Form factors of spin 1 analogue of the eight-vertex model

The twenty-one-vertex model, the spin $1$ analogue of the eight-vertex model is considered on the basis of free field representations of vertex operators in the $2\times 2$-fold fusion SOS model and vertex-face transformation. The tail operators, which translate corner transfer matrices of the twenty-one-vertex model into those of the fusion SOS model, are constructed by using free bosons and fermions for both diagonal and off-diagonal matrix elements with respect to the ground state sectors. Form factors of any local operators are therefore obtained in terms of multiple integral formulae, in principle. As the simplest example, the two-particle form factor of the spin operator is calculated explicitly.


Introduction
In this paper we consider the spin 1 analogue of Baxter's eight-vertex model [1], on the basis of vertex operator approach [2]. The model is often called twenty-one-vertex model since the R matrix has twenty one non-zero elements. The eight-vertex model is related to spin 1 2 anisotropic Heisenberg spin chain. It was found by Lashkevich and Pugai [3] that the correlation functions of the eight-vertex model can be obtained by using the free field realization of the vertex operators in the eight-vertex SOS model [4], with insertion of the nonlocal operator Λ, called 'the tail operator'. In [5] Lashkevich obtained integral formulae for form factors of the eight-vertex model.
There are some researches which generalize the study of [3,5]. The vertex operator approach for higher spin generalization of the eight-vertex model was presented in [6]. As for higher rank generalization, the integral formulae for correlation functions of Belavin's (Z/nZ)-symmetric model [7] were presented in [8], and those form factor formulae were presented in [9].
We are interested in the spontaneous polarization, a kind of one-point function of vertex models.
Baxter and Kelland [10] derived the expression of the spontaneous polarization of the eight-vertex model.
In [11] the same result was reproduced by solving a set of difference equations. Baxter-Kelland formula was also reproduced in [3] on the basis of vertex operator approach. The explicit expression for the spontaneous polarization of (Z/nZ)-symmetric model was found in [12], which was reproduced in [8], on the basis of vertex operator approach.
Let us mention on the trigonometric limit cases of elliptic vertex model. In [13] the spontaneous polarization formulae of the higher spin analogue of the six vertex model, the trigonometric limit of the eight-vertex model, were obtained by using Bethe Ansatz method. Idzumi [14] reproduced those formulae for spin 1 case in terms of vertex operator formalism. In the critical limit, the spin k 2 (isotropic) Heisenberg spin chain is described by level k Wess-Zumino-Witten model [15], whose central charge is given by c = 3k k+2 . Since c = 1 for the spin 1 2 case, the eight-vertex model can be described in terms of one boson. Spin 1 analogue of the eight-vertex model (twenty-one-vertex model) can be described in terms of one boson and one fermion, because c = 3 2 = 1 + 1 2 for k = 2. Actually, Idzumi [14], Bougourzi and Weston [16] constructed level 2 irreducible highest weight representations of the quantum affine Lie algebra U q ( sl 2 ) in terms of one boson and one fermion.
Let us turn to the elliptic case. Baxter [17,18,19] found the vertex-face transformation which relates the eight-vertex model and the SOS model. Boos et al. [20] proposed a conjectural formula for multipoint correlation functions of the Z-invariant (inhomogeneous) eight-vertex model. The restricted SOS (RSOS) model was constructed in [21]. The higher spin generalization of RSOS model was introduced in [22,23] on the basis of the fusion procedure. Kojima, Konno and Weston [6] constructed vertex operator formalism for the higher spin analogue of the eight-vertex model, by using vertex-face transformation onto k × k fusion SOS model.
The present paper is organized as follows. In section 2 we review the basic objects of the twenty-onevertex model, the corresponding fusion face model [22,23], the vertex-face correspondence of these two model, and the tail operators which translate correlation functions of fusion SOS model into those of the twenty-one-vertex model. Some detail definitions of the models concerned are listed in Appendix A. In section 3 we introduce a field representation for 2×2 fusion SOS model. The type I vertex operators, the tail operators and the CTM Hamiltonian can be realized in terms of bosons and fermions. Correlation functions of the twenty-one-vertex model can be obtained by these objects, in principle. Section 4 is devoted to derivation of the spontaneous polarization of the twenty-one-vertex model. Useful operator product expansion (OPE) formulae and commutation relations for basic operators are given in Appendix B. Some limiting cases are considered in order to examine the validity of the formula. In particular, the result is compared with that of the trigonometric model obtained by Idzumi [14]. Furthermore, we present the corresponding formula for the inhomogeneous case. In section 5 we give some concluding remarks.

