Correlation functions of the half-infinite XXZ spin chain with a triangular boundary

The half-infinite XXZ spin chain with a triangular boundary is considered in the massive regime. Two integral representations of correlation functions are proposed using bosonization. Sufficient conditions such that the expressions for triangular boundary conditions coincide with those for diagonal boundary conditions are identified. As an application, summation formulae of the boundary expectation values $\langle \sigma_1^a\rangle $ with $a=z,\pm$ are obtained. Exploiting the spin-reversal property, relations between $n$-fold integrals of elliptic theta functions are extracted.


Introduction
spin chain for more general boundary conditions has remained, up to now, essentially problematic. On one hand, even in a simpler case such as the XXZ spin chain with triangular boundary conditions for which the construction of the Bethe vector is feasible [24,25], it remains an open problem in the QISM. On the other hand, in the thermodynamic limit the application of the VOA requires the prior knowledge of the vacuum eigenvectors of the Hamiltonian. Since 1994 [6], even for the simpler case of triangular boundary conditions the solution to this problem has been unknown. However, a breakthrough was recently made in [27], which has opened the possibility of computing correlation functions in the thermodynamic limit of the XXZ open spin chain. Namely, based on the so-called Onsager's approach the structure of the eigenvectors of the finite XXZ spin chain for any type of boundary conditions was interpreted within the representation theory of the q-Onsager algebra [26,27]. In the thermodynamic limit, the q-Onsager algebra is realized by quadratics of the q-vertex operators associated with U q ( sl 2 ) [27] (see also [28] for an alternative derivation). As a consequence, vacuum eigenvectors of the Hamiltonian for a triangular boundary were constructed using the intertwining properties of the q-vertex operators with monomials of the q-Onsager basic generators [27]. The latter being expressed in terms of U q ( sl 2 ) generators, the VOA can be applied in a straightforward manner.

The purpose of this paper is to present the first examples of correlation functions of the half-infinite
XXZ open spin chain with a non-diagonal boundary, using the framework of the VOA. The results here presented extend the earlier studies [6,27]. Among the applications, closed formulae for the boundary expectation values of the spin operators are given and remarkable identities between n−fold integrals of elliptic theta functions are exhibited. Here we focus our attention on the simplest non-diagonal example, namely the half-infinite XXZ spin chain with upper or lower triangular boundary condition. We are interested in the Hamiltonian : (σ x k+1 σ x k + σ y k+1 σ y k + ∆σ z k+1 σ z k ) − 1 − q 2 4q Here we consider the model in the limit of the half-infinite spin chain, in the massive regime where Since under conjugation of H by the spin-reversal operatorν = ∞ j=1 σ x j the sign of the boundary term is reversed, we can restrict our discussion to the boundary term − 1−q 2 4q 1+r 1−r ≥ 0, or −1 ≤ r ≤ 1. Importantly, the two fundamental vacuum eigenvectors |±; i B (i = 0, 1) for the triangular boundary models H (±) B were constructed in a recent paper [27]. For instance, for the lower triangular boundary model H [n] q ! x n . (1.5) In this paper we give the dual vacuum eigenvectors B i; ±| (i = 0, 1) using the intertwining properties of the q-vertex operators of U q ( sl 2 ). For instance, for the lower triangular boundary model H Using these, we compute the integral representations of the correlation functions using the bosonizations.
As a special case, the summation formulae of the boundary expectation values of the spin operators are derived: This is one of the main result of this paper. Also, sufficient conditions such that the correlation functions for a triangular boundary coincide with those for a diagonal boundary are derived. As a special case, we have the following equation for the diagonal matrix σ z : (1.10) Finally, let us also mention that provided a suitable change of the boundary parameters, the Hamiltonian of the two triangular boundary models H B . Using this property, for instance we have the following identity of multiple integrals: (−q 2 ) n (z − z −1 ) − (1 + q 4n )/r + (z + z −1 )q 2n (1 − q 2n z/r)(1 − q 2n /rz) (1 − q 2 /zw a ) (1 − q 2 /rw a ) × Θ q 2 (w 1 w 2 )Θ q 2 (w 2 /w 1 )Θ q 2 (zw 3 /q)Θ q 2 (qw 3 /z) 3 a=1 Θ q 4 (w 2 a /q 2 ) 2 a=1 Θ q 2 (w a w 3 /q 2 )Θ q 2 (w a /qw 3 )Θ q 2 (w a z)Θ q 2 (w a /z) , (1.11) where we have used the elliptic theta function (1 − p n z). (1.12) Here the integration contours C l = C (+,1) l (l = 0, 1) are simple closed curves given below (4.58), (4.60).
The plan of this paper is as follows. In Section 2, the half-infinite XXZ spin chain with a triangular boundary is formulated using the q-vertex operator approach. In Section 3, we review the realizations of the vacuum eigenvectors and their duals [6,27]. In Section 4, two integral representations of the correlation functions are calculated using bosonizations. As a straightforward application, summation formulae of the boundary expectation values σ ± 1 are obtained. Also, we derive identities between multiple integrals of elliptic theta functions from spin-reversal property. For each type of integral representation, a sufficient condition such that the expression for a triangular boundary condition coincides with those for a diagonal boundary condition is identified. Concluding remarks are given in Section 5. In Appendix A we recall some basic facts about the quantum group U q ( sl 2 ) and fix the notations used in the main text. In Appendix B we recall the bosonizations of U q ( sl 2 ) and the q-vertex operators. In Appendix C we summarize convenient formulae for the calculations of the vacuum expectation values.

