Form factors in quantum integrable models with GL(3)-invariant R-matrix

We study integrable models solvable by the nested algebraic Bethe ansatz and possessing GL(3)-invariant R-matrix. We obtain determinant representations for form factors of off-diagonal entries of the monodromy matrix. These representations can be used for the calculation of form factors and correlation functions of the XXX SU(3)-invariant Heisenberg chain.


Introduction
The calculation of correlation functions in quantum integrable models is a very important and complex problem. A form factor approach is one of the most effective methods for solving this problem. For this reason, the study of form factors of local operators has attracted the attention of many authors. There are different methods to address the problem of the calculation of form factors. In the integrable models of the quantum field theory there exists the 'form factor bootstrap approach' [1][2][3][4][5][6][7]. It is based on a set of form factors axioms [2], which represent a set of difference equations that define specific analytic properties of form factors. These equations can be solved to provide the integral representations for form factors. The form factor bootstrap program is closely related to the approach based on the conformal field theory and its perturbation [8][9][10][11]. A purely algebraic method to calculate form factors in the infinite chain spin models was developed by the Kyoto group [12][13][14] using the representation theory of quantum affine algebras. This approach also yields integral formulas for the form factors of the local operators in such models. An alternative way to calculate form factors in the spin chain models was developed by the Lyon group after the inverse scattering problem was solved and local operators in the spin chain models were expressed in terms of the monodromy matrix elements [15]. In this framework one can obtain determinant formulas for the form factors of local spin operators. These determinant representations appeared to be very effective for the calculation of correlation functions [16][17][18].
In this article we try to address this problem and continue our study of form factors in GL(3)-invariant quantum integrable models solvable by the algebraic Bethe ansatz [19][20][21][22][23]. More precisely, we calculate matrix elements of the monodromy matrix entries T ij (z) with |i − j| = 1 between on-shell Bethe vectors (that is, the eigenstates of the transfer matrix). Recently, in the work [24], we obtained determinant representations for form factors of the diagonal elements T ii (z) (i = 1, 2, 3) of the monodromy matrix. Our method was based on the use of a twisted monodromy matrix [25][26][27]. However this approach fails if we deal with form factors of off-diagonal matrix elements. In this last case, one has to apply a more general method, which is based on the explicit calculation of the action of the monodromy matrix entries onto Bethe vectors. As we have shown in [28] this action gives a linear combination of Bethe vectors. Then the resulting scalar products can be evaluated in terms of sums over partitions of Bethe parameters [29].
The form factors of the monodromy matrix entries play a very important role. For a wide class of models, for which the inverse scattering problem can be solved [15,30], such matrix elements can be easily associated with form factors of local operators [15]. In particular, if E α,β m , α, β = 1, 2, 3, is an elementary unit ( E α,β jk = δ jα δ kβ ) associated with the m-th site of the SU (3)-invariant XXX Heisenberg chain, then E α,β m = (tr T (0)) m−1 T βα (0)(tr T (0)) −m . (1.1) Since the action of the transfer matrix tr T (0) on on-shell Bethe vectors is trivial, we see that the form factors of E α,β m are proportional to those of T βα . Thus, if we have an explicit and compact representations for form factors of T ij , we can study the problem of two-point and multi-point correlation functions, expanding them into series with respect to the form factors.
We have mentioned already that the problem considered in this paper is closely related to the calculation of Bethe vectors scalar products. In these scalar products, one of the vectors is on-shell, while the other one is off-shell (that is, this vector generically is not an eigenstate of the transfer matrix). A determinant representation for such type of scalar product was obtained in [31] for GL(2)-based models. This representation allows one to obtain various determinant formulas for form factors. Unfortunately, so far an analog of this determinant formula is not known in the case of integrable models based on the GL(3) symmetry. In our previous publications [24,32] we argued that such an analog hardly exists for the scalar products involving on-shell Bethe vector and a generic off-shell Bethe vector. However, calculating the form factors of the operators T ij we obtain scalar products involving very specific off-shell Bethe vectors. For such particular cases of scalar products we succeed to find representations in terms of a determinant, which is analogous to the determinant representation of [31].
The article is organized as follows. In section 2 we introduce the model under consideration and describe the notation used in the paper. We also give there explicit formulas for the scalar product of Bethe vectors obtained in [29] and explain the relationship between different form factors. In section 3 we formulate the main results of this paper. Section 4 is devoted to the derivation of the determinant representation for the form factor of the operator T 12 . In section 5 we prove the results for form factors of other operators T ij with |i−j| = 1. Appendix A contains the properties of the partition function of the six-vertex model with domain wall boundary conditions and several summation identities for it.
The models considered below are described by a GL(3)-invariant R-matrix acting in the tensor product of two auxiliary spaces In the above definition, I is the identity matrix in V 1 ⊗ V 2 , P is the permutation matrix that exchanges V 1 and V 2 , and c is a constant. The monodromy matrix T (w) satisfies the algebra 2) holds in the tensor product V 1 ⊗V 2 ⊗H, where V k ∼ C 3 , k = 1, 2, are the auxiliary linear spaces, and H is the Hilbert space of the Hamiltonian of the model under consideration. The matrices T k (w) act non-trivially in V k ⊗ H. The trace in the auxiliary space V ∼ C 3 of the monodromy matrix, tr T (w), is called the transfer matrix. It is a generating functional of integrals of motion of the model. The eigenvectors of the transfer matrix are called on-shell Bethe vectors (or simply on-shell vectors). They can be parameterized by sets of complex parameters satisfying Bethe equations (see section 2).

