NUPHB12973S0550-3213(14)00022-410.1016/j.nuclphysb.2014.01.011The AuthorsQuantum Field Theory and Statistical SystemsCorrelation functions of the half-infinite XXZ spin chain with a triangular boundaryP.Baseilhacabaseilha@lmpt.univ-tours.frT.Kojimab⁎kojima@yz.yamagata-u.ac.jpaLaboratoire de Mathématiques et Physique Théorique CNRS/UMR 7350, Fédération Denis Poisson FR2964, Université de Tours, Parc de Grammont, 37200 Tours, FranceLaboratoire de Mathématiques et Physique Théorique CNRS/UMR 7350Fédération Denis Poisson FR2964Université de ToursParc de GrammontTours37200FrancebDepartment of Mathematics and Physics, Faculty of Engineering, Yamagata University, Jonan 4-3-16, Yonezawa 992-8510, JapanDepartment of Mathematics and PhysicsFaculty of EngineeringYamagata UniversityJonan 4-3-16Yonezawa992-8510Japan⁎Corresponding author.AbstractThe 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 〈σ1a〉 with a=z,± are obtained. Exploiting the spin-reversal property, relations between n-fold integrals of elliptic theta functions are extracted.1IntroductionBeyond the Ising model, a large class of solvable lattice models have been discovered. In the context of quantum integrable systems on the lattice, spin chains are among the most studied examples with applications which range from condensed matter to high energy physics. Given a Hamiltonian, finding analytical expressions for the exact spectrum, identifying the structure of the space of the eigenvectors and deriving explicit expressions for correlation functions are essential steps in the non-perturbative characterization of the system's behavior which can be compared with experimental data.Among the simplest examples considered in the literature, the XXZ spin chain with different boundary conditions has received particular attention. Over the years, different approaches have been proposed in order to understand the Hamiltonian's spectral problem and derive the correlation functions. For models with periodic boundary conditions, the spectral problem can be handled by methods such as the Bethe ansatz (BA) [1], or the corner transfer matrix method (CTM) in the thermodynamic limit [2]. The computation of the correlation functions, however, is a much more difficult problem in general. Apart from the simplest example – namely the XXZ spin chain with periodic boundary condition – for which correlation functions have been proposed by the quantum inverse scattering method (QISM) [4,5] arising from the BA, the generalization of this result to models with higher symmetries requires a better understanding of mathematical structures, for instance, of determinant formulae of scalar products that involve the Bethe vectors [7] (see some recent progress in [8]). However, in the thermodynamic limit, this problem can be alternatively tackled using the q-vertex operator approach (VOA) [3] arising from the CTM. The space of states is identified with the irreducible highest weight representation of Uq(sl2ˆ) or higher rank quantum algebras. Correlation functions can be obtained using bosonizations of the q-vertex operators for Uq(sl2ˆ) or higher rank quantum algebras [3,9–13]. Either within the QISM or the VOA, correlation functions are obtained in the form of integrals of meromorphic functions in the thermodynamic limit.The situation for integrable spin chains with open boundaries is more difficult. On one hand, for the finite XXZ open chain with diagonal boundaries [14], related non-diagonal boundaries [15,16] or q a root of unity [17], the BA makes it possible to derive the spectrum and the eigenvectors11Note that recently, a modified BA approach has been considered which looks promising [18,19]. [16]. In each case, the corresponding models are studied using Sklyanin's general formulation of the BA applied to open boundary models [14]. On the other hand, in the thermodynamic limit, the VOA has been applied to the half-infinite XXZ spin chain with a diagonal boundary [6]. Although the hidden symmetry of this model was still unknown at that time,22Recently, the hidden symmetry of the spin chain with a diagonal and non-diagonal boundary has been identified: it is associated with the augmented q-Onsager algebra and q-Onsager algebra, respectively [27]. the diagonalization of the Hamiltonian could still be achieved. Based on these results, the computation of correlation functions has been achieved for diagonal boundary conditions either using the BA [20] or using the VOA in the thermodynamic limit [6]. Note that generalizations to models with higher symmetries have been studied for Uq(slNˆ), etc. [21–23]. In the thermodynamic limit, when the comparison is feasible the expressions obtained by both approaches essentially coincide.In spite of these important developments, the computation of correlation functions of the XXZ open 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 Uq(sl2ˆ) [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 Uq(sl2ˆ) 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:(1.1)HB(±)=−12∑k=1∞(σk+1xσkx+σk+1yσky+Δσk+1zσkz)−1−q24q1+r1−rσ1z−s1−rσ1±, where