Correlation functions of the integrable higher-spin XXX and XXZ spin chains through the fusion method

For the integrable higher-spin XXX and XXZ spin chains we present multiple-integral representations for the correlation function of an arbitrary product of Hermitian elementary matrices in the massless ground state. We give a formula expressing it by a single term of multiple integrals. In particular, we explicitly derive the emptiness formation probability (EFP). We assume $2s$-strings for the ground-state solution of the Bethe ansatz equations for the spin-$s$ XXZ chain, and solve the integral equations for the spin-$s$ Gaudin matrix. In terms of the XXZ coupling $\Delta$ we define $\zeta$ by $\Delta=\cos \zeta$, and put it in a region $0 \le \zeta<\pi/2s$ of the gapless regime: $-1<\Delta \le 1$ ($0 \le \zeta<\pi$), where $\Delta=1$ ($\zeta=0$) corresponds to the antiferromagnetic point. We calculate the zero-temperature correlation functions by the algebraic Bethe ansatz, introducing the Hermitian elementary matrices in the massless regime, and taking advantage of the fusion construction of the $R$-matrix of the higher-spin representations of the affine quantum group.


Introduction
The correlation functions of the spin-1/2 XXZ spin chain have been studied extensively through the algebraic Bethe-ansatz during the last decade [1,2,3,4,5,6]. The multiple-integral representations of the correlation functions for the infinite lattice at zero temperature first derived through the affine quantum-group symmetry [7,8] and also by solving the q-KZ equations [9,10] have been rederived and then generalized into those for the finite-size lattice under non-zero magnetic field. They are also extended into those at finite temperatures [11]. Furthermore, the asymptotic expansion of a correlation function has been systematically discussed [12]. Thus, the exact study of the correlation functions of the XXZ spin chain should be not only very fruitful but also quite fundamental in the mathematical physics of integrable models.
Recently, the correlation functions and form factors of the integrable higher-spin XXX spin chains and the form factors of the integrable higher-spin XXZ spin chains have been derived by the algebraic Bethe-ansatz method [13,14,15]. In the spin-1/2 XXZ chain the Hamiltonian under the periodic boundary conditions is given by Here σ a j (a = X, Y, Z) are the Pauli matrices defined on the jth site and ∆ denotes the XXZ coupling. We define parameter q by ∆ = (q + q −1 )/2 . (1.2) We define η and ζ by q = exp η and η = iζ, respectively. We thus have ∆ = cos ζ. In the massless regime: −1 < ∆ ≤ 1, we have 0 ≤ ζ < π for the spin-1/2 XXZ spin chain (1.1). At ∆ = 1 (i.e. q = 1), the Hamiltonian (1.1) corresponds to the antiferromagnetic Heisenberg (XXX) chain. The solvable higher-spin generalizations of the XXX and XXZ spin chains have been studied by the fusion method in several references [16,17,18,19,20,21,22,23]. The spin-s XXZ Hamiltonian is derived from the spin-s fusion transfer matrix (see also section 2.6). For instance, the Hamiltonian of the integrable spin-1 XXX spin chain is given by Here S j denotes the spin-1 spin-angular momentum operator acting on the jth site among the N s lattice sites of the spin-s chain. For the general spin-s case, the integrable spin-s XXX and XXZ Hamiltonians denoted H XXZ , respectively, can also be derived systematically. The correlation functions of integrable higher-spin XXX and XXZ spin chains are associated with various topics of mathematical physics. For the integrable spin-1 XXZ spin chain correlation functions have been derived by the method of q-vertex operators through some novel results of the representation theory of the quantum algebras [24,25,26,27,28]. They should be closely related to the higher-spin solutions of the quantum Knizhnik-Zamolodchikov equations [10]. For the fusion eight-vertex models, correlation functions have been discussed by an algebraic method [29]. Moreover, the partition function of the six-vertex model under domain wall boundary conditions have been extended into the higher-spin case [30].
In a massless region 0 ≤ ζ < π/2s, the low-lying excitation spectrum at zero temperature of the integrable spin-s XXZ chain should correspond to the level-k SU(2) WZWN model with k = 2s. By assuming the string hypothesis it is conjectured that the ground state of the integrable spin-s XXX Hamiltonian is given by N s /2 sets of 2s-strings [31]. It has also been extended into the XXZ case [32]. The ground-state solution of 2s-strings is derived for the spin-s XXX chain through the zero-temperature limit of the thermal Bethe ansatz [18]. The low-lying excitation spectrum is discussed in terms of spinons for the spin-s XXX and XXZ spin chains [31,32]. Numerically It was shown that the finite-size corrections to the ground-state energy of the integrable spin-s XXX chain are consistent with the conformal field theory (CFT) with c = 3s/(s + 1) [33,34,35,36]. Here c denotes the central charge of the CFT. It is also the case with the integrable spin-s XXZ chain in the region 0 ≤ ζ < π/2s [37,38,39]. The results are consistent with the conjecture that the ground state of the integrable spin-s XXZ chain with 0 ≤ ζ < π/2s is given by N s /2 sets of 2s-strings [22,32,37,38,39,40,41,42]. Furthermore, it was shown analytically that the low-lying excitation spectrum of the integrable spin-s XXZ chain in the region 0 ≤ ζ < π/2s is consistent with the CFT of c = 3s/(s + 1) [41,42]. In fact, the low-lying excitation spectrum of spinons for the spin-s XXX chain is described in terms of the level-k SU(2) WZWN model with k = 2s [43].
In the paper we calculate zero-temperature correlation functions for the integrable higherspin XXZ spin chains by the algebraic Bethe-ansatz method. For a given product of elementary matrices we present the multiple-integral representations of the correlation function in the region 0 ≤ ζ < π/2s of the massless regime near the antiferromagnetic point (ζ = 0). For an illustration, we derive the multiple-integral representations of the emptiness formation probability (EFP) of the spin-s XXZ spin chain, explicitly. Here the spin s is given by an arbitrary positive integer or half-integer. Assuming the conjecture that the ground-state solution of the Bethe ansatz equations is given by 2s-strings for the regime of ζ, we derive the spin-s EFP for a finite chain and then take the thermodynamic limit. We solve the integral equations associated with the spin-s Gaudin matrix for 0 ≤ ζ < π/2s, and express the diagonal elements in terms of the density of strings. Here we remark that the integral equations associated with the spin-s Gaudin matrix have not been explicitly solved, yet, even for the case of the integrable higher-spin XXX spin chains [13]. We also calculate the spin-s EFP for the homogeneous chain where all inhomogeneous parameters ξ p are given by zero. Here we shall introduce inhomogeneous parameters ξ p for p = 1, 2, . . . , N s , in §2. 4. Furthermore, we take advantage of the fusion construction of the spin-s R-matrix in the algebraic Bethe-ansatz derivation of the correlation functions [15].
Given the spin-s XXZ spin chain on the N s lattice sites, we define L by L = 2sN s and consider the spin-1/2 XXZ spin chain on the L sites with inhomogeneous parameters w j for j = 1, 2, . . . , L. In the fusion method we express any given spin-s local operator as a sum of products of operator-valued elements of the spin-1/2 monodromy matrix in the limit of sending inhomogeneous parameters w j to sets of complete 2s-strings as shown in Ref. [15]. Here, we apply the spin-1/2 formula of the quantum inverse scattering problem [4], which is valid at least for generic inhomogeneous parameters. Therefore, sending inhomogeneous parameters w j into complete 2s-strings, we can evaluate the vacuum expectation values or the form factors of spin-s local operators which are expressed in terms of the spin-1/2 monodromy matrix elements with generic inhomogeneous parameters w j . Here, the rapidities of the ground state satisfy the Bethe ansatz equations with inhomogeneous parameters w j . We assume in the paper that the Bethe roots are continuous with respect to inhomogeneous parameters w j , in particular, in the limit of sending w j to complete 2s-strings. We can construct higher-spin transfer matrices by the fusion method [22,23]. Here we recall that the spin-1/2 XXZ Hamiltonian (1.1) is derived from the logarithmic derivative of the rowto-row transfer matrix of the six-vertex model. We call it the spin-1/2 transfer matrix and denote it by t (1,1) (λ). Let us express by V (ℓ) an (ℓ + 1)-dimensional vector space. We denote by T (ℓ, 2s) (λ) the spin-ℓ/2 monodromy matrix acting on the tensor product of the auxiliary space V (ℓ) and the N s th tensor product of the quantum spaces, (V (2s) ) ⊗Ns . We call it of type (ℓ, (2s) ⊗Ns ), which we express (ℓ, 2s) in the superscript. Taking the trace of the spinℓ/2 monodromy matrix T (ℓ, 2s) (λ) over the auxiliary space V (ℓ) , we define the spin-ℓ/2 transfer matrix, t (ℓ, 2s) (λ). For ℓ = 2s, we have the spin-s transfer matrix t (2s, 2s) (λ), and we derive the spin-s Hamiltonian from its logarithmic derivative.
