integrable higher-spin XXZ spin chain correlation functions have been discussed in the massive regime by the method of q-vertex operators [48,49,50,23,51,52]. The purpose of this paper is to show some symmetry relations among the correlation functions of the integrable spin-s XXZ spin chain by explicitly calculating the multiple-integral representations for the spin-1 one-point functions. Associated with the quantum group U q (sl(2)) symmetry, there are several relations among the expectation values of products of the matrix elements of the monodromy matrices. For the spin-1 case, we confirm some of them by evaluating the multiple integrals analytically and explicitly. Here we should remark that the derivation of the multiple-integral representations for the spin-s XXZ correlation functions given in the previous papers [19,20,21] was not completely correct: the application of the formulas of the quantum inverse-scattering problem was not valid [53,54]. We thus review the revised derivation [53,54] in the paper. The spin-s correlation function of an arbitrary entry is now expressed in terms of a sum of multiple integrals, not as a single multiple integral. Furthermore, we show numerical results which confirm the analytical expressions of the spin-1 one-point functions.
Let us express by E 00 , E 11 and E 22 , the expectation values of S Z 1 = 1, S Z 1 = 0 and S Z 1 = −1, respectively, where S Z 1 denotes the Z-component of the spin operator defined on the first site. Then, we have the following: ζ − sin ζ cos ζ 2ζ sin 2 ζ , E 11 = cos ζ(sin ζ − ζ cos ζ) ζ sin 2 ζ .
We shall show the derivation of E 00 , E 11 and E 22 , in detail. Here we remark that the expressions of E 22 , the emptiness formation probability, and E 11 have been reported in [20] without an explicit derivation. In fact, although the derivation was not completely correct, the expressions of the spin-1 one-point functions are correct [53,54]. Here, the quantum group symmetry as well as the spin inversion symmetry play an important role, as we shall show explicitly in the present paper.
It is nontrivial to evaluate the multiple integral representations of the XXX and XXZ models analytically or even numerically. Let us now return to the spin-1/2 case. Boos and Korepin have analytically evaluated the emptiness formation probability P (n) of the XXX spin chain for up to n = 4 successive lattice sites [55]. Performing explicit evaluation of the multiple integrals, they successfully reproduced Takahashi's result that was obtained through the one-dimensional Hubbard model [56]. The method was applied to all the density matrix elements for up to n = 4 successive lattice sites in the XXX chain [57] and also in the XXZ chain [58,59,60]. Furthermore, the algebraic method to obtain the correlation functions of the XXX chain based on the qKZ equation has been developed [61] and the two-point functions up to n = 8 have been obtained so far [62,63,64,65]. At the special anisotropy ∆ = 1/2, some further results have been shown for the correlation functions through explicit evaluation [66,67,68,69].
The paper consists of the following. In Section 2 we review the Hermitian elementary matrices [20], and give the basis vectors and their conjugate vectors in the spin-1 case as an illustrative example. We also show a formula for expressing higher-spin local operators in terms of spin-1/2 local operators in the spin-1 case, which plays a central role in the revised method [53,54]. In Section 3 we summarize the notation of the fusion transfer matrices and the quantum inverse scattering problem for the spin-s operators. For an illustration, in Section 4, we show some relations among the expectation values of the Hermitian elementary matrices in the spin-1 XXX case and then in the spin-1 XXZ case. In particular, we show the spin inversion symmetry. We also show the transformation which maps the basis vectors of the spin-1 representation V (2) constructed in the tensor product of the spin-1/2 representations V (1) ⊗ V (1) to the basis of the three-dimensional vector space C 3 . The former basis is related to the fusion method, while the spin-1 XXZ Hamiltonian (1.1) is formulated in terms of the latter basis. In Section 5, we review the revised multiple-integral representations of correlation functions for the integrable spin-s XXZ spin chain [53,54]. Here we remark that necessary corrections to the previous papers [19] and [20] are listed in references [20] and [21] of the paper [54], respectively. In Section 6, we explicitly calculate the multiple integrals of the one-point functions for the spin-1 XXZ spin chain for a region in the massless regime. We show some details of the calculation such as shifting the integral paths. In Section 7 we show that the numerical estimates of the spin-1 one-point functions obtained through exact diagonalization of the spin-1 XXZ Hamiltonian (1.1) are consistent with the analytical expressions of the spin-1 one-point functions. Thus, we shall conclude that the analytical result of the spin-1 one-point functions should be valid.
In the massive regime where q = exp η with real η, conjugate vectors ℓ, n|| are also Hermitian conjugate to vectors ||ℓ, n .