we have used the standard Pauli matrices(1.2)σx=(0110),σy=(0−ii0),σz=(100−1),σ+=(0100),σ−=(0010). Here we consider the model in the limit of the half-infinite spin chain, in the massive regime where(1.3)Δ=q+q−12,−1<q<0,−1⩽r⩽1,s∈R. Since under conjugation of HB(±) by the spin-reversal operator νˆ=∏j=1∞σjx the sign of the boundary term is reversed, we can restrict our discussion to the boundary term −1−q24q1+r1−r⩾0, or −1⩽r⩽1. Importantly, the two fundamental vacuum eigenvectors |±;i〉B (i=0,1) for the triangular boundary models HB(±) were constructed in a recent paper [27]. For instance, for the lower triangular boundary model HB(−), the two fundamental vacuum eigenvectors33By definition [27], |±;i〉B (i=0,1) are eigenvectors of the Hamiltonian (1.1), not to be confused with the objects called the pseudo-vacuum vectors that arise in the algebraic BA approach. are given by q-exponentials of Uq(sl2ˆ) Chevalley generators acting on the vacuum eigenvectors |i〉B (i=0,1) of the model with a diagonal boundary (s=0) [6]:(1.4)|−;0〉B=expq(−sqe0q−h0)|0〉B,|−;1〉B=expq−1(−srf1)|1〉B. Here we have used the q-exponential function(1.5)expq(x)=∑n=0∞qn(n−1)2[n]q!xn. In this paper we give the dual vacuum eigenvectors 〈Bi;±| (i=0,1) using the intertwining properties of the q-vertex operators of Uq(sl2ˆ). For instance, for the lower triangular boundary model HB(−), the two fundamental dual vacuum eigenvectors are given by(1.6)〈B0;−|=〈B0|expq−1(sqe0q−h0),〈B1;−|=〈B1|expq(srf1). 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:(1.7)〈B0;−|σ1z|−;0〉B〈B0;−|−;0〉B=−1−2(1−r)2∑n=1∞(−q2)n(1−rq2n)2,(1.8)〈B0;−|σ1+|−;0〉B〈B0;−|−;0〉B=s(2+(1−r)∑n=1∞(−q2)n2q2n−r(1+q4n)(1−rq2n)2),(1.9)〈B0;−|σ1−|−;0〉B〈B0;−|−;0〉B=0. 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 σz:(1.10)〈Bi;±|σMz⋯σ2zσ1z|±;i〉B〈Bi;±|±;i〉B=〈Bi|σMz⋯σ2zσ1z|i〉B〈Bi|i〉B.Finally, let us also mention that provided a suitable change of the boundary parameters, the Hamiltonian of the two triangular boundary models HB(±) exchange each other under the action of the spin-reversal operator νˆ. As a consequence, correlation functions of the lower triangular model HB(−) are related to those of the upper triangular model HB(+). Using this property, for instance we have the following identity of multiple integrals:(1.11)(q4;q4)∞4(q2;q2)∞81−z2Θq4(z2)(2+1−z/rz∑n=1∞(−q2)n(z−z−1)−(1+q4n)/r+(z+z−1)q2n(1−q2nz/r)(1−q2n/rz))=(q2∫∫∫C0−∫∫∫C1)∏a=13dwa2π−1q2(1−1/rz)(1−q/rw3)∏a=12(1−q2/zwa)w22w33(1−q2w1/w2)(1−q4/w1w2)∏a=12(1−q2/rwa)Θq2(w1w2)Θq2(w2/w1)Θq2(zw3/q)Θq2(qw3/z)∏a=13Θq4(wa2/q2)∏a=12Θq2(waw3/q2)Θq2(wa/qw3)Θq2(waz)Θq2(wa/z), where we have used the elliptic theta function(1.12)Θp(z)=(p;p)∞(z;p)∞(p/z;p)∞,(z;p)∞=∏n=0∞(1−pnz). Here the integration contours Cl=Cl(+,1) (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 〈σ1±〉 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 Uq(sl2ˆ) and fix the notations used in the main text. In Appendix B we recall the bosonizations of Uq(sl2ˆ) and the q-vertex operators. In Appendix C we summarize convenient formulae for the calculations of the vacuum expectation values.2The half-infinite XXZ spin chain with a triangular boundaryIn 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].2.1Physical pictureIn this section we sketch a physical picture of our problem. In Sklyanin's framework [14], the transfer matrix TˆB(±,i)(ζ;r,s) that is a generating function of the Hamiltonian HB(±) (1.1) is introduced. Basically, 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:(2.1)R(ζ)=1κ(ζ)(1(1−ζ2)q1−q2ζ2(1−q2)ζ1−q2ζ2(1−q2)ζ1−q2ζ2(1−ζ2)q1−q2ζ21), where we have set(2.2)κ(ζ)=ζ(q4ζ2;q4)∞(q2/ζ2;q4)∞(q4/ζ2;q4)∞(q2ζ2;q4)∞,(z;p)∞=∏n=0∞(1−pnz). Let {v+,v−} denote the natural basis of V=C2. When viewed as an operator on V⊗V, the matrix elements of R(ζ)∈End(V⊗V) are given by R(ζ)vϵ1⊗vϵ2=∑ϵ1′,ϵ2′=±vϵ1′⊗vϵ2′R(ζ)ϵ1′ϵ2′ϵ1ϵ2, where the ordering of the index is given by v+⊗v+,v+⊗v−,v−⊗v+,v−⊗v−. As usual, when copies Vj of V are involved, Rij(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.(2.3)R12(ζ1/ζ2)R13(ζ1/ζ3)R23(ζ2/ζ3)=R23(ζ2/ζ3)R13(ζ1/ζ3)R12(ζ1/ζ2). The normalization factor (2.2) is determined by the following unitarity and crossing symmetry conditions:(2.4)R12(ζ)R21(ζ−1)=1,R(ζ)ϵ2ϵ1′ϵ2′ϵ1=R(−q−1ζ−1)−ϵ1ϵ2−ϵ1′ϵ2′. Also, we introduce the triangular K-matrix K(±)(ζ)=K(±)(ζ;r,s) [29,30] by(2.5)K(+)(ζ;r,s)=φ(ζ2;r)φ(ζ−2;r)(1−rζ2ζ2−rsζ(ζ2−ζ−2)ζ2−r01),(2.6)K(−)(ζ;r,s)=φ(ζ2;r)φ(ζ−2;r)(1−rζ2ζ2−r0sζ(ζ2−ζ−2)ζ2−r1), where we have set(2.7)φ(z;r)=(q4rz;q4)∞(q6z2;q8)∞(q2rz;q4)∞(q8z2;q8)∞. When viewed as an operator on V, the matrix elements of K(±)(ζ)∈End(V) are given by K(±)(ζ)vϵ=∑ϵ′=±vϵ′K(±)(ζ)ϵ′ϵ, where the ordering of the index is given by v+,v−. As usual, when copies Vj of V are involved, Kj(±)(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)K2(±)(ζ2)R21(ζ1ζ2)K1(±)(ζ1)R12(ζ1/ζ2)=R21(ζ1/ζ2)K1(±)(ζ1)R12(ζ1ζ2)K2(±)(ζ2). The normalization factor (2.7) is determined by the following boundary unitarity and boundary crossing symmetry [30]:(2.9)K(±)(ζ)K(±)(ζ−1)=1,K(±)(−q−1ζ−1)ϵ1ϵ2=∑ϵ1′,ϵ2′=±R(−qζ2)ϵ1′−ϵ2′−ϵ1ϵ2K(±)(ζ)ϵ2′ϵ1′. 