Notations
We use the same notations and conventions as in the paper [24]. Besides the function g(x, y) we also introduce a function f (x, y) Two other auxiliary functions will be also used .
The following obvious properties of the functions introduced above are useful: (2.5) Before giving a description of the Bethe vectors we formulate a convention on the notations. We denote sets of variables by bar:w,ū,v etc. Individual elements of the sets are denoted by subscripts: w j , u k etc. Notationx + c means that the constant c is added to all the elements of the setx. Subsets of variables are denoted by roman indices:ū I ,v iv ,w II etc. In particular, the notationū ⇒ {ū I ,ū II } means that the setū is divided into two disjoint subsetsū I andū II . We assume that the elements in every subset of variables are ordered in such a way that the sequence of their subscripts is strictly increasing. We call this ordering natural order.
In order to avoid too cumbersome formulas we use shorthand notations for products of functions depending on one or two variables. Namely, if functions g, f , h, t, as well as r 1 and r 3 (see (2.10)) depend on sets of variables, this means that one should take the product over the corresponding set. For example, In the last equation of (2.6) the setū is divided into two subsetsū I ,ū II , and the double product is taken with respect to all u k belonging toū I and all u j belonging toū II . We emphasize once more that this convention is only valid in the case of functions, which are by definition dependent on one or two variables. It does not apply to functions that depend on sets of variables. One of the central object in the study of scalar products of GL(3) invariant models is the partition function of the six-vertex model with domain wall boundary conditions (DWPF) [25,33]. We denote it by K n (x|ȳ). It depends on two sets of variablesx andȳ; the subscript shows that #x = #ȳ = n. The function K n has the following determinant representation [33] where ∆ ′ n (x) and ∆ n (ȳ) are It is easy to see that K n is symmetric overx and symmetric overȳ, however K n (x|ȳ) = K n (ȳ|x). Below we consider K n depending on combinations of sets of different variables, for example K n (ξ|{ᾱ,β +c}). Due to the symmetry properties of DWPF K n (ξ|{ᾱ,β +c}) = K n (ξ|{β +c,ᾱ}).