The half-infinite XXZ spin chain with a triangular boundary
In this Section we give a mathematical formulation of the half-infinite XXZ spin chain with a triangular boundary, based on the q-vertex operator approach [6].

Physical picture
In this Section we sketch a physical picture of our problem. In Sklyanin's framework [14], the transfer matrix T (±,i) B (ζ; r, s) that is a generating function of the Hamiltonian H it is built from two objects: the R-matrix and the K−matrix. For the model (1.1), one introduces the R-matrix R(ζ) defined as: where we have set When viewed as an operator on V ⊗ V , the matrix where the ordering of the index is given by As usual, when copies V j of V are involved, R ij (ζ) acts as R(ζ) on the i-th and j-th components and as identity elsewhere. The R-matrix R(ζ) satisfies the Yang-Baxter equation.
The normalization factor κ(ζ) is determined by the following unitarity and crossing symmetry conditions: Also, we introduce the triangular K-matrix K (±) (ζ) = K (±) (ζ; r, s) [29,30] by where we have set When viewed as an operator on V , the matrix elements of K (±) (ζ) ∈ End(V ) are given by where the ordering of the index is given by v + , v − . As usual, when copies V j of V are involved, K (±) j (ζ) acts as K (±) (ζ) on the j-th component and as identity elsewhere. The K-matrix K (±) (ζ) satisfies the boundary Yang-Baxter equation (also called the reflection equation): (2.8) The normalization factor (2.7) is determined by the following boundary unitarity and boundary crossing symmetry [30]: (2.9) The K (±) (ζ; r, s) defined in (2.5) and (2.6) give general scalar triangular solutions of (2.8) and (2.9).
In Sklyanin's framework, defined on a finite lattice the transfer matrix is built from a finite number of R−matrix [14]. In order to formulate the model (1.1), an infinite combination of R-matrices [6] is considered in T (±,i) B (ζ; r, s) . Generally speaking infinite combinations of the R-matrix are not free from the difficulty of divergence, however we know two useful concepts to study infinite combinations of the R-matrix. One is the corner transfer matrix (CTM) introduced by Baxter [2]. The other is the q-vertex operator introduced by Baxter [35] and Jimbo, Miwa, and Nakayashiki [37]. The CTM for U q ( sl 2 ) [38] gives a supporting argument for the mathematical formulation presented in the next Section, that is free from the difficulty of divergences. Following the strategy summarized in [6,37], let us recall the mathematical formulation of the q-vertex operators and the transfer matrix. Consider the infinite dimensional vector space · · · ⊗ V 3 ⊗ V 2 ⊗ V 1 on which the Hamiltonian (1.1) acts. Let us introduce the subspace H (i) (i = 0, 1) of the half-infinite spin chain by (ζ) for ǫ = ± which act on the space H (i) (i = 0, 1). Their matrix elements are given by products of the R-matrix as follows: where µ(0) = ǫ and µ(N ) = (−1) N +1−i . We expect that the q-vertex operators Φ give rise to well-defined operators. From heuristic arguments by using the R-matrix, the q-vertex opera- (ζ) satisfy the following relations: (2.14) From the property R(ζ) ǫ3,ǫ4 ǫ1,ǫ2 = R(ζ) −ǫ3,−ǫ4 −ǫ1,−ǫ2 , we havê where we have usedν = ∞ j=1 σ x j . Following the strategy [6] we introduce the transfer matrix T (±,i) B (ζ; r, s) using the q-vertex operators. (ζ; r, s), one follows the strategy that we call the qvertex operator approach [6]. As a guide for the structure of the eigenvectors of the Hamiltonian (1.1), the observation that the q-Onsager algebra -a coideal subalgebra of U q ( sl 2 ) -is the hidden symmetry of the Hamiltonian (1.1) plays a central role.
It is important to stress that the formulation (2.16) independently arises within the so-called Onsager's approach of the XXZ open spin chain [27]. Indeed, the transfer matrix of the model (1.1) can be formulated as the thermodynamic limit N → ∞ of the transfer matrix of the finite model, which, in this framework, is expressed in terms of generators of the q-Onsager algebra [26]. In this limit, the transfer matrix associated with (1.1) is a linear combination of q-Onsager currents. Either based on the central extension of the boundary Yang-Baxter equation [28] or using the closed relationship between the q-Onsager algebra and a certain coideal subalgebra of U q ( sl 2 ) [26], q-Onsager currents are realized as quadratic combinations of U q ( sl 2 ) q-vertex operators, providing an alternative derivation of the transfer matrix (2.16).

