On Bethe vectors in \( \mathfrak{g}{\mathfrak{l}}_3 \) -invariant integrable models

Journal of High Energy Physics, Jun 2018

Abstract We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing \( \mathfrak{g}{\mathfrak{l}}_3 \)-invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled.

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On Bethe vectors in \( \mathfrak{g}{\mathfrak{l}}_3 \) -invariant integrable models

JHE Bethe vectors in gl3-invariant integrable A. Liashyk 0 1 3 4 5 6 7 8 N.A. Slavnov 0 1 2 3 7 8 0 6 Usacheva str. , Moscow, 119048 , Russia 1 14b Metrolohichna str. , Kiev, 03143 , Ukraine 2 Theoretical Physics Department, Steklov Mathematical Institute of Russian Academy of Sciences 3 Bogoliubov Institute for Theoretical Physics , NAS of Ukraine 4 Faculty of Mathematics, National Research University Higher School of Economics 5 Center for Advanced Studies, Skolkovo Institute of Science and Technology 6 Department for Theory of Nuclei and Quantum Field Theory 7 8 Gubkina str. , Moscow, 119991 , Russia 8 3 Nobel str. , Moscow, 121205 , Russia We consider quantum integrable models solvable by the nested algebraic Bethe ansatz and possessing gl3-invariant R-matrix. We study a new recently proposed approach to construct on-shell Bethe vectors of these models. We prove that the vectors constructed by this method are semi-on-shell Bethe vectors for arbitrary values of Bethe parameters. They thus do become on-shell vectors provided the system of Bethe equations is fulfilled. Integrable Field Theories; Lattice Integrable Models - HJEP06(218) 1 Introduction 2 3 4 5 6 Basic notions of NABA 2.1 2.2 Notation Bethe vectors Special NABA-solvable models Main results 4.1 Multiple action of the operator Bg Proof of proposition 4.1 Action of Bg(z) on Bethe vectors 6.1 6.2 6.3 Action of T23(z2)Tb13(z1) Action of T13(z2)Tb12(z1) Action of Bg(z) on semi-on-shell Bethe vectors 7 Proof of proposition 4.2 7.1 Inductive basis 7.2 Inductive step 7.2.1 7.2.2 Contribution W1 Contributions W2 and W3 A Properties of DWPF B Proof of the connection between two types of Bethe vectors B.1 First step of induction B.2 Second step of induction C Action formulas C.2.1 C.2.2 Action of Tb13 Action of Tb12 C.1 Actions of the operators Tij on Bethe vectors Ba,b C.2 Actions of the operators Tbij on Bethe vectors Ba,b was proposed in [ 1 ]. In the present paper we study this method by the nested algebraic Bethe ansatz (NABA) in the case of quantum integrable models with gl3-invariant R-matrix. – 1 – There exist several ways to study quantum integrable models with a high rank of symmetry. A nested version of the Bethe ansatz [2] was proposed in [3–5]. In the context of the Quantum Inverse Scattering Method (QISM) [6–9], an algebraic version of this method (NABA) was developed in [10–12]. One more approach based on the qKZ equation and Jackson integrals was proposed in [13–15] and generalized to the superalgebras in [16]. We should also mention a method to construct Bethe vectors via certain projection of Drinfel’d currents, that was developed in a series of works [17–21]. The Separation of Variables (SoV) method [22, 23] was applied to the study of gl3-invariant quantum spin chains in [24]. The main task of the methods listed above is to construct the eigenfunctions of the quantum Hamiltonians. Traditionally they are called on-shell Bethe vectors. In distinction of the gl2 based models, a form of these eigenfunctions for the models with higher rank of symmetry is quite involved. This is due to the fact that these models describe physical systems with several types of particles. Respectively, one has to consider several creation operators corresponding to each type of excitations. For instance, within the framework of QISM, we deal with a quantum monodromy matrix T (u), whose trace plays the role of generating functional of the integrals of motion. The upper-triangular entries of the monodromy matrix Tij (u) with i < j are creation operators, and a physical space of states can be generated by successive action of these operators on a referent state |0i. In the case of the gl2 based models, there exits only one creation operator T12(u). Respectively, the eigenvectors of the quantum Hamiltonians have the form of products of these operators acting onto a referent state |0i. However, already in the case of the gl3 based models, we deal with three creation operators, and the form of on-shell Bethe vectors immediately becomes much more complex (see e.g. [25] and (2.14) for explicit formulas). It was observed in [ 1 ] that an operator used for constructing the SoV basis of the gl2-invariant spin chain can be also used for generating the basis of the on-shell Bethe vectors. It was conjectured in [ 1 ] that a similar effect might take place in the spin chains with higher rank of symmetry. In particular, in the gl3-invariant spin chain one should consider an operator1 (1.1) (1.2) Bg(u) = T23(u)T12(u − i)T23(u) − T23(u)T22(u − i)T13(u) + T13(u)T11(u − i)T23(u) − T13(u)T21(u − i)T13(u) for constructing the SoV basis [24]. Here Tij (u) are entries of a twisted monodromy matrix (see section 3 for more details). Then, in complete analogy with the case of gl2 based models, on-shell Bethe vectors can be presented as a successive action of Bg(ui) onto the referent state Bg(u1) . . . Bg(ua)|0i. This conjecture was justified by the computer calculation, however, an analytical proof is lacking so far. The goal of this paper is to find such the proof. Our proof of representation (1.2) is given within the framework of NABA. We show that representation (1.2) for on-shell Bethe vectors holds not only for spin chains, but for 1In [ 1 ] this operator was denoted as Bgood(u). We find this notation too heavy and reduce it to Bg(u). – 2 – a more wide class of integrable models possessing gl3-invariant R-matrix. In particular, we do not use the SoV method. The paper is organized as follows. We recall basic notions of NABA in section 2. There we also give a standard description of Bethe vectors within this method. Section 3 is devoted to special NABA-solvable models that usually are applied to the systems of physical interest. The main results of our paper are gathered in section 4. There we give explicit representation of the states (1.2) in terms of the monodromy matrix entries acting on the pseudovacuum vector. We also describe a relationship between the states (1.2) and the Bethe vectors obtained by the standard NABA approach. In the rest of the paper we give the proofs of the results of section 4. We identify the state (1.2) with a Bethe vector in section 5. In section 6 we compute the action of the operator Bg(u) on a generic Bethe vector. Finally, in section 7 we express the state (1.2) in terms of the monodromy matrix entries acting on the pseudovacuum vector. Several auxiliary identities for rational functions are gathered in appendix A. Appendix B contains a proof of connection between two types of Bethe vectors considered in the paper. Finally, the formulas of the action of the monodromy matrix entries onto the Bethe