Basic objects
The present section aims to formulate the problem, thereby fixing the notation.

Spin 1 analogue of the eight-vertex model
The twenty-one-vertex model is constructed from the original spin 1 2 eight-vertex model by fusion procedure. Let be the R-matrix of the eight-vertex model. Non-zero elements of the R-matrix are given as follows: be the twenty-one-vertex model. This R (1,1) (u) can be obtained from R(u) in terms of fusion procedure.
The following property is important in the fusion procedure. The explicit expressions of the matrix elements of R-matrix of the twenty-one-vertex model are given in Appendix A.
We assume that the parameters u, ǫ and r on (2.4) and (A.2) lie in the so-called principal regime: This is the antiferroelectric region of the parameters. The twenty-one-vertex model has three kinds of ground states labeled by i for i = 0, 1, 2. Accordingly, there are three spaces of physical states H (i) (i = 0, 1, 2). Here, the space H (i) is the C-vector space spanned by the half-infinite pure tensor vectors of the forms v s1 ⊗ v s2 ⊗ v s3 ⊗ · · · with s j ∈ {−1, 0, 1}, for j = 1, 2, 3, · · · (2.9) and Note that H (i) is isomorphic to the level 2 highest weight module of affine Lie algebra A 1 , with the highest wight respectively. Here, Λ i 's (i = 0, 1) denote the fundamental weights of A 1 . Let H * (i) be the dual of H (i) spanned by the half-infinite pure tensor vectors of the forms Let us consider the so-called low temperature limit x → 0 of (A.2) with ζ = x u be fixed. Then the R (1,1) (u) behaves as Thus, the South-East corner transfer matrix behaves 15) in the low temperature limit x → 0, where We assume that (2.15) is valid not only for low temperature limit x → 0 but also for finite 0 < x < 1 1 .
Likewise other three types of the corner transfer matrices are introduced as follows: where NE, NW and SW stand for the corners North-East, North-West and South-West. It seems to be rather general [1] that the product of four CTMs in the infinite lattice limit is independent of u: The trace of ρ (i) coincides with the principally specialized character of λ i , up to some factors [24]: Introduce the type I vertex operator by the following half-infinite transfer matrix Then the operator (2.20) is an intertwiner from H (i) to H (2−i) . The type I vertex operators satisfy the following commutation relation: Furthermore, the type I vertex operator Φ j (u) and ρ (i) introduced on (2.18) satisfy the homogeneity  Boltzmann weights are given as follows: Here we again assume that the parameters u, ǫ and r on (2.23) and (A.3) lie in (2.8). This region of the parameters is called regime III in the SOS-type model. For k, l ∈ Z and i = 0, 1, 2, let H (i) l,k be the space of admissible paths (k 0 , k 1 , k 2 , · · · ) such that k 0 = k, k j+1 ∼ k j for j = 0, 1, 2, 3, · · · , (2.24) and Also, let H * (i) l,k be the space of admissible paths (· · · , k −2 , k −1 , k 0 ) such that and After gauge transformation [22,23], the Boltzmann weights W 22 (u) in the so-called low temperature limit x → 0 behave as (2.28) Here we take the limit x → 0 with ζ = x u be fixed. Let A l.k and D (i) l.k be the SE, SW, NW, NE corner transfer matrix. Then the SE corner transfer matrix behaves as follows: Likewise other three types of the corner transfer matrices are introduced as follows: (2.31) It seems to be rather general [1] that the product of four CTMs in the infinite lattice limit is independent of u: l,k can be given as follows [25]: (2.33) Here c λi λj (x 4 ) is the string function [26], up to some factors. We change the definitions for the present purpose as follows: (2.34) Besides (2.34) we have the following symmetry Introduce the type I vertex operator by the following half-infinite transfer matrix Then the operator (2.37) is an intertwiner from H l,k ′ . The type I vertex operators satisfy the following commutation relation: (2.38) The free field realization of Φ(u) b a was constructed in [6]. See section 3.2. Furthermore, the type I vertex operator Φ(u) k ′ k and ρ (i) l,k introduced on (2.32) satisfy the homogeneity relation

Vertex-face correspondence
Baxter [17,18,19] introduced the intertwining vectors which relate the eight-vertex model and the SOS model. Let where the scalar function f (u) satisies Then the following relation holds: (cf. Figure 1) Note that the present intertwining vectors are different from the ones used in [17,18,19], which relate the R-matrix of eight-vertex model in the disordered phase and Boltzmann weights W of A (1) n−1 -model in the regime III.