Vertex operator approach
In this Section we give the mathematical formulation of our problem, the q-vertex operator approach to the half-infinite XXZ spin chain with a triangular boundary.
where the action of e 1 , f 1 , q h1 on V are given by representation with the fundamental weights Λ i (i = 0, 1). In other words, there exists a vector |i ∈ V (Λ i ) (i = 0, 1) such that for j = 0, 1. Let V * (Λ i ) the restricted dual representation of V (Λ i ). We introduce the type-I vertex for x ∈ U q ( sl 2 ). We set the elements of the type-I vertex operators The type-I vertex operators Φ (1−i,i) ǫ (ζ) and Φ * (1−i,i) ǫ (ζ) satisfy the following commutation relation, the duality, and the invertibility properties: where we have set Following the strategy of [6], as the generating function of the Hamiltonian H (ζ; r, s) using the q-vertex operators: Following the strategy in [3,6,37] and the CTM argument for U q ( sl 2 ) [38], as well as the alternative support within Onsager's framework [27], we study our problem upon the following identification: The point of using the q-vertex operators Φ is that they are well-defined objects, free from the difficulty of divergence. In addition, they are the unique solution of the intertwining relations defining the q-vertex operators of the q-Onsager algebra (see [27] for details), which characterizes the hidden non-Abelian symmetry of (1.1).
Finally, let us describe the spin-reversal property. Let ν : V (Λ 0 ) −→ V (Λ 1 ) be the vector-space isomorphism corresponding to the Dynkin diagram symmetry [3]. Then we have where we have used we find that In the next Section we describe the vacuum eigenvectors |±; i B and their duals B i; ±| (i = 0, 1) such Here we have set Once the vacuum eigenvectors are found, it is possible to create the excited states by an application of the type-II vertex operators. Recall that type-II vertex operators Ψ . Define the elements of the vertex operator The type-I vertex operators Φ (ζ) and the type-II vertex operators Ψ (i,1−i) µ (ξ) satisfy the following commutation relation: where we have set This commutation relation implies that Here the number i + N in the suffix (i + N, i + N − 1) of the q-vertex operator Ψ * (i+N,i+N −1) µ1 (ξ 1 ) should be understood modulo 2. Note that type-II vertex operators are intertwiners of the q-Onsager algebra, which justify their use in solving (1.1) [27].