vectors are given in appendix C. 2 Basic notions of NABA We consider quantum integrable models solvable by NABA and possessing the gl3-invariant R-matrix constant.2 R(u, v) = I ⊗ I + g(u, v)P, g(u, v) = c u − v Here I is the identity matrix in C3, P is the permutation matrix in C3 ⊗ C3, and c is a The monodromy matrix T (u) is a 3 × 3 matrix with operator-valued entries Tij (u) acting in a Hilbert space H. Their commutation relations are give by an RT T -relation (2.1) (2.2) (2.3) R(u, v) T (u) ⊗ I I ⊗ T (v) = I ⊗ T (v) T (u) ⊗ I R(u, v). It follows from (2.2) that an operator T (u) = tr T (u) = 3 i=1 X Tii(u) has the following property: [T (u), T (v)] = 0 for arbitrary u and v. This operator is called a transfer matrix. It plays the role of a generating functional of the integrals of motion of a quantum model under consideration. One of the main tasks of NABA is to find eigenvectors of this operator. If a 3 × 3 c-number matrix K is such that [R(x, y), K ⊗ K] = 0, then the matrices KT (u) and T (u)K also satisfy the RT T -relation (2.2). A peculiarity of the R-matrix (2.1) is that [R(x, y), K ⊗ K] = 0 holds for arbitrary K ∈ gl3. In particular, if K is invertible, 2To compare our presentation with the results of [ 1 ] one should set c = i. – 3 – then one can consider a transformation T (u) → T (K)(u) = KT (u)K−1. Obviously, this transformation preserves the transfer matrix. Besides the monodromy matrix T (u), we also will consider a matrix Tb(u) that is closely associated to a quantum comatrix [26, 27]. First, we introduce quantum minors tjk11,,jk22 (u) = Tj1,k1 (u)Tj2,k2 (u − c) − Tj2,k1 (u)Tj1,k2 (u − c). The entries of the quantum comatrix Tejk(u) then are given by where ¯j = {1, 2, 3} \ j. The quantum comatrix plays the role of the inverse monodromy HJEP06(218) where qdet T (u) is a quantum determinant of T (u) [26–29]. The matrix Tb(u) is defined as the transposition of Te(u) with respect to the secondary It is known [25–27, 30] that a mapping φ : T (u) 7→ Tb(u) is an automorphism of the RT T -algebra (2.2). Thus, the matrix Tb(u) satisfies the RT T -relation with the same R Using the matrix Tb(u) we can write down the operator Bg(u) (1.1) in a more compact The following obvious properties of the functions introduced above are useful: g(u, v) = −g(v, u), h(u − c, v) = g(u + c, v) = f (u − c, v) = u − v + c u − v , 1 g(u, v) , (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) 1 f (v, u) (2.10) Similar representation for Bg was used in [24]. 2.1 Notation Besides the function g(u, v) we also introduce two new functions f (u, v) = 1 + g(u, v) = Tejk(u) = (−1)j+kt¯k¯(u), j Te(u − c)T (u) = qdet T (u) I, Tbjk(u) = Te4−k,4−j (u). Before giving a description of the Bethe vectors we formulate a convention on the notation. We denote sets of variables by a bar: u¯, v¯, and so on. Individual elements of the sets are denoted by subscripts: uj , vk, and so on. Notation u¯ + c means that the constant c is added to all the elements of the set u¯. Subsets of variables are denoted by roman indices: u¯I, u¯II, v¯ii, and so on. In particular, we consider partitions of sets into subsets. Then the notation u¯ ⇒ {u¯I, u¯II} means that the set u¯ is divided into two disjoint subsets u¯I and u¯II. The order of the elements in each subset is not essential. A special notation u¯j is used for subsets complementary to the element uj , that is, u¯j = u¯ \ uj , v¯k = v¯ \ vk and so on. – 4 – In order to avoid too cumbersome formulas we use shorthand notation for products of functions depending on one or two variables. Namely, if the functions g, f , and h depend on sets of variables, this means that one should take the product over the corresponding set. For example, h(u¯, v) = Y h(uj , v); uj∈u¯ Y zj∈z¯ zj6=zi g(zi, z¯i) = g(zi, zj ); f (u¯II, u¯I) = f (uj , uk). Y Y uj∈u¯II uk∈u¯I In the last equation of (2.11) the set u¯ is divided into two subsets u¯I, u¯II, and the double product is taken with respect to all uk belonging to u¯I and all uj belonging to u¯II. We use the 1 if at least one of the sets is empty. the framework of NABA, it is assumed that successive action of these operators onto |0i generates vectors of the space H. Bethe vectors are special polynomials in Tij (u) with i < j acting on |0i. Their explicit form will be given later. Here we would like to mention that in the models with gl3-invariant R-matrix Bethe vectors depend on two sets of complex parameters u¯ = {u1, . . . , ua} and v¯ = {v1, . . . , vb} called Bethe parameters. We denote these vectors by Ba,b(u¯; v¯), where a and b respectively are the cardinalities of the sets u¯ and v¯. A characteristic property of the Bethe vectors is that they become eigenvectors of the transfer matrix T (z) = tr T (z) provided u¯ and v¯ enjoy ceratin constraint. In this case they are called on-shell Bethe vectors. Otherwise, if u¯ and v¯ are generic complex numbers, then the corresponding vector is called off-shell Bethe vector. In physical models, vectors of the space H describe states with quasiparticles (excitations) of two different types (colors). We say that a state has coloring {a, b}, if it contains a quasiparticles of the color 1 and b quasiparticles of the color 2. The vector |0i has zero coloring. The operator T12 is the creation operator of quasiparticles of the first color, while the operator T23 creates quasiparticles of the second color. The operator T13 creates one quasiparticle of the first color and one quasiparticle of the second color. The diagonal – 5 – (2.11) (2.12) (2.13) operators Tii are neutral, the matrix elements Tij with i > j play the role of annihilation operators. Generally, there are no restrictions on the coloring {a, b}, thus, the parameters a and b are arbitrary non-negative integers. In specific models, some restrictions may appear. Different methods to construct Bethe vectors were developed in [12, 15–17]. Several equivalent explicit representations were found in [25]. One of this representations reads Recall that here we use the shorthand notation (2.11), (2.12) for the products of the operators Tij and the functions λ2, f , and g. The sum in (2.14) is taken over partitions of the sets u¯ ⇒ {u¯I, u¯II} and v¯ ⇒ {v¯I, v¯II} such that #u¯I = #v¯I = n, where n = 0, 1, . . . , min(a, b). It is easy to see that each term of this sum has a fixed coloring {a, b}, and thus, Bethe vector Ba,b(u¯; v¯) has coloring that coincides with the cardinalities of the Bethe parameters. We would like to stress that generically there is no any restriction on the cardinalities of the Bethe parameters u¯ and v¯. In particular, one might have a < b, that is #u¯ < #v¯. The function Kn(v¯I|u¯I) in (2.14) is a partition function of the six-vertex model with domain wall boundary condition (DWPF) [31, 32]. It depends on two sets of variables v¯ and u¯; the subscript shows that #v¯ = #u¯ = n. The function Kn has the following determinant representation [32]: Kn(v¯|u¯) = h(v¯, u¯)   n  j<k Y g(vj, vk)g(uk, uj) dnet g(vj, uk) h(vj, uk) . (2.15) Some properties of Kn are gathered in appendix A. Observe that the normalization in (2.14) differs from the normalization of Bethe vectors used in [25]. The present normalization is chosen so that the Bethe vector does not have singularities for vj = uk and vj − c = uk. We also consider Bethe vectors Bba,b(u¯; v¯) which correspond to the monodromy matrix HJEP06(218) λˆ2(v¯II)λˆ2(u¯)g(v¯, u¯) Kn(v¯I|u¯I)f (u¯I, u¯II)f (v¯II, v¯I) Tb13(u¯I)Tb12(u¯II)Tb23(v¯II)|0i, (2.16) Tb(u). They have the form Bba,b(u¯; v¯) = where λˆ2(z) = λ1(z)λ3(z − c). and Bb: The automorphism T (u) 7→ Tb(u) generates a connection between the Bethe vectors B Bbb,a(v¯ + c; u¯) = (−1)a+b+ab λ2(u¯)λ2(v¯) Ba,b(u¯; v¯). a system of Bethe equations and has the following form: If the system (2.18) is fulfilled, then where , Below we will need the action formulas of the operators Tij (z) and Tbij (z) on the generic Bethe vectors. They were obtained in [25]. We give the list of necessary formulas in appendix C. 3 Special NABA-solvable models At the first sight, a method to construct on-shell Bethe vectors by means of the operator Bg(u) (1.1) contradicts to the content of the previous section. Indeed, according to the general scheme, the on-shell Bethe vector depends on two sets of variables subject to the equations (2.18). At the same time, vector (1.2) depends on only one set of variables. The solution of this contradiction lies in the fact that in some models there is a kind of hierarchy between the variables u¯ and v¯: the set u¯ plays a basic role, while the variables v¯ are auxiliary. In particular, the system of Bethe equations can be reformulated as a constraint on the Bethe parameters u¯ only (see (3.4) below). This class of models includes the XXX SU(3)-invariant Heisenberg chain, for which the operator Bg(u) was originally constructed in [ 1 ]. A characteristic property of these models is that only the operators T12(u) and T13(u) are true creation operators, while T23(u)|0i = 0. In spite of these models are a particular case of the models considered above, they find a wide application in physics.3 Consider a monodromy matrix T 0(u) such that T203(u)|0i = 0. This condition immediately implies a restriction on the vacuum eigenvalues λj (u). Indeed, it follows from the RT T -relation that [T302(u), T203(v)] = g(u, v) T202(v)T303(u) − T202(u)T303(v) . Acting with this equation onto |0i we obtain 0 = λ2(v)λ3(u) − λ2(u)λ3(v) |0i, 3One can also consider models, in which T12(u)|0i = 0, while T23(u) and T13(u) are true creation operators. This case is equivalent to the one considered in this paper, due to an automorphism of the RT T -algebra (2.2) with respect to the replacement Tij(u) → T4−j,4−i(−u). – 7 – (2.18) (2.19) (2.20) HJEP06(218) (3.1) (3.2) leading to λ2(u) = κλ3(u), where κ is a constant. Without loss of generality we can set λ2(z) = κ and λ3(z) = 1. At the same time, the vacuum eigenvalue λ1(z) still remains a free functional parameter. Below we omit the subscript and denote it λ1(z) = λ(z). Bethe equations (2.18) take the form One can show (see e.g. [33]) that this system implies λ(uj ) = κ this parameter. As both sides of (3.4) are polynomials in α of degree a, this condition is equivalent to a set of a equations for a variables u¯ = {u1, . . . , ua} (the free terms in both sides obviously are equal to 1). We see that the set of auxiliary variables v¯ is eliminated. According to the coloring prescriptions, quasiparticles of the second color now can be created by the action of the operator T103(u) only. Since this operator simultaneously creates a quasiparticle of the first color, we conclude that the coloring of any state in these models has a property b ≤ a. In particular, Bethe vectors B0a,b(u¯; v¯) for such the monodromy matrix possess this property. Their explicit form also simplifies: B0a,b(u¯; v¯) = X #u¯I=b κag(v¯, u¯) Kb(v¯|u¯I)f (u¯I, u¯II) T103(u¯I)T102(u¯II)|0i. these models. appropriate twist transformation In distinction of (2.14), here the sum is taken over partitions of the set u¯ ⇒ {u¯I, u¯II} such that #u¯I = b, while the set v¯ is not divided into subsets. We see that a generic off-shell Bethe vector B0a,b(u¯; v¯) still depends on the set of auxiliary Bethe parameters v¯. We will show, however, that the auxiliary parameters can be eliminated from on-shell Bethe vectors, as it was done for the system of Bethe equations. Thus, for the models with the monodromy matrix T 0(u), one can actually restrict himself with a one set of the Bethe parameters only. However, if we substitute the operators Ti0j (u) into equation (1.1) for Bg(u), then we see that Bg(u)|0i = 0. This is due to the fact that T203(u)|0i = 0. Thus, the operator (1.1) cannot be used as a creation operator in A nontrivial action of Bg(u) onto the pseudovacuum vector can be provided by an In paper [ 1 ], a generic twist matrix K was considered. We restrict ourselves with a ‘minimal’ twist, which provides a condition T23(u)|0i 6= 0, but does not change the action of other operators Tij onto |0i. Let T (u) = KT 0(u)K−1. K = I + β 1 − κ E23, – 8 – (3.3a) (3.3b) (3.4) where β 6= 0 is a complex number and E23 is an elementary unit matrix (E23)ij = δi2δj3. It is easy to see that the matrix T (u) has the same vacuum eigenvalues λ1(z) = λ(z), λ2(z) = κ, and λ3(z) = 1. However, now we have T23(u)|0i = β|0i provided κ 6= 1. Of course, the twist matrix (3.7) is not the only matrix, ensuring the condition T23(u)|0i 6= 0. We discuss more general twists in Conclusion. 4 Main results We are now in position to formulate our main results. Proposition 4.1. Let the vacuum eigenvalues of the monodromy matrix T (u) be given be and T23(u)|0i = β|0i. Let u¯ and v¯ be two sets of complex numbers such that #u¯ = a, #v¯ = b, and the constraint (3.3a) is fulfilled. Then Bethe vector Ba,b(u¯, v¯) has the following a X n=0 βb−n n X κa+bg(v¯, u¯) s=0 #u¯I=s X (−κ)n−sλ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III) × T13(u¯I)T13(u¯II)T12(u¯III)|0i. (4.2) Here the sum is taken over partitions of the set u¯ into three subsets u¯ ⇒ {u¯I, u¯II, u¯III}. The cardinalities of the subsets are shown explicitly by the subscripts of the sum symbol in (4.2). The proof of proposition 4.1 is based on the explicit representation for the Bethe vectors (2.14). This is done in section 5. Here we give several comments on this proposition. The condition (3.3a) is a part of Bethe equations, therefore, the corresponding Bethe vector can be called a semi-on-shell Bethe vector [34]. The constraint (3.3a) is a system of a equations for a + b variables. In particular, if4 b ≥ a, then we can consider (3.3a) as the system of equations for the parameters vk, k = 1, . . . , b. At the same time the parameters u¯ remain generic complex numbers, and one can easily show that the system is solvable. Furthermore, it follows from representation (4.2) that if vk, k = 1, . . . , b and vk′, k = 1, . . . , b′ are two different solutions to the system (3.3a), then κb′−bg(v¯′, u¯)Ba,b′ (u¯, v¯′) = βb′−bg(v¯, u¯)Ba,b(u¯, v¯). (4.3) This property is due to the very specific action of the operator T23(z) onto the pseudovacuum vector: T23(z)|0i = β|0i. Thus, two semi-on-shell Bethe vectors with different sets of the Bethe parameters v¯ and v¯′ actually are proportional to each other. In fact, for an appropriate normalization, semi-on-shell Bethe vectors (4.2) do not depend on the parameters of the set v¯. Proposition 4.1 implies that on-shell Bethe vectors also have representation (4.2). In this case the parameters u¯ and v¯ enjoy the additional set of equations (3.3b). We see, however, that the condition (3.3a) is already sufficient to eliminate the parameters v¯ from the representation for the Bethe vector. They are only included in the normalization factor. 