Algebraic Bethe ansatz for the trigonometric sl(2) Gaudin model with triangular boundary

In the derivation of the generating function of the Gaudin Hamiltonians with boundary terms, we follow the same approach used previously in the rational case, which in turn was based on Sklyanin's method in the periodic case. Our derivation is centered on the quasi-classical expansion of the linear combination of the transfer matrix of the XXZ Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. By defining the appropriate Bethe vectors which yield strikingly simple off-shell action of the generating function, we fully implement the algebraic Bethe ansatz, obtaining the spectrum of the generating function and the corresponding Bethe equations.

simplifies the commutation relations and facilitates the algebraic Bethe ansatz. Probably the simplest way to define the Bethe vectors is through the family of the creation operators C K (µ). These Bethe vectors ϕ M (µ 1 , µ 2 , . . . , µ M ) are symmetric functions of their arguments and they yield strikingly simple off-shell action of the generating function. Actually, it is as simple as it can be since it practically coincide with the corresponding formula in the case when the boundary matrix is diagonal [23]. The off-shell action of the generating function of the Gaudin Hamiltonians with the boundary terms yields the spectrum of the system and the corresponding Bethe equations. As usual, when the Bethe equations are imposed on the parameters of the Bethe vectors, the unwanted terms in the action of the generating function are annihilated.
This paper is organized as follows. In Section II we review some properties of the Lax operator and other fundamental tools in the study of the inhomogeneous XXZ Heisenberg spin chain, including the general solutions of the reflection equation and the dual reflection equation. As one of the main results of the paper, the generating function of the Gaudin Hamiltonians with boundary terms is derived in Section III, using the quasi-classical expansion of the linear combination of the transfer matrix of the inhomogeneous XXZ spin chain and the so-called Sklyanin determinant. The relevant algebraic structure as well as the implementation of the algebraic Bethe ansatz are presented in Section IV, including the definition of the Bethe vectors and the formulae of the off-shell action of the generating function of the Gaudin Hamiltonians. Our conclusions are presented in the Section V. Finally, in Appendix A are given some basic definitions for the convenience of the reader and some explicit formulas regarding the Bethe vector ϕ 3 (µ 1 , µ 2 , µ 3 ) are given in the Appendix B.

II Inhomogeneous XXZ Heisenberg spin chain
We briefly describe the inhomogeneous XXZ Heisenberg spin chain with N sites, characterised by the local space V m = C 2s+1 and inhomogeneous parameter α m . The Hilbert space of the system is (II.1) Here we follow [51] closely and set the Lax operator to be L 0m (λ) = (II. 2) Evidently this Lax operator is a two-by-two matrix in the auxiliary space V 0 = C 2 and it obeys where s m is the value of spin in the space V m [51]. The R-matrix of the XXZ Heisenberg spin chain is given by where λ is a spectral parameter, η is a quasi-classical parameter. Due to the commutation relations (A.2) it is straightforward to check the RLL-relations In this brief description of the XXZ Heisenberg spin chain with non-periodic boundary condition, we will follow Sklyanin's approach [30]. Boundary conditions on the left and right sites of the chain are encoded in the left and right reflection matrices K − and K + . The compatibility condition between the bulk and the boundary of the system takes the form of the so-called reflection equation. It is written in the following form for the left reflection matrix The general, spectral parameter dependent, solutions of the reflection equation (II.6) can be written as follows [56][57][58] The main tool in Sklyanin's framework is the corresponding monodromy matrix it consists of the reflection matrix K − 0 (λ) and the two monodromy matrices T 0 (λ) and T 0 (λ) From the definitions (II.9) and (II.10) and the relations (II.3) and (II.5) follow the RTT-relations Using these RTT-relations (II.11), (II.12), (II.13) and the reflection equation (II.6) it is straightforward to show that the exchange relations of the monodromy matrix T (II.14) These exchange relations (II.14) admit a central element, the so-called Sklyanin determinant, The open chain transfer matrix is given by the trace of the monodromy T (λ) over the auxiliary space V 0 with an extra reflection matrix K + (λ) [30], (II.16) The reflection matrix K + (λ) is the corresponding solution of the dual reflection equation [51] K Due to the fact that the reflection matrices K ∓ (λ) are defined up to multiplicative constants the values of parameters κ ∓ are not essential, as long as they are different from zero. Therefore they could be set to be one without any loss of generality. In particular, this will be evident throughout the Section IV. However, for completeness, we will keep them in our presentation.