Bethe vectors
Now we pass to the description of Bethe vectors. A generic Bethe vector is denoted by B a,b (ū;v). It is parameterized by two sets of complex parametersū = u 1 , . . . , u a andv = v 1 , . . . , v b with a, b = 0, 1, . . . . Dual Bethe vectors are denoted by C a,b (ū;v). They also depend on two sets of complex parametersū = u 1 , . . . , u a andv = v 1 , . . . , v b . The state withū =v = ∅ is called a pseudovacuum vector |0 . Similarly the dual state withū =v = ∅ is called a dual pseudovacuum vector 0|. These vectors are annihilated by the operators T ij (w), where i > j for |0 and i < j for 0|. At the same time both vectors are eigenvectors for the diagonal entries of the monodromy matrix where λ i (w) are some scalar functions. In the framework of the generalized model, λ i (w) remain free functional parameters. Actually, it is always possible to normalize the monodromy matrix T (w) → λ −1 2 (w)T (w) so as to deal only with the ratios If the parametersū andv of a Bethe vector 2 satisfy a special system of equations (Bethe equations), then it becomes an eigenvector of the transfer matrix (on-shell Bethe vector). The system of Bethe equations can be written in the following form: (2.11) These equations should hold for arbitrary partitions of the setsū andv into subsets {ū I ,ū II } and {v I ,v II } respectively. Obviously, it is enough to demand that the system (2.11) is valid for the particular case when the setsū I andv I consist of only one element. Then (2.11) coincides with the standard form of Bethe equations [23]. Ifū andv satisfy the system (2.11), then

Scalar products and form factors
The scalar products of Bethe vectors are defined as (2.14) We use here superscripts B and C in order to distinguish the sets of parameters entering these two vectors. In other words, unless explicitly specified, the variables {ū B ;v B } in B a,b and {ū C ;v C } in C a,b are not supposed to be related. Before giving an explicit formula for the scalar product we introduce the notion of highest coefficient Z a,b (t;x|s;ȳ). This function depends on four sets of variables with cardinalities #t = #x = a, #s = #ȳ = b, and a, b = 0, 1, . . . . There exist several explicit representations for the highest coefficient in terms of DWPF [34,35]. In this paper we use two of them. The first one reads The sum is taken with respect to all partitions of the setw into subsetsw I andw II with #w I = b and #w II = a.
The second representation has the following form: The sum is taken with respect to all partitions of the setᾱ into subsetsᾱ I andᾱ II with #ᾱ I = a and #ᾱ II = b. The scalar product (2.14) is a bilinear combination of the highest coefficients. It was calculated in the work [29] Here the sum is taken over the partitions of the setsū C ,ū B ,v C , andv B : (2.18) The partitions are independent except that #ū B I = #ū C I = k with k = 0, . . . , a, and #v B I = #v C I = n with n = 0, . . . , b.
In this formula the parametersū C ,ū B ,v C , andv B are arbitrary complex numbers, that is B a,b (ū B ;v B ) and C a,b (ū C ;v C ) are generic Bethe vectors. If one of these vectors, say C a,b (ū C ;v C ), is on-shell, then the parametersū C andv C satisfy the Bethe equations. In this case one can express the products r 1 (ū C II ) and r 3 (v C II ) in terms of the function f via (2.11). Form factors of the monodromy matrix entries are defined as The parameter z is an arbitrary complex number. Acting with the operator T ij (z) on B a,b (ū B ;v B ) via formulas obtained in [28] we reduce the form factor to a linear combination of scalar products, in which C a ′ ,b ′ (ū C ;v C ) is on-shell vector.

Relations between form factors
Obviously, there exist nine form factors of T ij (z) in the models with GL(3)-invariant R-matrix. However, not all of them are independent. In particular, due to the invariance of the R-matrix under transposition with respect to both spaces, the mapping defines an antimorphism of the algebra (2.2). Acting on the Bethe vectors this antimorphism maps them into the dual ones and vice versa and hence, the form factor F One more relationship between form factors arises due to the mapping ϕ: that defines an isomorphism of the algebra (2.2) [28]. This isomorphism implies the following transform of Bethe vectors: Since the mapping ϕ connects the operators T 11 and T 33 , it also leads to the replacement of functions r 1 ↔ r 3 . Therefore, if B a,b (ū;v) and C a,b (ū;v) are constructed in the representation V r 1 (u), r 3 (u) , when their images are in the representation V r 3 (−u), r 1 (−u) . Thus, (2.26)