Let
The explicit expressions of L are given in Appendix A.

Tail operators and commutation relations
Tail operators were originally introduced in [3,5], in order to translate correlation functions of the eight-vertex model into those of SOS model. Tail operators for higher spin case were constructed in [6], and those for higher rank case were constructed in [8,9].
Let us introduce the intertwining operators between H (i) and H (i) l,k : (2.47) From (A.5) and (A.8) the following intertwining relations hold: Tail operator is defined by the product of these two objects (see Figure 4): From (2.48), (2.49) and (2.50), we have In this paper we only need Λ(u 0 ) lk ′ lk , which is diagonal with respect to the boundary conditions. In what follows we suppress l-dependence to denote Λ(u 0 ) lk ′ lk by Λ(u 0 ) k ′ k . From (2.47), (2.50) and (2.44) we have It is obvious from (2.45), we have The relation (2.36) implies that (2.54) Insert unity (2.53) into the RHS of (2.54). Then we have Thus in what follows we assume that

Free filed realization
One of the most standard ways to calculate correlation functions and form factors is the vertex operator approach [2] on the basis of free field representation. The face type elliptic quantum group B q,λ ( sl 2 ) was introduced in [27]. The elliptic algebra U q,p ( sl 2 ) associated with fusion SOS models was defined in [28], and its free field representations were constructed in [28,29]. Using these representations we derive the free field representation of the tail operator in this section.

Bosons and fermions
Let us consider the bosons β m (m ∈ Z\{0}) with the commutation relations . The relation between the present β m and the bosons a m in [6] is as follows: We will deal with the bosonic Fock spaces F (i) l,k , (l, k ∈ Z) generated by β −m (m > 0) and e α over the vacuum vectors |l, k : Let K and L be the operators which act diagonally on F (i) l,k : Furthermore, let us consider the fermions with the anticommutation relations We refer to φ m 's for m ∈ Z + 1 2 as Neveu-Schwarz fermions, and φ m 's for m ∈ Z as Ramond fermions. Let be the fermionic Fock space.
Note that the following anticommutation relation holds: Here we use

Free field realization of type I vertex operators
Let us introduce the following basic operators where z = x 2u , w = x 2v . As for some useful OPE formulae and commutation relations, see Appendix B.
Then the type I vertex operators (half transfer matrices) on H (i) l,k can be realized in terms of bosons and fermions: where w j = x 2vj and Considering the denominators [v j − u − 1]'s together with the OPE formulae (B.2), the expressions (3.8) has poles at w j = x ±(2+2nr) z (n ∈ Z 0 ). The integral contour C for w j -integration is the anti-clockwise circle such that all integral variables lie in the common convergence domain; i.e., the contour C encircles the poles at w j = x 2+2nr z (n ∈ Z 0 ), but not the poles at w j = x −2−2nr z (n ∈ Z 0 ).
Note that These type I vertex operators satisfy the following commutation relations on H (i) l,k : (3.11) Dual vertex operators are likewise defined as follows: (3.12) Here the normalization factor can be determined as such that Φ(u) k ′ k and Φ * (u) k k ′ satisfy the inversion relation: As explained below (3.9), the integral contour C = C u actually depends on u. On eqs. (3.12) the w j -integration contour C u−1 of X(u − 1) encircles the poles at z j = x 2nr z (n ∈ Z 0 ), but not the poles (3.14) A level 2 representation of the elliptic algebra U x,p ( sl 2 ) was obtained in terms of one free boson and one free fermion in [30].

Free field realization of tail operators
Another ingredient of the present scheme is the tail operators Λ(u 0 ) k ′ k . In this paper we use a different normalization from the one used in [6]. Thus we briefly explain how to derive free field representations of Λ(u 0 ) k ′ k . When k ′ k − 2, let us consider (2.51) for (a, b, c) = (k, k + 2, k ′ ): It follows from (A.9) that L(u 0 − u) has simple poles at u 0 − u = ± 1 2 . Note that for k ′′ = k ′ , k ′ ± 2 are all equal. Thus if we assume that the LHS of (3.15) has no pole at u 0 = u − 1 2 , we have the following necessary conditions: i.e., Then the LHS of (3.17) contains Λ(u− 1 2 ) k k = 1. By changing k ′ = k−2, k−4, k−6, · · · , we can solve (3.17) iteratively as follows: Here we use the identity: Furthermore, we can check that (3.18) for generic u 0 satisfies (3.15).