We construct the ground state |ψ (2s) g of the spin-s XXZ Hamiltonian by the B operators of the 2-by-2 monodromy matrix T (1,2s) (λ). As shown by Babujan, the spin-s transfer matrix t (2s,2s) (λ) commutes with the spin-1/2 transfer matrix t (1,2s) (λ) due to the Yang-Baxter relations, and hence they have eigenvectors in common [18]. The ground state |ψ (2s) g of the spin-s XXZ spin chain is originally an eigenvector of the spin-s transfer matrix t (2s,2s) (λ), and consequently it is also an eigenvector of the spin-1/2 transfer matrix t (1,2s) (λ). Therefore, the ground state |ψ (2s) g of the spin-s XXZ spin chain can be constructed by applying the B operators of the 2-by-2 monodromy matrix T (1,2s) (λ) to the vacuum.
We can show that the fusion R-matrix corresponds to the R-matrix of the affine quantum group U q ( sl 2 ). We recall that by the fusion method, we can construct the R-matrix acting on the tensor product V (ℓ) ⊗ V (2s) [16,17,18,19,20,21,22,23]. We denote it by R (ℓ,2s) . In the affine quantum group, the R-matrix is defined as the intertwiner of the tensor product of two representations V and W [44,45,46]. Due to the conditions of the intertwiner the R-matrix of the affine quantum group is determined uniquely up to a scalar factor [47], which we denote by R V,W . Therefore, showing that the fusion R-matrix satisfies all the conditions of the intertwiner, we prove that the fusion R-matrix coincides with the R-matrix of the quantum group, R V,W . Consequently, for ℓ = 2s the fusion R-matrix, R (2s,2s) (λ), becomes the permutation operator when spectral parameter λ is given by zero. This property of the R-matrix plays a central role in the derivation of the integrable spin-s Hamiltonian. It is also fundamental in the inverse scattering problem in the spin-s case [48].
There are several relevant and interesting studies of the integrable spin-s XXZ spin chains. The expression of eigenvalues of the spin-s XXX transfer matrix t (2s,2s) (λ) was derived by Babujan [17,18,21] through the algebraic Bethe-ansatz method. It was also derived by solving the series of functional relations among the spin-s transfer matrices [22]. The functional relations are systematically generalized to the T systems [49]. Recently, the algebraic Bethe ansatz for the spin-s XXZ transfer matrix has been thoroughly reviewed and reconstructed from the viewpoint of the algebraic Bethe ansatz of the U(1)-invariant integrable model [50,51]. Quite interestingly, it has also been applied to construct the invariant subspaces associated with the Ising-like spectra of the superintegrable chiral Potts model [52].
The content of the paper consists of the following. In section 2, we introduce the Rmatrix for the spin-1/2 XXZ spin chain. We then introduce conjugate basis vectors in order to formulate Hermitian elementary matrices E m, n in the massless regime where |q| = 1. We define the massless higher-spin monodromy matrices T (ℓ, 2s) (λ) in terms of the conjugate vectors, after reviewing the fusion construction of the massive higher-spin monodromy matrices T (ℓ, 2s) (λ) and higher-spin XXZ transfer matrices, t (ℓ, 2s) (λ), for ℓ = 1, 2, . . ., as follows. We express the matrix elements of T (ℓ, 2s) 0,12···Ns (λ) in terms of those of the spin-1/2 monodromy matrix T (1,1) 0,12···L (λ). Here T (1,1) 0,12···L (λ) is defined on the tensor product of the two-dimensional auxiliary space V (1) 0 and the Lth tensor product of the 2-dimensional quantum space, (V (1) ) ⊗L . Here we recall L = 2s × N s . In the fusion construction [15], monodromy matrix T (1, 2s) 0,12···Ns (λ) acting on the N s lattice sites is derived from monodromy matrix T (1,1) 0,12···L (λ) acting on the 2sN s lattice sites by setting inhomogeneous parameters w j to N s sets of complete 2s-strings and by multiplying it by the N s th tensor product of projection operators which project (V (1) ) ⊗2s to V (2s) . In section 3, we explain the method for calculating the expectation values of given products of spin-s local operators. We express the local operators in terms of global operators with inhomogeneous parameters w (2s; ǫ) j , which are defined to be close to complete 2s-strings with small deviations of O(ǫ), and evaluate the scalar products and the expectation values for the Bethe state with the same inhomogeneous parameters w (2s; ǫ) j . Then, we obtain the expectation values, sending ǫ to 0. Here we note that the projection operators introduced in the fusion construction commute with the matrix elements of monodromy matrix T (1, 1) of inhomogeneous parameters w (2s; ǫ) j with O(ǫ) corrections. In section 4 we calculate the emptiness formation probability (EFP) for the spin-s XXZ spin chain for a large but finite chain, and then evaluate the matrix S which is introduced for expressing the EFP of an infinite spin-s XXZ chain in the massless regime with ζ < π/2s. Here we solve explicitly the integral equations for the spin-s Gaudin matrix, expressing the 2s-strings of the ground-state solution systematically in terms of the string centers. In section 5, we present explicitly the multiple-integral representation of the spin-s EFP. We also derive it for the inhomogeneous chain where all the inhomogeneous parameters ξ p are given by 0.
For an illustration, we calculate the multiple-integral representation of spin-1 EFP for m = 1, E 2, 2 , explicitly. In the XXX limit, the value of E 2, 2 approaches 1/3, which is consistent with the XXX result of Ref. [13]. In section 6, we present the multiple-integral representations of the integrable spin-s XXZ correlation functions. We express the correlation function of an arbitrary product of elementary matrices by a single term of multiple integrals. For instance, we calculate the multiple-integral representation of the spin-1 ground-state expectation value, E 1, 1 , explicitly, and show that it is consistent with the value of spin-1 EFP in section 5, i.e. we show E 1, 1 + 2 E 2, 2 = 1. Finally in section 7, we give concluding remarks.
2 Fusion transfer matrices 2.1 R-matrix and the monodromy matrix of type (1, 1 ⊗L ) Let us introduce the R-matrix of the XXZ spin chain [1,3,4,5]. 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. Let V 1 and V 2 be two-dimensional vector spaces. The R-matrix acting on V 1 ⊗ V 2 is given by where u = λ 1 − λ 2 , b(u) = sinh u/ sinh(u + η) and c ± (u) = exp(±u) sinh η/ sinh(u + η). In the massless regime, we set η = iζ by a real number ζ, and we have ∆ = cos ζ. In the paper we mainly consider the region 0 ≤ ζ < π/2s. In the massive regime, we assign η a real nonzero number and we have ∆ = cosh η > 1. Here we remark that the R + (λ 1 − λ 2 ) is compatible with the homogeneous grading of U q ( sl 2 ), which is explained in Appendix A [15]. We denote by R (p) (u) or simply by R(u) the symmetric R-matrix where c ± (u) of (2.1) are replaced by c(u) = sinh η/ sinh(u + η) [15]. The symmetric R-matrix is compatible with the affine quantum group U q ( sl 2 ) of the principal grading [15].
Let s be an integer or a half-integer. We shall mainly consider the tensor product V (2s) 1 Ns of (2s+1)-dimensional vector spaces V (2s) j with L = 2sN s . In general, we consider the tensor product V have spectral parameters λ j for j = 1, 2, . . . , r. For a given set of matrix elements A a, α b, β for a, b = 0, 1, . . . , 2s j and α, β = 0, 1, . . . , 2s k , we define operator A j,k by We now consider the (L + 1)th tensor product of spin-1/2 representations, which consists of the tensor product of auxiliary space V (1) 0 and the Lth tensor product of quantum spaces . We call it the tensor product of type (1, 1 ⊗L ) and denote it by the following symbol: Applying definition (2.2) for matrix elements R(u) ab cd of a given R-matrix, we define Rmatrices 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. . (2.6) 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 (1,1+) 0,12···L (λ; {w j } L )] a,b for a, b = 0, 1. We derive the monodromy matrix consistent with principal grading, T (1,1 p) 0,12···L (λ; {w j } L ), from that of homogeneous grading via similarity transformation χ 01···L as follows [15].
ab (λ a − λ b ) for a, b = 1, 2, . . . , n, where w = + and w = p in superscripts show the homogeneous and the principal grading, respectively. Then, the monodromy matrix of type (1, 1 ⊗L w) is expressed as follows. (2.9) For instance we have B (1 w)