4Recall that due to T23(z)|0i 6= 0 we have no restriction b ≤ a. – 9 – Now we give an explicit representation for the multiple action of the operator Bg onto pseudovacuum vector |0i. It was shown in [ 1 ] that [Bg(u), Bg(v)] = 0 for arbitrary u and v. Thus, given a set u¯ = {u1, . . . , ua}, then the notation Bg(u¯) = Y Bg(uj ) a j=1 is well defined. #u¯ = a. Then Proposition 4.2. Let the vacuum eigenvalues of the monodromy matrix T (u) be as in proposition 4.1 and T23(u)|0i = β|0i. Let a set u¯ consist of generic complex numbers and a n=0 n (−κ)n−sλ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III) (4.4) HJEP06(218) × T13(u¯I)T13(u¯II)T12(u¯III)|0i. (4.5) Here the sum over partitions of u¯ is taken as in proposition 4.1. This proposition gives the result of multiple action of the operator Bg onto |0i in terms of multiple actions of the creation operators T12 and T13. The proof of proposition 4.2 is given in section 7. Comparing (4.5) and (4.2) we immediately arrive at Corollary 4.1. Under the conditions of propositions 4.1 and 4.2 Bg(u¯)|0i = β2a−bκa+bg(v¯, u¯)Ba,b(u¯, v¯). (4.6) Thus, if the Bethe parameters u¯ and v¯ satisfy Bethe equations (3.3a), (3.3b), then the vector Bg(u¯)|0i is on-shell Bethe vector, as it is proportional to the on-shell Bethe vector Ba,b(u¯, v¯). One can also consider the vector Bg(u¯)|0i for generic complex u¯. Equation (4.6) remains true in this case, if the set v¯ satisfies the system (3.3a). Due to the property (4.3) one can always provide the solvability of this system for generic complex u¯. 5 Proof of proposition 4.1 We begin with an explicit form of Bethe vectors corresponding to the twisted monodromy matrix T (u) (3.6). This form follows from the general representation (2.14), where one should take into account the condition T23(u)|0i = β|0i. Then Ba,b(u¯; v¯) = βb−nKn(v¯I|u¯I)f (u¯I, u¯II)f (v¯II, v¯I) T13(u¯I)T12(u¯II)|0i. Here, like in (2.14), the sum is taken over partitions of the sets u¯ ⇒ {u¯I, u¯II} and v¯ ⇒ {v¯I, v¯II}. The subscripts of the sums show that the partitions satisfy restrictions #u¯I = #v¯I = n, where n = 0, 1, . . . , min(a, b). The sum over partitions v¯ ⇒ {v¯I, v¯II} can be transformed into a sum over additional partitions of the subset u¯I via (A.2), in which one should set x¯ = v¯ and y¯ = u¯I. Then X #v¯I=n n s=0 #u¯i=s Kn(v¯I|u¯I)f (v¯II, v¯I) = X X (−1)n−sf (u¯i, u¯ii)f (v¯, u¯i). (5.2) Here in the lhs, the sum is taken over partitions v¯ ⇒ {v¯I, v¯II} so that #v¯I = n. In the rhs, the sum is taken over all possible partitions u¯I ⇒ {u¯i, u¯ii}. Substituting this into (5.1) we find Ba,b(u¯, v¯) = X a βb−n n X n=0 κa+b−ng(v¯, u¯) s=0 #u¯i=s X #u¯ii=n−s (−1)n−sf (u¯i, u¯ii)f (v¯, u¯i)f (u¯i, u¯II)f (u¯ii, u¯II) × T13(u¯i)T13(u¯ii)T12(u¯II)|0i. (5.3) In (5.3), the sum is taken over partitions of the set u¯ into three subsets u¯ ⇒ {u¯i, u¯ii, u¯II}. The cardinalities of subsets are shown explicitly by the subscripts of the sum. Note that we have replaced the upper summation limit min(a, b) with a in the sum over n. If a ≤ b, then this replacement certainly is possible. If a > b, then all the terms in the sum over n with n > b vanish due to proposition A.2. Indeed, due to this proposition the sum in the r.h.s. of (5.2) gives a determinant (A.14). The latter vanishes for n > b. Thus, if a > b, then the sum in (5.3) actually breaks at n = b. Suppose that Ba,b(u¯, v¯) is a semi-on-shell Bethe vector whose Bethe parameters satisfy the condition (3.3a). Then taking the product of equations (3.3a) over subset u¯i we find f (v¯, u¯i) = κ−sλ(u¯i) f (u¯i, u¯ii)f (u¯i, u¯II) f (u¯II, u¯i)f (u¯ii, u¯i) . (5.4) Substituting this into (5.3) we arrive at Ba,b(u¯, v¯) = X a βb−n n X n=0 κa+bg(v¯, u¯) s=0 #u¯i=s X #u¯ii=n−s (−κ)n−sλ(u¯i)f (u¯ii, u¯i)f (u¯II, u¯i)f (u¯ii, u¯II) × T13(u¯i)T13(u¯ii)T12(u¯II)|0i. (5.5) This representation coincides with (4.2) up to the labels of the subsets. Thus, proposition 4.1 is proved. 6 Action of Bg(z) on Bethe vectors We use induction over a in order to prove proposition 4.2. However, before doing this, we find the action of the operator Bg(z) on an arbitrary Bethe vector Ba,b(u¯, v¯). This will give us a necessary tool for the proof. Below, for some time, we do not use restrictions T23(z)|0i = β|0i, λ2(z) = κ, and λ3(z) = 1. Instead, we consider the most general case of the monodromy matrix. In order to avoid new notation, we still denote this monodromy matrix by T (z). However, we do not assume that the action of T23(z) has some peculiarity, nor do we impose any restrictions on the eigenvalues λj(z). We simply consider the action of the operator Bg(z) (2.8) on an arbitrary Bethe vector Ba,b(u¯; v¯) using (2.17) and action formulas (C.2)–(C.4). We also replace the expression for Bg(z) (2.8) by T23(z2)Tb13(z1) − T13(z2)Tb12(z1) and consider the limit zk → z (k = 1, 2) in the end of the calculations. Then we specify the obtained result to the semi-on-shell Bethe vectors described in section 4. In this section we study the action of T23(z2)Tb13(z1): Then due to (C.2) we obtain Λ1 = T23(z2)Tb13(z1)Ba,b(u¯; v¯). Λ1 = (−1)a+b+ab λ1(u¯)λ3(v¯) λ2(u¯)λ2(v¯) T23(z2)Tb13(z1)Bbb,a(v¯ + c; u¯). Λ1 = (−1)a+b+abλ1(u¯)λ3(v¯)λˆ2(z1) h(z1, v¯ + c)h(u¯, z1)λ2(u¯)λ2(v¯) T23(z2)Bbb+1,a+1({v¯ + c, z1}; {u¯, z1}). Turning back from Bb to B and using λˆ2(z) = λ1(z)λ3(z − c) we arrive at Λ1 = (−1)a+b+1λ2(z1)λ2(z1 − c)g(z1, v¯) T23(z2)Ba+1,b+1({u¯, z1}; {v¯, z1 − c}). It remains to act with T23(z2) onto the obtained vector via (C.4): Λ1 = × X f (η¯I, η¯II)h(v¯, η¯I)h(z2, η¯I) Ba+1,b+2(η¯II; {v¯, z1 − c, z2}), where η¯ = {u¯, z1, z2} and the sum is taken over partitions η¯ ⇒ {η¯I, η¯II} so that #η¯I = 1. We see that η¯I 6= z1 due to the function g(z1, η¯I) in the denominator of (6.5). Thus, either η¯I = z2 or η¯I = uj, where j = 1, . . . , a. Respectively, we can present Λ1 in the following form Λ1 = Λ(10) + X Λ(1j). a j=1 Here Λ(10) corresponds to the case η¯I = z2: (6.1) (6.2) (6.3) (6.4) (6.5) (6.6) (−1)a+b+1λ2(z1)λ2(z2)λ2(z1 − c)g(z1, v¯)g(z1, z2) × f (uj, u¯j)h(uj, z1)h(v¯, uj)f (z2, uj)Ba+1,b+2({u¯j, z1, z2}; {v¯, z1 − c, z2}). (6.8) Due to (C.2) we can present the vector Ba+1,b+2({u¯j, z1, z2}; {v¯, z1 − c, z2}) as a result Ba+1,b+2({u¯j, z1, z2}; {v¯, z1 − c, z2}) = h(z2, u¯j)h(z2, z1)h(v¯, z2) λ2(z2)g(z1, z2) × T13(z2)Ba,b+1({u¯j, z1}; {v¯, z1 − c}). (6.9) Observe that here we can take the limit z1 = z2 = z: h(uj, z1)g(z2, uj)f (uj, u¯j)h(v¯, uj) Λ(j) 1 z1=z2=z = (−1)aλ2(z)λ2(z − c)g(v¯, z) Now we study the action