Affine quantum group
In order to define the R-matrix in terms of algebraic relations we now introduce the affine quantum group U q ( sl 2 ). It is an infinite-dimensional algebra generalizing the quantum group U q (sl 2 ).
The algebra U q ( sl 2 ) is an associative algebra over C generated by X ± i , K ± i for i = 0, 1 with the following defining relations: The algebra U q ( sl 2 ) is also a Hopf algebra over C with comultiplication: and antipode: and counit: ε(X ± i ) = 0 and ε(K i ) = 1 for i = 0, 1.
The quantum group U q (sl 2 ) gives a Hopf subalgebra of U q ( sl 2 ) generated by X ± i , K i with either i = 0 or i = 1. Thus, the affine quantum group generalizes the quantum group U q (sl 2 ).

Evaluation representations with principal and homogeneous gradings
We shall introduce two types of representations of U q ( sl 2 ): evaluation representations associated with principal grading and that with homogeneous grading. The former is related to the symmetric R-matrix which leads to the most concise expression of the integrable quantum spin Hamiltonian, while the latter is related to the asymmetric R-matrix R + (u) which we shall define in Section 3.2 and suitable for an explicit construction of representations of the quantum group.
Here and hereafter we denote by X ± and K the generators of U q (sl 2 ).
Let us now introduce a representation of U q ( sl 2 ) associated with homogeneous grading [2]. With a nonzero complex number λ we define a homomorphism of algebras ϕ (p) Thus, from a given finite-dimensional representation (π (ℓ) , V (ℓ) ) of the quantum group U q (sl 2 ), we derive a representation of the affine quantum group U q ( sl 2 ) by π (ℓ) (ϕ (p) λ (a)) for a ∈ U q ( sl 2 ), where ϕ (p) λ (·) is given by (2.1). We call it an evaluation representation of the affine quantum group; more specifically, the spin-ℓ/2 evaluation representation with evaluation parameter λ associated with principal grading. We denote it by (π Similarly in the case of principal grading, we now introduce a representation associated with homogeneous grading [2]. With a nonzero complex number λ we define a homomorphism of algebras ϕ (+) λ : U q ( sl 2 ) → U q (sl 2 ) by the following: From a given finite-dimensional representation (π (ℓ) , V (ℓ) ) of the quantum group U q (sl 2 ) we derive a representation of the affine quantum group U q ( sl 2 ) by π (ℓ) (ϕ λ (·) is given by (2.2). We call it the spin-ℓ/2 evaluation representation with evaluation parameter λ associated with homogeneous grading. We denote it by (π (ℓ+) λ , V (ℓ) (λ)) or V (ℓ+) (λ).

Defining relations of the R-matrix
Let us now define the R-matrix for any given pair of finite-dimensional representations of the affine quantum group U q ( sl 2 ). Let (π 1 , V 1 ) and (π 2 , V 2 ) be finite-dimensional representations of U q ( sl 2 ). We define the R-matrix R 12 for the tensor product V 1 ⊗ V 2 by the following relations: Here τ denotes the permutation operator: For an illustration, let us write down relations (2.3) of the R-matrices associated with evaluation representations. We call them intertwining relations. Associated with principal grading we have for a = X ± 0 , X ± 1 and K 1 , respectively, the following relations: Associated with homogeneous grading we have (2.5) Here λ 1 and λ 2 correspond to the "string centers" of the sets of the evaluation parameters associated with the evaluation representations π 1 and π 2 . We have λ 1 = ξ 1 − (ℓ − 1)η/2, if π 1 is given by the spin-ℓ/2 evaluation representation derived from the tensor product (V (1) ) ⊗ℓ with complete ℓ-string w (ℓ) j for j = 1, 2, . . . , ℓ. Here we shall define complete strings in Section 3.6. We can show that the solution of intertwining relations (2.3) is unique. We may therefore define the R-matrix in terms of relations (2.3).
We remark that relations (2.4) for the evaluation representation associated with principal grading are mapped into those of (2.5) associated with homogeneous grading through a similarity transformation, which we call the gauge transformation. We shall formulate it in Section 3.4.
Proposition 1 ( [53,54]). The spin-ℓ/2 symmetric elementary matrices associated with principal grading are decomposed into a sum of products of the spin-1/2 elementary matrices as follows Here the sum is taken over all sets of ε β s such that the number of integers β satisfying ε β = 1 for 1 ≤ β ≤ ℓ is equal to j. We take a set of ε ′ α s such that the number of integers α satisfying ε ′ α = 1 for 1 ≤ α ≤ ℓ is equal to i. The expression (2.10) is independent of the order of ε ′ α s with respect to α.
The formula (2.10) plays a central role in the revised derivation of the spin-ℓ/2 form factors and the spin-ℓ/2 XXZ correlation functions [53,54]. We shall derive (2.10) in Appendix A. We recall that the derivation of the multiple-integral representations of the integrable spin-s XXZ spin chain given in the previous papers [19,20,21] was not completely correct [53,54]. In fact, the transfer matrix becomes non-regular at λ = ξ k [54], and hence the straightforward application of the QISP formula was not valid.