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 TˆB(±,i)(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 Uq(sl2ˆ) [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 ⋯⊗V3⊗V2⊗V1 on which the Hamiltonian (1.1) acts. Let us introduce the subspace H(i) (i=0,1) of the half-infinite spin chain by(2.10)H(i)=Span{⋯⊗vp(N)⊗⋯⊗vp(2)⊗vp(1)|p(N)=(−1)N+i (N≫1)}, where p:N→{±}. We introduce the q-vertex operator Φˆϵ(1−i,i)(ζ) and the dual q-vertex operator Φˆϵ⁎(1−i,i)(ζ) for ϵ=± which act on the space H(i) (i=0,1). Their matrix elements are given by products of the R-matrix as follows:(2.11)(Φˆϵ(1−i,i)(ζ))⋯p(N)⋯p(2) p(1)⋯p(N)′⋯p(2)′p(1)′=limN→∞∑μ(1),μ(2),…,μ(N)=±∏j=1NR(ζ)μ(j−1) p(j)μ(j) p(j)′,(2.12)(Φˆϵ⁎(1−i,i)(ζ))⋯p(N)⋯p(2) p(1)⋯p(N)′⋯p(2)′p(1)′=limN→∞∑μ(1),μ(2),…,μ(N)=±∏j=1NR(ζ)p(j) μ(j)p(j)′ μ(j−1), where μ(0)=ϵ and μ(N)=(−1)N+1−i. We expect that the q-vertex operators Φˆϵ(1−i,i)(ζ) and Φˆϵ⁎(1−i,i)(ζ) give rise to well-defined operators. From heuristic arguments by using the R-matrix, the q-vertex operators Φˆϵ(1−i,i)(ζ) and Φˆϵ⁎(1−i,i)(ζ) satisfy the following relations:(2.13)Φˆϵ2(i,1−i)(ζ2)Φˆϵ1(1−i,i)(ζ1)=∑ϵ1′,ϵ2′=±R(ζ1/ζ2)ϵ1ϵ2ϵ1′ϵ2′Φˆϵ1′(i,1−i)(ζ1)Φˆϵ2′(1−i,i)(ζ2),(2.14)Φˆϵ⁎(1−i,i)(ζ)=Φˆ−ϵ(1−i,i)(−q−1ζ). From the property R(ζ)ϵ1,ϵ2ϵ3,ϵ4=R(ζ)−ϵ1,−ϵ2−ϵ3,−ϵ4, we have(2.15)νˆΦˆϵ(i,1−i)(ζ)νˆ=Φˆ−ϵ(1−i,i)(ζ), where we have used νˆ=∏j=1∞σjx. Following the strategy [6] we introduce the transfer matrix TˆB(±,i)(ζ;r,s) using the q-vertex operators.(2.16)TˆB(±,i)(ζ;r,s)=∑ϵ1,ϵ2=±Φˆϵ1⁎(i,1−i)(ζ−1)K(±)(ζ;r,s)ϵ1ϵ2Φˆϵ2(1−i,i)(ζ). From heuristic arguments by using the boundary Yang–Baxter equation (2.8) and relations of the q-vertex operators (2.13) and (2.14), we have the commutativity of the transfer matrix:(2.17)[TˆB(±,i)(ζ1;r,s),TˆB(±,i)(ζ2;r,s)]=0for any ζ1,ζ2. The Hamiltonian HB(±) (1.1) is obtained as(2.18)ddζTˆB(±,i)(ζ;r,s)|ζ=1=4q1−q2HB(±)+const. In order to diagonalize the transfer matrix TˆB(±,i)(ζ;r,s), one follows the strategy that we call the q-vertex 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 Uq(sl2ˆ) – 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 Uq(sl2ˆ) [26], q-Onsager currents are realized as quadratic combinations of Uq(sl2ˆ) q-vertex operators, providing an alternative derivation of the transfer matrix (2.16).2.2Vertex operator approachIn 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. Let Vζ=V⊗C[ζ,ζ−1]=Vζ(+)⊕Vζ(−) where Vζ(±)=span{v±⊗ζ2n,v∓⊗ζ2n−1 (n∈Z)}. Let Vζ the evaluation representation of Uq(sl2ˆ) by setting(2.19)e0⋅vϵ⊗ζm=(f1vϵ)⊗ζm+1,e1⋅vϵ⊗ζm=(e1vϵ)⊗ζm+1,f0⋅vϵ⊗ζm=(e1vϵ)⊗ζm−1,f1⋅vϵ⊗ζm=(f1vϵ)⊗ζm−1,qh0=q−h1,qh1⋅vϵ⊗ζm=(qh1vϵ)⊗ζm,ρ=ζddζ±12 on Vζ(±), where the action of e1,f1,qh1 on V are given by e1v+=0, e1v−=v+, f1v+=v−, f1v−=0, and qh1v±=q±1v±. Let V(Λi) the irreducible highest weight Uq(sl2ˆ) representation with the fundamental weights Λi (i=0,1). In other words, there exists a vector |i〉∈V(Λi) (i=0,1) such that(2.20)ej|i〉=0,qhj|i〉=q(hj,Λi)|i〉,fj(hj,Λi)+1|i〉=0,V(Λi)=Uq(sl2ˆ)|i〉, for j=0,1. Let V⁎(Λi) the restricted dual representation of V(Λi). We introduce the type-I vertex operators Φϵ(1−i,i)(ζ) as the intertwiner of Uq(sl2ˆ):(2.21)Φ(1−i,i)(ζ):V(Λi)→V(Λ1−i)⊗Vζ,Φ(1−i,i)(ζ)⋅x=Δ(x)⋅Φ(1−i,i)(ζ),(2.22)Φ⁎(1−i,i)(ζ):V(Λi)⊗Vζ→V(Λ1−i),Φ⁎(1−i,i)(ζ)⋅Δ(x)=x⋅Φ⁎(1−i,i)(ζ), for x∈Uq(sl2ˆ). We set the elements of the type-I vertex operators(2.23)Φ(1−i,i)(ζ)=∑ϵΦϵ(1−i,i)(ζ)⊗vϵ,Φϵ⁎(1−i,i)(ζ)|v〉=Φ⁎(1−i,i)(ζ)(|v〉⊗vϵ). The type-I vertex operators Φϵ(1−i,i)(ζ) and Φϵ⁎(1−i,i)(ζ) satisfy the following commutation relation, the duality, and the invertibility properties:(2.24)Φϵ2(i,1−i)(ζ2)Φϵ1(1−i,i)(ζ1)=∑ϵ1′,ϵ2′=±R(ζ1/ζ2)ϵ1ϵ2ϵ1′ϵ2′Φϵ1′(i,1−i)(ζ1)Φϵ2′(1−i,i)(ζ2),(2.25)Φϵ⁎(1−i,i)(ζ)=Φ−ϵ(1−i,i)(−q−1ζ),(2.26)g∑ϵ=±Φϵ⁎(i,1−i)(ζ)Φϵ(1−i,i)(ζ)=id,gΦϵ1(i,1−i)(ζ)Φϵ2⁎(1−i,i)(ζ)=δϵ1ϵ2id, where we have set(2.27)g=(q2;q4)∞(q4;q4)∞. Following the strategy of [6], as the generating function of the Hamiltonian HB(±) (1.1) we introduce the “renormalized” transfer matrix TB(±,i)(ζ;r,s) using the q-vertex operators:(2.28)TB(±,i)(ζ;r,s)=g∑ϵ1,ϵ2=±Φϵ1⁎(i,1−i)(ζ−1)K(±)(ζ;r,s)ϵ1ϵ2Φϵ2(1−i,i)(ζ). From the boundary Yang–Baxter equation and the properties of the q-vertex operators (2.8), (2.9), (2.24), (2.25), (2.26), we have the following properties of the “renormalized” transfer matrix:(2.29)[TB(±,i)(ζ1;r,s),TB(±,i)(ζ2;r,s)]=0for any ζ1,ζ2,(2.30)TB(±,i)(1;r,s)=id,TB(±,i)(ζ;r,s)TB(±,i)(ζ−1;r,s)=id.(2.31)TB(±,i)(−q−1ζ−1;r,s)=TB(±,i)(ζ;r,s). Following the strategy in [3,6,37] and the CTM argument for Uq(sl2ˆ) [38], as well as the alternative support within Onsager's framework [27], we study our problem upon the following identification:(2.32)TB(±,i)(ζ;r,s)=TˆB(±,i)(ζ;r,s),Φϵ(1−i,i)(ζ)=Φˆϵ(1−i,i)(ζ),Φϵ⁎(1−i,i)(ζ)=Φˆϵ⁎(1−i,i)(ζ). The point of using the q-vertex operators Φϵ(1−i,i)(ζ), Φϵ⁎(1−i,i)(ζ) associated with Uq(sl2ˆ) 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(Λ0)→V(Λ1) be the vector-space isomorphism corresponding to the Dynkin diagram symmetry [3]. Then we have(2.33)ν−1ejν=e1−j,ν−1fjν=f1−j,ν−1qhjν=qh1−j. Moreover we have(2.34)νΦϵ(0,1)(ζ)ν=Φ−ϵ(1,0)(ζ), which gives the same relation as (2.15). The q-vertex operator Φϵ(1−i,i)(ζ) has a bosonization summarized in Appendix B. Note that we have two bosonizations of the q-vertex operator based on (2.34). Noting the relation(2.35)σxK(±)(ζ;r,s)σx=Λ(ζ;r)K(∓)(ζ;1/r,−s/r), where we have used(2.36)Λ(ζ;r)=1ζ2Θq4(rζ2)Θq4(q2rζ−2)Θq4(rζ−2)Θq4(q2rζ2), we find that(2.37)ν−1TB(±,1)(ζ;r,s)ν=Λ(ζ;r)TB(∓,0)(ζ;1/r,−s/r). In the next section we describe the vacuum eigenvectors |±;i〉B and their duals 〈Bi;±| (i=0,1) such that(2.38)〈Bi;±|TB(±,i)(ζ;r,s)=〈Bi;±|Λ(i)(ζ;r),(2.39)TB(±,i)(ζ;r,s)|±;i〉B=Λ(i)(ζ;r)|±;i〉B. Here we have set(2.40)Λ(i)(ζ;r)={1(i=0),Λ(ζ;r)(i=1).