of T13(z2)Tb12(z1): Using again (2.17) we have and due to (C.3) we obtain Λ2 = T13(z2)Tb12(z1)Ba,b(u¯; v¯). Λ2 = (−1)a+b+ab λ1(u¯)λ3(v¯) λ2(u¯)λ2(v¯) T13(z2)Tb12(z1)Bbb,a(v¯ + c; u¯), Λ2 = (−1)a+b+abλˆ2(z1)λ1(u¯)λ3(v¯)g(z1, v¯) λ2(u¯)λ2(v¯)h(u¯, z1) T13(z2) × X fg((ξ¯ξ¯III,, vξ¯¯I))hh((zξ¯1I,, ξz¯I1)) Bbb+1,a({v¯ + c, z1}; ξ¯II), where ξ¯ = {u¯, z1} and the sum is taken over partitions ξ¯ ⇒ {ξ¯I, ξ¯II} so that #ξ¯I = 1. Turning back to the vector B we find Λ2 = (−1)a+1λ2(z1)λ2(z1 − c)g(z1, v¯) λ1(ξ¯I)f (ξ¯II, ξ¯I)h(ξ¯I, z1) Ba,b+1(ξ¯II; {v¯, z1 − c}). (6.10) (6.11) (6.12) (6.13) (6.14) (6.15) There is no problem to compute the action of T13(z2), however, we do not do this. Instead we present the obtained result in the form similar to (6.6) Λ2 = Λ(20) + X Λ(2j). a j=1 Here Λ(20) corresponds to the partition ξ¯I = z1: Observe that here we can take the limit z1 = z2 = z: Λ(20) = (−1)a+1λ1(z1)λ2(z1 − c)g(u¯, z1)T13(z2)Ba,b+1(u¯; {v¯, z1 − c}). Λ(0) 2 z1=z2=z = (−1)a+1λ1(z)λ2(z − c)g(u¯, z)T13(z)Ba,b+1(u¯; {v¯, z − c}). The contributions Λ(2j) correspond to the partitions ξ¯I = uj and have the following form: Λ(2j) = (−1)aλ2(z1)λ2(z1 − c)g(z1, v¯) λ1(uj)f (u¯j, uj)f (uj, z1) Ba,b+1({u¯j, z1}; {v¯, z1 − c}). Here we also can take the limit Λ(j) 2 z1=z2=z = (−1)aλ2(z)λ2(z − c)g(z, v¯) λ1(uj)f (u¯j, uj)f (uj, z) Ba,b+1({u¯j, z}; {v¯, z − c}). Action of Bg(z) on semi-on-shell Bethe vectors Consider the difference of the contributions Λ(1j) and Λ(2j) at z1 = z2 = z. Using (6.11) and (6.20) we find Λ(1j) − Λ(2j) z1=z2=z = (−1)aλ2(z)λ2(z − c) f (uj, z)g(v¯, z) g(v¯, uj)h(u¯, z) × f (uj, u¯j)f (v¯, uj) − λ2(uj) λ1(uj) f (u¯j, uj) T13(z)Ba,b+1({u¯j, z}; {v¯, z − c}). If Ba,b(u¯; v¯) is a semi-on-shell Bethe vector such that λ2(uj) λ1(uj) f (u¯j, uj) = f (uj, u¯j)f (v¯, uj), then this difference vanishes. In particular, if we impose the constraint (3.3a) (setting λ1(uj) = λ(uj) and λ2(uj) = κ), then the contributions Λ(j) and Λ(j) cancel each other. 1 2 It is remarkable, however, that the cancellation of these terms takes place in the most general case of the semi-on-shell Bethe vectors, for which λ1(z) and λ2(z) are free functional parameters. (6.16) (6.17) (6.18) (6.19) (6.20) (6.21) (6.22) HJEP06(218) Now we are able to prove proposition 4.2 via induction over a. For this, we specify the action formulas of section 6 to the case λ1(z) = λ(z) and λ2(z) = κ. Consider the action of Bg(z) onto |0i = B0,0(∅; ∅). Then due to (6.7), (6.18) we have , Using (5.3) we easily find B0,1(∅; z − c) = βκ−1|0i. In the case a = 1, b = 2, equation (5.3) gives B1,2(u; {v1, v2}) = β2 κ3g(v¯, u) κ β T12(u)|0i + (f (v¯, u) − 1)T13(u)|0i . Setting here u = z1, v1 = z1 − c, and v2 = z2 we obtain leading to Thus, we arrive at B1,2(z1; {z1 − c, z2}) = β2 κ3g(z1, z2) T12(z1)|0i − κ β T13(z1)|0i , Λ(0) = β2T12(z)|0i − κβT13(z)|0i. 1 Bg(z)|0i = Λ(0) − Λ(0) = β2T12(z)|0i + β(λ(z) − κ)T13(z)|0i. 1 2 It is easy to see that representation (4.5) gives the same result for a = 1: 1 n=0 n s=0 #z¯I=s #z¯II=n−s Bg(z)|0i = X β2−n X X (−κ)n−sλ(z¯I)T13(z¯I)T13(z¯II)T12(z¯III)|0i. Here the sum is taken over partitions of the set z¯ (consisting on one element z) into three subsets z¯I, z¯II, and z¯III. Clearly, two of these subsets are empty. Because of this reason we did not write the product of the f -functions in (7.6) (see (4.5)), as these products are taken at least over one empty set. Setting successively in (7.6) z¯I = z, z¯II = z, and z¯III = z we obtain three contributions coinciding with (7.5). Thus, the induction basis is checked. It is interesting to write down this result in terms of the entries of the original monodromy matrix T 0(u). Using (3.6) and (3.7) we find T13(u) = T103(u) − β 1 − κ T102(u), T12(u) = T102(u). Then replacing z with u in representation (7.5) we obtain Bg(u)|0i = β(λ(u) − κ)T103(u)|0i + β 2 1 − λ(u) 1 − κ T102(u)|0i. (7.1) HJEP06(218) (7.2) (7.3) (7.4) (7.5) (7.6) (7.7) (7.8) The monodromy matrix T 0(u) has two on-shell Bethe vectors in the case a = 1: B10,0(u, ∅) and B10,1(u, v). In the first case, there is only one Bethe equation λ(u) = κ, and hence, (7.8) yields Bg(u)|0i = β2T102(u)|0i. In the second case we have a system of two Bethe equations what implies λ(u) = 1. Then (7.8) yields Both vectors T102(u)|0i and T103(u)|0i indeed are on-shell Bethe vectors respectively for λ(u) = κ and λ(u) = 1. Thus, the action of Bg(u) onto the pseudovacuum vector does give the on-shell Bethe vectors, if u is a root of Bethe equations. We assume that (4.5) holds for some a ≥ 1. Then due to corollary 4.1 the action Bg(u¯)|0i with u¯ = {u1, . . . , ua} is proportional to the semi-on-shell Bethe vector Ba,b(u¯, v¯) (4.2), where the set v¯ enjoys the constraint (3.3a). Hence, we have Bg(z)Bg(u¯)|0i = β2a−bκa+bg(v¯, u¯)Bg(z)Ba,b(u¯, v¯). The action of Bg(z) onto Ba,b(u¯, v¯) is given by the terms Λ(20) (6.18) and Λ(10) (6.7) (in the limit z1 = z2 = z). Thus, Bg(z)Bg(u¯)|0i = β2a−bκa+bg(v¯, u¯) Λ(10) − Λ(20) z1=z2=z . Now we should set λ1(z) = λ(z) and λ2(z) = κ in (6.7) and (6.18) for Λ(k0) and substitute these expressions into (7.13). We obtain (7.9) (7.10) (7.11) (7.12) (7.13) (7.14) (7.15) z1=z2=z (7.16) where and Bg(z)Bg(u¯)|0i = M1 + M2, M1 = β2a−bκa+b+1g(v¯, u¯)λ(z)g(z, u¯)T13(z)Ba,b+1(u¯; {v¯, z − c}), M2 = β2a−bκa+b+3g(v¯, u¯) g(v¯, z1)g(u¯, z2)g(z1, z2) Ba+1,b+2({u¯, z1}; {v¯, z1 − c, z2}) It remains to substitute explicit expression (5.3) for the Bethe vectors Ba,b+1(u¯; {v¯, z − c}) and Ba+1,b+2({u¯, z1}; {v¯, z1 − c, z2}) into (7.15) and (7.16). However, before doing this, it is convenient to describe an expected form of the result. We expect that multiple action Bg(z)Bg(u¯)|0i is given by (4.5), in which one should replace u¯ with η¯ = {u¯, z} and a → a + 1. That is, Bg(z)Bg(u¯)|0i = X β2a+2−n X X (−κ)n−sλ(η¯I)f (η¯II, η¯I)f (η¯III, η¯I)f (η¯II, η¯III) Let us give more details on the expected form of the result (7.17). z ∈ η¯II; z ∈ η¯III. Respectively, there are three contributions There are three possibilities in the sum over partitions in the r.h.s. of (7.17): z ∈ η¯I; HJEP06(218) Bg(z)Bg(u¯)|0i = W1 + W2 + W3. (7.18) × T13(η¯I)T13(η¯II)T12(η¯III)|0i. (7.17) a+1 n=0 a n=0 a n=0 n In the first case we have W1 = λ(z)T13(z) X β2a+1−n X X Indeed, we can set η¯I = {z, u¯I}, η¯II = u¯II, and η¯III = u¯III. The set η¯I is not empty, thus, s ∈ [1, . . . , n]. This implies n ∈ [1, . . . , a + 1]. Shifting n → n + 1 and s → s + 1 in (7.17) we arrive at (7.19). In the second case η¯II = {z, u¯II}, η¯I = u¯I, and η¯III = u¯III. The set η¯II is not empty, thus, s ∈ [0, . . . , n − 1]. We also have n ∈ [1, . . . , a + 1], because the union {η¯I, η¯II} is not empty. Shifting n → n + 1 in (7.17) we arrive at W2 = −κT13(z) X β2a+1−n X X (−κ)n−sλ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III) × f (z, u¯I)f (z, u¯III)T13(u¯I)T13(u¯II)T12(u¯III)|0i. (7.20) Finally, in the third case η¯I = u¯I, η¯II = u¯II, and η¯III = {z, u¯III}. The set η¯III is not empty, thus, n ∈ [0, . . . , a]. We obtain a n=0 n W3 = X β2a+2−n X X (−κ)n−sλ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III) × f (z, u¯I)f (u¯II, z)T13(u¯I)T13(u¯II)T12(u¯III)T12(z)|0i. (7.21) Thus, our goal is to check that equations (7.14)–(7.16) give all three contributions Wj, j = 1, 2, 3. 