Vacuum eigenvectors and their duals
In the VOA, the computation of correlation functions essentially relies on the prior knowledge of realizations of the vacuum eigenvectors and their duals in terms of q-bosons. For the model (1.1) with diagonal boundary conditions s = 0, they have been proposed in [6]. For the model (1.1) with s = 0, vacuum eigenvectors have been proposed using the representation theory of the spectrum generating algebra (q-Onsager algebra). These objects playing a central role in the analysis of further Sections, below we recall the basic notations and main results of [6,27]. For completeness, dual vacuum eigenvectors of the model (1.1) are described in details.

Diagonal boundary
Upon the specialization s = 0 the Hamiltonian H degenerate to the one of the XXZ spin-chain with a diagonal boundary: Let us recall the structure of the the vacuum eigenvectors |i B and their duals B i| for the XXZ spinchain with a diagonal boundary, obtained in [6]. The spectral problem for the Hamiltonian is identical with the one for the "renormalized" transfer matrix T (±,i) B (ζ; r, 0), namely: Realizations of the vacuum eigenvectors and their duals follow from the bosonization of the q-vertex operators. Indeed, recall that for i = 0, 1 the bosonization of the irreducible highest weight representation V (Λ i ) and the restricted dual representation V * (Λ i ) with the fundamental weights Λ i is [3] (see also Appendix B): Note that the highest weight vector |i and the lowest weight vector i| are given by [3]: In [6], solving directly (3.3) using the bosonization of the transfer matrix that comes from the one of the q-vertex operators (B.9) and (B.10), realizations of the vacuum eigenvectors and their duals were obtained. The results of [6] are summarized as follows: Here we have set and Note that the spin-reversal property of the "renormalized" transfer matrix T that the two vacuum eigenvectors |i B (i = 0, 1) and their duals B i| (i = 0, 1) should be related by However, since the spin-reversal symmetry is obscured in the bosonization, we do not know how to verify this directly from the bosonization formulae.

Triangular boundary
In this Section we recall the structure of the vacuum eigenvectors and their duals for the half-infinite XXZ spin chain with a triangular boundary (1.1). For s ∈ R we are interested in the vacuum eigenvectors Clearly, the Hamiltonian (1.1) can be considered as an integrable perturbation of the diagonal boundary case s = 0. In addition, the spectrum generating algebra associated with (1.1) is the q-Onsager algebra [27]. As a consequence, any eigenvector of the transfer matrix T In the model (1.1), recall that realizations of the q-Onsager fundamental generators are known in terms of U q ( sl 2 ) Drinfeld's basic generators [26]. For the vacuum eigenvectors |±; i B , it is thus natural to look for a combinations of monomials in terms of basic Drinfeld generators acting on |i B . The results of [27] are summarized as follows:

15)
where |i B are the vacuum eigenvectors (3.7) of the diagonal boundary s = 0 [6]. Here we have used the q-exponential function exp q (x) given in (1.5). In order to show the relation (3.14), the following intertwining property of the q-vertex operators are needed: Here we have set Once we assume the relation (3.13), the spin-reversal property of the vacuum eigenvectors follows directly from the q-exponential formula. Indeed, which is consequence of the following reversal property For s ∈ R, we are now interested in the dual vacuum eigenvectors given by Above arguments also hold for the construction of the dual vacuum eigenvectors, from which we obtain the following realizations: Here B i| are the dual vacuum eigenvectors (3.8) of the diagonal boundary s = 0 [6]. In order to show the relation (3.30), we need the following intertwining property of the dual q-vertex operators: Once we assume the relation (3.13), the spin-reversal property of the dual vacuum eigenvectors for a triangular boundary condition follows directly from the q-exponential formula: Finally, from the relation for the q-exponential function exp Hence we have Here we have used the formulae of the norms of B i|i B given in [6].