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 Ψϵ⁎(1−i,i)(ξ) are the intertwiners of Uq(sl2ˆ):(2.41)Ψ⁎(1−i,i)(ξ):Vξ⊗V(Λi)→V(Λ1−i),Ψ⁎(1−i,i)(ξ)⋅Δ(x)=x⋅Ψ⁎(1−i,i)(ξ), for x∈Uq(sl2ˆ). Define the elements of the vertex operator(2.42)Ψμ⁎(1−i,i)(ξ)|v〉=Ψ⁎(1−i,i)(ξ)(vμ⊗|v〉). The type-I vertex operators Φϵ(1−i,i)(ζ) and the type-II vertex operators Ψμ(i,1−i)(ξ) satisfy the following commutation relation:(2.43)Φϵ(i,1−i)(ζ)Ψμ⁎(1−i,i)(ξ)=τ(ζ/ξ)Ψμ⁎(i,1−i)(ξ)Φϵ(1−i,i)(ζ), where we have set(2.44)τ(ζ)=ζ−1Θq4(qζ2)Θq4(qζ−2). This commutation relation implies that(2.45)Ψμ1⁎(i+N,i+N−1)(ξ1)⋯ΨμN−1⁎(i+2,i+1)(ξN−1)ΨμN⁎(i+1,i)(ξN)|±;i〉B is an eigenvector of TB(±,i)(ζ;r,s) with eigenvalue(2.46)Λ(i)(ζ;r)∏j=1Nτ(ζ/ξj)τ(ζξj). Here the number i+N in the suffix (i+N,i+N−1) of the q-vertex operator Ψμ1⁎(i+N,i+N−1)(ξ1) 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].3Vacuum eigenvectors and their dualsIn 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.3.1Diagonal boundaryUpon the specialization s=0 the Hamiltonian HB(±) degenerate to the one of the XXZ spin-chain with a diagonal boundary:(3.1)HB(±)|s=0=−12∑k=1∞(σk+1xσkx+σk+1yσky+Δσk+1zσkz)−1−q24q1+r1−rσ1z. Let us recall the structure of the vacuum eigenvectors |i〉B and their duals 〈Bi| for the XXZ spin-chain with a diagonal boundary, obtained in [6]. The spectral problem for the Hamiltonian is identical with the one for the “renormalized” transfer matrix TB(±,i)(ζ;r,0), namely:(3.2)TB(±,i)(ζ;r,0)|i〉B=Λ(i)(ζ;r)|i〉B,(3.3)〈Bi|TB(±,i)(ζ;r,0)=Λ(i)(ζ;r)B〈i|.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(Λi) and the restricted dual representation V⁎(Λi) with the fundamental weights Λi is [3] (see also Appendix B):(3.4)V(Λi)=C[a−1,a−2,a−3,…]⊗(⨁n∈ZCeΛi+nα),(3.5)V⁎(Λi)=C[a1,a2,a3,…]⊗(⨁n∈ZCe−Λi−nα). Note that the highest weight vector |i〉 and the lowest weight vector 〈i| are given by [3]:(3.6)〈i|=1⊗e−Λi,|i〉=1⊗eΛi. 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:(3.7)|i〉B=exp(Fi)|i〉,Fi=12∑n=1∞nαn[2n]q[n]qa−n2+∑n=1∞βn(i)a−n,(3.8)〈Bi|=〈i|exp(Gi),Gi=12∑n=1∞nγn[2n]q[n]qan2+∑n=1∞δn(i)an. Here we have set(3.9)αn=−q6n,γn=−q−2n, and(3.10)βn(i)=−θnq5n/2(1−qn)[2n]q+{−q7n/2rn[2n]q(i=0),+q3n/2r−n[2n]q(i=1),(3.11)δn(i)=θnq−3n/2(1−qn)[2n]q+{−q−5n/2rn[2n]q(i=0),+q−n/2r−n[2n]q(i=1), where(3.12)θn={1for n even,0for n odd. Note that the spin-reversal property of the “renormalized” transfer matrix TB(±,i)(ζ;r,s) (2.37) suggests that the two vacuum eigenvectors |i〉B (i=0,1) and their duals 〈Bi| (i=0,1) should be related by(3.13)ν(〈B0|)|r→1/r=〈B1|,ν(|0〉B)|r→1/r=|1〉B. However, since the spin-reversal symmetry is obscured in the bosonization, we do not know how to verify this directly from the bosonization formulae.3.2Triangular boundaryIn 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 |±;i〉B which satisfy(3.14) TB(±,i)(ζ;r,s)|±,i〉B=Λ(i)(ζ;r)|±;i〉B. 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 TB(±,i)(ζ;r,s) associated with the Hamiltonian (1.1) can be potentially written in terms of monomials of the q-Onsager generators acting on |i〉B. In the model (1.1), recall that realizations of the q-Onsager fundamental generators are known in terms of Uq(sl2ˆ) 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:(3.15)|+;0〉B=expq−1(sf0)|0〉B,(3.16)|+;1〉B=expq(srqe1q−h1)|1〉B,(3.17)|−;0〉B=expq(−sqe0q−h0)|0〉B,(3.18)|−;1〉B=expq−1(−srf1)|1〉B, where |i〉B are the vacuum eigenvectors (3.7) of the diagonal boundary s=0 [6]. Here we have used the q-exponential function expq(x) given in (1.5). In order to show the relation (3.14), the following intertwining property of the q-vertex operators are needed:(3.19)Φ−(1−i,i)(ζ)B+,0n=q−nB+,0nΦ−(1−i,i)(ζ),(3.20)Φ+(1−i,i)(ζ)B+,0n=qnB+,0nΦ+(1−i,i)(ζ)+s ζ−1[n]qB+,0n−1Φ−(1−i,i)(ζ),(3.21)Φ−(1−i,i)(ζ)B+,1n=qnB+,1nΦ−(1−i,i)(ζ),(3.22)Φ+(1−i,i)(ζ)B+,1n=q−nB+,1nΦ+(1−i,i)(ζ)+srζ[n]qB+,1n−1Φ−(1−i,i)(ζ),(3.23)Φ−(1−i,i)(ζ)B−,0n=q−nB−,0nΦ−(1−i,i)(ζ)+s ζ[n]qB−,0n−1Φ+(1−i,i)(ζ),(3.24)Φ+(1−i,i)(ζ)B−,0n=qnB−,0nΦ+(1−i,i)(ζ),(3.25)Φ−(1−i,i)(ζ)B−,1n=qnB−,1nΦ−(1−i,i)(ζ)+srζ−1[n]qB−,1n−1Φ+(1−i,i)(ζ),(3.26)Φ+(1−i,i)(ζ)B−,1n=q−nB−,1nΦ+(1−i,i)(ζ). Here we have set(3.27)B+,0=sf0,B+,1=srqe1q−h1,B−,0=sqe0q−h0,B−,1=srf1. Once we assume the relation (3.13), the spin-reversal property of the vacuum eigenvectors follows directly from the q-exponential formula. Indeed,(3.28)ν(|±;0〉B)|r→1/r,s→−s/r=|∓;1〉B, which is consequence of the following reversal property(3.29)ν−1B±,1ν|r→1/r,s→−s/r=−B∓,0.For s∈R, we are now interested in the dual vacuum eigenvectors given by(3.30)〈Bi;±|TB(±,i)(ζ;r,s)=Λ(i)(ζ;r)〈Bi;±|. Above arguments also hold for the construction of the dual vacuum eigenvectors, from which we obtain the following realizations:(3.31)〈B0;+|=〈B0|expq(−sf0),(3.32)〈B1;+|=〈B1|expq−1(−srqe1q−h1),(3.33)〈B0;−|=〈B0|expq−1(sqe0q−h0),(3.34)〈B1;−|=〈B1|expq(srf1). Here 〈Bi| 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:(3.35)Φ+⁎(1−i,i)(ζ)B+,0n=q−nB+,0nΦ+⁎(1−i,i)(ζ),(3.36)Φ−⁎(1−i,i)(ζ)B+,0n=qnB+,0nΦ−⁎(1−i,i)(ζ)−qsζ−1[n]qB+,0n−1Φ+⁎(1−i,i)(ζ),(3.37)Φ+⁎(1−i,i)(ζ)B+,1n=qnB+,1nΦ+⁎(1−i,i)(ζ),(3.38)Φ−⁎(1−i,i)(ζ)B+,1n=q−nB+,1nΦ−⁎(1−i,i)(ζ)−srqζ[n]qB+,1n−1Φ+⁎(1−i,i)(ζ),(3.39)Φ+⁎(1−i,i)(ζ)B−,0n=q−nB−,0nΦ+⁎(1−i,i)(ζ)−sqζ[n]qB−,0n−1Φ−⁎(1−i,i)(ζ),(3.40)Φ−⁎(1−i,i)(ζ)B−,0n=qnB−,0nΦ−⁎(1−i,i)(ζ),(3.41)Φ+⁎(1−i,i)(ζ)B−,1n=qnB−,1nΦ+⁎(1−i,i)(ζ)−qsrζ−1[n]qB−,1n−1Φ−⁎(1−i,i)(ζ),(3.42)Φ−⁎(1−i,i)(ζ)B−,1n=q−nB−,1nΦ−⁎(1−i,i)(ζ). 