7.2.1 Consider the term M1. Using (5.3) for Ba,b+1(u¯; {v¯, z − c}) we obtain M1 = λ(z)f (u¯, z)T13(z) X κnβ2a+1−n X X (−1)n−s a n=0 n × f (u¯I, u¯II) f (u¯I, z) f (v¯, u¯I) f (u¯I, u¯III)f (u¯II, u¯III) T13(u¯I)T13(u¯II)T12(u¯III)|0i. (7.22) Taking into account (3.3a) we arrive at HJEP06(218) M1 = λ(z)T13(z) X β2a+1−n X X (−κ)n−sf (u¯II, z)f (u¯III, z) Using (3.3a) we arrive at M2(1) = −κT13(z) X β2a+1−n X X (−κ)n−sf (z, u¯I)f (z, u¯III) and we see that this is exactly W2 (7.20). × f (u¯I, u¯II)f (v¯, u¯I)f (u¯I, u¯III)f (u¯II, u¯III) T13(u¯I)T13(u¯II)T12(u¯III)|0i. (7.25) × λ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III) T13(u¯I)T13(u¯II)T12(u¯III)|0i, (7.26) a+1 n=0 We see that this is exactly W1 (7.19). Contributions W2 and W3 Consider the term M2. {z1, u¯} = η¯ we obtain M2 = X κnβ2a+2−n X X n n n × λ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III) T13(u¯I)T13(u¯II)T12(u¯III)|0i. (7.23) Using (5.3) for Ba+1,b+2({u¯, z1}; {v¯, z1 − c, z2}) and setting (−1)n−s f (v¯, η¯I)f (z2, η¯I) f (η¯I, z1) × f (η¯I, η¯II)f (η¯I, η¯III)f (η¯II, η¯III) T13(η¯I)T13(η¯II)T12(η¯III)|0i. (7.24) We see that z1 ∈/ η¯I, otherwise 1/f (η¯I, z1) = 0. Thus, either z1 ∈ η¯II or z1 ∈ η¯III. Respectively, M2 consists of two contributions: M2(1) corresponding to the case z1 ∈ η¯II and M2(2) corresponding to the case z1 ∈ η¯III. Let z1 ∈ η¯II. Then we can set η¯I = u¯I, η¯III = u¯III, and η¯II = {z1, u¯II}. We also have n − s > 0, and thus, s ∈ [0, . . . , n − 1] and n ∈ [1, . . . , a + 1]. Shifting n → n + 1 and setting z1 = z2 = z we obtain M2(1) = −κT13(z) X κnβ2a+1−n X X (−1)n−sf (z, u¯I)f (z, u¯III) a n=0 a n=0 × f (u¯I, u¯II)f (v¯, u¯I)f (u¯I, u¯III)f (u¯II, u¯III) T13(u¯I)T13(u¯II)T12(u¯III)T12(z)|0i. (7.27) HJEP06(218) M2(2) = X κnβ2a+2−n X X (−1)n−sf (z, u¯I)f (u¯II, z) n Let now z1 ∈ η¯III. Then we can set η¯I = u¯I, η¯II = u¯II, and η¯III = {z1, u¯III}. We also have n ∈ [0, . . . , a]. Setting z1 = z2 = z we obtain × λ(u¯I)f (u¯II, u¯I)f (u¯III, u¯I)f (u¯II, u¯III) T13(u¯I)T13(u¯II)T12(u¯III)T12(z)|0i. (7.28) We see that this is exactly W3 (7.21). Thus, the multiple action Bg(z)Bg(u¯)|0i is given by the formulas (7.19)–(7.21), leading to (7.17). Hence, proposition 4.2 is proved. Conclusion. In this paper we have proved one of conjectures of [ 1 ]. Namely we have shown that the successive action of the operator Bg (1.1) on the pseudovacuum vector generates on-shell Bethe vectors in gl3-invariant models, provided the arguments of these operators satisfy Bethe equations. Furthermore, if the arguments of the Bg operators are generic complex numbers, then the successive action of Bg gives a semi-on-shell Bethe vector. This property holds not only for gl3-invariant spin chains, but for a wider class of models, for instance, for the two-component generalization of the Lieb-Liniger model [3, 35– 37]. At the same time, we would like to emphasize that the operator Bg can not be used to construct on-shell Bethe vectors in generic NABA-solvable models. The restriction T203(u)|0i = 0 is crucial. On the other hand, the existence of this restriction was clear from the outset, since within the framework of the new approach Bethe vectors depend only on one set of variables by construction, rather than two sets, as is the case of the Bethe vectors of the general form. In this paper, we considered the minimal twist (3.7). A general twist Kgen can be treated as further twisting of the matrix T (u). It is quite natural to expect that the effect of the general twist must be similar to what one has in the case of gl2 based models [38]. Namely, we saw that for the minimal twist, the multiple action of Bg was equivalent to the one semi-on-shell Bethe vector Ba,b(u¯, v¯). Most probably, that the multiple action of Bg in the case of the general twist is equivalent to a linear combination of semi-onshell Bethe vectors with different sets of the Bethe parameters. However, as soon as we impose Bethe equations, only one term in this linear combination should survive. The proof of this property in the case of gl2-invariant models is very simple (see [38]). However, a generalization of this proof to the models with gl3-invariant R-matrix meats certain technical difficulties. Therefore, we did not consider the case of the general twist. Despite the fact that we have proved the property of Bg(u) to generate on-shell Bethe vectors, we still do not have a clear understanding of why this is happening. In this context, the most intrigues looks the cancellation of ‘unwanted’ terms (6.21). Recall that this cancellation takes place for a general semi-on-shell Bethe vector. We do not need to assume any specific form of λj(u) and specific action of T23(u) onto |0i. Perhaps this is due to some hidden structure of the operator Bg(u), which is not yet clear. It would be very interesting to find this structure. Finally it is worth mentioning that a generalization of the operator Bg(u) to the glN invariant spin chains (N > 3) was also proposed in [ 1 ]. It was conjectured that this operator also generates on-shell Bethe vectors, similarly to the gl3 case. Basing on the results of this paper we can assume that the successive action of Bg(u) is equivalent to a semi-on-shell Bethe vector of a certain glN -invariant integrable model. However, the method that we used in this paper hardly can be applied to the case N > 3, as it becomes very bulky. Acknowledgments We would like to thank I. Kostov, F. Levkovich-Maslyuk, S. Pakuliak, and E. Ragoucy for numerous and fruitful discussions. The work of A.L. has been funded by Russian Academic Excellence Project 5-100 and by Young Russian Mathematics award. A part of this work, section 6, was performed in Steklov Mathematical Institute of Russian Academy of Sciences by N.A.S. and he was supported by the Russian Science Foundation under grant 14-50-00005. A Properties of DWPF The DWPF Kn(x¯|y¯) defined by (2.15) is a rational function of x¯ and y¯. It is symmetric over x¯ and symmetric over y¯. If xj → ∞ (or yj → ∞) and all other variables are fixed, then Kn(x¯|y¯) → 0. This function has simple poles at xj = yk, j, k = 1, . . . , n. The residues in these poles can be expressed in terms of Kn−1. Due to the symmetry of Kn over x¯ and over y¯, it is enough to consider the residue at xn = yn: xn→yn Kn(x¯|y¯) = g(xn, yn)f (x¯n, xn)f (yn, y¯n)Kn−1(x¯n|y¯n) + reg, (A.1) where reg means regular part. the function Kn(x¯|y¯) unambiguously [31, 32]. Proposition A.1. Let #x¯ = m and #y¯ = n so that m ≥ n. Then The properties listed above, together with the initial condition K1(x|y) = g(x, y) fix X Kn(x¯I|y¯)f (x¯II, x¯I) = X X (−1)n−kf (y¯I, y¯II)f (x¯, y¯I). (A.2) Here in the lhs, the sum is taken over partitions x¯ ⇒ {x¯I, x¯II} so that #x¯I = n. In the rhs, the sum is taken over all possible partitions y¯ ⇒ {y¯I, y¯II}. n Proof. We use induction over n. For n = 1, equation (A.2) takes the form m j=1 X g(xj, y)f (x¯j, xj) = f (x¯, y) − 1. Obviously, the