Free field realization of CTM Hamiltonian
We can realize the CTM Hamiltonian of 2 × 2 fusion SOS model in terms free fields as follows: Secondly, the traces on the bosonic/fermionic Fock space are given as follows: which implies (2.54). From these checks we conclude that H 4 Spontaneous polarization

Integral formulae
We are now in a position to calculate the spontaneous polarization, i.e., the vacuum expectation value of S z at the center site of the lattice: for i = 0, 1, 2. It is difficult to simplify the sum of (4.2). We thus use another representation of (4.1): Using this trick, we have for a = 1, 2.
First consider S z

(1)
i . Since the twenty-one-vertex model does not contain the parameter u 0 , a correlation function such as (4.5) should be u 0 -independent. For simplicity of calculation, let u 0 = u − r+1 2 . Then from j1=±1 we can express (4.5) as follows: ±l is a contribution from k 2 = k ∓ 2 2 . By using j1=±1 , the H (i) l after taking the sum over k 1 reduces to where w j = x 2vj , and is symmetric with respect to v 1 and v 2 , and also note that the integral contours for w 1 and w 2 are the same. Owing to the commutation relation (B.8), the term proportional to h(v 1 , v 2 ; u, k) on (4.8) vanishes after the integrals. Thus we have whereg The expression (4.7) is due to the symmetry of the space of states H Let us recall (3.6), and first perform the trace on the fermionic Fock space F φ . Simple calculation shows Secondly, let us perform the trace on the bosonic Fock space F (i) l,k . By using OPE formulae (B.1-B.5) we have

and A(v) denotes the fermion contraction
What we want to calculate is the following trace:  Multiply x 4mn by the coefficient of β n −m |l, k on (4.14), and take the sum with respect to n. Then we obtain Hence, by performing the trace on the F and F (i) (w 1 , w 2 ) denotes the Fermionic trace derived on (4.10).
Next, let us take the sum over k on (4.17).  Here, the order of the sum over k and n can be exchanged because the double sums absolutely converge.
When i = 0, 2, the explicit form of (4.18) is as follows: Here, add the corresponding term on H  Then we obtain the following expression for i = 0, 2: . .
Thus we obtain the following integral formulae of S z 1 i for i = 0, 2: .

Some limiting cases
Let i = 2 and perform the integral once on (4.22). Then we have A very analogous result to (4.25) was reported by H. Konno in [31]. Konno's formula is given in terms of a contour integral of a monomial of a product of elliptic functions, whereas our formula (4.25) is that of a binomial of the products of elliptic functions.
In order to examine the validity of the formulae, let us consider some limiting cases. We derived (4.25) under the assumption of r > 2. However, this expression is well defined even for r = 2. Let r = 2. Then the model describes the spin 1 analogue of the Ising model, and the expression (4.25) can be simplified as follows: . (4.26) Note that by transforming w 1 = x −2 w ′ 1 , the integral (4.26) reduces to the one along the contour x 2 C with changing the sign of the integrand. Thus (4.26) can be evaluated by the half of the residue at The expression (4.25) can be also simplified when r is an odd integer. In this case the sign of the integrand changes under the transformation v 1 → v 1 + r. Thus the (4.25) can be evaluated by the half of the sum of residues on the annulus between C and x 2r C. On this annulus, there exist simple poles at x 2n z (1 n r − 2), ±x r z and ±x r+2 z. Thus we obtain 'the sum of products formulae' for odd integers r's.
When r → 2 + 0 (ν → 1 2 ), we should let β be located above the real axis. The expression (4.28) in the limit r → 2 + 0 reduces to 29) which is consistent with (4.27) in the limit ǫ → +0. This is due to the following approximation property Thus, the order of the double limits r → 2 + 0 and ǫ → +0 are commutative.
Finally, let us consider the trigonometric limit r → ∞. This limit describes the nineteen-vertex model in the antiferroelectric regime, whose integral formulae for correlation functions were obtained by Idzumi [14], Bougourzi and Weston [16]. The spontaneous polarization of the twenty-one-vertex model (4.25) in the limit r → ∞ reduces to: This expression should give another integral formula for the spontaneous polarization of the nineteenvertex model. The terms proportional to −(1 + x 2 ) on (4.31) can be evaluated by the half of the residue at w 1 = z. Furthermore, the term proportional to x 2 w1 z on (4.31) can be formally evaluated by the sum of the residues at w 1 = x 2n z for n 0. Concerning the term proportional to z w1 , the sequence of the residues inside the contour C diverge so that we can evaluate the integral formally by the sum of the residues outside C, i.e., those at w 1 = x −2n z for n 1. Thus we have Idzumi performed his integral formula up to x 150 to obtain the following infinite product representation [14]: Unfortunately, (4.32) done not coincide with (4.33). We have no resolution of this mystery at present.