Conjugate vectors and the massless fusion R-matrices
In order to construct Hermitian elementary matrices in the massless regime where |q| = 1, we now introduce vectors ||ℓ, n which are Hermitian conjugate to ℓ, n|| when |q| = 1 for positive integers ℓ with n = 0, 1, . . . , ℓ. Setting the norm of ||ℓ, n such that ℓ, n|| ||ℓ, n = 1, we have Here we have denoted the binomial coefficients for integers ℓ and n with 0 ≤ n ≤ ℓ as follows.
We define the massless fusion R-matrix R (ℓ, 2s +) 0, j , applying projection operators P consisting of conjugate vectors to the product of R-matrices, as follows.
We shall define the monodromy matrix of type (1, (2s) ⊗Ns ) associated with homogeneous grading. We first define the massless monodromy matrix by Here, the (0,0) element is given by 12···L . We then define the massive monodromy matrix by Let us introduce a set of 2s-strings with small deviations from the set of complete 2s-strings.
Here ǫ is very small and r are generic parameters. We express the elements of the monodromy matrix T (1,1) with inhomogeneous parameters given by w (2s; ǫ) j for j = 1, 2, . . . , L as follows. . (2.30) Here we recall that A . 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 The trace of A over V (ℓ) is equivalent to the trace of A over the ℓth tensor product of V (1) , (V (1) ) ⊗ℓ , multiplied by a projector P (ℓ) (or P (ℓ) ) as follows.
3 Higher-spin expectation values