Main results
The main example considered in this paper is the form factor F In order to describe the determinant representation for this form factor we first of all introduce a set of variablesx = {x 1 , . . . , x a ′ +b } as the union of three sets and define a scalar function H Then for general a and b we introduce an a,b (z) admits the following determinant representation: The proof of this Proposition is given section 4. Remark 1. The order of the elements in the setx is not essential, because the prefactor ∆ a ′ +b (x) and det a ′ +b N (1,2) are antisymmetric under permutations of any two elements ofx. We used the ordering as in (3.2), because it is more convenient for the derivation of the determinant representation (3.6) Remark 2. It is straightforward to check that due to (2.13) the entries of the matrix N (1,2) are proportional to the Jacobians of the transfer matrix eigenvalues In this sense the representation (3.6) is an analog of the determinant representations for form factors in the GL(2)-based models [15]. In particular, at b = 0 the equation (3.6) reproduces the result of [15]. Determinant representations for other form factors F a,b (z) with |i − j| = 1 can be derived from (3.6) by the mappings (2.23), (2.26). First, we give the explicit formulas for the form factor of the operator T 23 where a ′ = a and b ′ = b + 1.
We introduce a set of variablesȳ = {y 1 , . . . , y a+b ′ } as the union of three sets and a function H Then for general a and b we define (3.14) a,b (z) admits the following determinant representation: Similarly to the case considered in Proposition 3.1, the order of the elements in the setȳ is not essential, and the entries of the matrix N (2,3) can be expressed in terms of the Jacobians of the transfer matrix eigenvalues. a,b (z) admits the following determinant representation: a ′ ,b and N (1,2) are given by (3.3) and (3.6) respectively. The form factor F (2,1) a,b (z) admits the following determinant representation: a,b ′ and N (2,3) are given by (3.12) and (3.15) respectively. The proofs of Proposition 3.2 and Proposition 3.3 are given section 5. Remark. We would like to stress that although the representations (3.17) and (3.18) formally coincide with (3.6) and (3.15), the values of a ′ and b ′ in these formulas are different. Indeed, one has a ′ = a + 1 and (3.18). Therefore, in particular, the matrices N (1,2) and N (2,3) in (3.6) and (3.15) have a size (a + b + 1) × (a + b + 1), while in the equations (3.17) and (3.18) the same matrices have a size (a + b) × (a + b).

Derivation of the determinant representation
In this section we prove the determinant representation (3.6) for the form factor of the operator T 12 (z). We use the same technique as in the work [32].
First of all we need a formula for the action of T 12 on the Bethe vectors [28] Here {v, z} =ξ and {ū, z} =η. The sum is taken over partitionsξ a,b (z) to a linear combination of scalar products Now we can substitute here the expression (2.17) for the scalar product, replacing there the set u B byη and the setv B byξ ii . Using the Bethe equations for the setū C The sum is taken with respect to partitions: where #η I = #ū C I = k with k = 0, . . . , a + 1; #ξ i = 1; and #ξ I = #v C I = n with n = 0, . . . , b. Substituting here (2.15) for Z a+1−k,n and (2.16) for Z k,b−n we find The sum is taken with respect to the partitions (4.5) and two additional partitions:w ⇒ {w I ,w II } andᾱ ⇒ {ᾱ I ,ᾱ II } with #w I = n and #ᾱ I = k.
Remark. Note that the restrictions on the cardinalities of subsets are explicitly specified by the subscripts of DWPF. For example, the DWPF K k (ū C I |ᾱ I ) is defined only if #ū C I = #ᾱ I = k. Therefore below we do not specify the cardinalities of subsets in separate comments. Now we can apply (A.4) to the terms in the square brackets in the second line of (4.6). The sum with respect to the partitionsū C ⇒ {ū C I ,ū C II } gives Similarly, setting {ξ i ,ξ II } =ξ III we calculate the sum with respect to the partitionsξ III ⇒ {ξ i ,ξ II }: where we have used (2.5). Then (4.6) turns into Now one should distinguish between two cases: z ∈ξ III or z ∈ξ I . In the first case the contribution to the form factor does not depend on r 3 (z), while in the second case it is proportional to r 3 (z). Thus, we can write F (4.10) We will calculate Ω 1 and Ω 2 separately.