Example: spin-1 case
We shall show reduction formula (2.10) for the spin-1 case.

Fusion transfer matrices and higher-spin expectation values
We construct the monodromy matrices of the integrable higher-spin XXZ spin chains through the fusion method. We then evaluate the form factor of a given product of the higher-spin operators by reducing them into a sum of products of the spin-1/2 operators and calculate their scalar products of the spin-1/2 operators through Slavnov's formula. When we reduce the higher-spin operators, we make use of the fusion construction where all the elements are constructed from a sum of products of the spin-1/2 operators multiplied by the projection operators.

Tensor product notation
Let s be an integer or a half-integer. We shall mainly consider the tensor product V have spectral parameters λ j for j = 1, 2, . . . , N s . We denote by E a, b a unit matrix that has only one nonzero element equal to 1 at entry (a, b) where a, b = 0, 1, . . . , 2s. For a given set of matrix elements A a, α b, β for a, b = 0, 1, . . . , 2s and α, β = 0, 1, . . . , 2s, we define operators A j,k for 1 ≤ j < k ≤ N s by In the tensor product space, (V (2s) ) ⊗Ns , we define E m,n(2sw) i for i = 1, 2, . . . , N s and w = +, p by The elementary matrices E n,n(2sw) for n = 0, 1, . . . , 2s and w = +, p, are Hermitian in the massless regime.
We remark that the R + (λ 1 − λ 2 ) is compatible with the homogeneous grading of U q ( sl 2 ). In fact, it is straightforward to see that the asymmetric R-matrix satisfies the intertwining relations associated with homogeneous grading (2.5) for the tensor product of the spin-1/2 representations of U q (sl 2 ), V We denote by R (p) (u) or simply by R(u) the symmetric R-matrix where c ± (u) of (3.2) are replaced by c(u) = sinh η/ sinh(u + η) [19]. The symmetric R-matrix is compatible with evaluation representation associated with principal grading for the affine quantum group U q ( sl 2 ) [19]. Hereafter we express R + and R (p) by R (1w) with w = + and p, respectively.

Monodromy matrix of type (1, 1 ⊗L )
We now consider the (L + 1)th tensor product of the spin-1/2 representations, which consists of the tensor product of auxiliary space V (1) 0 and the Lth tensor product of quantum spaces V . We call it the tensor product of type (1, 1 ⊗L ) and denote it by the following symbol: Applying definition (3.1) for matrix elements R(u) ab cd of a given R-matrix such as R (1w) with w = + and p, we define R-matrices R jk (λ j , λ k ) = R jk (λ j − λ k ) for integers j and k with 0 ≤ j < k ≤ L. For integers j, k and ℓ with 0 ≤ j < k < ℓ ≤ L, the R-matrices satisfy the Yang-Baxter equations We define the monodromy matrix of type (1, 1 ⊗L ) associated with homogeneous grading by Here we have set λ j = w j for j = 1, 2, . . . , L, where w j are arbitrary parameters. We call them inhomogeneous parameters. We have expressed the symbol of type (1, 1 ⊗L ) as (1, 1) in superscript. The symbol (1, 1+) denotes that it is consistent with homogeneous grading. We express operator-valued matrix elements of the monodromy matrix as follows .
Here {w j } L denotes the set of L parameters, w 1 , w 2 , . . . , w L . We also denote the matrix elements of the monodromy matrix by [T
We denote by Φ (2s) Here ξ b denote the inhomogeneous parameters of the spin-s XXZ spin chains, which will be given in equation (3.4) of Section 3.6. We note that Λ b corresponds to the string center of the 2s-string,
Applying projection operator P (ℓ) a 1 a 2 ···a ℓ to the vectors in the tensor product V (1) Ns , which gives the quantum space for the higher-spin transfer matrices. We construct the bth component V of the quantum space from the 2sth tensor product of the spin-1/2 representations: V Here we recall L = 2sN s .
3.6 Higher-spin monodromy matrix of type (ℓ, (2s) ⊗Ns ) Let us now introduce complete strings. For a positive integer ℓ we call the following set of rapidities λ j a complete ℓ-string: Here we call parameter Λ the string center.
Here we recall that V Here we remark that it is associated with homogeneous grading.
Let us now construct the higher-spin monodromy matrices associated with principal grading. From the higher-spin monodromy matrices associated with homogeneous grading we derive them through the inverse of the gauge transformation as follows [54] Here χ (ℓ,2s) a 1 ···a ℓ ,12...Ns denote the following: where Λ 0 denotes the string center, Λ 0 = λ a 1 − (ℓ − 1)η/2. For an illustration, let us consider the case of ℓ = 1. For type (1, (2s) ⊗N s ) the monodromy matrix associated with homogeneous grading and that with principal grading are related to each other as follows In terms of the operator-valued matrix elements we have We shall now introduce the spin-1/2 monodromy matrices with special inhomogeneous parameters. Let us introduce a set of 2s-strings with small deviations from the set of complete 2s-strings Here ǫ is a infinitesimally small generic number and r are generic parameters. We call the set of rapidities w (2s;ǫ) 2s(b−1)+β for β = 1, 2, . . . , 2s "almost complete 2s-strings". We denote by T (1,2s+;ǫ) (λ) the spin-1/2 monodromy matrix T (1,1+) with inhomogeneous parameters w j being given by the set of almost complete 2s-strings: We express the elements of T (1,2s+;ǫ) (λ) as follows .