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:(3.43)ν(〈B0;±|)|r→1/r,s→−s/r=〈B1;∓|.Finally, from the relation for the q-exponential function expq(x)expq−1(−x)=1 we deduce(3.44)〈Bi;±|±;i〉B=〈Bi|i〉B. Hence we have(3.45)〈Bi;±|±;i〉B={(q4r2;q8)∞(q6;q8)∞(q2r2;q8)∞(i=0),(q4/r2;q8)∞(q6;q8)∞(q2/r2;q8)∞(i=1). Here we have used the formulae of the norms of 〈Bi|i〉B given in [6].4Correlation functionsIn 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.4.1DefinitionsIn this section, we focus our attention on the vacuum expectation values of products of the q-vertex operators given by(4.1)Pϵ1,ϵ2,…,ϵM(±,i)(ζ1,ζ2,…,ζM;r,s)=〈Bi;±|Φϵ1(i,1−i)(ζ1)Φϵ2(1−i,i)(ζ2)⋯ΦϵM(1−i,i)(ζM)|±;i〉B〈Bi;±|±;i〉B, 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 ∑j=1Mϵj=0, the vacuum expectation value of triangular boundary coincides with the one of diagonal boundary. Namely,(4.2)〈Bi;±|Φϵ1(i,1−i)(ζ1)⋯ΦϵM(1−i,i)(ζM)|±;i〉B〈Bi;±|±;i〉B=〈Bi|Φϵ1(i,1−i)(ζ1)⋯ΦϵM(1−i,i)(ζM)|i〉B〈Bi|i〉B.Note that from the spin-reversal properties (2.34), (3.28), (3.43), we have(4.3)Pϵ1,…,ϵM(±,i)(ζ1,…,ζM;r,s)=P−ϵ1,…,−ϵM(∓,1−i)(ζ1,…,ζM;1/r,−s/r), which will be used in the last subsection. We have two bosonizations of the q-vertex operator, which are based on the relation: νΦϵ(0,1)(ζ)ν=Φ−ϵ(1,0)(ζ). Hence we have two formulae of the correlation functions in (4.3).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 Vn⊗⋯⊗V2⊗V1. The corresponding local operator L acting on V(Λi) 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(4.4)Eϵϵ′⊗id⊗⋯⊗id︸n−1, the corresponding local operator Eϵϵ′ is given by(4.5)Eϵϵ′=Eϵϵ′(1,1,…,1), where we have set(4.6)Eϵϵ′(ζ1,ζ2,…,ζn)=gn∑ϵ1,…,ϵn−1=±Φϵ1⁎(i,i+1)(ζ1)⋯Φϵn−1⁎(i+n−2,i+n−1)(ζn−1)Φϵ⁎(i+n−1,i+n)(ζn)Φϵ′(i+n,i+n−1)(ζn)Φϵn−1(i+n−2,i+n−1)(ζn−1)⋯Φϵ1(i+1,i)(ζ1). From the inversion property of the q-vertex operators (2.26), we have(4.7)〈Bi;±|Eϵnϵn′(ζ1,…,ζn)Eϵn−1ϵn−1′(ζ1,…,ζn−1)⋯Eϵ1ϵ1′(ζ1)|±;i〉B〈Bi;±|±;i〉B=gnP−ϵ1,−ϵ2,…,−ϵn,ϵn′,…,ϵ2′,ϵ1′(±,i)(−q−1ζ1,−q−1ζ2,…,−q−1ζn,ζn,…,ζ2,ζ1). As a consequence, correlation functions of local operators are given by:(4.8)〈Bi;±|Eϵnϵn′Eϵn−1ϵn−1′⋯Eϵ1ϵ1′|±;i〉B〈Bi;±|±;i〉B=gnP−ϵ1,…,−ϵn,ϵn′,…,ϵ1′(±,i)(−q−1,…,−q−1,1,…,1). In what follows we calculate the vacuum expectation values explicitly using bosonizations.4.2First integral representationConsider the M-point functions with M an even integer:(4.9)Pϵ1⋯ϵM(+,i)(ζ1,…,ζM;r,s)=P−ϵ1⋯−ϵM(−,1−i)(ζ1,…,ζM;1/r,−s/r). Here we calculate the vacuum expectation value using bosonizations associated with the upper-triangular boundary model HB(+). In what follows, it is convenient to set zj=ζj2 and(4.10)A={1⩽a⩽M|ϵa=+}. The normal ordering of products of the type-I vertex operators are given by(4.11)Φϵ1(i,1−i)(ζ1)⋯ΦϵM(1−i,i)(ζM)=(−q3)14M(M−1)−∑a∈Aa(1−q2)|A|∏j=1Mζj1+ϵj2+M−j∏1⩽j<k⩽M(q2zk/zj;q4)∞(q4zk/zj;q4)∞∏a∈A∮Cdwa2π−1wa∏a,b∈Aa<b(wa−wb)(wa−q2wb)∏a∈A{∏1⩽j⩽a(zj−q−2wa)∏a⩽j⩽M(wa−q4zj)}:Φ−(i,1−i)(ζ1)⋯Φ−(1−i,i)(ζM)∏a∈AX−(wa):. Here the integration contour C encircles wa=0 (a∈A) in such a way that q4zj (a⩽j⩽M) is inside and q2zj (1⩽j⩽a) is outside. From the bosonizations of the Drinfeld's realization in Appendix B, we have(4.12)f0=qh1x−1+=q∂∮dv2π−1vX+(v),e1q−h1=x0+q−h1=∮dv2π−1X+(v)q−∂. From the normal orderings in Appendix B we have(4.13)(f0)l=ql(l+1)∏b=1l∮dvb2π−1vb∏1⩽a<b⩽l(va−vb)(va−q−2vb):∏b=1lX+(vb):ql∂,(4.14)(e1q−h1)l=q−l(l−1)∏b=1l∮dvb2π−1∏1⩽a<b⩽l(va−vb)(va−q−2vb):∏b=1lX+(vb):q−l∂. Hence we have the following normal orderings(4.15)expq(−sf0)⋅Φϵ1(0,1)(ζ1)Φϵ2(1,0)(ζ2)⋯ΦϵM(1,0)(ζM)⋅expq−1(sf0)=(−q3)14M(M−1)−∑a∈Aa(1−q2)|A|∏j=1Mζj1+ϵj2+M−j∏1⩽j<k⩽M(q2zk/zj;q4)∞(q4zk/zj;q4)∞∑n=0∞sn{∑l,m⩾0l+m=n(−1)l[l]q![m]q!q32l2+l2+12m2+32m+l(2m+M−2|A|)}∏b=1n∮dvb2π−1vb∏a∈A∮dwa2π−1wa∏b=1n∏j=1M(vb−q3zj)∏a∈A{∏j⩽a(zj−q−2wa)∏a⩽j(wa−q4zj)}∏1⩽a<b⩽n(va−vb)(va−q−2vb)∏a,b∈Aa<b(wa−wb)(wa−q2wb)∏b=1n∏a∈A(vb−qwa)(vb−q−1wa):Φ−(0,1)(ζ1)Φ−(1,0)(ζ2)⋯Φ−(1,0)(ζM)∏a∈AX−(wa)∏b=1nX+(vb):qn∂, and(4.16)expq−1(−srqe1q−h1)⋅Φϵ1(1,0)(ζ1)Φϵ2(0,1)(ζ2)⋯ΦϵM(0,1)(ζM)⋅expq(srqe1q−h1)=(−q3)14M(M−1)−∑a∈Aa(1−q2)|A|∏j=1Mζj1+ϵj2+M−j∏1⩽j<k⩽M(q2zk/zj;q4)∞(q4zk/zj;q4)∞∑n=0∞(s/rq)n{∑l,m⩾0l+m=n(−1)l[l]q![m]q!q−32l(l−1)−12m(m−1)−l(2m+M−2|A|)}∏b=1n∮dvb2π−1∏a∈A∮dwa2π−1wa∏b=1n∏j=1M(vb−q3zj)∏a∈A{∏j⩽a(zj−q−2wa)∏a⩽j(wa−q4zj)}∏1⩽a<b⩽n(va−vb)(va−q−2vb)∏a,b∈Aa<b(wa−wb)(wa−q2wb)∏b=1n∏a∈A(vb−qwa)(vb−q−1wa):Φ−(1,0)(ζ1)Φ−(0,1)(ζ2)⋯Φ−(0,1)(ζM)∏a∈AX−(wa)∏b=1nX+(vb):q−n∂. The zero-mode eα part of the operator :Φ−(i,1−i)(ζ1)Φ−(1−i,i)(ζ2)⋯Φ−(1−i,i)(ζM)∏a∈AX−(wa)∏b=1nX+(vb):q(1−2i)n∂ is given by e(n−|A|+M2)α. Then, the condition for which the vacuum expectation value is non-vanishing reads:(4.17)〈Bi|:Φ−(i,1−i)(ζ1)⋯Φ−(1−i,i)(ζM)∏a∈AX−(wa)∏b=1nX+(vb):q(1−2i)n∂|i〉B≠0,⟺〈Bi|e(n−|A|+M2)α|i〉B≠0⟺|A|−M2=n⩾0.Hence, provided the condition |A|⩾M2 is satisfied, non-zero vacuum expectation values of type-I vertex operators are given by:(4.18)〈Bi;+|Φϵ1(i,1−i)(ζ1)Φϵ2(1−i,i)(ζ2)⋯ΦϵM(1−i,i)(ζM)|+;i〉B=(qs)|A|−M2(−q2r)(M2−|A|)i(−q3)14M2+12Mi−∑a∈Aa(1−q2)|A|{∑l,m⩾0l+m=|A|−M2(−1)l[l]q![m]q!q−12l(l+1)+12m(m+1)}∏j=1Mζj1+ϵj2+M−j+i∏1⩽j<k⩽M(q2zk/zj;q4)∞(q4zk/zj;q4)∞∏b=1|A|−M2∮dvb2π−1vb2i−1∏a∈A∮dwa2π−1wa1−i∏b=1|A|−M2∏j=1M(vb−q3zj)∏a∈A{∏j⩽a(zj−q−2wa)∏a⩽j(wa−q4zj)}∏1⩽a<b⩽|A|−M2(va−vb)(va−q−2vb)∏a,b∈Aa<b(wa−wb)(wa−q2wb)∏b=1|A|−M2∏a∈A(vb−qwa)(vb−q−1wa)〈Bi|e∑j=1MP(zj)+∑a∈AR−(wa)+∑b=1|A|−M2R+(vb)e∑j=1MQ(zj)+∑a∈AS−(wa)+∑b=1|A|−M2S+(vb)|i〉B. Next we calculate the following vacuum expectation value more explicitly.(4.19)〈Bi|e∑j=1MP(zj)+∑a∈AR−(wa)+∑b=1|A|−M2R+(vb)e∑j=1MQ(zj)+∑a∈AS−(wa)+∑b=1|A|−M2S+(vb)|i〉B=〈i|eG(i)e∑n=1∞a−nXne−∑n=1∞anYneF(i)|i〉. Here we have used(4.20)Xn=q7n/2[2n]q∑j=1Mzjn−qn/2[n]q∑a∈Awan+q−n/2[n]q∑b=1|A|−M2vbn,(4.21)Yn=q−5n/2[2n]q∑j=1Mzj−n−qn/2[n]q∑a∈Awa−n+q−n/2[n]q∑b=1|A|−M2vb−n. Using the relation 〈B+;i|i;+〉B =〈Bi|i〉B, we have the following formula in [6]:(4.22)〈i|eG(i)e∑n=1∞a−nXne−∑n=1∞anYneF(i)|i〉〈B+;i|i;+〉B=exp(∑n=1∞[2n]q[n]qn11−αnγn{12γnXn2−αnγnXnYn+12αnYn2+(δn(i)+γnβn(i))Xn−(βn(i)+αnδn(i))Yn}). Here αn=−q6n, γn=−q−2n, βn(i), δn(i) are given in (3.9), (3.10), (3.11), respectively. Note that the following infinite product relation(4.23)exp(−∑n=1∞1nzn(1−p1n)(1−p2n)⋯(1−pn))=(z;p1,p2,…,pN)∞ has been used, where we denote(4.24)(z;p1,p2,…,pN)∞=∏n1,n2,…,nN=0∞(1−p1n1p2n2⋯pNnNz). Below, we introduce the double-infinite products(4.25){z}∞=(z;q4,q4)∞,[z]∞=(z;q8,q8)∞. We have following infinite product formula of the vacuum expectation value. Note that the formulae summarized in Appendix C are convenient for these calculations.(4.26)〈i|eG(i)e∑n=1∞a−nXne−∑n=1∞anYneF(i)|i〉〈B+;i|i;+〉B=({q6}∞{q8}∞)M{(q4;q2)∞}|A|{(q2;q2)∞}|A|−M2∏1⩽j<k⩽M{q6zjzk}∞{q2/zjzk}∞{q6zj/zk}∞{q6zk/zj}∞{q8zjzk}∞{q4/zjzk}∞{q8zj/zk}∞{q8zk/zj}∞∏j=1M[q10zj2]∞[q14zj2]∞[q10/zj2]∞[q6/zj2]∞[q12zj2]∞[q16zj2]∞[q12/zj2]∞[q8/zj2]∞∏j=1M∏b=1|A|−M2(qzjvb;q4)∞(q7zj/vb;q4)∞(qvb/zj;q4)∞(q3/zjvb;q4)∞∏j=1M∏a∈A(q2zjwa;q4)∞(q8zj/wa;q4)∞(q2wa/zj;q4)∞(q4/zjwa;q4)∞∏a∈A(wa2/q2;q4)∞(q6/wa2;q4)∞∏b=1|A|−M2(vb2/q2;q4)∞(q6/vb2;q4)∞∏b=1|A|−M2∏a∈A(vbwa/q3;q2)∞(q3vb/wa;q2)∞(q3wa/vb;q2)∞(q5/vbwa;q2)∞∏a,b∈Aa<b(wawb/q2;q2)∞(q4wa/wb;q2)∞(q4wb/wa;q2)∞(q6/wawb;q2)∞∏1⩽a<b⩽|A|−M2(vavb/q4;q2)∞(q2va/vb;q2)∞(q2vb/va;q2)∞(q4/vavb;q2)∞{∏j=1M(q2rzj;q4)∞(q4rzj;q4)∞∏b=1|A|−M2(1−rvb/q3)∏a∈A(1−q−2rwa)(i=0),∏j=1M(1/rzj;q4)∞(q2/rzj;q4)∞∏b=1|A|−M2(1−q/rvb)∏a∈A(1−q2/rwa)(i=1). Summarizing the above calculations, we finally obtain the following first integral representation of the M-point functions with M even:(4.27)Pϵ1,…,ϵM(+,i)(ζ1,…,ζM;r,s)=P−ϵ1,…,−ϵM(−,1−i)(ζ1,…,ζM;1/r,−s/r)=(qs)|A|−M2(−q3)14M2−∑a∈Aa({q6}∞{q8}∞)M(q2;q2)∞2|A|−M2∏1⩽j<k⩽M{q6zjzk}∞{q2/zjzk}∞{q6zj/zk}∞{q2zk/zj}∞{q8zjzk}∞{q4/zjzk}∞{q8zj/zk}∞{q4zk/zj}∞∏j=1M[q10zj2]∞[q14zj2]∞[q10/zj2]∞[q6/zj2]∞[q12zj2]∞[q16zj2]∞[q12/zj2]∞[q8/zj2]∞∏j=1Mζj1+ϵj2+M−j∑l,m⩾0l+m=|A|−M2(−1)lq−12l(l+1)+12m(m+1)[l]q![m]q!∮⋯∮Cl(+,i)∏a∈Adwa2π−1wa1−i∏b=1|A|−M2dvb2π−1vb2i−1∏b=1|A|−M2∏j=1M(vb−q3zj)∏a∈A{∏1⩽j⩽a(zj−q−2wa)∏a⩽j⩽M(wa−q4zj)}∏j=1M∏b=1|A|−M2(qzjvb;q4)∞(q7zj/vb;q4)∞(qvb/zj;q4)∞(q3/zjvb;q4)∞∏j=1M∏a∈A(q2zjwa;q4)∞(q8zj/wa;q4)∞(q2wa/zj;q4)∞(q4/zjwa;q4)∞∏a∈A(wa2/q2;q4)∞(q6/wa2;q4)∞∏b=1|A|−M2(vb2/q2;q4)∞(q6/vb2;q4)∞∏b=1|A|−M2∏a∈Avb2(vbwa/q3;q2)∞(q3vb/wa;q2)∞(wa/qvb;q2)∞(q5/vbwa;q2)∞∏a,b∈Aa<bwa2(wawb/q2;q2)∞(q4wa/wb;q2)∞(wb/wa;q2)∞(q6/wawb;q2)∞∏1⩽a<b⩽|A|−M2va2(vavb/q4;q2)∞(q2va/vb;q2)∞(q−2vb/va;q2)∞(q4/vavb;q2)∞{∏j=1M(q2rzj;q4)∞(q4rzj;q4)∞∏b=1|A|−M2(1−rvb/q3)∏a∈A(1−q−2rwa)(i=0),(−q2r)M2−|A|(−q3)M2∏j=1Mζj(1/rzj;q4)∞(q2/rzj;q4)∞∏b=1|A|−M2(1−q/rvb)∏a∈A(1−q2/rwa)(i=1).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 Cl(+,0) is a simple closed curve that satisfies the following conditions for s=0,1,2,… . We set L=|A|−M2. The wa (a∈A) encircles q8+4szj (1⩽j<a), q4+4szj (a⩽j⩽M), q4+4s/zj (1⩽j⩽M), q3+2svb (1⩽b⩽l), q−1+2svb (l<b⩽L), q5+2s/vb (1⩽b⩽L), but not q2−4szj (1⩽j⩽a), q−2−4szj (a<j⩽M), q−2−4s/zj (1⩽j⩽M), q1−2svb (1⩽b⩽l), q−3−2svb (l<b⩽L), q3−2s/vb (1⩽b⩽L), q2/r. The vb (1⩽b⩽l) encircles q−1+2swa, q5+2s/wa (a∈A) but not q−3−2swa, q3−2s/wa (a∈A). The vb (l<b⩽L) encircles q3+2swa, q5+2s/wa (a∈A) but not q1−2swa, q3−2s/wa (a∈A). Similarly, the integration contour Cl(+,1) is a simple closed curve such that wa (a∈A) encircles q2/r in addition the same points as Cl(+,0) does.4.3Second integral representationIn this section we consider M-point functions with M an even integer:(4.28)Pϵ1,…,ϵM(−,i)(ζ1,…,ζM;r,s)=P−ϵ1,…,−ϵM(+,1−i)(ζ1,…,ζM;1/r,−s/r). Our aim is to calculate the vacuum expectation values using bosonizations associated with the lower-triangular boundary model HB(−). Recall that A={1⩽a⩽M|ϵa=+}. From the bosonizations of the Drinfeld realization in Appendix B, we have(4.29)e0q−h0=q−1x1−=q−1∮dv2π−1vX−(v),f1=x0−=∮dv2π−1X−(v). From the normal orderings in Appendix B, we have(4.30)(e0q−h0)l=q−l∏b=1l∮dvb2π−1vb∏1⩽a<b⩽l(va−vb)(va−q2vb):∏b=1lX−(vb):,(4.31)(f1)l=∏b=1l∮dvb2π−1∏1⩽a<b⩽l(va−vb)(va−q2vb):∏b=1lX−(vb):. We have the following normal orderings(4.32)expq−1(sqe0q−h0)Φϵ1(0,1)(ζ1)Φϵ2(1,0)(ζ2)⋯ΦϵM(1,0)(ζM)expq(−sqe0q−h0)=(−q3)14M(M−1)−∑a∈Aa(1−q2)|A|∏j=1Mζj1+ϵj2+M−j∏1⩽j<k⩽M(q2zk/zj;q4)∞(q4zk/zj;q4)∞∑n=0∞(s/q2)n∑l,m⩾0l+m=n(−1)mq−12l(l−1)+12m(m−1)(−q3)−Mm[l]q![m]q!