Algebraic Bethe ansatz
In terms of the vacuum vector |0 where all spins are up, we define functions a(λ) and d(λ) by We have a(λ; {w j } L ) = 1 and . For the homogeneous grading (w = +) and the principal grading (w = p), it is easy to show the following relations: where a (2s) (λ; {ξ k }) and d (2s) (λ; {ξ k }) are given by In the massless regime, we define the Bethe vectors | {λ α } | for w = + and p, as follows.
It gives an eigenvector of the transfer matrix with the following eigenvalue: (3.13) Let us assume that in the limit of ǫ going to 0, the set of Bethe roots {λ j (ǫ)} M is given by {λ j } M . Then, we have (3.14) 3.2 Hermitian elementary matrices E m , n (2s +) i in the massless regime We define massless elementary matrices E m, n (2s+) for m, n = 0, 1, . . . , 2s, in the spin-s representation of U q (sl 2 ) as follows.
In the tensor product space, (V (2s) ) ⊗Ns , we define E m, n (2s +) i Elementary matrices E n, n (2s +) for n = 0, 1, . . . , 2s, are Hermitian in the massless regime. In fact, when |q| = 1, for m, n = 0, 1, . . . , 2s, we have We can express any given spin-s local operator of the massless case in terms of the spin-1/2 global operators by a method similar to the massive case [15]. For m = n, we have Formulas expressing E m, n (2s+) for m > n or m < n are given in Appendix E. When we evaluate expectation values, we want to remove the projection operators introduced in order to express the spin-s local operator in terms of spin-1/2 global operators such as in (3.18). Then, we shall make use of the following lemma. } L ) such as A (2s +;ǫ) (λ) in the limit of ǫ going to 0.
Proof. Taking derivatives with respect to inhomogeneous parameters w j , we can show 12···L as follows [15].
3.3 Expectation value of a local operator through the limit: ǫ → 0 In the massless regime, we define the expectation value of product of operators m k=1 E We evaluate the expectation value of a given spin-s local operator for a Bethe-ansatz eigenstate |{λ α } (2s) M , as follows. We first assume that the Bethe roots {λ α (ǫ)} M are continuous with respect to small parameter ǫ. We express the spin-s local operator in terms of spin-1/2 global operators such as formula (3.18) with generic inhomogeneous parameters w (2s;ǫ) j . Applying (3.19) and (3.19) we remove the projection operators out of the product of global operators. We next calculate the scalar product for the Bethe state |{λ k (ǫ)} (2s; ǫ) M which has the same inhomogeneous parameters w (2s;ǫ) j , making use of the formulas of the spin-1/2 case. Then we take the limit of sending ǫ to 0, and obtain the expectation value of the spin-s local operator.
We then move the projection operators P (2s) in the leftward direction, making use of (3.19). Thus, the right-hand side of (3.26) is now given by the following: After applying the gauge transformation χ −1 1···L inverse to (2.7) [15], we obtain Here A (2s; ǫ) and D (2s; ǫ) denote matrix elements A (2s p; ǫ) and D (2s p; ǫ) of the monodromy matrix with principal grading, respectively. In the last line of (3.27), we have evaluated the eigenvalue of transfer matrix Before sending ǫ to 0, we expand the products of C operators multiplied by operators A and D by the commutation relations between C and A as well as C and D, respectively. We then evaluate the scalar product of B and C operators with inhomogeneous parameters w (2s;ǫ) j . Finally, we derive the expectation value in the limit of sending ǫ to 0.
Sending ǫ to 0, we calculate the expectation value of ) on the vacuum |0 as follows.
after sending ǫ to 0, the result is different from (3.30) as follows. We shall introduce ℓ-strings for an integer ℓ. Let us shift rapidities λ j by sη such asλ j = λ j +sη. Then, the Bethe ansatz equations (3.9) are given by We define an ℓ-string by the following set of rapidities.
We call µ a the center of the ℓ-string and ǫ (α) a string deviations. We assume that ǫ If they are zero, then we call the set of rapidities of (4.2) a complete ℓ-string. The string center µ a corresponds to the central position among the ℓ complex numbers:λ a , . . . ,λ a . Furthermore we assume that µ a are real. If inhomogeneous parameters, ξ p , are small enough, then the Bethe ansatz equations should have an ℓ-strings as a solution.
In terms of rapidities λ j which are not shifted, an ℓ-string is expressed in the following form: We denote λ (α) a also by λ (a,α) . Let us now introduce the conjecture that the ground state of the spin-s case |ψ (2s) g is given by N s /2 sets of 2s-strings:  Hereafter we set M = 2sN s /2 = sN s .
According to analytic and numerical studies [40,41,42], we may assume the following properties of string deviations ǫ (α) a s. When N s is very large, the deviations are given by In the thermodynamic limit: N s → ∞, the Bethe ansatz equations for the ground state of the higher-spin XXZ chain become the integral equation for the string centers, as shown in Appendix F [53]. The density of string centers, ρ tot (µ), is given by Thus, the sum over all the Bethe roots of the ground state is evaluated by integrals in the thermodynamic limit, N s → ∞, as follows.