Here we recall that
. We also remark the following:

Series of commuting higher-spin transfer matrices
Suppose that |ℓ, m for m = 0, 1, . . . , ℓ, are the orthonormal basis vectors of V (ℓ) , and their dual vectors are given by ℓ, m| for m = 0, 1, . . . , ℓ. We define the trace of operator A over the space V (ℓ) by We define the massless transfer matrix of type (ℓ, (2s) ⊗Ns ) by It follows from the Yang-Baxter equations that the higher-spin transfer matrices commute in the tensor product space V Ns , which is derived by applying projection operator L . For instance, for the massless transfer matrices, making use of (2.8) and (2.9) we show Consequently, for the massless transfer matrices, the eigenvectors of t (1,2s+) 12···Ns (λ) constructed by applying B (2s+) (λ) to the vacuum |0 also diagonalize the higher-spin transfer matrices, in particular, the spin-s massless XXZ transfer matrix, t (2s,2s+) 12···Ns (λ). Thus, we construct the ground state of the higher-spin XXZ Hamiltonian in terms of operators B (2s+) (λ), which are the (0, 1)-element of the monodromy matrix T (1,2s+) .
In the massless regime, we define the Bethe vectors | {λ α } 1···L . The Bethe vector (3.5) gives an eigenvector of the massless transfer matrix for w = + and w = p with the following eigenvalue: if rapidities {λ j } M satisfy the Bethe ansatz equations Let us denote by |{λ α (ǫ)} (2sw;ǫ) M the Bethe vector of M Bethe roots {λ j (ǫ)} M for w = +, p: where rapidities {λ j (ǫ)} M satisfy the Bethe ansatz equations with inhomogeneous parameters w (2s;ǫ) j as follows It gives an eigenvector of the transfer matrix with the following eigenvalue: Let us assume that in the limit of ǫ going to 0, the set of Bethe roots {λ j (ǫ)} M approaches {λ j } M . Assuming the continuity of the limiting procedure, we have Thus, the expectation value with respect to the Bethe state of {λ j } M is given by the limit of that of {λ j (ǫ)} M sending ǫ to zero. For the B operators associated with principal grading, we have Let us introduce symbols for the ground state of the integrable spin-s XXZ spin chain. We denote it by |ψ (2sp) g associated with principal grading. It is given by multiplying the projection operator to such a product of the spin-1/2 B operators with inhomogeneous parameters being given by the set of complete 2s-strings that acts on the vacuum: We denote by |ψ (2sp;0) g the product of the spin-1/2 B operators with inhomogeneous parameters given by complete 2s-strings w (2s) j which acts on the vacuum state:

Commutation relations with projection operators
Let us discuss an application of the fusion construction of projection operators (3.3). Hereafter we assume that rapidity λ does not take such discrete values at which the transfer matrix becomes singular or non-regular, such as w For instance we have P 12···L thanks to the fusion construction of projection operators (3.3) [19]. We derive (3.7) making use of (2.8).