∏b=1n∮dvb2π−1vb∏a∈A∮dwa2π−1wa∏a,b∈Aa<b(wa−wb)(wa−q2wb)∏a∈A{∏j⩽a(zj−q−2wa)∏a⩽j(wa−q4zj)}∏1⩽a<b⩽n(va−vb)(va−q2vb)∏j=1M{∏b=1l(vb−q4zj)∏b=l+1n(zj−q−2vb)}∏b=1l∏a∈A(vb−wa)(vb−q2wa)∏a∈A∏b=l+1n(wa−vb)(wa−q2vb):Φ−(0,1)(ζ1)Φ−(1,0)(ζ2)⋯Φ−(1,0)(ζM)∏a∈AX−(wa)∏b=1nX−(vb):, and(4.33)expq(srf1)Φϵ1(1,0)(ζ1)Φϵ2(0,1)(ζ2)⋯ΦϵM(0,1)(ζM)expq−1(−srf1)=(−q3)14M(M−1)−∑a∈Aa(1−q2)|A|∏j=1Mζj1+ϵj2+M−j∏1⩽j<k⩽M(q2zk/zj;q4)∞(q4zk/zj;q4)∞∑n=0∞(s/r)n∑l,m⩾0l+m=n(−1)mq12l(l−1)−12m(m−1)(−q3)−Mm[l]q![m]q!∏b=1n∮dvb2π−1∏a∈A∮dwa2π−1wa∏a,b∈Aa<b(wa−wb)(wa−q2wb)∏a∈A{∏j⩽a(zj−q−2wa)∏a⩽j(wa−q4zj)}∏1⩽a<b⩽n(va−vb)(va−q2vb)∏j=1M{∏b=1l(vb−q4zj)∏b=l+1n(zj−q−2vb)}∏a=1l∏b∈A(va−wb)(va−q2wb)∏a∈A∏b=l+1n(wa−vb)(wa−q2vb):Φ−(0,1)(ζ1)Φ−(1,0)(ζ2)⋯Φ−(1,0)(ζM)∏a∈AX−(wa)∏b=1nX−(vb):. The zero-mode eα part of the operator :Φ−(i,1−i)(ζ1)⋯Φ−(1−i,i)(ζM)∏a∈AX−(wa)×∏b=1nX−(vb): is given by eα(M2−|A|−n). Hence, the condition for which the vacuum expectation value is non-vanishing reads(4.34)〈Bi|:Φ−(i,1−i)(ζ1)⋯Φ−(1−i,i)(ζM)∏a∈AX−(wa)∏b=1nX−(vb):|i〉B≠0,⟺〈Bi|e(M2−|A|−n)α|i〉B≠0⟺M2−|A|=n⩾0. For M2⩾|A|, it implies that non-zero vacuum expectation value takes the form(4.35)〈B−;i|Φϵ1(i,1−i)(ζ1)Φϵ2(1−i,i)(ζ2)⋯ΦϵM(i,1−i)(ζM)|i;−〉B=(−q3)14M2+i2M−∑a∈Aa(1−q2)|A|(s/q2)12M−|A|(q2/r)(M2−|A|)i∏j=1Mζj1+ϵj2+M−j+i∏1⩽j<k⩽M(q2zk/zj;q4)∞(q4zk/zj;q4)∞∑l,m⩾0l+m=M2−|A|(−1)m(M+1)q{−12l(l−1)+12m(m−1)}(1−2i)−3Mm[l]q![m]q!∏b=1M2−|A|∮dvb2π−1vb1−2i∏a∈A∮dwa2π−1wa1−i∏a,b∈Aa<b(wa−wb)(wa−q2wb)∏a∈A{∏1⩽j⩽a(zj−q−2wa)∏a⩽j⩽M(wa−q4zj)}∏1⩽a<b⩽M2−|A|(va−vb)(va−q2vb)∏j=1M{∏b=1l(vb−q4zj)∏b=l+1M2−|A|(zj−q−2vb)}∏b=1l∏a∈A(vb−wa)(vb−q2wa)∏a∈A∏b=l+1M2−|A|(wa−vb)(wa−q2vb)〈Bi|e∑j=1MP(zj)+∑a∈AR−(wa)+∑b=1M2−|A|R−(vb)e∑j=1MQ(zj)+∑a∈AS−(wa)+∑b=1M2−|A|S−(vb)|i〉B. Next, we calculate the vacuum expectation value more explicitly.(4.36)〈Bi|e∑j=1MP(zj)+∑a∈AR−(wa)+∑a=1M2−|A|R−(va)e∑j=1MQ(zj)+∑a∈AS−(wa)+∑a=1M2−|A|S−(va)|i〉B=〈i|eG(i)e∑n=1∞a−nXne−∑n=1∞anYneF(i)|i〉. Here we have defined(4.37)Xn=q7n/2[2n]q∑j=1Mzjn−qn/2[n]q∑a∈Awan−qn/2[n]q∑b=1M2−|A|vbn,(4.38)Yn=q−5n/2[2n]q∑j=1Mzj−n−qn/2[n]q∑a∈Awa−n−qn/2[n]q∑b=1M2−|A|vb−n. By straightforward calculations, the following infinite product formula of the vacuum expectation value is obtained:(4.39)〈i|eG(i)e∑n=1∞a−nXne−∑n=1∞anYneF(i)|i〉〈B+;i|i;+〉B=({q6}∞{q8}∞)M{(q4;q2)∞}M2∏1⩽j<k⩽M{q6zjzk}∞{q2/zjzk}∞{q6zj/zk}∞{q6zk/zj}∞{q8zjzk}∞{q4/zjzk}∞{q8zj/zk}∞{q8zk/zj}∞∏j=1M[q10zj2]∞[q14zj2]∞[q10/zj2]∞[q6/zj2]∞[q12zj2]∞[q16zj2]∞[q12/zj2]∞[q8/zj2]∞∏a∈A(wa2/q2;q4)∞(q6/wa2;q4)∞∏j=1M∏a∈A(q2zjwa;q4)∞(q8zj/wa;q4)∞(q2wa/zj;q4)∞(q4/zjwa;q4)∞∏b=1M2−|A|(vb2/q2;q4)∞(q6/vb2;q4)∞∏j=1M∏b=1M2−|A|(q2zjvb;q4)∞(q8zj/vb;q4)∞(q2vb/zj;q4)∞(q4/zjvb;q4)∞∏b=1M2−|A|∏a∈A(vbwa/q2;q2)∞(q4vb/wa;q2)∞(q4wa/vb;q2)∞(q6/vbwa;q2)∞∏a,b∈Aa<b(wawb/q2;q2)∞(q4wa/wb;q2)∞(q4wb/wa;q2)∞(q6/wawb;q2)∞∏1⩽a<b⩽L(vavb/q2;q2)∞(q4va/vb;q2)∞(q4vb/va;q2)∞(q6/vavb;q2)∞{∏j=1M(q2rzj;q4)∞(q4rzj;q4)∞1∏a∈A(1−rwa/q2)∏b=1M2−|A|(1−rvb/q2)(i=0),∏j=1M(1/rzj;q4)∞(q2/rzj;q4)∞1∏a∈A(1−q2/rwa)∏b=1M2−|A|(1−q2/rvb)(i=1). Summarizing the above calculations, we finally obtain the second integral representation of the M-point functions with M even.(4.40)Pϵ1,…,ϵM(−,i)(ζ1,…,ζm;r,s)=P−ϵ1,…,−ϵM(+,1−i)(ζ1,…,ζM;1/r,−s/r)=(−q3)14M2−∑a∈Aa(s/q2)M2−|A|(1−q2)|A|({q6}∞{q8}∞)M{(q4;q2)∞}M2∏1⩽j<k⩽M{q6zjzk}∞{q2/zjzk}∞{q6zj/zk}∞{q2zk/zj}∞{q8zjzk}∞{q4/zjzk}∞{q8zj/zk}∞{q4zk/zj}∞∏j=1M[q10zj2]∞[q14zj2]∞[q10/zj2]∞[q6/zj2]∞[q12zj2]∞[q16zj2]∞[q12/zj2]∞[q8/zj2]∞∏j=1Mζj1+ϵj2+M−j∑l,m⩾0l+m=M2−|A|(−1)m(M+1)q{−12l(l−1)+12m(m−1)}(1−2i)−3Mm[l]q![m]q!∮⋯∮Cl(−,i)∏b=1M2−|A|dvb2π−1vb1−2i∏a∈Adwa2π−1wa1−i∏a∈A(wa2/q2;q4)∞(q6/wa2;q4)∞∏b=1M2−|A|(vb2/q2;q4)∞(q6/vb2;q4)∞∏a∈A{∏1⩽j⩽a(zj−q−2wa)∏a⩽j⩽M(wa−q4zj)}∏j=1M{∏b=1l(vb−q4zj)∏b=l+1M2−|A|(zj−q−2vb)}∏a,b∈Aa<bwa2(wawb/q2;q2)∞(q4wa/wb;q2)∞(wb/wa;q2)∞(q6/wawb;q2)∞∏j=1M∏a∈A(q2zjwa;q4)∞(q8zj/wa;q4)∞(q2wa/zj;q4)∞(q4/zjwa;q4)∞∏1⩽a<b⩽M2−|A|va2(vavb/q2;q2)∞(q4va/vb;q2)∞(vb/va;q2)∞(q6/vavb;q2)∞∏j=1M∏b=1M2−|A|(q2zjvb;q4)∞(q8zj/vb;q4)∞(q2vb/zj;q4)∞(q4/zjvb;q4)∞∏b=1l∏a∈Avb2(vbwa/q2;q2)∞(q4vb/wa;q2)∞(wa/vb;q2)∞(q6/vbwa;q2)∞∏b=l+1M2−|A|∏a∈Awa2(vbwa/q2;q2)∞(vb/wa;q2)∞(q4wa/vb;q2)∞(q6/vbwa;q2)∞{∏j=1M(q2rzj;q4)∞(q4rzj;q4)∞1∏a∈A(1−rwa/q2)∏b=1M2−|A|(1−rvb/q2)(i=0),(−q3)M2(q2/r)M2−|A|×∏j=1Mζj(1/rzj;q4)∞(q2/rzj;q4)∞1∏a∈A(1−q2/rwa)∏b=1M2−|A|(1−q2/rvb)(i=1). 