Correlation functions
In this Section, two integral representations for correlation functions of q-vertex operators are proposed.
In particular, it is shown that the expressions obtained for a subset of correlation functions for the triangular boundary case coincide with the ones associated with a diagonal boundary. Based on the exact relation between local spin operators and q-vertex operators [3,6], summation formulae for the boundary expectation value of the spin operator in the models (1.1) are derived. In the last subsection, using the spin-reversal property we deduce linear relations between certain multiple integrals involving elliptic theta functions. The simplest examples are presented.

Definitions
In this Section, we focus our attention on the vacuum expectation values of products of the q-vertex operators given by with M an even integer. Our purpose is to derive them as integrals of meromorphic functions involving infinite products (4.27) and (4.40), which will be detailed in the next two subsections. In particular, as we will show in the next Section, upon the condition M j=1 ǫ j = 0, the vacuum expectation value of triangular boundary coincides with the one of diagonal boundary. Namely, Upon specialization of the spectral parameters (see [3,6]), the vacuum expectation values (4.1) give multi-point correlation functions of local spin operators of the half-infinite XXZ spin chain with a triangular boundary (1.1). Let L be a linear operator on the n-fold tensor products of the two-dimensional space V n ⊗ · · · ⊗ V 2 ⊗ V 1 . The corresponding local operator L acting on V (Λ i ) can be defined in terms of the type-I vertex operators in exactly the same way as in the bulk theory [3]. Explicitly, if L is the matrix at the n-th site the corresponding local operator E ǫǫ ′ is given by where we have set From the inversion property of the q-vertex operators (2.26), we have (−q −1 ζ 1 , −q −1 ζ 2 , · · · , −q −1 ζ n , ζ n , · · · , ζ 2 , ζ 1 ). (4.7) As a consequence, correlation functions of local operators are given by: In what follows we calculate the vacuum expectation values explicitly using bosonizations.

First integral representation
Consider the M -point functions with M an even integer : The normal ordering of products of the type-I vertex operators are given by Here the integration contour C encircles w a = 0 (a ∈ A) in such a way that q 4 z j (a ≤ j ≤ M ) is inside and q 2 z j (1 ≤ j ≤ a) is outside. From the bosonizations of the Drinfeld's realization in Appendix B, we From the normal orderings in appendix B we have Hence we have the following normal orderings and The zero-mode e α part of the operator : Φ : q (1−2i)n∂ is given by e (n−|A|+ M 2 )α . Then, the condition for which the vacuum expectation value is nonvanishing reads: Hence, provided the condition |A| ≥ M 2 is satisfied, non-zero vacuum expectation values of type-I vertex operators are given by: Next we calculate the following vacuum expectation value more explicitly. Here we have used Using the relation B +; i|i; + B = B i|i B , we have the following formula in [6]: Here α n = −q 6n , γ n = −q −2n , β has been used, where we denote (z; p 1 , p 2 , · · · , p N ) ∞ = ∞ n1,n2,···,nN =0 (1 − p n1 1 p n2 2 · · · p nN N z).