Quantum inverse scattering problem (QISP) for the spin-s operators
We can express any given spin-s local operator in terms of the spin-1/2 global operators such as A, B, C and D; i.e. we have the QISP formulas for the spin-s local operators [54]. For an illustration, we show the case of b = 1, i.e., we express one of the spin-s elementary matrices in terms of the spin-1/2 global operators.
Lemma 2 ( [53,54]). For a pair of integers i and j satisfying 1 ≤ i, j ≤ ℓ, the spin-ℓ/2 elementary matrix associated with principal grading is decomposed into a sum of products of the matrix elements of the spin-1/2 monodromy matrix as follows Here the sum is taken over all sets of ε β such that the number of integers β satisfying ε β = 1 and 1 ≤ β ≤ ℓ is given by j. We take a set of ε ′ α such that the number of integers α satisfying ε ′ α = 1 and 1 ≤ α ≤ ℓ is given by i. We have expressed the element of (α, β) in the monodromy matrix T (1,ℓp;ǫ) (λ) by T (1,ℓp;ǫ) α,β (λ) for α, β = 0, 1.
For an illustration, let us consider the spin-1/2 formula [9,11] (see also [10,73]): Here we recall that the spin-1/2 transfer matrix t Here we note that we have R 0n (0) = Π 0,n from the normalization condition of the R-matrices, where Π 0,n denotes the permutation operator acting on the 0th and nth sites (see also Section 3.5). Thus, we have (3.10) We note that the QISP formulas (3.8) hold if the inhomogeneous parameters are generic. If we send them to a set of complete 2s-strings such as w (2s) j , then the transfer matrix becomes non-regular or singular, and relations such as (3.10) do not hold. Instead of complete 2s-strings, we therefore put "almost complete 2s-strings", w (3.11) In order to evaluate (3.11) we make use of the following formulas.
Proposition 2 ( [53,54]). Let us take a pair of integers i 1 and j 1 satisfying 1 ≤ i 1 , j 1 ≤ ℓ. For arbitrary parameters {µ α } N and (3.12) Here we take the sum over all sets of ε β such that the number of integers β with ε β = 1 for 1 ≤ β ≤ ℓ is given by j 1 . We take a set of ε ′ α such that the number of integers α satisfying ε ′ α = 1 for 1 ≤ α ≤ ℓ is given by i 1 . Each summand is symmetric with respect to exchange of ε ′ α ; i.e., the following expression is independent of any permutation π ∈ S ℓ : (3.13) Here we remark that S n denotes the symmetric group of n elements. We evaluate the expectation value of a given spin-s local operator for a Bethe-ansatz eigen- M , as follows. We first express the spin-s local operators in terms of the spin-1/2 local operators via formula (2.10). Through Proposition 2 the expectation value of the spin-s local operators is reduced into those of the spin-1/2 local operators. We now assume that the Bethe roots {λ α (ǫ)} M are continuous with respect to small parameter ǫ. It follows from the assumption that each entry of the Bethe eigenstate |{λ k (ǫ)} (2s;ǫ) M is continuous with respect to ǫ. Then, we apply the spin-1/2 QISP formula with generic inhomogeneous parameters w (2s;ǫ) j such as formula (3.9). We next calculate the scalar product for the Bethe state |{λ k (ǫ)} (2s;ǫ) M . It has the same inhomogeneous parameters w (2s;ǫ) j as the global operators appearing in the QISP formula, so that we can make use of Slavnov's formula of scalar products for the spin-1/2 case. Calculating explicitly the determinant of the scalar product with Slavnov's formula, we can show that the expression of the scalar product is continuous with respect to ǫ at ǫ = 0. Thus, sending ǫ to 0, we obtain the expectation value of the spin-s local operator (3.11).
Here we take the sum over all sets of ε β s such that the number of integers β satisfying ε β = 1 for 1 ≤ β ≤ ℓ is given by j 1 . We take a set of ε ′ α such that the number of integers α satisfying ε ′ α = 1 for 1 ≤ α ≤ ℓ is given by We can evaluate the form factors and the expectation values of a spin-ℓ/2 operator through Corollary 1 [54]. The corrections of the form factors given in the paper [19] are listed in reference [20] of the paper [54] (see also [53]). Corrections for the paper [20] are listed in reference [21] of the paper [54].
For an illustration, let us consider the spin-1 case. We calculate the one-point function

Quantum group symmetry relations in the spin 1 case
We show some important topics. We derive symmetry relations among the expectation values of products of the spin-1/2 operators from the spin inversion symmetry. In particular, we show how to transform the basis vectors constructed in the 2sth tensor product space of the spin-1/2 representations to the 2s + 1-dimensional vectors in C 2s+1 .

Rotation symmetry of the XXX spin chain and irreducible components of operators
Let us consider the XXX case where the SU(2) symmetry holds for the total spin operators. The tensor product of two spin-1/2 representations of sl (2)  Here we recall that || − + denotes |1 1 ⊗ |0 2 .
We thus have In terms of irreducible components, we have We thus have We shall evaluate the expectation values of spin-s local operators by reducing them into those of the spin-1/2 local operators. Applying formula (2.10) to the case of ε ′ 1 = 0 and ε ′ 2 = 1, which correspond to + and −, respectively, we have Here we remark that the vector |ψ