Derivation of the spin-s EFP for a finite chain
We define the emptiness formation probability (EFP) for the spin-s case by We shall denote τ (2s +) (m) by τ (2s) (m). Let us assume that Bethe roots {λ α (ǫ)} M with inhomogeneous parameters w (2s; ǫ) j (j = 1, 2, . . . , L; L = 2sN s ) become the ground-state solution of the spin-s XXZ spin chain, {λ α } M , in the limit of sending ǫ to 0. We denote the Bethe vector with Bethe roots {λ α (ǫ)} M by (4.13) Here we recall the transformation inverse to (2.7). We now calculate the norm of the spin-s ground state from that of the spin-1/2 case through the limit of sending ǫ to 0 as follows.
where matrix elements of the spin-s Gaudin matrix for j, k = 1, 2, . . . , M, are given by .
(4.18) We remark that from (4.18) the set of integers c 1 , . . . , c 2sm of the most dominant terms in (4.17) are given by m sets of 2s-strings. If they are not, the numerator of (4.18) and hence the right-hand-side of (4.17) becomes smaller at least by the order of 1/N s in the large N s limit. However, each of the most dominant terms diverges with respect to N s in the large-N s limit, and they should cancel each other so that the final result becomes finite. We therefore calculate all possible contributions with respect to the set of integers, c 1 , c 2 , . . . , c 2sm .
Let us take a sequence of distinct integers c j satisfying 1 ≤ c j ≤ M for j = 1, 2, . . . , 2sm. . . , c 2sm }, and put the elements in increasing order such as z 1 < z 2 < · · · < z M −2sm . We then extend the sequence z n of M − 2sm integers into that of M integers by setting z j+M −2sm = c j for j = 1, 2, . . . , 2sm. We shall denote z n also by z(n) for n = 1, 2, . . . , M.
In terms of sequence (z n ) M we express the scalar product in the last line of (4.17) as follows.
Proof. We first consider the case of positive n. Let us recall the Lieb equation Shifting variable λ analytically to λ + iζ − iǫ in (4.35) we have Combining ρ(λ + iζ) = −ρ(λ) we obtain eq. (4.33) for n = 1. Making analytic continuation with respect to λ we derive eq.   Relations (4.39) are expressed in terms of integrals as follows.