The first particular case
Here we consider the case z ∈ξ III . The corresponding contribution Ω 1 to the form factor does not depend on r 3 (z). Therefore without loss of generality below we will set r 3 (z) = 0. We can setξ I =v B I andξ III = {z,v B II }. Then the product f −1 (z,ξ III ) vanishes, however this zero is compensated by the pole of K b−n+1 ({z,ᾱ II − c}|ξ III ) (see (A.1)): Substituting this into (4.9) and using Bethe equations for r 3 (ξ I ) = r 3 (v B I ) we obtain after simple algebra The sum with respect to the partitionsv B ⇒ {v B I ,v B II } (see the terms in the square brackets in (4.12)) can be calculated via (A.4) Thus, we obtain . (4.14) Now it is necessary to specify the partitions of the setsᾱ andw. We set We denote the cardinalities of these subsets as #η j = k j and #v C j = n j , where j = i, ii, iii, iv. Evidently iv j=i k j = a + 1 and iv j=i n j = b. It is also easy to see that (4. 16) In terms of the subsets introduced above the equation (4.14) takes the form Then one should express r 1 (η i ) and r 3 (v C i ) in terms of the Bethe equations. Observe that z / ∈η i , due to the factor f −1 (z,η i ). Therefore the subsetη i consists of the elements u B j only, and one do can use the Bethe equations for r 1 (η i ). Therefore and These expressions should be substituted into (4.17).
Remark. Formally one can also use the Bethe equations for the product r 3 (v C ii ). However it is more convenient to keep this product as it is.
Finally, we introduce new subsets and we denote n I = #η I = #v C I . We draw the readers attention that these new subsets have nothing to do with the subsets used in (4.14). We use, however, the same notation, as we deal with the sum over partitions, and therefore it does not matter how we denote separate terms of this sum.
Then the equation (4.17) can be written in the following form: where we have introduced three new functions: G n I , L a+1 , and M b . Originally all of them are defined as sums over partitions. The function G n I is given by where the sum is taken over partitionsη I ⇒ {η i ,η iv } andv C 25) where the sum is taken over partitionsη II ⇒ {η ii ,η iii }.
Finally, the function M b ({η I ,v C II }|v B ) is given by where the sum is taken over partitionsv C It is straightforward to check that substituting the definitions (4.24)-(4.26) into (4.23) we reproduce (4.17).
It is remarkable that all the sums with respect to partitions in (4.24)-(4.26) can be explicitly computed. The function G n I (η I |v C I ) can be calculated via (A.15) .
Observe that C 1 (v C k ) = 0 due to the factor f −1 (v C , γ). Therefore, dividing in (A.5) the setγ into two subsets {γ I ,γ II } one should consider only the partitions for whichv C I ⊂γ II . It means that actually we deal with the partitions of the subsetη II only. Namely, we can setγ I =η ii and γ II = {v C I ,η iii }. Then the sum in (A.5) coincides with the sum (4.25) and we obtain where the matrix L (1,2) is given by (3.4).
Similarly for the calculation (4.26) one should set in the sum (A.5) . (4.30) Then C 2 (η k ) = 0 either due to the product f −1 (γ,ū B ) or due to the condition r 3 (z) = 0 (that we freely imposed in this subsection). Therefore we can setγ I = {v C iii ,η I } andγ II =v C ii in (A.5). Then the sum (A.5) turns into (4.26) and we obtain where the matrix M (1,2) is given by (3.5). IntroducingL we obtain after simple algebra (4.33) Define a setx as in (3.2) For arbitrary partitionx ⇒ {x I ,x II } with #x I = a + 1 and #x II = b we have where P I,II is the parity of the permutation mapping the sequence {x I ,x II } into the ordered sequence x 1 , . . . , x a+b+1 . Settingx I = {η II ,v C I } andx II = {η I ,v C II } we obtain after elementary algebra . (4.36) Thus, the equation (4.33) can be written in the form Here x I k (resp. x II k ) is the k-th element of the subset x I (resp. x II ). It is easy to see that the prefactor in the first line of (4.37) coincides with the function H (1,2) a+1,b (see (3.3)). The sum (4.37) is nothing but the expansion of the determinant of the (a + b + 1) × (a + b + 1) matrix N (1,2) with respect to the first (a + 1) rows. Thus, we finally obtain . (4.38)

The second particular case
Now we turn back to the equation (4.9) and consider the case z ∈ξ I , that is we compute the term Ω 2 in (4.10). The general idea of the calculation is the same as in the case of Ω 1 , however there are several subtleties. Sinceη = {z,ū B }, the product f −1 (ξ I ,η) vanishes. The only possible way to obtain non-zero contribution to Ω 2 is to compensate this zero by the pole of K n (ξ I |w I ). The last one occurs if and only if z ∈w I , which implies z ∈η II . Thus, we can set Substituting this into (4.9) we obtain where we have also used the Bethe equations for r 3 (v B I ): Applying (A.4) to the terms in the last line of (4.40) we take the sum over partitionsv B ⇒ {v B I ,v B II }: Thus, we arrive at Now we should specify the subsets similarly to (4.15) (4.44) We again denote the cardinalities of the subsets above as #ū B j = k j and #v C j = n j . Now iv j=i k j = a, iv j=i n j = b and (4.45) Using the new subsets we obtain an analog of (4.17) Now one should make the same transforms as before. Namely, we should simplify K n and K b−n via (A.2), (A.3); express r 1 (ū B i ) and r 3 (v C i ) in terms of Bethe equations; introduce new subsetsū Pay attention that now n I = #ū B I + 1 = #v C I . We also introduce z ′ = z + c. Then the equation (4.46) can be written in the following form: Here where the sum is taken over partitionsū B where the sum is taken over partitionsv C II ⇒ {v C ii ,v C iii }. The function G n I (ū B I |v C I ) can be calculated via (A.16) . (4.52) The calculation of L a+1 ({ū B II ,v C I }|ū C ) is the same as the one of L a+1 ({η II ,v C I }|ū C ) (one should only replace everywhereη II byū B II ). The calculation of M b ({ū B I ,v C II , z ′ }|v B ) also is almost the same as before. The difference is that now it depends on additional parameter z ′ . However this difference does not make sense, if we set by definition r 3 (z ′ ) = 0. We can always do it, because the form factor anyway does not depend on r 3 (z ′ ). Thus, we find and Formally, the obtained representations coincide with (4.29), (4.31). However the setsγ are different. In (4.53) the setγ does not contain z, while in (4.29) it could contain z. Respectively, in (4.54) the setγ contains z ′ , while in (4.31) it was z-independent.
and substituting (4.52)-(4.54) into (4.48) we after simple algebra arrive at the analog of (4. 33) . Similarly to (4.34) we introduce a setx ′ as Then the analog of (4.36) has the following form: . (4.58) It is important that unlike the previous case we have #v C I = n I and #ū B I = n I − 1. Therefore, in particular, Thus, the equation (4.56) can be written in the form provided r 3 (z ′ ) = 0. This formula is almost the expansion of the (a+b+1)×(a+b+1) determinant with respect to the first (a + 1) rows. We should take care only about the condition z ′ ∈x ′ II . This can be done if we set by definition L (1,2) (z ′ , u C j ) ≡ 0. We do can impose this constraint, since L (1,2) (z ′ , u C j ) does not enter the formula (4.60). Then we obtain j,a+1 = 0, j = 1, . . . , a + 1, , j = 1, . . . , b.

(4.62)
We see that all the columns of the obtained matrix coincide with the ones of the matrix in (4.38), except the (a + 1)-th column (associated with the parameter z ′ ), where non-zero matrix elements are It is easy to see that Thus we obtain for all the elements of the (a + 1)-th column Hence, we arrive at It remains to combine (4.38) and (4.66). This can be easily done, because for any linear function of φ(ζ) = Aζ + B one has

Other form factors
Consider again the form factor of the operator T 12 Applying the mapping ϕ (2.24) we obtain Thus, in order to obtain the determinant representation for the form factor F (2,3) a,b (z) one should take the resulting formulas for F (1,2) a,b (z), set therẽ and replace the function r 1 by r 3 and vice versa. One can say that the mapping ϕ actually acts on the determinant representation (3.6) via the replacements described above.
Consider how ϕ acts on the prefactor H Hence, we obtain Similarly one can convince oneself that a,b (z) we arrive at (3.15) for F (2,3) a,b (z). Since the mapping ψ yields the same replacements of the parameters, we conclude that applying ψ to F (1,2) a,b (z) and F (2,3) a,b (z) we obtain the determinant representations for F (3,2) a,b (z) and F (2,1) a,b (z) respectively. In this way we prove Proposition 3.3.

Conclusion
In this paper we considered the form factors of the monodromy matrix entries in the models with GL(3)-invariant R-matrix. We obtained determinant representations for the form factors F (i,j) a,b (z) of the operators T ij (z) with |i−j| = 1. In our previous publication [24] we have already calculated the form factors of the diagonal entries T ii (z). Thus, the only unknown remains the form factor F Another possible way to solve the problem is to use the multiple integral representation for scalar products of the Bethe vectors obtained recently in [36]. This representation might be useful for the study of the form factor F Concluding this paper we would like to say few words about possible applications of the results obtained. Models with higher rank symmetries play an important role in condensed matter physics. They appear for instance in two-component Bose (or Fermi) gas and in the study of models of cold atoms (for e.g. ferromagnetism or phase separation). One can also mention 2-band Hubbard models (mostly in the half-filled regime), in the context of strongly correlated electronic systems. In that case, the symmetry increases when spin and orbital degrees of freedom are supposed to play a symmetrical role leading to an SU (4) or even an SO(8) symmetry (see e.g. [37,38]). All these studies require to look for integrable models with SU (N ) symmetry, the first step being the SU (3) case. Compact determinant representations for form factors of the monodromy matrix entries give a possibility to study correlation functions of such models. We have mentioned already in the Introduction that these representations allow one to calculate the correlation functions of integrable spin chains via their form factor expansion. Furthermore, the explicit representations for the form factors also play an important role in the models, for which the solution of the inverse scattering problem is not known (see e.g. [18,39]). In this context it is worth mentioning the work [40], where the form factors in the model of two-component Bose gas were studied.
Apart from condensed matter physics, let us also mention super-Yang-Mills theories. Integrability has proved to be a very efficient tool for the calculation of scattering amplitudes in these models. The calculation of these amplitudes can be related to scalar products of Bethe vectors. In particular, in the SU (3) subsector of the theory, one just needs the SU (3) Bethe vectors. Hence, the knowledge of the form factors is also essential in this context.
Finally, in view of the potential applications, there is reason to wonder whether the results obtained in the present paper could be generalized to the models based on GL(N ) group with N > 3. However, the structure of the obtained determinant representations does not provide obvious clues about their possible generalization to the case N > 3. We would like to be very cautious with any 'obvious' predictions in this field. It is sufficient to recall some conjectures formulated previously on the basis of the results obtained in GL(2)-based models. Indeed, in the case N = 2 the analogs of the form factors considered in the present paper are proportional to the Jacobian of the transfer matrix eigenvalue on one of the vectors. The natural hypothesis was that this structure is preserved in the case N > 2. We see, however, that already for N = 3 the determinant representations have a more complicated structure. In particular they contain the Jacobians of the transfer matrix eigenvalues on both vectors. It is very possible that in the case N > 3, the determinant representations for form factors (if they exist) have more sophisticated structure, that is difficult to see in the case of N = 3. Therefore we believe that the systematic study of the problem of generalization is the only way to solve it. In this context let us quote the work [41] where some preliminary results for GL(N )-based models were obtained.
The proofs of these lemmas is given in [32].
Lemma A.3. Letᾱ andβ be two sets of generic complex numbers with #ᾱ = #β = m, and z is an arbitrary complex. Then where the sum is taken over all possible partitions of the setsᾱ andβ with #ᾱ I = #β I = m I , m I = 0, . . . , m, and #ᾱ II = #β II = m II = m − m I . This lemma is a generalization of the lemma 6.3 of the work [32]. In particular, the statement of the latter can be obtained from (A.6) in the limit z → ∞.