l.h.s. of (A.3) is partial fraction decomposition of the r.h.s. Thus, identity (A.2) is valid for n = 1 and arbitrary m ≥ 1. Assume that (A.2) holds for some n − 1 and arbitrary m ≥ n − 1. Let X X (−1)n−kf (y¯I, y¯II)f (x¯, y¯I). Consider properties of Hnℓ,m and Hnr,m as functions of yn at other variables fixed. Both functions are rational functions of yn. Due to the properties of Kn(x¯I|y¯), the function Hnℓ,m(x¯; y¯) vanishes as yn → ∞. Let us show that Hnr,m(x¯; y¯) has the same property. We use the fact that for arbitrary finite z the functions f (z, yn) and f (yn, z) go to 1 as yn → ∞. Clearly, we have either yn ∈ y¯I or yn ∈ y¯II in the sum over partitions over y¯. Consider the first case. Then k > 0 and we can set y¯I = {yn, y¯i}. We obtain X (−1)n−kf (yn, y¯II)f (y¯i, y¯II)f (x¯, yn)f (x¯, y¯i) = X X (−1)n−k−1f (y¯i, y¯II)f (x¯, y¯i). ynli→m∞ n X k=1 #y¯i=k−1 lim yn→∞ n−1 X In the second case k < n and we can set y¯II = {yn, y¯ii}. We obtain X (−1)n−kf (y¯I, yn)f (y¯I, y¯ii)f (x¯, y¯I) = X X (−1)n−kf (y¯I, y¯ii)f (x¯, y¯I). n−1 k=0 #y¯i=k n−1 The remaining sum over partitions gives Hnℓ−1,m−1(x¯j; y¯n), and we finally arrive at yn→xj Hnℓ,m(x¯; y¯) = g(xj, yn)f (x¯j, xj)f (yn, y¯n)Hnℓ−1,m−1(x¯j; y¯n) + reg. Relabeling y¯i → y¯I in (A.5) and y¯ii → y¯II in (A.6) we see that the obtained sums over partitions cancel each other. Thus Hnr,m(x¯; y¯) → 0 as yn → ∞. It remains to compare the residues of two rational functions in the poles yn = xj, j = 1, . . . , m. Let yn → xj in the function Hnℓ,m(x¯; y¯). The pole occurs if and only if xj ∈ x¯I. Setting x¯I = {xj, x¯I′} and using (A.1) we find Hnℓ,m(x¯; y¯) yn→xj = X #x¯I′=n−1 g(xj, yn)f (x¯I′, xj)f (yn, y¯n)Kn−1(x¯I′|y¯n)f (x¯II, x¯I′)f (x¯II, xj)+reg, where reg means regular part. Obviously f (x¯I′, xj)f (x¯II, xj) = f (x¯j, xj). Hence, Hnℓ,m(x¯; y¯) yn→xj = g(xj, yn)f (x¯j, xj)f (yn, y¯n) Kn−1(x¯I′|y¯n)f (x¯II, x¯I′) + reg. X #x¯I′=n−1 (A.3) (A.4) Consider now the behavior of Hnr,m(x¯; y¯) at yn → xj. The pole occurs if and only if yn ∈ y¯I. Setting y¯I = {yn, y¯I′} we obtain Hnr,m(x¯; y¯) yn→xj = X X k=1 #y¯I′=k−1 (−1)n−kf (yn, y¯II)f (y¯I′, y¯II) Using f (yn, y¯I′)f (yn, y¯II) = f (yn, y¯n) and changing k → k + 1 we find × f (x¯j, y¯I′)g(xj, yn)f (x¯j, xj)f (yn, y¯I′) + reg. (A.10) Hnr,m(x¯; y¯) yn→xj = g(xj, yn)f (x¯j, xj)f (yn, y¯n) X X (−1)n−1−kf (y¯I′, y¯II)f (x¯j, y¯I′)+reg. The remaining sum over partitions gives Hnr−1,m−1(x¯j; y¯n), and we finally arrive at yn→xj Hnr,m(x¯; y¯) = g(xj, yn)f (x¯j, xj)f (yn, y¯n)Hnr−1,m−1(x¯j; y¯n) + reg. Due to the induction assumption Hnr−1,m−1(x¯j; y¯n) = Hnℓ−1,m−1(x¯j; y¯n). Hence, the residues of Hnr,m(x¯; y¯) and Hnℓ,m(x¯; y¯) in the poles at yn = xj coincide. Since both functions vanish at yn → ∞ we conclude that Hnr,m(x¯; y¯) = Hnℓ,m(x¯; y¯). Proposition A.2. Let #x¯ = m and #y¯ = n. Then X Here the sum is taken over all possible partitions y¯ ⇒ {y¯I, y¯II}. If m < n, then det n f (yj, y¯j)f (x¯, yj) − δjk = 0. Proof. Expanding the determinant in the r.h.s. of (A.13) over diagonal minors we find  s  s (A.11) (A.12) (A.13) (A.14) (A.15) (A.16) det n h(yj, yk) f (yj, y¯j)f (x¯, yj) − δjk = (−1)n + X(−1)n−s X Thus, we obtain det n h(yj, yk) f (yj, y¯j)f (x¯, yj) − δjk = (−1)n + X(−1)n−s X n s=1 n s=1 The determinant in the r.h.s. is the Cauchy determinant, hence, 1≤j1<···<js≤n p=1 1 det s h(yji, yjk ) = s Y p6=q p,q=1 f (yjp, yjq ) 1 .  Y f (yjp, y¯jp)f (x¯, yjp) dset h(yji, yjk ) . 1 Y f (yjp, y¯jp)f (x¯, yjp) 1≤j1<···<js≤n p=1  s Y p6=q 1 p,q=1 f (yjp, yjq ) . (A.17) This is exactly the sum over partitions in the l.h.s. of (A.13). Let now m < n. Obviously, det f (yj, y¯j)f (x¯, yk) − δjk , because both matrices are related by a similarity transformation. It is easy to see that the matrix in the r.h.s. of (A.18) has an eigenvector with zero eigenvalue: where Indeed, consider a function . Due to the condition m < n this function vanishes as z → ∞. Hence, it has the following partial fraction decomposition cn−m Qpm=1(xp −z) Qqn=1(z −yq +c) = X cn−m Qpm=1(xp −yk +c) k=1 (z −yk +c) Qqn=1,q6=k(yk −yq) k=1 = Xn g(yk, y¯k)h(x¯, yk) . h(z, yk) Setting here z = yj we arrive at k=1 Xn g(yk, y¯k)h(x¯, yk) − = 0. On the other hand, substituting νj from (A.20) into (A.19) we immediately obtain the l.h.s. of (A.23). B Proof of the connection between two types of Bethe vectors The proof of (2.17) is based on the double induction, first on a, and then on b. B.1 First step of induction We first assume that b = 0. Then (2.17) takes the form Bb0,a(∅; u¯) = (−1)a λλ12((uu¯¯)) Ba,0(u¯; ∅). For a = 0, (B.1) turns into a trivial identity: |0i = |0i. It is easy to see that (B.1) also holds for a = 1: Bb0,1(∅; u) = Tb23(u)|0i λˆ2(u) = − T12(u)|0i λ1(u) = − λλ21((uu)) B1,0(u; ∅), where we used (C.8) for Tb23(u). (A.18) (A.19) (A.20) (A.21) (A.22) (A.23) (B.1) (B.2) HJEP06(218) Assume now that (B.1) holds for some a ≥ 1. Then we have for #u¯ = a Bb0,a+1(∅; {u¯, z}) = Tb23(z) λˆTb22(3z()u¯λˆ)2|(0u¯i) = (−1)aTb23(z) λˆ2(z)λ1(u¯) T12(u¯)|0i . Substituting here Tb23(z) from (C.8) we find Bb0,a+1(∅; {u¯, z}) = (−1)a T13(z)T32(z − c) − T12(z)T33(z − c) λˆ2(z)λ1(u¯) . T12(u¯)|0i To calculate the obtained action we use commutation relations of the monodromy matrix [Tij(u), Tkl(v)] = g(u, v) Tkj(v)Til(u) − Tkj(u)Til(v) . In particular, we have T32(u)T12(v) = T12(v)T32(u)f (u, v) − T12(u)T32(v)g(u, v). We see that permuting the operators T32 and T12 we obtain the annihilation operator T32 on the right. Eventually, this operator approaches the vector |0i and annihilates it. Thus, the contribution from the term T13(z)T32(z − c) vanishes. The commutation relations (B.5) also imply T33(u)T12(v) = T12(v)T33(u) + g(u, v) T13(v)T32(u) − T13(u)T32(v) . (B.7) We see that when the operator T33 is permuted with the operator T12, it either commutes or generates the operator T32. As we have already seen, the latter annihilates the state T12(u¯)|0i. Thus, the operator T33(z − c) acts on the state T12(u¯)|0i as T33(z − c)T12(u¯)|0i = λ3(z − c)T12(u¯)|0i. Substituting this into (B.4), we arrive at Bb0,a+1(∅; {u¯, z}) = (−1)a+1λ3(z −c) T12(z)T12(u¯)|0i λ1(u¯)λˆ2(z) = (−1)a+1 λλ21((uu¯¯))λλ21((zz)) Ba+1,0({u¯, z}; ∅), what completes the first step of the induction. Thus, equation (2.17) holds for b = 0 and a arbitrary non-negative. B.2 Second step of induction We pass to the second step of induction. This time we use a recursion for the Bethe vectors Bb [25] λˆ2(z)g(u¯, z)Bbb+1,a({v¯ + c, z}; u¯) = Tb12(z)Bbb,a(v¯ + c; u¯) (B.3) (B.4) (B.5) (B.6) (B.8) (B.9) a j=1 + X g(uj, z) fg((u¯ujj,,uv¯j)) Tb13(z)Bbb,a−1(v¯ + c; u¯j). (B.10) This recursion allows us to uniquely construct the Bethe vector Bbb+1,a, knowing the Bethe Assume that (2.17) holds for some b ≥ 0 and a arbitrary. Then we can replace the Bethe vectors Bb by B in the r.h.s. of (B.10). We obtain λˆ2(z)g(u¯, z)Bbb+1,a({v¯+c, z}; u¯) = (−1)a+b+ab λ2(u¯)λ2(v¯) ( Tb12(z) Ba,b(u¯; v¯) j=1 We should compute the action of the operator Tb12(z) on Ba,b(u¯; v¯) and the action of the operator Tb13(z) on Ba−1,b(u¯j; v¯). This is done in sections C.2.2 and C.2.1 respectively. The