III Trigonometric Gaudin model with boundary
The study of the open Gaudin model requires the following condition to be imposed on the reflection matrices [33,53] lim η→0 K + (λ)K − (λ) = κ 2 sinh(ξ − λ) sinh(ξ + λ) − φψ sinh 2 (λ) ½. (III.1) In particular, this implies that the parameters of the reflection matrices on the left and on the right end of the chain are the same. In general this is not the case in the study of the open spin chain. However, this condition is essential for the Gaudin model. Then we will write Moreover, it is straightforward to check the following useful identities Analogously to the rational case [53], our first step in the derivation of the Gaudin model is to obtain first few terms in the power series in η of the Lax operator (II.2) It is important to notice that the spin operators S α m , with α = +, −, 3, on the right hand side of (III.6) satisfy the usual commutation relations As expected, the term linear in η in the expansion above defines the Gaudin Lax matrix [51,55] When the quasi-classical property of the R-matrix (II.4) is taken into account, then substitution of the expansion (III.6) into the RLL-relations (II.5) yields the so-called Sklyanin linear bracket The classical r-matrix (III.10) has the unitarity property and satisfies the classical Yang-Baxter equation Thus the Sklyanin linear bracket (III.11) is anti-symmetric and it obeys the Jacobi identity. It follows that the entries of the Lax matrix (III.8) generate a Lie algebra, the so-called Gaudin algebra, in particular, relevant for the trigonometric Gaudin model with periodic boundary conditions [18].
Our next step is to consider the expansion of the monodromy matrix (II.9) with respect to the quasi-classical parameter η Analogously, it is straightforward to obtain the expansion of the monodromy matrix (II.10) in the powers the quasi-classical parameter η Using the formulae above (III.14) and (III.15) as well as the first three terms in the power series of the K-matrix (III.3) we can deduce the expansion of t(λ) (IV. 19) in powers of η. Similarly, the expansion of ∆ [T (λ)] (II.15) in powers of η is obtained. However these formulas are long and cumbersome, therefore we will not present them here. Instead we will give the expansion of the difference between the transfer matrix of the chain and the so-called Sklyanin determinant. In order to simplify these formulae we introduce the following notation where the Gaudin Lax matrix L 0 (λ) and the reflection matrix K 0 (λ) are given in (III.8) and (III.2), respectively. Therefore, it can be shown that, using the notation above, the expansion in powers of η of the difference between t(λ) and ∆ [T (λ)] is given by Actually, a straightforward calculation shows that the terms in the second line of the expression above vanish (III.18) Also, it is important to notice that using the following identity 19) the first term in the third line of (III.17) can be simplified Finally, the expansion (III.17) reads As expected [53], this shows that commutes for different values of the spectral parameter, Therefore τ(λ) (III.22) can be considered to be the generating function of Gaudin Hamiltonians with boundary terms. In the next section we will obtain these Gaudin Hamiltonians explicitly as well as the spectrum and the corresponding Bethe vectors of the generating function.

IV Linear bracket and algebraic Bethe ansatz
With the aim of implementing the algebraic Bethe ansatz to the trigonometric Gaudin model with triangular K-matrix we seek the linear bracket relations for the Lax operator (III.16). As in the rational case [53], the classical r-matrix (III.10) satisfies the classical Yang-Baxter equation (III.13) and has the unitarity property (III.12). Moreover, the classical r-matrix and the reflection matrix (III.2) satisfy the classical reflection equation With the aim of obtaining the suitable linear bracket relations of the Lax operator (III.16) we define the non-unitary r-matrix It is straightforward to check that this r-matrix satisfies the classical Yang-Baxter equation The linear bracket for the he Lax operator (III.16) reads This linear bracket is obviously anti-symmetric. It obeys the Jacobi identity because the rmatrix (IV.2) satisfies the classical Yang-Baxter equation. Implementation of the algebraic Bethe ansatz requires triangularity of the K-matrix (III.2). As opposed to the rational case [53] were the triangularity of the K-matrix can be guaranteed by the similarity transformation independent of the spectral parameter, in the present case the reflection matrix cannot be brought to the upper triangular form by the U(1) symmetry transformations. On the contrary, we have to impose an extra condition on the parameters of K(λ). By setting φ = 0 the reflection matrix becomes upper triangular Evidently, the inverse matrix is Direct substitution of the formulae above into (III.16), yields the following local realisation for the entries of the Lax matrix Similarly, substituting (IV.5) and (IV.6) together with (III.10) into (IV.2) we obtain r K 00 ′ (λ, µ) explicitly. This non-unitary, classical r-matrix together with the Lax matrix (IV.7) defines the Lie algebra relevant for the open trigonometric Gaudin model. The relation (IV.4) is a compact matrix form of the following commutation relations for the generators E(λ), H(λ) and F(λ) where The generating function of the Gaudin Hamiltonians (III.22) in terms of the entries of the Lax matrix is given by From (IV.13) we have that the last term is (IV.20) and therefore the final expression is (IV.21) Our next step is to introduce the new generators e(λ), h(λ) and f (λ) as the following linear combinations of the original ones The key observation is that in the new basis we have Our aim is to implement the algebraic Bethe ansatz based on the Lie algebra (IV.25) -(IV.28). To this end we need to obtain the expression for the generating function τ(λ) in terms of the generators e(λ), h(λ) and f (λ). The first step is to invert the relations (IV.22) -(IV.24) In particular, we have Substituting (IV.29) -(IV.32) into (IV.21) we obtain the desired expression for the generating function (IV. 33) In the Hilbert space H (II.1), in every V m = C 2s+1 there exists a vector ω m ∈ V m such that We define a vector Ω + to be Ω + = ω 1 ⊗ · · · ⊗ ω N ∈ H.
(IV.37) and by substituting the function ρ(λ) (IV.36) the eigenvalue χ 0 (λ) can be expressed as (IV. 38) An essential step in the algebraic Bethe ansatz is the definition of the corresponding Bethe vectors. In this case, they are symmetric functions of their arguments and are such that the off-shell action of the generating function of the Gaudin Hamiltonians is as simple as possible. With this aim we proceed to show that the Bethe vector ϕ 1 (µ) has the form where c 1 (µ) is given by Evidently, the action of the generating function of the Gaudin Hamiltonians reads A straightforward calculation shows that the commutator in the first term of (IV.41) is given by Therefore the action of the generating function τ(λ) on ϕ 1 (µ) is given by .