Applying proposition 4.2 and corollary 4.3 to (4.46) we have
where integer a is given by a = [(z(j) − 1)/2s] + 1. In terms of function a(z) = [(z − 1)/2s] + 1 we have (4.50) Here we have replaced the sum over t by the sum over c and γ where z(t) = C = 2s(c − 1) + γ with 1 ≤ γ ≤ 2s. Expressing z(j) and k by A and B, respectively, we have the following equations: (4.51) Let us introduce b and β by k = 2s(b − 1) + β + M ′ with 1 ≤ β ≤ 2s and 1 ≤ b ≤ N s /2. In terms of string center µ a we express (or approximate) ϕ z(j), k ({ξ p }) by a continuous function of µ a and ξ b , as follows.
By taking the large-N s limit, the discrete equations (4.51) are now expressed as follows.
Let us recall the assumption that function ϕ (β) α (µ a , ξ b ) is continuous with respect to µ a and ξ b . Then, for any given set of integers, c 1 , . . . , c 2sm satisfying 0 < c j ≤ M for j = 1, 2, . . . , 2sm, we may approximate the matrix elements of (Φ (2s) ′ ) −1 Ψ (2s) ′ as follows. For integers j and k with 1 ≤ j, k ≤ 2sm, we define a j , α j , b k and β k as follows. (4.58) Then, we have Here a j , α j , b k and β k are given by (4.58). Then, we obtain and we have

Fourier transform in the cases of spin-1 and general spin-s
The integral equations (4.53) for the spin-1 case are given by .
Here, we first subtract the 2nd row by the 1st row multiplied by e −ζω . We next add the 2nd column to the 1st column. Finally, the determinant is factorized and we have the result. By the same method we can calculate the determinant of M (2s) K 2 for the spin-s case. We have detM (2s) The determinant is nonzero generically, and hence the solution to matrix equation (4.73) is unique. Therefore, we obtain the solution of integral equation (4.53).
5 The EFP of the spin-s XXZ spin chain near AF point

Multiple-integral representations of the spin-s EFP
Let us derive multiple-integral representations for the emptiness formation probability of the spin-s XXZ spin chain. We shall take the large N s limit of the EFP (4.30) for a finite-size system, and we replace rapidity λ c j with complex variable λ j for j = 1, 2, . . . , 2sm, as follows. For a given rapidity of 2s-string, λ A = µ a − (α − 1/2)η + ǫ A , we define its regular part λ are complex integral variables corresponding to complete strings such as λ ′ k = µ k − (β − 1/2)η for some integer β with 1 ≤ β ≤ 2s where η = iζ with 0 < ζ < π and µ k is real. We express λ ′ k and ǫ c k simply by λ k and ǫ k , respectively, and then we obtain multiple-integral representations.
Applying (4.63) we derive the emptiness formation probability for arbitrary spin-s in the thermodynamic limit N s → ∞, as follows.
In the homogeneous case we have ǫ p = 0 for p = 1, 2, . . . , N s . We have thus defined inhomogeneous parameters ξ p . We recall that in the homogeneous case, the spin-s Hamiltonian is derived from the logarithmic derivative of the spin-s transfer matrix.
Here we should remark that an expression of the matrix elements of S similar to (5.3) has been given in eq. (6.14) of Ref. [13] for the correlation functions of the integrable spin-s XXX spin chain.