Inhomogeneous model
In this subsection let us consider the inhomogeneous model, and let us calculate the following quantity: (4.34) As was done in section 4.1, we divide S z 1 i (u 1 , u 2 ) into two parts, S z For the former part we put u 0 = u 2 − r+1 2 , and for the latter part we put ǫr . By repeating the same procedure we obtain where u 12 = u 1 − u 2 . By transforming v 1 → v 1 + u 2 , (4.35) actually depends only on u 12 , i.e., S z 1 2 (u 1 , u 2 ) = S z 1 2 (u 12 ). Furthermore, S z 1 2 (0) coincides the expression for spontaneous polarization (4.25), as expected.

Calculation of S x i
In this paper we presented vertex operator approach for correlation functions of spin 1 analogue of the eight-vertex model, the twenty-one-vertex model. Using the present formalism, we can obtain the integral formulae for any correlation functions of the twenty-one-vertex model, in principle. However, it is very difficult to perform the sum over k's, the local states of the fusion SOS model, for general n-point functions with n 2.
In order to show the relevance of our formalism, let us calculate S x i for i = 0, 1, 2, the vacuum expectation value of S x at the center site of the lattice: and We repeat the similar calculation as was done for S z for the calculation i . In this case we should replaceg(v 1 , v 2 ; u, k) on (4.9) bỹ .
for the calculation of S x 2 (1) i . In this case we should replace g(v 1 , v 2 ; u, k) on (4.9) byg Thus, the sum over k on (4.38) can be reduced to (4.18) with {s − k} replaced by Here note that s = v 1 + v 2 − 2u + 1. Accordingly, the corresponding equality (4.20) for the calculation of S x 1 i should be given by for p = 0, 1. Therefore we obtain the expected result for S x 1 i : (4.39)

Concluding remarks
In this paper we have derived integral formulae for the spontaneous polarization of the twenty-one-vertex model. For that purpose we constructed the free field representations of type I vertex operators Φ(u) k ′ k in 2 × 2 fusion SOS model, the tail operators Λ(u 0 ) k ′ k and the corner transfer Hamiltonian H (i) l,k . Our integral formulae are given by (4.22-4.25). The formula (4.22) is given in terms of the two-fold integral. By performing the integral once, we further obtain the one-fold integral formula (4.25). We examined the validity of our results by considering some limiting case. When r → 2 + 0 we obtain an infinite product representation by performing the remained one-fold integral. When r is an odd integer our formula (4.25) reduces to the sum of infinite product formulae. The critical limit ǫ → +0 was considered to obtain the spontaneous polarization of the twenty-one-vertex model in the massless regime. We take the trigonometric limit r → ∞ to compare with the spontaneous polarization of the nineteen-vertex model obtained by Date et. al. [13] and Idzumi [14]. It is a future issue to show that our formulae reproduce their results in the trigonometric limit. Furthermore, we presented the corresponding formula for the inhomogeneous twenty-one-vertex model.
Our approach is based on some assumptions. We assumed that the vertex operator algebra (2.48-2.49) and (2.56) correctly describes the intertwining relation between the twenty-one vertex model and 2 × 2 fusion SOS model. We also assumed that the free field representations (3.18) and (3.20-3.21) provide relevant representations of the vertex operator algebra. Using the present formalism, we can obtain the integral formulae for any correlation functions of the twenty-one-vertex model, in principle.
However, it is very difficult to perform the sum over k's, the local states of the fusion SOS model, for general n-point functions with n 2. As a consistency check of our formalism, we thus calculate the vacuum expectation value S x 1 i to derive that S x 1 i = 0, as expected. In order to derive integral formulae for form factors of the twenty-one-vertex model, we need a free field representation of Λ(u 0 ) l ′ k ′ lk , non-diagonal components of the tail operator with respect to the ground state sectors. We wish to address this issue in a separate paper.
Note that some of components are modified by symmetrization of the R-matrix.
In this article we assume that the parameters v, ǫ and r lie in the so-called principal regime (2.8).

A.2 Boltzmann weights of 2 × 2 fusion SOS model
In what follows we use the following symbols:   Note that some of weights are modified by symmetrization of the Boltzmann weights. In this paper we consider so-called Regime III in the model, i.e., 0 < u < 1.

A.3 Fused intertwining vectors
For k ′ = k, k ± 2, let Then the following relation holds: