GL(3) -Based Quantum Integrable Composite Models. II. Form Factors of Local Operators

We study integrable models solvable by the nested algebraic Bethe ansatz and possessing the GL(3)-invariant R-matrix. We consider a composite model where the total monodromy matrix of the model is presented as a product of two partial monodromy matrices. Assuming that the last ones can be expanded i...

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Datum:	2015
Hauptverfasser:	Pakuliak, S., Ragoucy, E., Slavnov, N.A.
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Sprache:	English
Veröffentlicht:	Інститут математики НАН України 2015
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1471352019-02-14T01:25:15Z GL(3) -Based Quantum Integrable Composite Models. II. Form Factors of Local Operators Pakuliak, S. Ragoucy, E. Slavnov, N.A. We study integrable models solvable by the nested algebraic Bethe ansatz and possessing the GL(3)-invariant R-matrix. We consider a composite model where the total monodromy matrix of the model is presented as a product of two partial monodromy matrices. Assuming that the last ones can be expanded into series with respect to the inverse spectral parameter we calculate matrix elements of the local operators in the basis of the transfer matrix eigenstates. We obtain determinant representations for these matrix elements. Thus, we solve the inverse scattering problem in a weak sense. 2015 Article GL(3) -Based Quantum Integrable Composite Models. II. Form Factors of Local Operators / S. Pakuliak, E. Ragoucy, N.A. Slavnov // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 17B37; 81R50 DOI:10.3842/SIGMA.2015.64 http://dspace.nbuv.gov.ua/handle/123456789/147135 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We study integrable models solvable by the nested algebraic Bethe ansatz and possessing the GL(3)-invariant R-matrix. We consider a composite model where the total monodromy matrix of the model is presented as a product of two partial monodromy matrices. Assuming that the last ones can be expanded into series with respect to the inverse spectral parameter we calculate matrix elements of the local operators in the basis of the transfer matrix eigenstates. We obtain determinant representations for these matrix elements. Thus, we solve the inverse scattering problem in a weak sense.
format	Article
author	Pakuliak, S. Ragoucy, E. Slavnov, N.A.
spellingShingle	Pakuliak, S. Ragoucy, E. Slavnov, N.A. GL(3) -Based Quantum Integrable Composite Models. II. Form Factors of Local Operators Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Pakuliak, S. Ragoucy, E. Slavnov, N.A.
author_sort	Pakuliak, S.
title	GL(3) -Based Quantum Integrable Composite Models. II. Form Factors of Local Operators
title_short	GL(3) -Based Quantum Integrable Composite Models. II. Form Factors of Local Operators
title_full	GL(3) -Based Quantum Integrable Composite Models. II. Form Factors of Local Operators
title_fullStr	GL(3) -Based Quantum Integrable Composite Models. II. Form Factors of Local Operators
title_full_unstemmed	GL(3) -Based Quantum Integrable Composite Models. II. Form Factors of Local Operators
title_sort	gl(3) -based quantum integrable composite models. ii. form factors of local operators
publisher	Інститут математики НАН України
publishDate	2015
url	http://dspace.nbuv.gov.ua/handle/123456789/147135
citation_txt	GL(3) -Based Quantum Integrable Composite Models. II. Form Factors of Local Operators / S. Pakuliak, E. Ragoucy, N.A. Slavnov // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 39 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT pakuliaks gl3basedquantumintegrablecompositemodelsiiformfactorsoflocaloperators AT ragoucye gl3basedquantumintegrablecompositemodelsiiformfactorsoflocaloperators AT slavnovna gl3basedquantumintegrablecompositemodelsiiformfactorsoflocaloperators
first_indexed	2025-07-11T01:25:41Z
last_indexed	2025-07-11T01:25:41Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 064, 18 pages GL(3)-Based Quantum Integrable Composite Models. II. Form Factors of Local Operators Stanislav PAKULIAK abc, Eric RAGOUCY d and Nikita A. SLAVNOV e a Institute of Theoretical & Experimental Physics, 117259 Moscow, Russia b Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow reg., Russia E-mail: pakuliak@jinr.ru c Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Moscow reg., Russia d Laboratoire de Physique Théorique LAPTH, CNRS and Université de Savoie, BP 110, 74941 Annecy-le-Vieux Cedex, France E-mail: eric.ragoucy@lapth.cnrs.fr e Steklov Mathematical Institute, Moscow, Russia E-mail: nslavnov@mi.ras.ru Received February 18, 2015, in final form July 22, 2015; Published online July 31, 2015 http://dx.doi.org/10.3842/SIGMA.2015.064 Abstract. We study integrable models solvable by the nested algebraic Bethe ansatz and possessing the GL(3)-invariant R-matrix. We consider a composite model where the to- tal monodromy matrix of the model is presented as a product of two partial monodromy matrices. Assuming that the last ones can be expanded into series with respect to the in- verse spectral parameter we calculate matrix elements of the local operators in the basis of the transfer matrix eigenstates. We obtain determinant representations for these matrix elements. Thus, we solve the inverse scattering problem in a weak sense. Key words: Bethe ansatz; quantum affine algebras, composite models 2010 Mathematics Subject Classification: 17B37; 81R50 1 Introduction The algebraic Bethe ansatz was found to be a powerful method for describing the spectrum of various quantum integrable models [7, 23, 33, 36]. In this approach, quantum Hamiltonians and all other integrals of motion are generated by a transfer matrix. The eigenstates of the latest can be found in a systematic way, leading to a set of equations determining the spectrum (Bethe equations). Despite the significant progress of the algebraic Bethe ansatz in calculating the spectrum, the application of this technique to the problem of calculating correlation functions for a long time led to much more limited results. It is worth mentioning the papers [10, 11, 21], where series representations for correlations of the model of one-dimensional bosons were obtained. Later, Fredholm determinant representations for correlation functions of this model were derived by the method of dual fields (see [22] and references therein). Solution of the quantum inverse scattering problem [20, 25] has opened up new opportunities of the algebraic Bethe ansatz. Using this result correlation functions of the XXZ spin chain were studied in series of works [8, 9, 15, 18, 19]. The explicit formulas for the local operators provided by the inverse scattering problem also had played an important role in calculating their form factors [5, 16, 17]. It is worth mentioning, however, that the results of [20, 25] essentially were based on the fact that the monodromy matrix of the model could be constructed from the mailto:pakuliak@jinr.ru mailto:eric.ragoucy@lapth.cnrs.fr mailto:nslavnov@mi.ras.ru http://dx.doi.org/10.3842/SIGMA.2015.064 2 S. Pakuliak, E. Ragoucy and N.A. Slavnov R-matrix. This is true for various spin chains, but not in general. In the present paper we use the approach of [10] for calculating form factors of local operators in quantum GL(3)-invariant models. Let us briefly describe the main idea of this method. The key equation of the quantum inverse scattering method is the RTT -relation [33, 36] R12(u, v)T1(u)T2(v) = T2(v)T1(u)R12(u, v). (1.1) Here T (u) is the monodromy matrix, R(u, v) is the R-matrix. In GL(3)-invariant models the R-matrix acts in the tensor product of two auxiliary spaces V1⊗ V2 (Vk ∼ C3, k = 1, 2) and has the form R(u, v) = I + g(u, v)P, g(u, v) = c u− v . (1.2) Here I is the identity matrix in V1⊗V2, P is the permutation matrix that exchanges V1 and V2, and c is a constant. The monodromy matrix T (u) acts in C3 ⊗H, where H is the Hilbert space of the Hamiltonian of the model under consideration. Equation (1.1) holds in the tensor product V1 ⊗ V2 ⊗H, and the matrices Tk(w) act non-trivially in Vk ⊗H. The monodromy matrix T (u) of a lattice quantum model is equal to the product of local L-operators T (u) = LM (u) · · ·L1(u), (1.3) where M is the number of lattice sites, and every L-operator satisfies the RTT -relation with the R-matrix (1.2). Continuous quantum models appear in the limit M →∞. Let us fix some site m (1 ≤ m < M) and define two partial monodromy matrices T (1)(u) and T (2)(u) as T (1)(u) = Lm(u) · · ·L1(u), T (2)(u) = LM (u) · · ·Lm+1(u). (1.4) Then obviously T (u) = T (2)(u)T (1)(u). (1.5) We call such model composite generalized model1 [10]. We assume that local L-operators in (1.3) depend on the spectral parameter u as follows2: Ln(u) = 1 + c u Ln[0] + o ( u−1 ) , u→∞. Here 1 is the identity operator in C3 ⊗H, and the matrix elements of Ln[0] are local operators of the model. Then it is easy to see that both partial monodromy matrices T (l)(u) have the standard expansion over c/u: T (l)(u) = 1 + ∞∑ n=0 T (l)[n] ( c u )n+1 , l = 1, 2, (1.6) where the partial zero mode T (1)[0] is equal to T (1)[0] = m∑ n=1 Ln[0]. (1.7) 1The authors of [10] used the terminology two-site model. We think that this terminology becomes misleading in the case of spin chains. The terminology two-component model used in [34] also becomes misleading in the case of multi-component Bose or Fermi gases. We choose to use the terminology introduced in [29]. 2See [24] for concrete examples of L-operators satisfying the RTT -relation with the GL(3)-invariant R-matrix. GL(3)-Based Quantum Integrable Composite Models. II. Form Factors of Local Operators 3 In this paper we develop a method of calculating form factors of matrix elements T (1) ij [0]. We reduce them to the form factors of the monodromy matrix entries Tij(z) studied in our previous publications. In this way we obtain determinant representations for the form factors of T (1) ij [0] and extract explicitly their dependence on the lattice site number m. Then, taking the lattice derivative of the results obtained, we find determinant representations for the form factors of the local operators (Lm[0])ij . We do not give here explicit determinant formulas, but the reader can find them in [2, 4, 27, 28, 30]. The paper is organized as follows. In Section 2 we introduce basic notions of the generalized model. In Section 3 we consider composite generalized model. Section 4 contains the main results of this paper. There we reduce form factors of local operators to the ones of the monodromy matrix entries for the GL(3) case and we conjecture a form for the GL(N) case. The following sections contain the proofs for GL(3). In Section 5 we consider the form factors of the diagonal partial zero modes T (1) ii [0]. Finally, in Section 6 we study the form factors of the partial zero modes T (1) ij [0] for i 6= j. 2 Generalized model 2.1 Bethe vectors In the framework of the algebraic Bethe ansatz the entries of T (w) act in a Hilbert space H and its dual H∗ that possess a pseudovacuum vector \|0〉 and a dual pseudovacuum vector 〈0\| respectively. They are normalized by the condition 〈0\|0〉 = 1. These vectors are annihilated by the operators Tij(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 Tii(w)\|0〉 = λi(w)\|0〉, 〈0\|Tii(w) = λi(w)〈0\|, i = 1, 2, 3, where λi(w) are some scalar functions. In the framework of the generalized model, λi(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 r1(w) = λ1(w) λ2(w) , r3(w) = λ3(w) λ2(w) . (2.1) Below we assume that λ2(w) = 1. Bethe vectors are certain polynomials in the operators Tij(u) with i < j acting on the pseudovacuum vector \|0〉 [3, 12, 13, 23, 37]. In the GL(3)-invariant models they depend on two sets of variables called Bethe parameters. We denote the Bethe vectors Ba,b(ū; v̄). Here the Bethe parameters are ū = {u1, . . . , ua} and v̄ = {v1, . . . , vb}. The subscripts a and b (a, b = 0, 1, . . . ) respectively denote the cardinalities of the sets ū and v̄. Similarly we can construct dual Bethe vectors in the dual space as polynomials in the opera- tors Tij(u) with i > j acting on the dual pseudovacuum vector 〈0\|. We denote them Ca,b(ū; v̄) with the same meaning of the arguments and subscripts. 2.2 Notation Besides the function g(u, v) we also introduce a function f(u, v) f(u, v) = u− v + c u− v . (2.2) We denote sets of variables by bar: ū, v̄ etc. If necessary, the cardinalities of the sets are given in special comments. Individual elements of the sets are denoted by subscripts: wj , uk 4 S. Pakuliak, E. Ragoucy and N.A. Slavnov etc. We say that x̄ = x̄′, if #x̄ = #x̄′ and xi = x′i (up to a permutation) for i = 1, . . . ,#x̄. We say that x̄ 6= x̄′ otherwise. Below we consider partitions of sets into subsets. The notation ū⇒ {ūI, ūII} means that the set ū is divided into two disjoint subsets. As a rule, we use roman numbers for subscripts of subsets: ūI, v̄ii etc. However, if we deal with a big quantity of subsets, then we use standard arabic numbers for their notation. In such cases we give special comments to avoid ambiguities. Similarly to our previous papers (see, for instance, [29]) we use a shorthand notation for products of some functions. Namely, if the functions rk (2.1) or the function f (2.2) depend on sets of variables, this means that one should take the product over the corresponding set. For example, r1(ū) = ∏ uk∈ū r1(uk), f(z, w̄) = ∏ wj∈w̄ f(z, wj), f(ū, v̄) = ∏ uj∈ū ∏ vk∈v̄ f(uj , vk). (2.3) By definition any product with respect to the empty set is equal to 1. If we have a double product, then it is also equal to 1 if at least one of sets is empty. In Section 3 we shall introduce several new scalar functions and will extend the conven- tion (2.3) to their products. 2.3 On-shell Bethe vectors In the algebraic Bethe ansatz the role of a quantum Hamiltonian is played by the transfer matrix. It is the trace in the auxiliary space of the monodromy matrix: trT (u). The eigenstates of the transfer matrix are called on-shell Bethe vectors. The eigenstates of the transfer matrix in the dual space are called dual on-shell Bethe vectors3. We will denote usual and dual Bethe vectors as Ba,b(ū, v̄) and Ca,b(ū, v̄) respectively. Different equivalent formulas for these Bethe vectors were presented in the paper [3]. In what follows we will not use these explicit expressions. Instead, we will use formulas which relate Bethe vectors of the composite model with the ones for the components of the model (see formulas (3.4) and (3.6) which are proved in the first part of this paper [29]) and the action of the monodromy matrix elements onto Bethe vectors obtained in [3]. A (dual) Bethe vector becomes on-shell, if the Bethe parameters satisfy the system of Bethe equations. We give this system in a slightly unusual form r1(ūI) = f(ūI, ūII) f(ūII, ūI) f(v̄, ūI), r3(v̄I) = f(v̄II, v̄I) f(v̄I, v̄II) f(v̄I, ū). (2.4) These equations should hold for arbitrary partitions of the sets ū and v̄ into subsets {ūI, ūII} and {v̄I, v̄II} respectively. Obviously, it is enough to demand that the system (2.4) is valid for the particular case, when the sets ūI and v̄I consist of only one element. Then it turns into the standard system of Bethe equations. If the sets ū and v̄ satisfy (2.4), then trT (w)Ba,b(ū, v̄) = τ(w\|ū, v̄)Ba,b(ū, v̄), Ca,b(ū, v̄) trT (w) = τ(w\|ū, v̄)Ca,b(ū, v̄), with τ(w\|ū, v̄) = r1(w)f(ū, w) + f(w, ū)f(v̄, w) + r3(w)f(w, v̄). Besides usual on-shell Bethe vectors it is also convenient to consider twisted on-shell Bethe vectors (see, e.g., [2]). They are eigenstates of a twisted transfer matrix, that in its turn, is the 3For simplicity here and below we do not distinguish between vectors and dual vectors, because their properties are completely analogous to each other. GL(3)-Based Quantum Integrable Composite Models. II. Form Factors of Local Operators 5 trace in the auxiliary space of the twisted monodromy matrix Tκ̄(u). The last one is defined as Tκ̄(u) = κ̂T (u), where κ̂ = diag(κ1, κ2, κ3). The matrix elements κi (i = 1, 2, 3) are called twist parameters. A (dual) Bethe vector becomes twisted on-shell vector, if the Bethe parameters satisfy the system of twisted Bethe equations: r1(ūI) = ( κ2 κ1 )kI f(ūI, ūII) f(ūII, ūI) f(v̄, ūI), r3(v̄I) = ( κ2 κ3 )nI f(v̄II, v̄I) f(v̄I, v̄II) f(v̄I, ū), (2.5) where kI = #ūI and nI = #v̄I. These equations also should hold for arbitrary partitions of the sets ū and v̄ into subsets. 2.4 Description of scalar products Recall a formula for the scalar product of generic Bethe vectors [32]: Sa,b ≡ Ca,b(ūC ; v̄C)Ba,b(ūB; v̄B) = ∑ r1(ūBI )r1(ūCII )r3(v̄CII )r3(v̄BI )f(ūCI , ū C II )f(ūBII , ū B I ) (2.6) × f(v̄CII , v̄ C I )f(v̄BI , v̄ B II ) f(v̄CI , ū C I )f(v̄BII , ū B II ) f(v̄C , ūC)f(v̄B, ūB) ZaII,bI(ū C II ; ūBII \|v̄CI ; v̄BI )ZaI,bII(ū B I ; ūCI \|v̄BII ; v̄CII ). Here all the Bethe parameters are generic complex numbers and the sum is taken over the partitions of the sets ūC , ūB, v̄C , and v̄B ūC ⇒ {ūCI , ūCII }, v̄C ⇒ {v̄CI , v̄CII }, ūB ⇒ {ūBI , ūBII }, v̄B ⇒ {v̄BI , v̄BII }. The partitions are independent except that #ūBI = #ūCI = aI with aI = 0, . . . , a, and #v̄BI = #v̄CI = bI with bI = 0, . . . , b. From this we find #ūBII = #ūCII = aII = a − aI and #v̄BII = #v̄CII = bII = b − bI. The functions ZaII,bI and ZaI,bII are so-called highest coefficients. They are equal to a partition function of 15-vertex model with special boundary conditions [32]. The reader can find their explicit representations in [1, 38]. We do not use these explicit formulas in the present paper except Z0,0(∅;∅\|∅;∅) = 1. This condition is needed to satisfy the normalization S0,0 = 〈0\|0〉 = 1. If Ca,b(ūC ; v̄C) is a twisted on-shell vector and Ba,b(ūB; v̄B) is a usual on-shell vector, then we can express the functions r1(uC,Bj ) and r3(vC,Bj ) in (2.6) in terms of (twisted) Bethe equa- tions (2.4), (2.5). We denote such scalar product by S(κ̄) a,b . It is easy to see that S(κ̄) a,b = ∑( κ2 κ1 )aII ( κ2 κ3 )bII f(ūCII , ū C I )f(ūBI , ū B II )f(v̄CI , v̄ C II )f(v̄BII , v̄ B I ) × f(v̄CII , ū C II )f(v̄BI , ū B I )ZaII,bI(ū C II ; ūBII \|v̄CI ; v̄BI )ZaI,bII(ū B I ; ūCI \|v̄BII ; v̄CII ). (2.7) Remark 2.1. One should be careful using Bethe equations in the sums over partitions, be- cause there might be problems if some parameters from the sets ūC , v̄C coincide with the ones from ūB, v̄B (i.e., ūC∩ ūB 6= ∅ or/and v̄C∩ v̄B 6= ∅). The matter is that after imposing the Bethe equations we cannot consider the limit where one solution of Bethe equations goes to another solution. Instead one should first take the limit in (2.6) (what leads to the appearance of the derivatives r′1(u) and r′3(v)) and only then impose Bethe equations. However, if we consider the scalar product of twisted and usual on-shell vectors, then we can use the twisted and usual Bethe equations from the very beginning. The matter is that in this case the parameters ūC and v̄C are functions of κ̄ = {κ1, κ2, κ3}: ūC = ūC(κ̄) and v̄C = v̄C(κ̄). Therefore we always can take κ̄ such that ūC ∩ ūB = ∅ and v̄C ∩ v̄B = ∅. Then, if necessary, we can consider the limit where some parameters coincide. In this case we should treat the variables ūC and v̄C as functions of κ̄. 6 S. Pakuliak, E. Ragoucy and N.A. Slavnov The sum over partitions (2.7) was studied in [2, 4] for arbitrary values of the Bethe parameters. There this sum was reduced to a single determinant in the case κ1 = κ3. It was proved that the determinant vanishes at4 κ̄ = 1. Thus, if ūC,B 6= ∅ or v̄C,B 6= ∅, then setting κ̄ = 1 in (2.7) we obtain an identity 0 = ∑ f(ūCII , ū C I )f(ūBI , ū B II )f(v̄CI , v̄ C II )f(v̄BII , v̄ B I )f(v̄CII , ū C II )f(v̄BI , ū B I ) × ZaII,bI(ū C II ; ūBII \|v̄CI ; v̄BI )ZaI,bII(ū B I ; ūCI \|v̄BII ; v̄CII ). (2.8) In the exceptional case ūC,B = v̄C,B = ∅ we have S(κ̄) 0,0 ∣∣∣ κ̄=1 = 1. (2.9) Remark 2.2. One should not be surprised that the r.h.s. of (2.8) does not give the norm of an on-shell Bethe vector in the case {ūC , v̄C} = {ūB, v̄B}. Indeed, as we explained above, in order to obtain the norm one should consider the limit {ūC(κ̄), v̄C(κ̄)} → {ūB, v̄B} at κ̄→ 1 in (2.7). Instead we simply set κ̄ = 1 in (2.7) for generic values of the Bethe parameters. In this case we obtain that the sum (2.8) vanishes [2], and then this result can be continued to the point {ūC , v̄C} = {ūB, v̄B}. Identity (2.8) plays a central role in the calculation of form factors of local operators. 2.5 Universal form factors Form factors of the monodromy matrix entries are defined as F (i,j) a,b (z) ≡ F (i,j) a,b (z \| ūC , v̄C ; ūB, v̄B) = Ca ′,b′(ūC ; v̄C)Tij(z)Ba,b(ūB; v̄B), where both Ca′,b′(ūC ; v̄C) and Ba,b(ūB; v̄B) are on-shell Bethe vectors, and a′ = a+ δi1 − δj1, b′ = b+ δj3 − δi3. The parameter z is an arbitrary complex number. It was proved in [30] that if {ūC , v̄C} 6= {ūB, v̄B}, then the combination F (i,j) a,b (ūC , v̄C ; ūB, v̄B) = F (i,j) a,b (z \| ūC , v̄C ; ūB, v̄B) τ(z \| ūC , v̄C)− τ(z \| ūB, v̄B) does not depend on z. We call F (i,j) a,b (ūC , v̄C ; ūB, v̄B) the universal form factor of the operator Tij(z). If ūC ∩ ūB = ∅ and v̄C ∩ v̄B = ∅, then the universal form factor is determined by the R-matrix only. It does not depend on a specific model, in particular, on the functions r1(z) and r3(z). 3 Composite generalized model Consider a composite generalized model defined by (1.4), (1.5). Every T (l)(u) satisfies RTT - relation (1.1) and has its own vacuum state \|0〉(l). Hereby \|0〉 = \|0〉(1) ⊗ \|0〉(2). The opera- tors T (2) ij (u) and T (1) kl (v) commute with each other, as they act in different spaces. Let T (l) ii (u)\|0〉(l) = λ (l) i (u)\|0〉(l), l = 1, 2. (3.1) 4Here and below the notation κ̄ = 1 means κi = 1, i = 1, 2, 3. GL(3)-Based Quantum Integrable Composite Models. II. Form Factors of Local Operators 7 We also introduce r (l) k (u) = λ (l) k (u) λ (l) 2 (u) l = 1, 2, k = 1, 3. Obviously λi(u) = λ (1) i (u)λ (2) i (u), rk(u) = r (1) k (u)r (2) k (u). Below we express form factors in terms of r (1) k (u), therefore we introduce a special notation for these functions r (1) k (u) = `k(u), and hence, r (2) k (u) = rk(u) `k(u) , k = 1, 3. (3.2) Observe that (1.6) implies `k(u) = 1 + `k[0] c u + o ( u−1 ) , (3.3) and due to (1.5) rk(u) = 1 + rk[0] c u + o ( u−1 ) . We extend convention (2.3) to the products of the functions r (l) k (u) and `k(u). Namely, whenever these functions depend on sets of variables (for instance r (l) k (ū) or `k(v̄II)) this means the product over the corresponding set. 3.1 Bethe vectors and partial Bethe vectors We can introduce partial Bethe vectors B(l) a,b(ū; v̄) for both partial monodromy matrices T (l)(u). Then Bethe vectors of the total monodromy matrix can be expressed in terms of the partial Bethe vectors as follows [6, 29]: Ba,b(ū; v̄) = ∑ r (2) 1 (ūI)r (1) 3 (v̄II) f(ūII, ūI)f(v̄II, v̄I) f(v̄II, ūI) B(1) aI,bI (ūI; v̄I)B (2) aII,bII (ūII; v̄II). (3.4) The sum is taken over partitions ū⇒ {ūI, ūII} and v̄ ⇒ {v̄I, v̄II}. The cardinalities of the subsets are given by the subscripts of the Bethe vectors. We can present the product of functions r (2) 1 (ūI) as r (2) 1 (ūI) = r1(ūI)` −1 1 (ūI), see (3.2). More- over, if we deal with an on-shell Bethe vector, we can express r1(ūI) in terms of the function f , thanks to the Bethe equations (2.4). Then we obtain Ba,b(ū; v̄) = ∑ `3(v̄II) `1(ūI) f(ūI, ūII)f(v̄II, v̄I)f(v̄I, ūI)B (1) aI,bI (ūI; v̄I)B (2) aII,bII (ūII; v̄II). (3.5) Similarly, dual Bethe vectors can be expressed in terms of partial dual Bethe vectors Ca,b(ū; v̄) = ∑ r (1) 1 (ūII)r (2) 3 (v̄I) f(ūI, ūII)f(v̄I, v̄II) f(v̄I, ūII) C(1) aI,bI (ūI; v̄I)C (2) aII,bII (ūII; v̄II), (3.6) where the sum is taken again over partitions ū⇒ {ūI, ūII} and v̄ ⇒ {v̄I, v̄II}. If Ca,b(ū; v̄) is a twisted on-shell Bethe vector, then we can use again (3.2) as r (2) 3 (v̄I) = r3(v̄I)` −1 3 (v̄I) and express r3(v̄I) through the twisted Bethe equations (2.5). We get C(κ̄) a,b (ū; v̄) = ∑( κ2 κ3 )bI `1(ūII) `3(v̄I) f(ūI, ūII)f(v̄II, v̄I)f(v̄I, ūI)C (1) aI,bI (ūI; v̄I)C (2) aII,bII (ūII; v̄II). (3.7) Here we have added the superscript (κ̄) to the vector C(κ̄) a,b (ū; v̄) in order to stress that it is a twisted dual on-shell Bethe vector. 8 S. Pakuliak, E. Ragoucy and N.A. Slavnov 3.2 The action of total and partial zero modes The action of the operators T (l) ij (z) on the corresponding partial Bethe vectors B(l) a,b(ū; v̄) is the same as the action of total Tij(z) on the total Bethe vectors Ba,b(ū; v̄) [3]. One should only replace in the formulas the functions rk(z) by their partial analogs r (l) k (z). The same replacement should be done in the action of the partial zero modes on the partial Bethe vectors [30]. In this section we give some of those actions used below. The action of the total zero modes Tij [0] (with i < j) on the total Bethe vectors Ba,b can be easily extracted from the formulas given in Appendix A of the first part of this paper [29] using expansion of the monodromy matrix elements (1.6). They are T13[0]Ba,b(ū; v̄) = lim w→∞ w c Ba+1,b+1({w, ū}; {w, v̄}), T12[0]Ba,b(ū; v̄) = lim w→∞ w c Ba+1,b({w, ū}; v̄), T23[0]Ba,b(ū; v̄) = lim w→∞ w c Ba,b+1(ū; {w, v̄}). (3.8) The right action of the operators Tji[0] with i < j on dual Bethe vectors is quite analogous. One should replace in (3.8) Tij [0] by Tji[0] and Ba,b(ū; v̄) by Ca,b(ū; v̄). The action of the partial zero modes T (1) ij [0] (with i < j) on the partial Bethe vectors B(1) a,b is similar to (3.8): T (1) 13 [0]B(1) a,b(ū; v̄) = lim w→∞ w c B (1) a+1,b+1({w, ū}; {w, v̄}), T (1) 12 [0]B(1) a,b(ū; v̄) = lim w→∞ w c B (1) a+1,b({w, ū}; v̄), T (1) 23 [0]B(1) a,b(ū; v̄) = lim w→∞ w c B (1) a,b+1(ū; {w, v̄}). (3.9) The action of the partial zero modes T (1) ii [0] has the following form T (1) 11 [0]B(1) a,b(ū; v̄) = (`1[0]− a)B(1) a,b(ū; v̄), T (1) 22 [0]B(1) a,b(ū; v̄) = (a− b)B(1) a,b(ū; v̄), T (1) 33 [0]B(1) a,b(ū; v̄) = (`3[0] + b)B(1) a,b(ū; v̄), (3.10) where `k[0] are determined by (3.3). In all the formulas above Bethe vectors (partial or total) are generic. In Section 6 we will also use singular properties of on-shell (dual) Bethe vectors Ca,b(ūC ; v̄C)Tij [0] = 0, Tji[0]Ba,b(ūC ; v̄C) = 0, i < j. (3.11) Here Ca,b(ūC ; v̄C) and Ba,b(ūC ; v̄C) are on-shell Bethe vectors. This property was found in [26] for GL(N)-invariant models. In the GL(3) case it also follows from the explicit formulas of the action of the operators Tij(z) onto Bethe vectors [3]. 4 Main results Theorem 4.1. Let Ca′,b′(ūC ; v̄C) and Ba,b(ūB; v̄B) be total on-shell vectors such that {ūC , v̄C} 6= {ūB, v̄B} (that is, these on-shell vectors have different eigenvalues). Then Ca′,b′(ūC ; v̄C)T (1) ij [0]Ba,b(ūB; v̄B) = ( `1(ūC)`3(v̄B) `1(ūB)`3(v̄C) − 1 ) F (i,j) a,b (ūC , v̄C ; ūB, v̄B), (4.1) where F (i,j) a,b is the universal form factor of the total operator Tij(z) and a′ = a + δi1 − δj1, b′ = b+ δj3 − δi3. GL(3)-Based Quantum Integrable Composite Models. II. Form Factors of Local Operators 9 Theorem 4.2. Let Ba,b(ū; v̄) be a total on-shell vector and Ca,b(ū; v̄) its dual on-shell vector. Let Ca,b(ū(κ̄); v̄(κ̄)) be a deformation of Ca,b(ū; v̄) such that the parameters ū(κ̄) and v̄(κ̄) satisfy twisted Bethe equations (2.5), and ū(κ̄) = ū, v̄(κ̄) = v̄ at κ̄ = 1. Then Ca,b(ū; v̄)T (1) ii [0]Ba,b(ū; v̄) = ( δi,1`1[0] + δi,3`3[0] + d dκi log `1 ( ū(κ̄) ) `3 ( v̄(κ̄) ) ∣∣∣ κ̄=1 ) ‖Ba,b(ū; v̄)‖2.(4.2) The proofs of these theorems will be given in the next sections. If the partial monodromy matrix T (1)(u) has the structure (1.4), then the functions `k(u) actually depend also on the number m: `k(u) = m∏ n=1 `k(u\|n), k = 1, 3, where `k(u\|n) are the local ratios `1(u\|n) = λ1(u\|n) λ2(u\|n) , `3(u\|n) = λ3(u\|n) λ2(u\|n) . (4.3) In (4.3), we introduced the vacuum eigenvalues of local L-operators Ln(u) (Ln(u))ii\|0〉 = λi(u\|n)\|0〉, i = 1, 2, 3. Using (1.7) and Theorems 4.1, 4.2 we can find form factors of the local operators (Lm[0])ij , i, j = 1, 2, 3. Namely, one has simply to consider the difference of two T (1)(u) based on m and m− 1 respectively. If {ūC , v̄C} 6= {ūB, v̄B}, then we have Ca′,b′(ūC ; v̄C) ( Lm[0] ) ij Ba,b(ūB; v̄B) = ( `1(ūC \|m)`3(v̄B\|m) `1(ūB\|m)`3(v̄C \|m) − 1 )(m−1∏ n=1 `1(ūC \|n)`3(v̄B\|n) `1(ūB\|n)`3(v̄C \|n) ) F (i,j) a,b (ūC , v̄C ; ūB, v̄B). If {ūC , v̄C} = {ūB, v̄B} = {ū, v̄}, then Ca,b(ū; v̄) ( Lm[0] ) ij Ba,b(ū; v̄) = d dκi log `1 ( ū(κ̄)\|m ) `3 ( v̄(κ̄)\|m ) ∣∣∣ κ̄=1 ‖Ba,b(ū; v̄)‖2. If we deal with a continuum model, then form factors of local operators can be found directly from (4.1), (4.2). In this case the integer number m turns into a continuous variable x. This parameter enters only the functions `k, and taking the x-derivative of (4.1), (4.2) we find form factors of local operators in the point x. Thus, we obtain form factors of local operators in the generalized model without use of a specific representation of the algebra (1.1). In fact, it means that we have a solution of the quantum inverse scattering problem in the weak sense. We cannot express the local operators in terms of the monodromy matrix entries as it was done in [20, 25], but we can find all their matrix elements in the basis of the transfer matrix eigenstates. Furthermore, we have determinant formulas for all these matrix elements [2, 4, 27, 28, 30]. GL(N) generalisation. We would like to mention that Theorems 4.1 and 4.2 admit a direct generalization to GL(N)-invariant models with N > 3. Indeed, Bethe vectors (and dual ones) of GL(N) models depend on N − 1 sets of parameters t̄j = {tj1, t j 2, . . . , t j aj}, j = 1, 2, . . . , N − 1 and N − 1 integers aj that correspond to the cardinalities of each set: Bā(t̄) = Ba1,a2,...,aN−1 ( t̄1, t̄2, . . . , t̄N−1 ) , Cā(t̄) = Ca1,a2,...,aN−1 ( t̄1, t̄2, . . . , t̄N−1 ) . 10 S. Pakuliak, E. Ragoucy and N.A. Slavnov The action of the diagonal entries T (l) ii on the vacuum vectors is similar to (3.1) T (l) ii (t)\|0〉(l) = λ (l) i (t)\|0〉(l), l = 1, 2, i = 1, . . . , N, and we can introduce αi(t) = λ (1) i (t) λ (1) i+1(t) , i = 1, . . . , N − 1. Note that in the case N = 3 we have α1(t) = `1(t), while α2(t) = `−1 3 (t). Conjecture 4.3. Form factors of the partial zero modes T (1) ij [0] in GL(N)-invariant models are given by Cb̄(s̄)T (1) ij [0]Bā(t̄) = ( N−1∏ k=1 αk(s̄ k) αk(t̄k) − 1 ) F (i,j) ā (s̄; t̄), for s̄ 6= t̄ Cā(t̄)T (1) ii [0]Bā(t̄) = ( λ (1) i [0] + N−1∑ k=1 d dκi logαk ( t̄k(κ̄) )∣∣∣ κ̄=1 ) ‖Bā(t̄)‖2. where F (i,j) ā is the universal (z-independent) form factor of the total operator Tij(z) and we extended the convention (2.3) to the functions αk. t̄(κ̄) is a set of κ-twisted on-shell Bethe parameters, coinciding with t̄ when κ̄ = 1. This conjecture generalizes Theorems 4.1 and 4.2, proved for N = 2 and N = 3. Remark that since Cb̄(s̄)Tij [0]Bā(t̄) = 0 when s̄ 6= t̄, the above conjecture and theorems provide also the form factors for T (2) ij [0]. It is worth mentioning that in the cases N = 2 and N = 3, compact determinant representa- tions for the universal form factors are known. In contrast, in the caseN > 3 such representations are missing up to now. Nevertheless, if Conjecture 4.3 is valid in the GL(N) case, then it gives explicit dependence on the lattice site m of the partial zero modes form factors. 5 Form factors of diagonal operators We begin our consideration with the form factors of the diagonal partial zero modes T (1) ii [0]. It is convenient to construct a special generating functional for these form factors [10]. Consider an operator Qβ̄ = 3∑ i=1 βiT (1) ii [0], where βi are some complex numbers. The generating functional is M (κ̄) a,b = C(κ̄) a,b (ū C ; v̄C)eQβ̄Ba,b(ūB; v̄B). (5.1) Here Ba,b(ūB; v̄B) is an on-shell Bethe vector, C(κ̄) a,b (ū C ; v̄C) is a dual twisted on-shell Bethe vector with the twist parameters κi = eβi . Lemma 5.1. Let M (κ̄) a,b is defined as in (5.1). Then M (κ̄) a,b = eβ1`1[0]+β3`3[0] `1(ūC)`3(v̄B) `1(ūB)`3(v̄C) S(κ̄) a,b , (5.2) where S(κ̄) a,b is the scalar product of the twisted and the usual on-shell Bethe vectors (2.7). GL(3)-Based Quantum Integrable Composite Models. II. Form Factors of Local Operators 11 It is worth mentioning that an analog of (5.2) for GL(2)-based models was obtained in [14]. We will give a proof of Lemma 5.1 in Section 5.1. Now we show how equation (5.2) implies some statements of Theorems 4.1 and 4.2. Differentiating (5.1) over κi at κ̄ = 1 we obtain d dκi M (κ̄) a,b ∣∣∣ κ̄=1 = d dκi S(κ̄) a,b ∣∣∣ κ̄=1 + C(κ̄) a,b (ū C ; v̄C) ∣∣∣ κ̄=1 T (1) ii [0]Ba,b(ūB; v̄B). (5.3) Pay attention that the dual vector Ca,b(ūC ; v̄C) = C(κ̄) a,b (ū C ; v̄C) ∣∣∣ κ̄=1 is an on-shell vector. Thus, the second term in the r.h.s. of (5.3) is a form factor of the partial zero mode T (1) ii [0]. On the other hand, differentiating the r.h.s. of (5.2) over κi at κ̄ = 1 we find d dκi eβ1`1[0]+β3`3[0] `1(ūC)`3(v̄B) `1(ūB)`3(v̄C) S(κ̄) a,b ∣∣∣ κ̄=1 = `1(ūC)`3(v̄B) `1(ūB)`3(v̄C) ∣∣∣ κ̄=1 d dκi S(κ̄) a,b ∣∣∣ κ̄=1 + δC,B ( δi,1`1[0] + δi,3`3[0] + d dκi log `1(ū) `3(v̄) ∣∣∣ κ̄=1 ) ‖Ba,b(ū; v̄)‖2, (5.4) where δC,B = { 1, if {ūC , v̄C} ∣∣ κ̄=1 = {ūB, v̄B}, 0, if {ūC , v̄C} ∣∣ κ̄=1 6= {ūB, v̄B}. Deriving this formula we used the orthogonality of on-shell Bethe vectors depending on different Bethe parameters. Comparing equations (5.3) and (5.4) we immediately arrive at the statement of Theorem 4.2. If {ūC , v̄C} ∣∣ κ̄=1 6= {ūB, v̄B}, then we obtain Ca,b(ūC ; v̄C)T (1) ii [0]Ba,b(ūB; v̄B) = ( `1(ūC)`3(v̄B) `1(ūB)`3(v̄C) − 1 )∣∣∣ κ̄=1 d dκi S(κ̄) a,b ∣∣∣ κ̄=1 . It was proved in [4] that the κi-derivative of the scalar product S(κ̄) a,b at κ̄ = 1 is equal to the universal form factor of the operator Tii(z). Thus, the statement of Theorem 4.1 is proved for partial zero modes T (1) ii [0]. 5.1 Proof of Lemma 5.1 The proof of Lemma 5.1 is lengthy but straightforward. Let us first sketch the general strategy before going into details. Knowing the action of the operator eQβ̄ on the partial Bethe vectors we find its action on the total Bethe vectors. Then we can can calculate the matrix element M (κ̄) a,b in terms of scalar products of partial Bethe vectors, for which we use equation (2.6). The resulting formula becomes rather cumbersome. In particular, it contains a sum over partitions of every set of the original Bethe parameters into four subsets. Therefore in this section we use standard arabic indices in order to label these subsets. New subsets of the Bethe parameters can be easily recombined into new sets of variables, and after this the proof reduces to the use of identity (2.8) and equation (2.7). Let us now give the details. Using (3.10) and (3.5) we find the action of eQβ̄ on the total on-shell Bethe vector eQβ̄Ba,b(ūB; v̄B) = ∑ eβ1(`1[0]−aI)+β2(aI−bI)+β3(`3[0]+bI) `3(v̄BII ) `1(ūBI ) × f(ūBI , ū B II )f(v̄BII , v̄ B I )f(v̄BI , ū B I )B(1) aI,bI (ūBI ; v̄BI )B(2) aII,bII (ūBII ; v̄BII ). 12 S. Pakuliak, E. Ragoucy and N.A. Slavnov Multiplying this equation from the left by C(κ̄) a,b (ū C ; v̄C) and using (3.7) we obtain M (κ̄) a,b = ∑ eβ1(`1[0]−aI)+β2aI+β3`3[0] `1(ūCII )`3(v̄BII ) `1(ūBI )`3(v̄CI ) × f(ūCI , ū C II )f(ūBI , ū B II )f(v̄CII , v̄ C I )f(v̄BII , v̄ B I )f(v̄BI , ū B I )f(v̄CI , ū C I ) × C(1) aI,bI (ūCI ; v̄CI )B(1) aI,bI (ūBI ; v̄BI ) · C(2) aII,bII (ūCII ; v̄CII )B(2) aII,bII (ūBII ; v̄BII ). (5.5) Thus, we have obtained the expression for M (κ̄) a,b in terms of scalar products of partial Bethe vectors. Note that in spite of the sets ūB and v̄B satisfy the Bethe equations (2.4), and the sets ūC and v̄C satisfy the twisted Bethe equations (2.5), the partial Bethe vectors in (5.5) are not (twisted) on-shell vectors. In other words we deal with the scalar products of generic Bethe vectors in (5.5). Therefore, we do not write the additional superscript (κ̄) for the dual vectors and we should use (2.6) for the calculation of their scalar products. Hereby, for the scalar product of the vectors C(1) and B(1) we should replace in (2.6) the functions rk by `k, while for the scalar product of the vectors C(2) and B(2) we should replace in (2.6) the functions rk by rk` −1 k . The use of (2.6) introduces new partitions of the subsets of Bethe parameters, so that, as mentioned above (see Section 2.2), we use now arabic numbers to label the numerous subsubsets. Thus, we have for the first scalar product C(1) a,b(ū C I ; v̄CI )B(1) a,b(ū B I ; v̄BI ) = ∑ `1(ūB1 )`1(ūC3 )`3(v̄C3 )`3(v̄B1 )f(ūC1 , ū C 3 )f(ūB3 , ū B 1 ) × f(v̄C3 , v̄ C 1 )f(v̄B1 , v̄ B 3 ) f(v̄C1 , ū C 1 )f(v̄B3 , ū B 3 ) f(v̄CI , ū C I )f(v̄BI , ū B I ) Za3,b1(ūC3 ; ūB3 \|v̄C1 ; v̄B1 )Za1,b3(ūB1 ; ūC1 \|v̄B3 ; v̄C3 ). The summation is taken with respect to the partitions ūC,BI ⇒ {ūC,B1 , ūC,B3 }, v̄C,BI ⇒ {v̄C,B1 , v̄C,B3 }. The cardinalities of the subsubsets are an = #ūC,Bn , bn = #v̄C,Bn , n = 1, 3. Similarly C(2) a,b(ū C II ; v̄CII )B(2) a,b(ū B II ; v̄BII ) = ∑ r1(ūB2 )r1(ūC4 )r3(v̄C4 )r3(v̄B2 ) `1(ūB2 )`1(ūC4 )`3(v̄C4 )`3(v̄B2 ) f(ūC2 , ū C 4 )f(ūB4 , ū B 2 ) (5.6) × f(v̄C4 , v̄ C 2 )f(v̄B2 , v̄ B 4 ) f(v̄C2 , ū C 2 )f(v̄B4 , ū B 4 ) f(v̄CII , ū C II )f(v̄BII , ū B II ) Za4,b2(ūC4 ; ūB4 \|v̄C2 ; v̄B2 )Za2,b4(ūB2 ; ūC2 \|v̄B4 ; v̄C4 ). Here the sum is taken over partitions ūC,BII ⇒ {ūC,B2 , ūC,B4 }, v̄C,BII ⇒ {v̄C,B2 , v̄C,B4 }. The cardinalities of the subsubsets are still denoted by an = #ūC,Bn , and bn = #v̄C,Bn , n = 2, 4. Now we should express the products of the functions rk in (5.6) via the (twisted) Bethe equations for the full sets {ūB, v̄B} and {ūC , v̄C}. We have r1(ūB2 ) = f(ūB2 , ū B 1 )f(ūB2 , ū B 3 )f(ūB2 , ū B 4 ) f(ūB1 , ū B 2 )f(ūB3 , ū B 2 )f(ūB4 , ū B 2 ) f(v̄B, ūB2 ), r3(v̄B2 ) = f(v̄B1 , v̄ B 2 )f(v̄B3 , v̄ B 2 )f(v̄B4 , v̄ B 2 ) f(v̄B2 , v̄ B 1 )f(v̄B2 , v̄ B 3 )f(v̄B2 , v̄ B 4 ) f(v̄B2 , ū B), r1(ūC4 ) = ea4(β2−β1) f(ūC4 , ū C 1 )f(ūC4 , ū C 2 )f(ūC4 , ū C 3 ) f(ūC1 , ū C 4 )f(ūC2 , ū C 4 )f(ūC3 , ū C 4 ) f(v̄C , ūC4 ), r3(v̄C4 ) = eb4(β2−β3) f(v̄C1 , v̄ C 4 )f(v̄C2 , v̄ C 4 )f(v̄C3 , v̄ C 4 ) f(v̄C4 , v̄ C 1 )f(v̄C4 , v̄ C 2 )f(v̄C4 , v̄ C 3 ) f(v̄C4 , ū C). GL(3)-Based Quantum Integrable Composite Models. II. Form Factors of Local Operators 13 All these expressions should be substituted into (5.5). After simple but exhausting algebra we obtain M (κ̄) a,b = ∑ eβ1`1[0]+β3`3[0]+(β2−β1)(a−a2)+(β2−β3)b4 `1(ūC2 )`1(ūC3 )`3(v̄B1 )`3(v̄B4 ) `1(ūB2 )`1(ūB3 )`3(v̄C1 )`3(v̄C4 ) × FCuuFCvvFCvuFBuuFBvvFBvuZ. (5.7) Here the sum is taken over partitions of every set of Bethe parameters into four subsets ūC,B ⇒ {ūC,B1 , ūC,B2 , ūC,B3 , ūC,B4 }, v̄C,B ⇒ {v̄C,B1 , v̄C,B2 , v̄C,B3 , v̄C,B4 }. We have #ūBn = #ūCn = an and #v̄Bn = #v̄Cn = bn, n = 1, . . . , 4, but the values an and bn are free. Note that a2 and b4 explicitly appear as coefficients in (5.7), so that manipulations with ūC,B2 and v̄C,B4 should include these coefficients. The factor Z in (5.7) is the product of four highest coefficients Z = Za3,b1(ūC3 ; ūB3 \|v̄C1 ; v̄B1 )Za1,b3(ūB1 ; ūC1 \|v̄B3 ; v̄C3 )Za4,b2(ūC4 ; ūB4 \|v̄C2 ; v̄B2 )Za2,b4(ūB2 ; ūC2 \|v̄B4 ; v̄C4 ). The other factors in (5.7) denoted by F with different subscripts and superscripts are products of f functions: FCuu = f(ūC4 , ū C 1 )f(ūC3 , ū C 2 )f(ūC4 , ū C 2 )f(ūC4 , ū C 3 )f(ūC1 , ū C 2 )f(ūC1 , ū C 3 ), FBuu = f(ūB1 , ū B 4 )f(ūB2 , ū B 3 )f(ūB2 , ū B 1 )f(ūB2 , ū B 1 )f(ūB3 , ū B 4 )f(ūB3 , ū B 4 ), FCvv = f(v̄C1 , v̄ C 4 )f(v̄C2 , v̄ C 3 )f(v̄C2 , v̄ C 1 )f(v̄C2 , v̄ C 4 )f(v̄C3 , v̄ C 1 )f(v̄C3 , v̄ C 4 ), FBvv = f(v̄B4 , v̄ B 1 )f(v̄B3 , v̄ B 2 )f(v̄B1 , v̄ B 3 )f(v̄B4 , v̄ B 3 )f(v̄B1 , v̄ B 2 )f(v̄B4 , v̄ B 2 ), FCvu = f(v̄C1 , ū C 4 )f(v̄C4 , ū C 4 )f(v̄C1 , ū C 1 )f(v̄C4 , ū C 1 )f(v̄C3 , ū C 4 )f(v̄C4 , ū C 3 ), FBvu = f(v̄B3 , ū B 3 )f(v̄B2 , ū B 2 )f(v̄B3 , ū B 2 )f(v̄B2 , ū B 3 )f(v̄B1 , ū B 2 )f(v̄B2 , ū B 1 ). It remains to combine the subsubsets into new groups: {ūC,B1 , ūC,B4 } = ūC,Bi , {ūC,B2 , ūC,B3 } = ūC,Bii , {v̄C,B1 , v̄C,B4 } = v̄C,Bi , {v̄C,B2 , v̄C,B3 } = v̄C,Bii . Then we recast (5.7) as follows: M (κ̄) a,b = ∑ ūC,B⇒{ūC,Bi ,ūC,Bii } v̄C,B⇒{v̄C,Bi ,v̄C,Bii } `1(ūCii )`3(v̄Bi ) `1(ūBii )`3(v̄Ci ) f(ūCi , ū C ii )f(ūBii , ū B i )f(v̄Cii , v̄ C i )f(v̄Bi , v̄ B ii ) × f(v̄Ci , ū C i )f(v̄Bii , ū B ii )G1(ūCi , ū B i ; v̄Cii , v̄ B ii )G2(ūCii , ū B ii ; v̄ C i , v̄ B i ), (5.8) where factors G1 and G2 are given as sums over partitions G1(ūCi , ū B i ; v̄Cii , v̄ B ii ) = ∑ ūC,Bi ⇒{ūC,B1 ,ūC,B4 } v̄C,Bii ⇒{v̄C,B2 ,v̄C,B3 } f(ūC4 , ū C 1 )f(ūB1 , ū B 4 )f(v̄C2 , v̄ C 3 )f(v̄B3 , v̄ B 2 ) × f(v̄C3 , ū C 4 )f(v̄B2 , ū B 1 )Za4,b2(ūC4 ; ūB4 \|v̄C2 ; v̄B2 )Za1,b3(ūB1 ; ūC1 \|v̄B3 ; v̄C3 ), (5.9) and G2(ūCii , ū B ii ; v̄ C i , v̄ B i ) = ∑ ūC,Bii ⇒{ūC,B2 ,ūC,B3 } v̄C,Bi ⇒{v̄C,B1 ,v̄C,B4 } eβ1`1[0]+β3`3[0]+(β2−β1)(a−a2)+(β2−β3)b4 14 S. Pakuliak, E. Ragoucy and N.A. Slavnov × f(ūC3 , ū C 2 )f(ūB2 , ū B 3 )f(v̄C1 , v̄ C 4 )f(v̄B4 , v̄ B 1 )f(v̄C4 , ū C 3 )f(v̄B1 , ū B 2 ) × Za3,b1(ūC3 ; ūB3 \|v̄C1 ; v̄B1 )Za2,b4(ūB2 ; ūC2 \|v̄B4 ; v̄C4 ). (5.10) It is not difficult to see that the sum over partitions in (5.9) coincides with the sum in (2.8) up to relabeling of the subsets: ūC,B1 → ūC,BI , ūC,B4 → ūC,BII , v̄C,B2 → v̄C,BI , and v̄C,B3 → v̄C,BII . Thus, we conclude that G1 = 0 unless ūC,Bi = ∅ and v̄C,Bii = ∅. Hence, ūC,Bii = ūC,B, v̄C,Bi = v̄C,B, and a1 = a4 = 0, b2 = b3 = 0. Then due to (2.9) G1 = 1. Looking now at (5.10) and comparing it with (2.7) we see that they coincide up to the common prefactor eβ1`1[0]+β3`3[0] and relabeling of the subsets: ūC,B2 → ūC,BI , ūC,B3 → ūC,BII , v̄C,B1 → v̄C,BI , and v̄C,B4 → v̄C,BII . Hence, G2 = eβ1`1[0]+β3`3[0]S(κ̄) a,b . Substituting this into (5.8) and setting there ūC,Bi = ∅ and v̄C,Bii = ∅ we immediately arrive at (5.2). 6 Form factor for off-diagonal partial zero modes Now we study the form factors of off-diagonal partial zero modes. We apply a strategy similar to the one used in [27], using the commutation relations of the zero modes and properties of the type (3.9), (3.10). First, we note that the RTT -relation (1.1) implies in particular for the (partial) zero modes [Tii[0], Tji[0]] = Tji[0], [T (l) ii [0], T (l) ji [0]] = T (l) ji [0], i 6= j, [Tij [0], Tii[0]] = Tij [0], [T (l) ij [0], T (l) ii [0]] = T (l) ij [0], i 6= j, [Tij [0], Tki[0]] = Tkj [0], [T (l) ij [0], T (l) ki [0]] = T (l) kj [0], i 6= j 6= k, l = 1, 2. Now, using Tkl[0] = T (1) kl [0] + T (2) kl [0] and [T (1) ij [0], T (2) kl [0]] = 0 we conclude [T (1) ii [0], Tji[0]] = T (l) ji [0], i 6= j, [Tij [0], T (1) ii [0]] = T (1) ij [0], i 6= j, [Tij [0], T (1) ki [0]] = T (1) kj [0], i 6= j 6= k, (6.1) which are the central relations that we will use for our calculations. As a notation, we will note the form factor of T (1) ij [0] as M (i,j) a,b (ūC , v̄C ; ūB, v̄B) = Ca′,b′(ūC ; v̄C)T (1) ij [0]Ba,b(ūB; v̄B), i, j = 1, 2, 3, with a′ = a+ δi1 − δj1 and b′ = b+ δj3 − δi3, and where both vectors are on-shell. 6.1 Form factor of T (1) 12 [0] We start with M (1,2) a,b , the form factor of T (1) 12 [0]. We make the calculation in details, the other ones following the same steps. To get M (1,2) a,b , we start with the form factor M (2,2) a+1,b and send one of the parameters in ūB to infinity (keeping all the other parameters finite), using relations (3.8): lim w→∞ w c M (2,2) a+1,b(ū C , v̄C ; {ūB, w}, v̄B) = lim w→∞ w c Ca+1,b(ū C ; v̄C)T (1) 22 [0]Ba+1,b({ūB, w}; v̄B) = Ca+1,b(ū C ; v̄C)T (1) 22 [0]T12[0]Ba,b(ūB; v̄B). GL(3)-Based Quantum Integrable Composite Models. II. Form Factors of Local Operators 15 Now, since Ca+1,b(ū C ; v̄C) is on-shell with all parameters finite, it is annihilated by T12[0] due to (3.11). Thus, we can replace the product T (1) 22 [0]T12[0] by its commutator which in turn gives T (1) 12 [0] through (6.1). It leads to lim w→∞ w c M (2,2) a+1,b(ū C , v̄C ; {ūB, w}, v̄B) = M (1,2) a,b (ūC , v̄C ; ūB, v̄B). (6.2) It remains to compute the limit of the explicit expression (4.1) for M (2,2) a+1,b, that was proved in the previous section. It is obvious that lim w→∞ `k(w) = 1, k = 1, 3, and it has been shown in [27] that5 lim w→∞ w c F (2,2) a+1,b(ū C , v̄C ; {ūB, w}, v̄B) = F (1,2) a,b (ūC , v̄C ; ūB, v̄B). (6.3) Hence, we get expression (4.1) for M (1,2) a,b . 6.2 Other form factors As already mentioned, the calculation for other form factors follows the same steps, so we just sketch the proofs. Form factor of T (1) 23 [0]. To get M (2,3) a,b , we start with M (2,2) a,b+1(ūC , v̄C ; ūB, {v̄B, w}) and take the limit w →∞. It makes appear T23[0] that annihilates Ca,b+1(ūC ; v̄C). It leads to lim w→∞ w c M (2,2) a,b+1(ūC , v̄C ; ūB, {v̄B, w}) = −M (2,3) a,b (ūC , v̄C ; ūB, v̄B). (6.4) Form factor of T (1) 21 [0]. To get M (2,1) a,b , we start with M (2,2) a,b ({ūC , w}, v̄C ; ūB, v̄B) and take the limit w →∞. It makes appear T21[0] on the left, and it annihilates Ba,b(ūC ; v̄C) due to (3.11). Then, we can again replace the product T21[0]T (1) 22 [0] by its commutator, and we obtain lim w→∞ w c M (2,2) a,b ({ūC , w}, v̄C ; ūB, v̄B) = M (2,1) a,b (ūC , v̄C ; ūB, v̄B). (6.5) Form factor of T (1) 32 [0]. The form factor M (3,2) a,b is obtained through the limit lim w→∞ w c M (2,2) a,b (ūC , {v̄C , w}; ūB, v̄B) = −M (3,2) a,b (ūC , v̄C ; ūB, v̄B). (6.6) Form factor of T (1) 13 [0]. To get M (1,3) a,b , it is convenient to start with the already known form factor M (2,3) a+1,b(ū C , v̄C ; {ūB, w}, v̄B). Taking the limit w → ∞ we obtain T12[0] that annihilates Ca+1,b(ū C ; v̄C). Hence, due to the last equation (6.1) we obtain lim w→∞ w c M (2,3) a+1,b(ū C , v̄C ; {ūB, w}, v̄B) = M (1,3) a,b (ūC , v̄C ; ūB, v̄B). (6.7) Form factor of T (1) 31 [0]. To get M (3,1) a,b , we start with M (3,2) a,b ({ūC , w}, v̄C ; ūB, v̄B) and take the limit w →∞. We obtain lim w→∞ w c M (3,2) a,b ({ūC , w}, v̄C ; ūB, v̄B) = M (3,1) a,b (ūC , v̄C ; ūB, v̄B). (6.8) Thus, starting from one initial form factor M (2,2) a,b we can obtain all other form factors M (i,j) a,b in the special limits of the Bethe parameters. This property is a direct consequence of the property of the form factors of the monodromy matrix entries F (i,j) a,b [30]. 5Strictly speaking the proof of [28] was done for the full form factor, but it extends straightforwardly to the universal form factor. 16 S. Pakuliak, E. Ragoucy and N.A. Slavnov Consistency with morphisms. It is worth mentioning that there exist also other relations between different form factors M (i,j) a,b [28, 30]. These relations appear due to morphisms of the algebra (1.1). It was shown in [3] that the mappings ψ : Tij(u) 7→ Tji(u), ϕ : Tij(u) 7→ T4−j,4−i(−u), (6.9) define morphisms of the algebra (1.1). Hereby, the mapping ϕ is an isomorphism, while ψ is an antimorphism. Both mappings (6.9) exchange the partial monodromy matrices of the composite model [29] ψ : T (l) ij (u) 7→ T (3−l) ji (u), ϕ : T (l) ij (u) 7→ T (3−l) 4−j,4−i(−u). Transformations (6.9) induce relations between different form factors of the monodromy matrix entries F (i,j) a,b (see [27, 28] for details). The latest, in their turn, give us relations between the universal form factors F (i,j) a,b F (i,j) a,b (ūC , v̄C ; ūB, v̄B) = −F(j,i) a′,b′(ū B, v̄B; ūC , v̄C), F (i,j) a,b (ūC , v̄C ; ūB, v̄B) = F (4−j,4−i) b,a (−v̄C ,−ūC ;−v̄B,−ūB). (6.10) Thus, actually it is enough to compute only four form factors of the partial zero modes T (1) 1j [0] (j = 1, 2, 3) and T (1) 22 [0]. All other form factors can be obtained by the mappings described above. Using the explicit determinant representations for the from factors one can check that they enjoy both the limiting procedures (6.2), (6.4)–(6.8), and the transformations (6.10). Generalisation to GL(N). One can apply the same procedure to the GL(N) case. In particular, the commutation relations (6.1) are still valid, and the limits of the type (3.9), (3.10) and (6.3) have been proven for GL(N) in [27]. The singular vector properties of the type (3.11) were obtained for GL(N) in [26]. All that allows us to relate the different form factors in the same way we did for GL(3). It is easy to check that the Conjecture (4.3) is consistent with these relations. 7 Conclusion In this paper we have studied form factors of the partial zero modes in a composite generalized model with GL(3)-invariant R-matrix. We have reduced these form factors to the ones of the monodromy matrix entries Tij(u) considered in our previous publications. As we have mentioned already, it means that we have a solution of the inverse scattering problem in the weak sense. Apparently the same type of the formulas remains true for GL(N)-invariant models with N > 3. We are planning to apply these results to the study of form factors of local operators in the model of one-dimensional two-component Bose gas with δ-function interaction [35, 39]. Several form factors in this model were studied already in the framework of the coordinate Bethe ansatz for some particular cases of Bethe vectors [31]. We are going to apply the algebraic Bethe ansatz techniques in order to compute the form factors in the general case. This model possesses GL(3)- invariant R-matrix (1.2) and can be considered in the framework of the scheme described in the present paper. However the asymptotic expansion of the monodromy matrices (1.6) in the case of Bose gas should be modified. In its turn, this modification leads to a modification of the partial zero modes. Nevertheless the derivation of determinant representations for the form factors of the diagonal partial zero modes T (1) ii [0] does not change. 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GL(3) -Based Quantum Integrable Composite Models. II. Form Factors of Local Operators

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