(4.24)
Below, we introduce the double-infinite products We have following infinite product formula of the vacuum expectation value. Note that the formulae summarized in Appendix C are convenient for these calculations.
anYn e F (i) |i Here we have used {z} ∞ and [z] ∞ defined in (4.25). Recall that the set A is given in (4.10). Here the integration contour C (+,0) l is a simple closed curve that satisfies the following conditions for s = 0, 1, 2, · · ·. We set L = |A| − M 2 . The w a (a ∈ A) encircles q 8+4s z j (1 ≤ j < a), q 4+4s z j (a ≤ j ≤ M ), 1+2s w a , q 5+2s /w a (a ∈ A) but not q −3−2s w a , q 3−2s /w a (a ∈ A). The v b (l < b ≤ L) encircles q 3+2s w a , q 5+2s /w a (a ∈ A) but not q 1−2s w a , (a ∈ A). Similarly, the integration contour C (+,1) l is a simple closed curve such that w a (a ∈ A) encircles q 2 /r in addition the same points as C (+,0) l does.
From the normal orderings in appendix B, we have We have the following normal orderings and exp q s r The zero-mode e α part of the operator : Φ . Hence, the condition for which the vacuum expectation value is non-vanishing reads For M 2 ≥ |A|, it implies that non-zero vacuum expectation value takes the form Next, we calculate the vacuum expectation value more explicitly.
anYn e F (i) |i . Here we have defined By straightforward calculations, the following infinite product formula of the vacuum expectation value is obtained: anYn e F (i) |i to the formula (4.18). We note that M j=1 ǫ j = 0 ⇔ n = |A| − M 2 = 0. Upon the specialization n = |A| − M 2 = 0, we have The same argument holds for (4.35). We conclude that upon the parity preserving condition we have the same integral representation as the one for the diagonal boundary conditions [6]. Here we note that we have revised misprints in (4.8) of [6]. (4.43) Here we have used {z} ∞ and [z] ∞ defined in (4.25). The integration contour C 0 is a simple closed curve that satisfies the following conditions for s = 0, 1, 2, · · ·. The w a (a ∈ A) encircles q 8+4s z j (1 ≤ j < a), 0 is a simple closed curve such that w a (a ∈ A) encircles q 2 /r in addition the same points as C

Relations between multiple integrals
Linear relations between n-fold integrals are known in the mathematical literature [39,40,41]. In the context of conformal field theory, some examples also arise in the calculation of correlation functions which contain screening operators. According to the spin-reversal property (4.3), infinitely many relations of this kind can be exhibited based on previous results. Note that our relations can not be reduced to the relations between n-fold integrals of elliptic gamma functions summarized in [40,41]. Also, note that we understand the RHS of the spin-reversal property (4.3) as an analytic continuation of the parameter r.
Secondly, from P (+,1) we have the following identity: (4.60) The integration contour C (+,1) l is a simple closed curve such that w a (a = 1, 2) encircles q 2 /r in addition the same points as the integration contour C (+,0) l does. The integration contour C (+,0) l is given below (4.58). Obviously, using the spin-reversal property of the correlation functions, we can write down infinitely many identities between multiple integrals of elliptic theta functions.

Conclusion
In the present paper, based on the q-vertex operator approach developed in [3,6], two integral representations for the correlation functions of the half-infinite XXZ spin chain with a triangular boundary have been derived in the massive regime. In the special case of diagonal boundary condition, known results are recovered. Due to the presence of a non-diagonal boundary field coupled to the system, the number of particle excitations is no longer conserved in this model. Here, expectation values of the spin operators σ ± 1 which characterize this phenomena have been explicitly proposed. Note that accordingly to [34], the analysis presented here may be extended to the massless regime in a similar way. For the diagonal case, integral representations of the correlation functions are already known [31,20]. As mentioned in the Introduction, for a triangular boundary the construction of the transfer matrix' eigenvectors within the BA approach has been recently achieved [25]. Based on it, an alternative derivation of the correlation functions here presented would be highly desirable. problems. It would be interesting to extend the analysis to models with higher symmetry -for instance U q ( sl(M |N )) -or the ABF model governed by the elliptic quantum group B q,λ ( sl 2 ) [35,36]. Besides, having a better understanding of the space of states through the representation theory is highly desirable.
Indeed, extracting interesting physical data -except in some special cases -from integral representations of correlation functions is a rather complicated problem. In this direction, the remarkable connection between the q-Onsager algebra and the theory of special functions [42] may be promising, as well as the link between solutions of the reflection quantum Knizhnik-Zamolochikov equations [6] and Koornwinder polynomials [43,44] (see also [45]).
Finally, we would like to point out that promising routes have been explored recently. Within Sklyanin's framework, let us mention for instance the functional approach of Galleas [32], the extension of Sklyanin's separation of variable approach [33] or the modified algebraic Bethe ansatz approach proposed in [19] which may provide an alternative derivation of above results.