results have the following form: Tb13(z)Ba−1,b(u¯j; v¯) = (−1)a+bλ2(z)λ2(z − c) hg((u¯zj,,v¯z)) Ba,b+1({u¯j, z}; {v¯, z − c}), (B.12) and ( Tb12(z)Ba,b(u¯; v¯) = (−1)a+1λ2(z − c) λ1(z)g(u¯, z)Ba,b+1(u¯; {v¯, z − c}) j=1 λ2(uj)g(v¯, uj)h(u¯j, z) ) + λ2(z)g(v¯, z) Xa λ1(uj)g(z, uj)f (u¯j, uj) Ba,b+1({u¯j, z}; {v¯, z − c}) . Λ(z) = λ2(z) h(v¯, z)h(z, u¯) Substituting these formulas into (B.11) we immediately arrive at λˆ2(z)Bbb+1,a({v¯ + c, z}; u¯) = (−1)1+b+ab λ2(u¯)λ2(v¯) λ1(u¯)λ3(v¯) λ1(z)λ2(z − c)Ba,b+1(u¯; {v¯, z − c}). Finally, using λˆ2(z) = λ1(z)λ3(z − c) we obtain This completes the second step of the induction. Bbb+1,a({v¯+c, z}; u¯) = (−1)a+(b+1)+a(b+1) λλ21((uu¯¯))λλ23((vv¯¯))λλ23((zz −−cc)) Ba,b+1(u¯; {v¯, z −c}). (B.15) C C.1 also set 5We set by definition Bbb,−1 = 0. Action formulas Actions of the operators Tij on Bethe vectors Ba,b In this section we give a list of formulas for the actions of the operators Tij(z) on the Bethe vectors Ba,b(u¯; v¯). These formulas were obtained in [25]. Here they are adopted to the new normalization of the Bethe vectors. In all action formulas η¯ = {z, u¯} and ξ¯ = {z, v¯}. We (B.13) (B.14) (C.1) • Action of T12(z): • Action of T23(z): T13(z)Ba,b(u¯; v¯) = Λ(z)Ba+1,b+1(η¯; ξ¯). T12(z)Ba,b(u¯; v¯) = Λ(z) X h(z, ξ¯I) f (ξ¯II, ξ¯I)h(ξ¯I, η¯) Ba+1,b(η¯; ξ¯II). The sum is taken over partitions ξ¯ ⇒ {ξ¯I, ξ¯II} so that #ξ¯I = 1. T23(z)Ba,b(u¯; v¯) = Λ(z) f (η¯I, η¯II)h(ξ¯, η¯I) Ba,b+1(η¯II; ξ¯). The sum is taken over partitions η¯ ⇒ {η¯I, η¯II} so that #η¯I = 1. (C.2) (C.3) (C.4) • Action of T22(z): T22(z)Ba,b(u¯; v¯) = Λ(z) • Action of T11(z): T11(z)Ba,b(u¯; v¯) = Λ(z) X #ξ¯I=#η¯I=1 X #ξ¯I=#η¯I=1 h(z, ξ¯I)h(η¯I, z) f (η¯I, η¯II)f (ξ¯II, ξ¯I)h(ξ¯I, η¯)h(ξ¯II, η¯I) Ba,b(η¯II; ξ¯II). (C.5) λ2(η¯I)g(ξ¯II, η¯I)h(z, ξ¯I) λ1(η¯I)f (η¯II, η¯I)f (ξ¯II, ξ¯I)h(ξ¯I, η¯II) Ba,b(η¯II; ξ¯II). (C.6) The sum is taken over partitions ξ¯ ⇒ {ξ¯I, ξ¯II} and η¯ ⇒ {η¯I, η¯II} so that #ξ¯I = #η¯I = 1. The sum is taken over partitions ξ¯ ⇒ {ξ¯I, ξ¯II} and η¯ ⇒ {η¯I, η¯II} so that #ξ¯I = #η¯I = 1. • Action of T21(z): T21(z)Ba−1,b(u¯; v¯) = Λ(z) X × h(ξ¯I, η¯II)h(ξ¯I, η¯III)h(ξ¯II, η¯II)Ba−1,b(η¯III; ξ¯II). (C.7) The sum is taken over partitions ξ¯ ⇒ {ξ¯I, ξ¯II} and η¯ ⇒ {η¯I, η¯II, η¯III} so that #ξ¯I = #η¯I = #η¯II = 1. The actions of Tbij onto Bba,b(u¯; v¯) are given by the same formulas, where we should put hats for the operators, the vacuum eigenvalues λk(z), and the Bethe vectors. The action formulas (C.2)–(C.7) allow us to derive the actions of the operators Tbij onto the Bethe vectors Ba,b(u¯; v¯). For this we should express Tbij in terms of the original entries Tij via (2.5)–(2.7). In particular, we have Tb12(z) = T21(z)T13(z − c) − T23(z)T11(z − c), Tb23(z) = T13(z)T32(z − c) − T12(z)T33(z − c), Tb13(z) = T12(z)T23(z − c) − T13(z)T22(z − c). (C.8) HJEP06(218) Then the actions of Tbij onto Ba,b(u¯; v¯) can be obtained via successive application of the formulas (C.2)–(C.7). Below we give some details of this derivation for the action of Tb13 and Tb12. C.2.1 Action of Tb13 The operator Tˆ13(z) is given by the last equation (C.8). It is convenient to consider the following combination T12(x)T23(y) − T13(x)T22(y) (C.9) and set x = z, y = z − c in the end. Such the replacement of Tb13(z) allows us to avoid singular expressions in the intermediate computations. Applying successively, first (C.5) and (C.2), and then (C.4) and (C.3) we obtain T13(x)T22(y)Ba,b(u¯; v¯) = and T12(x)T23(y)Ba,b(u¯; v¯) = Λ(x, y) h(y, x) Λ(x, y) h(y, x) X #ξ¯I=1 X f (η¯I, η¯II)f (ξ¯II, ξ¯I) h(ξ¯I, η¯)h(ξ¯II, η¯I) Ba+1,b+1(η¯II; ξ¯II), h(η¯I, y)h(x, ξ¯I)f (η¯I, x) f (η¯I, η¯II)f (ξ¯II, ξ¯I)h(ξ¯I, η¯)h(ξ¯II, η¯I) Ba+1,b+1(η¯II; ξ¯II). Here η¯ = {u¯, x, y} and ξ¯ = {v¯, x, y}. The sum is taken over partitions η¯ ⇒ {η¯I, η¯II} and ξ¯ ⇒ {ξ¯I, ξ¯II} so that #η¯I = #ξ¯I = 1. Here we also introduced Λ(x, y) = λ2(x)λ2(y) h(x, y)h(v¯, x)h(v¯, y)h(x, u¯)h(y, u¯) . Taking the difference of (C.10) and (C.11) we arrive at Tb13(z)Ba,b(u¯; v¯) = Λ(x, y) X h(η¯I, y)h(x, ξ¯I)h(y, ξ¯I)f (η¯I, x) f (η¯I, η¯II)f (ξ¯II, ξ¯I)h(ξ¯I, η¯)h(ξ¯II, η¯I) Ba+1,b+1(η¯II; ξ¯II) (C.10) (C.11) (C.12) x=z . y=z−c (C.13) Now we should consider several cases. First of all, we see that η¯I 6= x, because otherwise the factor 1/f (η¯I, x) in (C.13) is equal to zero. Thus, either η¯I = y or η¯I = uj, j = 1, . . . , a. Consider the first case η¯I = y and denote this contribution by G. Then G = Λ(x, y)f (y, u¯)h(v¯, y)h(x, y) X f (ξ¯II, ξ¯I)h(ξ¯I, x)h(ξ¯I, u¯) Ba+1,b+1({u¯, x}; ξ¯II) This case respectively should be divided into subcases. • ξ¯I = x, hence, ξ¯II = {v¯, y}. Then, substituting (C.12) in (C.14) we find x=z . y=z−c (C.14) (C.15) (C.16) (C.17) G1 = (−1)a+b+1λ2(z)λ2(z − c) hg((zu¯,, v¯z)) Ba+1,b+1({u¯, z}; {v¯, z − c}). • ξ¯I = y, hence, ξ¯II = {v¯, x}. Then h(ξ¯I, x) = h(y, x) → 0, as x → z and y → z − c. Thus, this contribution vanishes. • ξ¯I = vj, j = 1, . . . , b, hence, ξ¯II = {v¯j, x, y}. Then G(12) = λ2(x)λ2(y)g(y, u¯)f (v¯j, vj)g(x, vj)g(y, vj)h(vj, u¯) Ba+1,b+1({u¯, x}; {v¯j, x, y}) y=x=z−zc . h(v¯j, x)h(x, u¯) In this case the Bethe vector Ba,b({u¯, x}; {v¯j, x, y}) vanishes in the limit x = z and y = z − c. Indeed, we have due to (C.2) Ba+1,b+1({u¯, x}; {v¯j, x, y}) = h(y, x)h(v¯j, x)h(x, u¯)T13(x)Ba,b(u¯; {v¯j, y}), 1 and the r.h.s. of (C.17) vanishes, because h(z − c, z) = 0. Similarly, one should consider the case η¯I = uj, j = 1, . . . , a. The analysis of this case Bethe vector Ba,b(u¯; v¯) is given by (C.15): shows that all the corresponding contributions vanish. Thus, the action of Tb13(z) on the Tb13(z)Ba,b(u¯; v¯) = (−1)a+b+1λ2(z)λ2(z − c) hg((zu¯,, v¯z)) Ba+1,b+1({u¯, z}; {v¯, z − c}). (C.18) C.2.2 The action of Tb12(z) can be considered exactly in the same manner. Using (C.8) and the action formulas (C.2)–(C.7) we obtain Tb12(z)Ba,b(u¯; v¯) = Λ(x, y) X × h(ξ¯, η¯II)h(ξ¯I, η¯III)Ba,b+1(η¯III; ξ¯II) x=z . y=z−c (C.19) Here η¯ = {u¯, x, y} and ξ¯ = {v¯, x, y}. The sum is taken over partitions η¯ ⇒ {η¯I, η¯II, η¯III} and ξ¯ ⇒ {ξ¯I, ξ¯II} so that #η¯I = #η¯II = #ξ¯I = 1. Again one should consider several cases. The analysis shows that non-vanishing contributions arise if and only if ξ¯I = x and η¯II = y. Then Tb12(z)Ba,b(u¯; v¯) = λ2(x)λ2(y)g(y, u¯)g(v¯, x) h(x, u¯) × X λ1(η¯I)f (η¯III, η¯I)h(x, η¯III) Ba,b+1(η¯III; {v¯, y}) x=z , y=z−c (C.20) HJEP06(218) where η¯ = {u¯, x} and the sum is taken over partitions η¯ ⇒ {η¯I, η¯III} so that #η¯I = 1. Then we should consider two cases. First, we can set η¯I = x and η¯III = u¯. Then we obtain the first term in (B.13). The second case is η¯I = uj and η¯III = {u¯j , x}, j = 1, . . . , a. Then we obtain the second term in (B.13). Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Separation of Variables for SU(N ) Quantum Spin Chains, JHEP 09 (2017) 111 [arXiv:1610.08032] [INSPIRE]. [2] H. Bethe, On the theory of metals. 1. 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A. Liashyk, N. A. Slavnov. On Bethe vectors in \( \mathfrak{g}{\mathfrak{l}}_3 \) -invariant integrable models, Journal of High Energy Physics, 2018, 18, DOI: 10.1007/JHEP06(2018)018