Symmetric expression of the spin-s EFP
We shall express the spin-s EFP (5.1) in a simpler way, making use of permutations of 2sm integers, 1, 2, · · · , 2sm, and the formula of the Cauchy determinant.
Reformulating the sum over c j s in (4.30) in terms of a j 's, in the large N s limit we have detS(λ π1 , . . . , λ π(2sm) ) H (2s) (λ π1 , . . . , λ π(2sm) ) , where symbols λ j denote the following We calculate detS applying the Cauchy determinant formula and we obtain the symmetric expression of the spin-s EFP as follows. .

The spin-s EFP for the homogeneous chain
Sending ξ b to zero for b = 1, 2, . . . , m, we derive the spin-s EFP in the homogeneous limit. Here we remark that the spin-s Hamiltonian is derived from the logarithmic derivative of the spin-s transfer matrix t (2s, 2s) (λ; {ξ b } Ns ) in the homogeneous case where ξ b = 0 for b = 1, 2, . . . , N s .

(5.13)
Here we recall definition (5.10) of λ j . Let us discuss that expression (5.13) gives the spin-s EFP for the homogeneous chain. First, we remark that τ (2s) Ns (m; {ξ p } m ) does not depend on ξ p with p > m. Hence we may consider that inhomogeneous parameters ξ p with p > m are all set to be zero, after computing the EFP: τ (2s) We now show that the order of the homogeneous limit: ξ p → 0 and the thermodynamic limit N s → ∞ can be reversed. We can show the following relation: Ns (m; {ξ p } m ) is continuous with respect to ξ p at ξ p = 0 for p = 1, 2, . . . , m. We first reformulate the sum over c j in (4.30) into the sum over a (j,k) by relation (5.8).
We then apply the Cauchy determinant formula to evaluate detS(ĉ j ) as follows.
Making use of (5.17) we show that such factors in the denominator of (4.30) that vanish in the limit of sending ξ p to zero are canceled by the factors in the numerator of (5.17). We thus have shown that the EFP for the finite system, τ Therefore, expression (5.13) gives the spin-s EFP for the homogeneous chain. That is, we have the following equality: Here we note that detS(λ 1 , λ 2 ) = 0 for λ 1 = µ 1 − η/2 and λ 2 = µ 2 − 3η/2, or for λ 1 = µ 1 − 3η/2 and λ 2 = µ 2 − η/2. Showing the following relations of integrals we thus have Evaluating the integral we obtain the spin-1 EFP with m = 1 as follows.
In the second line of (5.26) the first term corresponds to the first term of (5.19), while the second term to the second term of (5.19) with µ 1 and µ 2 being exchanged.
6 Spin-s XXZ correlation functions near AF point 6.1 Finite-size correlation functions of the integrable spin-s XXZ spin chain We now calculate correlation functions other than EFP for the spin-s XXZ spin chain by the method of §3.3, making use of the formulas of Hermitian elementary matrices such as (3.18). We define the correlation function of the spin-2s XXZ spin chain for a given product of (2s + 1) × (2s + 1) elementary matrices such as E as follows. (6.1) By the method of expressing spin-s local operators in terms of spin-1/2 global operators [15], we express the mth product of (2s + 1) × (2s + 1) elementary matrices in terms of a 2smth product of 2 × 2 elementary matrices with entries {ǫ j , ǫ ′ j } as follows. We evaluate the spin-2s XXZ correlation function F (2s +) ({i k , j k }) by We denote the right-hand side of (6.3) by F (2s +) ({ǫ j , ǫ ′ j }). Let us introduce some symbols. We denote by α + the set of suffices j such that ǫ j = 0, and by α − the set of suffices j such that ǫ ′ j = 1: We denote by α + and α − the number of elements of the set α + and α − , respectively. Due to charge conservation, we have α + + α − = 2sm. (6.5) We denote by j min and j max the smallest element and the largest element of α − , respectively. We also denote by j ′ min and j ′ max the smallest element and the largest element of α + , respectively. Let c j (j ∈ α − ) and c =1 . (6.7) Extending the derivation of the spin-s EFP we can rigorously derive the following expression of the spin-s XXZ correlation functions in the massless regime with 0 ≤ ζ < π/2s.
We recall that a(j) and β(j) have been defined in terms of the Gauss' symbol [·] by a(j) = [(j − 1)/2s] + 1 and β(j) = j − 2s[(j − 1)/2s], respectively. We remark that under the limit of sending ǫ to zero, the sum over variable c j is restricted up to M.