Spin inversion symmetry
For even L we may assume the spin inversion symmetry: U |ψ Here we recall that associated with the ground state of the integrable spin-s XXZ spin chain the vector |ψ (2sp;0) g is given by |ψ We derive symmetry relations as follows [53,54] (4.2) Applying the spin-inversion symmetry (4.2) we derive symmetry relations among the expectation values of local or global operators [53,54].
For an illustration, let us evaluate the one-point function in the spin-1 case with i 1 = j 1 = 1, E 1,1(2p) 1 . Setting ε ′ 1 = 0 and ε ′ 2 = 1 we decompose the spin-1 elementary matrix in terms of a sum of products of the spin-1/2 ones Through the symmetry relations (3.13) with respect to ε ′ α we have the following equalities: g and hence we have the equalities of the four terms. We therefore obtain the following: We thus derive the double-integral representation of the one-point function E of [20], as we shall show in Section 6.
In Section 3.6 we have defined the spin-s XXZ transfer matrix through the fusion method. It is expressed in terms of operators defined on the Lth tensor product space of the spin-1/2 representations, (V (1) ) ⊗L , and given by a 2 L ×2 L matrix. We have constructed them by applying the projection operators to the spin-1/2 XXZ transfer matrix with inhomogeneous parameters given by complete strings w (2s) j . We now formulate the spin-s XXZ transfer matrix in terms of the basis of the (2s + 1)dimensional vector space C 2s+1 such as |2s, m)) for m = 0, 1, . . . , 2s. As the basis vectors of the (2s + 1)-dimensional representation of U q (sl 2 ) we introduce vectors |2s, m with the following normalization: |2s, m = ||2s, m / 2s m for m = 0, 1, . . . , 2s.
In the massless regime where q is complex with |q| = 1, explicitly we have Taking the Hermitian conjugate of S we have It is straightforward to show the following: In terms of the bras and kets we have |2, m)) 2, m| Similarly, we have |2, m 2, m|.
In order to transform the conjugate vectors ||2, m it is also useful to introduce the complex conjugates of transformations S and S † : They are related to the projection operator P (2) . We have |2, m 2, m| = P (2) .
The spin-1 elementary matrices E i,j(2p) are transformed into the 3 × 3 unit matrices E i,j as For instance we have We have thus confirmed relations (4.3). Let us introduce the transformation which maps the tensor product of the spin-s representations: Ns to the tensor product of the (2s + 1)-dimensional representations: Ns . We define it by the tensor product of transformation S as follows Ns . We also define its complex conjugate Ns .
Let us consider the spin-s ground state with (2s + 1)-dimensional entries, |Ψ (2s) G . For the spin-1 case, it gives the ground state of the spin-1 XXZ Hamiltonian (1.1). In terms of the ground state constructed by the fusion method, |ψ (2sp) g , it is given by Here we recall that |ψ (2sp) g denotes the ground state of the integrable spin-s XXZ spin chain constructed through the fusion method, where the evaluation representations are associated with principal grading. In terms of the eigenvector with (2s + 1)-dimensional entries, the expectation value of a given local operator E with (2s + 1)-dimensional entries is given by Therefore, the operator E corresponds to the operator E (2sp) in the fusion construction as follows For instance, from (4.3) we have the following: Similarly, we have the following relations for the spin-s XXZ transfer matrices defined as (2s + 1) Ns × (2s + 1) Ns matrices t (ℓ,2s) 12···Ns , to those of the fusion construction: 12···Ns (λ)S 1 ⊗ · · · ⊗ S Ns for ℓ = 1, 2, . . . , 2s.