(IV.44)
The unwanted term in (IV.43) vanishes when the following Bethe equation is imposed on the parameter µ, Thus we have shown that ϕ 1 (µ) (IV.39) is the desired Bethe vector of the generating function τ(λ) corresponding to the eigenvalue χ 1 (λ, µ) (IV.44). We seek the Bethe vector ϕ 2 (µ 1 , µ 2 ) as the following symmetric function (IV.48) One way to obtain the action of τ(λ) on ϕ 2 (µ 1 , µ 2 ) is to write Then, we substitute (IV.42) in the second and third term above and use the relations which follow from the definition (IV.46). After expressing appropriately the first term on the right-hand side of (IV.49) and using twice the expression for the action of the commutator of τ(λ) with the generator f (λ) on the vector Ω + (IV.42) as well as the identities (IV.50) and (IV.51), a straightforward calculation shows that the off-shell action of the generating function τ(λ) on ϕ 2 (µ 1 , µ 2 ) is given by with the eigenvalue The two unwanted terms in the action above (IV.52) vanish when the Bethe equations are imposed on the parameters µ 1 and µ 2 , with i = 1, 2. Therefore ϕ 2 (µ 1 , µ 2 ) is the Bethe vector of the generating function of the Gaudin Hamiltonians with the eigenvalue χ 2 (λ, µ 1 , µ 2 ).
Therefore we look for the Bethe vector ϕ 4 (µ 1 , . . . , µ 4 ) in the form With the aim of calculating the action of the generating function of the Gaudin Hamiltonians on the vector above we calculate [[[[τ(λ), C 1 (µ 1 )] , C 2 (µ 2 )] , C 3 (µ 3 )] , C 4 (µ 4 )] Ω + , expressing it appropriately as a linear combination of all the previous Bethe vectors. This formula is very long and cumbersome and for this reason, is not presented in the text. Using this result it is possible to obtain the desired off-shell action in the following form where the eigenvalue is This result we have confirmed also by symbolic computing. The four unwanted terms in (IV.64) vanish when the Bethe equation are imposed on the parameters µ i , We readily proceed to define ϕ M (µ 1 , µ 2 , . . . , µ M ), for an arbitrary positive integer M, ϕ M (µ 1 , µ 2 , . . . , µ M ) = C 1 (µ 1 )C 2 (µ 2 ) · · · C M (µ M )Ω + , (IV.67) and the operators C K (µ) are given in (IV.58). Although the operators C K (µ) do not commute, the Bethe vector ϕ M (µ 1 , µ 2 , . . . , µ M ) is a symmetric function of its arguments, since these operators satisfy the following identity, with the eigenvalue The M unwanted terms in (IV.69) vanish when the Bethe equation are imposed on the parameters µ i ,

V Conclusions
We have derived the generating function of the Gaudin Hamiltonians with boundary terms following the same approach used previously in the rational case, which in turn was based on Sklyanin's method in the periodic case. Our derivation is centered on the quasi-classical expansion of the linear combination of the transfer matrix of the XXZ Heisenberg spin chain and the central element, the so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with boundary terms are obtained as the residues of the generating function. Our next step was the implementation of the algebraic Bethe ansatz for the trigonomet- Probably the simplest way to define the relevant Bethe vectors is using the family of the creation operators C K (µ) (IV.58). These Bethe vectors ϕ M (µ 1 , µ 2 , . . . , µ M ) (IV.67) are symmetric functions of their arguments and they yield strikingly simple off-shell action of the generating function. In this sense, we have fully implemented the algebraic Bethe ansatz, obtaining the spectrum of the generating function and the corresponding Bethe equations.
It would be of considerable interest to establish a relationship between these Bethe vectors of the trigonometric Gaudin model and solutions to the corresponding generalized Knizhnik-Zamolodchikov equations, analogously as it was done in the case when the boundary matrix is diagonal [23] but these results will be reported elsewhere.