Multiple-integral representations of the spin-s XXZ correlation function for an arbitrary product of elementary matrices
Let us formulate matrix S for the correlation function of an arbitrary product of elementary matrices. We define the (j, k) element of matrix S = S (λ j ) 2sm ; (w Here δ(α, β) denotes the Kronecker delta. We define α(λ j ) by α( . We remark that µ j correspond to the centers of complete 2s-strings λ j . We also remark that the above definition of matrix S generalizes that of (5.3) since α(λ j ) is now also defined also for λ j = w (2s) k . Let Γ j be a small contour rotating counterclockwise around λ = w (2s) j . Since the detS has simple poles at λ = w (2s) j with residue 1/2πi, we therefore have For sets α − and α + we defineλ j for j ∈ α − andλ ′ j for j ∈ α + , respectively, by the following sequence: Thus, from the expression of the correlation function in terms of a finite sum (6.8) we obtain the multiple-integral representation as follows.
It is straightforward to take the homogeneous limit: ξ k → 0. Here we recall that (sgn σ) denotes the sign of permutation σ ∈ S 2sm /(S m ) 2s .

An example of the spin-1 correlation function
Applying formula (3.18) to the spin-1 case with m = n = 1, we have 1···L .
(6.16) Therefore, we evaluate it sending ǫ to zero, as follows.
where Q(λ 1 , λ 2 ) is given by and matrix S(λ 1 , λ 2 ) is given by . (6.19) We thus note note that the correlation function is now expressed in terms of a single product of the multiple-integral representation.
Let us now evaluate the double integral (6.17), explicitly. The integral over λ 1 is decomposed into the following: (6.20) Thus, the integral (6.17) is calculated as We have thus confirmed (5.24) directly evaluating the integrals.

Concluding remarks
In the paper we have explicitly shown the multiple-integral representation of the emptiness formation probability for the integrable spin-s XXZ spin chain in a region of the massless regime of η = iζ with 0 ≤ ζ < π/2s. We have also calculated the emptiness formation probability for the homogeneous case of the integrable spin-s XXZ spin chain.
In the XXX limit where we send ζ to zero, the expression of EFP for the spin-s XXZ case reduces to that of the spin-s XXX case.
Moreover, we have presented a formula for the multiple-integral representation of the spin-s XXZ correlation function of an arbitrary product of elementary matrices in the massless regime where η = iζ with 0 ≤ ζ < π/2s. We have also presented the symmetric expression of the multiple-integral representations of the spin-s XXZ correlation functions.
Finally, we have introduced conjugate vectors ||2s, n in order to formulate Hermitian elementary matrices E m, n (2s +) and Hermitian projection operators P (ℓ) in the massless regime. We have also defined the massless fusion R-matrices R (ℓ, 2s w) for w = + and p.

Acknowledgment
One of the authors (T.D.) would like to thank B.M. McCoy and other participants of the workshop on the integrable chiral Potts model, organized by M.T. Batchelor, Kioloa, NSW, Australia, Dec. [7][8][9][10][11]2008, for many useful comments. He is grateful to K. Motegi, J. Sato, and Y. Takeyama for bringing him useful references. Furthermore, the authors would like to thank S. Miyashita for encouragement and keen interest in this work. This work is partially supported by Grant-in-Aid for Scientific Research (C) No. 20540365.

A Affine quantum group with homogeneous grading
The affine quantum algebra U q ( sl 2 ) is an associative algebra over C generated by X ± i , K ± i for i = 0, 1 with the following relations: Here the symbol [n] q denotes the q-integer of an integer n: 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 algebra U q (sl 2 ) is given by the Hopf subalgebra of U q ( sl 2 ) generated by X ± i , K i with either i = 0 or i = 1. Hereafter we denote by X ± and K the generators of U q (sl 2 ).
For a given complex number λ we define a homomorphism of algebras ϕ λ : U q ( sl 2 ) → U q (sl 2 ).

B Fusion projection operators being idempotent
We give the derivation [23] of (P (ℓ) 12···ℓ making use of the Yang-Baxter equations.
12···ℓ defined by (2.12) have the following two expressions: Proof. Applying notation (2.2) to permutation operator Π 1,2 we define permutation operators Π j, k for integers j and k satisfying 0 ≤ j < k ≤ L. The form of the left-hand side of (B.1) is expressed in terms of R-matrices as follows (see also eq. (3.7) of [15]).
Making use of the Yang-Baxter equations (2.4) we reformulate (B.2) as follows.
which gives the expression of the right-hand side of (B.1).
1 , respectively. Making use of matrix elements of the monodromy matrix of type (1, 2) we derive the fusion R matrix of type (2,2). For an illustration, let us calculate R  The fusion R-matrix becomes permutation Π 0, 1 at u = 0. In fact, we have R

G Some formulas of the algebraic Bethe ansatz
Applying the commutation relations between C and D operators we have sinh(λ α − λ a j ) sinh(λ α − w j ) 1≤j<k≤m sinh(λ a j − λ a k ) sinh(w j − w k ) .(G.2)