Multiple-integral representations for spin-s case
We introduce some useful symbols for expressing the correlation functions of the integrable spin-s XXZ spin chain. We derive the multiple-integral representation of the spin-s correlation functions by mainly following the procedures of [20] except for the formula of reducing the higher-spin form factors into the spin-1/2 scalar products such as in Corollary 1. Let us sketch the main procedures for deriving the multiple-integral representation of the spins XXZ correlation functions. First, we introduce the spin-s elementary operators as the basic blocks for constructing the local operators of the integrable spin-s XXZ spin chain. Secondly, we reduce them into a sum of products of the spin-1/2 elementary operators, which we express through the spin-1/2 QISP formula in terms of the matrix elements of the spin-1/2 monodromy matrix, and evaluate their scalar products through Slavnov's formula of the Bethe-ansatz scalar products. Here, the expectation value of a physical quantity is expressed as a sum of the ratios of the Bethe-ansatz scalar products to the norm of the Bethe-ansatz eigenvector. Furthermore, the ratios are expressed in terms of the determinants of some matrices. Thirdly, solving the integral equations for the matrices in the thermodynamic limit, we derive the multiple-integral representation of the correlation functions.
Let us summarize the multiple-integral representations of correlation functions for the integrable spin-s XXZ spin chain in a region of the massless regime with 0 ≤ ζ < π/2s [20]. We show the revised expression [53,54]. Here we recall that in the massless regime we set η = iζ with 0 ≤ ζ < π.
We express the mth product of (2s + 1) × (2s + 1) elementary matrices in terms of a sum of 2smth products of the 2 × 2 elementary matrices with entries {ǫ j , ǫ ′ j }; i.e., e ε ′ 1 ,ε 1 1 · · · e ε ′ 2sm ,ε 2sm 2sm [20,54]. For given sets of ε j and ε ′ j for j = 1, 2, . . . , 2sm we define α − by the set of integers j satisfying ε ′ j = 1 and α + by the set of integers j satisfying ε j = 0: We denote by α − and α + the number of elements of the set α − and α + , respectively. Due to the "charge conservation", we have Precisely, we have α − = m k=1 i k and α + = 2sm − m k=1 j k . Here we recall that for the R-matrix of the XXZ spin chain matrix elements R(u) ab cd vanish if a + b = c + d, which we call the charge conservation. It follows from the charge conservation that the correlation function F We remark that the charge conservation of the R-matrix corresponds to the "ice rule" of the six-vertex model, which is defined as a two-dimensional ferro-electric lattice model.
Here we also assume that string deviations δ (α) a are very small for large N s . In terms of rapidities forming strings, λ (α) a , the spin-s ground state associated with the principal grading is given by Here we have M Bethe roots with M = 2sN s /2 = sN s . The density of string centers, ρ(λ), is given by which has simple poles at λ = iζ(n + 1/2), for n ∈ Z with the residues (−1) n /(2πi). We define the (j, k) element of a matrix S = S (λ j ) 2sm ; (w (2s) j ) 2sm by for j, k = 1, 2, . . . , 2sm.
Here δ(α, β) denotes the Kronecker delta. We define β(j) by where the Gauss symbol [[x]] is defined by the greatest integer less than or equal to a real number x. We define α(λ j ) by α( [20]. We remark that µ j correspond to the centers of complete 2s-strings λ j . When we evaluate α(λ j ), we assume that the integral paths of With the above notations, we express correlation functions for the massless spin-s XXZ chain in the form of multiple integrals as follows Here we have defined Q({ε j , ε ′ j }; λ 1 , . . . , λ 2sm ) by Here we have set ǫ k,ℓ = iǫ for Im(λ k − λ ℓ ) > 0 and ǫ k,ℓ = −iǫ for Im(λ k − λ ℓ ) < 0, where ǫ is an infinitesimally small positive real number: 0 < ǫ ≪ 1. The normalization factor C is given by where q = e η = e iζ .
Here we should remark that in (5.2) the sum of α + ({ε j }) is taken over all sets {ε j } corresponding to {ε  We calculate analytically the integrals for the spin-1 one-point functions. Considering the residues which are derived when we shift the integral paths, we explicitly evaluate the double integrals expressing the spin-1 one-point functions. Hereafter, we shall often denote the spin-s elementary matrices E i,j(2sp) by E i,j for simplicity.

E 22 : The emptiness formation probability
Let us evaluate the emptiness formation probability (EFP) E 2,2(2p) . In this case we have Here the symbol ∅ denotes the empty set. We evaluate EFP as follows Let us denote the integral path ∞+iα −∞+iα by C iα . The multiple-integral formula reads where Q(λ 1 , λ 2 ) and S(λ 1 , λ 2 ) are expressed in terms of ϕ(x) = sinh(x) as We now shift the integral paths C −iǫ and C −η−iǫ into C −η/2 and C −3η/2 , respectively. During the contour deformation each of the integral paths does not cross any pole of the integrand, and hence we have We now denote C −η/2 and C −3η/2 by C 1 and C 2 , respectively. After expanding the above expression with respect to the types of integral paths, we have four terms. However, only two of them survive due to the Kronecker deltas in the matrix S 2 + η/2) 2 + η/2) 2 + η/2)ρ(λ 2 − w 1 + η/2), Substituting w where x 1 = µ 1 − ξ 1 , x 2 = µ 2 − ξ 1 , and Q 12 and Q 21 are given by Thus, we have where I 12 and I 21 are given by The integrand Q 12 is transformed into Q 21 when we shift the integral path as x 1 → x 1 − η and x 2 → x 2 + η. First we shift the integral path in I 12 as x 1 → x 1 − η.
Here we note that due to the sign in front of ǫ in the denominator of (6.1), the integrand Q 12 has a pole at x 1 = x 2 − iǫ as a function of x 1 . Here we recall that ǫ is an infinitesimally small positive real number. We therefore express the integral I 12 in terms of a sum of two integrals, J 1 + J 2 , as follows Here we have made use of the anti-periodicity: ρ(x + nη) = (−1) n ρ(x). We also remark that the simple pole at x 1 = −η/2 due to ρ(x 1 ) is canceled by the factor ϕ(x 1 + η/2) in Q 12 (x 1 , x 2 ).
Let us first consider the single integral J 2 derived from the pole at x 1 = x 2 − ǫ. Explicitly evaluating the integral J 2 we have Here we have made use of formula (B.2).
Let us next consider the double integral J 1 . We shift the integral path in J 1 as x 2 → x 2 + η. We derive the wanted integral I 21 as follows Here we note that the simple pole at x 2 = η/2 due to ρ(x 2 ) is canceled by the factor ϕ(x 2 − η/2) in Q 12 .
Here we shall give definitions of integrals J 1 , J 2 , K 1 and K 2 and calculate them shortly in the following. For K 1 and K 2 , making use of the formula: 2πiRes [ ρ(λ − w + η/2)| λ=w ] = 1, we have We have defined the integrals J 1 and J 2 by As in the case of E 22 , we transform the integral J 1 into J 2 by shifting the integral path as x 1 → x 1 − η and x 2 → x 2 + η. First we shift the integral path in J 1 as x 1 → x 1 − η. There are two simple poles at x 1 = x 2 − iǫ and x 1 = −η/2. Using 2πiRes ρ(x)| x=−η/2 = −1, we can calculate the residues as Thus we have Next we shift the integral path as x 2 → x 2 + η. Here we remark that the simple pole at x 2 = η/2 of ρ(x 2 ) has zero residue due to the factor ϕ(x 2 − η/2) of the integrand. Thus we have where we have omitted the infinitesimal ǫ since we can shift the integral path without crossing the poles. Thus, we have where we have used the fact that K 2 = I 1 . Using the formula (B.1), we have K 1 = 1/2. Next we consider the integral I 2 . Shifting the integral path of x as x → x + iπ, we have Making use of the formula: sinh(x + η/2) sinh(x − η/2) = (cosh 2x − cosh η)/2 we have where we have used the formula (B.2). Finally, we obtain E 11 = 2(−K 1 − I 2 ) = cos ζ(sin ζ − ζ cos ζ) ζ sin 2 ζ .
Finally in Section 6 we give an important remark: through an explicit evaluation of the multiple integrals of E 1,1(2p) we have shown the following relations: It follows that in the spin-1 case, every one-point function is expressed in terms of a single multiple integral, which corresponds to the expectation value of a single product of the local spin-1/2 operators. In general, however, the spin-s correlation function of an arbitrary entry is expressed in terms of the expectation values of a sum of products of the local spin-1/2 operators such as shown in (5.2). Here we recall that the sum over sets α + ({ε β }) in (5.2) corresponds to the sum over sequences {ε β } in the reduction formula of Corollary 1. 7 Consistency with numerical estimates of the spin-1 one-point functions We now show that the analytical expressions of the spin-1 one-point functions are consistent with their numerical estimates, which are obtained by the method of numerical exact diagonalization of the integrable spin-1 XXZ Hamiltonian. Let us fist summarize the analytical results derived in Section 6. Evaluating the multiple integrals explicitly, we have obtained all the one-point function for the integrable spin-1 XXZ chain as E 2,2(2p) = E 0,0(2p) = ζ − sin ζ cos ζ 2ζ sin 2 ζ , E 1,1(2p) = cos ζ(sin ζ − ζ cos ζ) ζ sin 2 ζ , which are shown in Fig. 1. In particular, via evaluation of the multiple integrals, we have confirmed the uniaxial symmetry relation: Through the direct evaluation of the multiple integrals we confirm the identity: E 22 + E 11 + E 00 = 1. Here we recall that assuming the uniaxial symmetry (7.1) the analytical expression of E 00 has been given in [20]. Furthermore, we have confirmed the relations among the correlation functions due to the quantum group U q (sl 2 ) symmetry and the spin inversion symmetry as follows = 2 e 1,0 1 e 0,1 2 . In the XXX limit ∆ → 1 we have E 22 = E 11 = E 00 = 1/3, which has been shown by Kitanine in the XXX case [17]. In the free Fermion limit ∆ → 0 we have E 22 = E 00 = 1/2, and E 11 = 0. Here we should remark that we consider the region 0 ≤ ζ < π/(2s) with s = 1, namely, 0 < ∆ ≤ 1.
Finally, we confirm the analytical results by comparing them with the numerical results of exact diagonalization, which are shown in Fig. 2. In Fig. 2, the red and blue lines represent the analytical results obtained by evaluating the multiple integrals of the one-point functions, E 22 = E 00 and E 11 , respectively. The black dotted lines represent the numerical estimates of the one-point functions which are obtained by the method of exact diagonalization of the integrable spin-1 XXZ Hamiltonian with the system size of N s = 8. We numerically obtain the ground-state eigenvector of the integrable spin-1 XXZ Hamiltonian, and calculate the numerical estimates of the one-point functions.