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 Cl(−,0) is a simple closed curve that satisfies the following conditions for s=0,1,2,… . The wa (a∈A) encircles q8+4szj (1⩽j<a), q4+4szj (a⩽j⩽M), q4+4s/zj (1⩽j⩽M), but not q2−4szj (1⩽j⩽a), q−2−4szj (a<j⩽M), q−2−4s/zj (1⩽j⩽M), q2/r. The vb (1⩽b⩽l) encircles q4+4szj, q4+4s/zj (1⩽j⩽M), but not q−2−4szj, q−2−4s/zj (1⩽j⩽M), q2/r. The vb (l<b⩽M2−|A|) encircles q8+4szj, q4+4s/zj (1⩽j⩽M) but not q2−4szj, q−2−4s/zj (1⩽j⩽M), q2/r. The integration contour Cl(−,1) is a simple closed curve such that wa (a∈A) encircles q2/r and vb (1⩽b⩽M2−|A|) encircles q2/r in addition the same points as Cl(+,0) does.4.4Diagonal degenerationThe purpose of this subsection is to identify a sufficient condition such that the expressions for a triangular boundary condition coincide with those associated with a diagonal boundary condition. Let us go back to the formula (4.18). We note that ∑j=1Mϵj=0⇔n=|A|−M2=0. Upon the specialization n=|A|−M2=0, we have(4.41)〈Bi;±|Φϵ1(i,1−i)(ζ1)⋯ΦϵM(1−i,i)(ζM)|±;i〉B=〈Bi|Φϵ1(i,1−i)(ζ1)⋯ΦϵM(1−i,i)(ζM)|i〉B. The same argument holds for (4.35). We conclude that upon the parity preserving condition(4.42)ϵ1+ϵ2+⋯+ϵM=0, 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)〈Bi;±|Φϵ1(i,1−i)(ζ1)⋯ΦϵM(1−i,i)(ζM)|±;i〉B〈Bi;±|±;i〉B=(−q3)14M2−∑a∈Aa({q6}∞{q8}∞)M(q2;q2)∞M2∏j=1Mζj1+ϵj2−j+M∏1⩽j<k⩽M{q6zjzk}∞{q2/zjzk}∞{q6zj/zk}∞{q2zk/zj}∞{q8zjzk}∞{q4/zjzk}∞{q8zj/zk}∞{q4zk/zj}∞∏j=1M[q10zj2]∞[q14zj2]∞[q10/zj2]∞[q6/zj2]∞[q12zj2]∞[q16zj2]∞[q12/zj2]∞[q8/zj2]∞∏a∈A∮C0(i)dwa2π−1∏a,b∈Aa<bwa2(q−2wawb;q2)∞(q4wa/wb;q2)∞(wb/wa;q2)∞(q6/wawb;q2)∞∏a∈A∏j=1M(q2zjwa;q4)∞(q8zj/wa;q4)∞(q2wa/zj;q4)∞(q4/zjwa;q4)∞∏a∈A(q−2wa2;q4)∞(q6/wa2;q4)∞∏a∈A{∏1⩽j⩽a(zj−wa/q2)∏a⩽j⩽M(wa−q4zj)}{∏j=1M(q2rzj;q4)∞(q4rzj;q4)∞∏a∈Awa(1−rwa/q2)(for i=0),(−q3)M2∏j=1Mζj(1/rzj;q4)∞(q2/rzj;q4)∞∏a∈A1(1−q2/rwa)(for i=1). Here we have used {z}∞ and [z]∞ defined in (4.25). The integration contour C0(0) is a simple closed curve that satisfies the following conditions for s=0,1,2,… . The wa (a∈A) encircles q8+4szj (1⩽j<a), q4+4szj (a⩽j⩽M), q4+4s/zj (1⩽j⩽M), but not q2−4szj (1⩽j⩽a), q−2−4szj (a<j⩽M), q−2−4s/zj (1⩽j⩽M), q2/r. The integration contour C0(1) is a simple closed curve such that wa (a∈A) encircles q2/r in addition the same points as C0(0) does.To conclude, let us mention that according to the zero-mode operators (4.17) and (4.34), we have(4.44)Pϵ1,…,ϵM(±,i)(ζ1,…,ζM;r,s)=0for ±(M2−|A|)>0.4.5Boundary expectation values of spin operatorsIn this section we study simple examples. First, let us consider the boundary expectation value given by(4.45)〈Bi;−|σ1+|−;i〉B〈Bi;−|−;i〉B=gP−,−(−,i)(−q−1ζ,ζ;r,s)|ζ=1, where, according to (4.40), we have:(4.46)P−,−(−,i)(−q−1ζ,ζ;r,s)=(−s)ζΘq4(ζ4)(1−ζ4)(q2;q2)∞3∮C(−,i)dv2π−1v(1+v/q2ζ2)(1−q2/vζ2)Θq4(q2v2)Θq2(vζ2)Θq2(v/ζ2){(1−rζ2)(1−rv/q2)q2(i=0),(1−rζ2)(1−rv/q2)rv(i=1). Here the integration contour C(−,0) encircles q2kζ2, q2k+2/ζ2 (k=1,2,…). The integration contour C(−,1) encircles q2kζ2, q2k+2/ζ2 (k=1,2,…), and q2/r. Here we have used following simplification(4.47)(q4;q4)∞1−z2Θq4(z2)=({q6}∞{q8}∞)2{q6z1z2}∞{q2/z1z2}∞{q6z1/z2}∞{q2z2/z1}∞{q8z1z2}∞{q4/z1z2}∞{q8z1/z2}∞{q4z2/z1}∞∏j=12[q10zj2]∞[q14zj2]∞[q10/zj2]∞[q6/zj2]∞[q12zj2]∞[q16zj2]∞[q12/zj2]∞[q8/zj2]∞|z1=q−2z,z2=z. Using properties of the theta function: Θq4(q2v2)Θq2(vζ2)Θq2(v/ζ2)=(−q−2)kΘq4(q−4kq2v2)Θq2(q−2kvζ2)Θq2(q−2kv/ζ2) and Θp(1/z)=−1zΘp(z), we calculate the residues. We have(4.48)gP−,−(−,0)(−q−1ζ,ζ)=sζ(2+1−rzz∑k=1∞(−q2)k(z−z−1)−r(1+q4k)+(z+z−1)q2k(1−q2krz)(1−q2kr/z)), and(4.49)gP−,−(−,1)(−q−1ζ,ζ)=sζr(2z+1−rzz∑k=1∞(−1)k(z−z−1)rq2k+q2k+q−2k−(z+z−1)r(1−rq2kz)(1−rq2k/z))+sζq2r2(q2;q2)∞4(q4;q4)∞2(1+1/rz)(1−r/z)(1−rz)1−z2Θq4(z2)Θq4(q6/r2)Θq2(q2z/r)Θq2(q2/zr). Upon the specialization ζ=1, the boundary expectation value of the spin operator σ1+ in the model HB(−) is obtained:(4.50)〈B0;−|σ1+|−;0〉B〈B0;−|−;0〉B=s(2+(1−r)∑k=1∞(−q2)k2q2k−r(1+q4k)(1−rq2k)2),(4.51)〈B1;−|σ1+|−;1〉B〈B1;−|−;1〉B=sr(2+(1−r)∑k=1∞(−1)kq2k+q−2k−2r(1−rq2k)2)+sq2r2(1+1/r)(q4;q4)∞2(q2;q2)∞2(q6/r2;q4)∞(r2/q2;q4)∞(q2/r;q2)∞2(q2r;q2)∞2. Then, using the spin-reversal property (4.3) the boundary expectation value of the spin operator σ1− in the model HB(+) immediately follows:(4.52)〈B1;+|σ1−|+;1〉B〈B1;+|+;1〉B=−sr(2+(1−1/r)∑k=1∞(−q2)k2q2k−(1+q4k)/r(1−q2k/r)2),(4.53)〈B0;+|σ1−|+;0〉B〈B0;+|+;0〉B=−s(2+(1−1/r)∑k=1∞(−1)kq2k+q−2k−2/r(1−q2k/r)2)−sq2r(1+r)(q4;q4)∞2(q2;q2)∞2(q6r2;q4)∞(1/q2r2;q4)∞(q2/r;q2)∞2(q2r;q2)∞2. Let us now turn to the boundary expectation values of σ1z in HB(±). According to (4.42), it reduces to the known result for a diagonal boundary [6]. Namely,(4.54)〈B0;±|σ1z|±;0〉B〈B0;±|±;0〉B=−1−2(1−r)2∑k=1∞(−q2)k(1−rq2k)2,(4.55)〈B1;±|σ1z|±;1〉B〈B1;±|±;1〉B=−1−2(1−r)2∑k=1∞(−q2)k(1−rq2k)2+2(q4;q4)∞2(q2;q2)∞2(q2r2;q4)∞(q2/r2;q4)∞(q2r;q2)∞2(q2/r;q2)∞2. Finally, note that the remaining expectation values are vanishing:(4.56)〈Bi;±|σ1±|±;i〉B〈Bi;±|±;i〉B=0for i=0,1. Simplifications occur upon the specializations r=±1,0,∞: such cases correspond to free (Neumann) boundary condition r=−1 (h=0), fixed (Dirichlet) boundary condition r=1 (h=∞) whereas for r=0,∞ the Hamiltonian enjoys formal Uq(sl2) invariance.4.6Relations between multiple integralsLinear relations between n-fold integrals are known in the mathematical literature [39–41]. 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 cannot 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. Here, we focus on the simplest examples: a relation between a triple integral and a single integral that has been computed explicitly in previous subsection is exhibited in two different cases. First, from(4.57)P+,+(+,0)(−q−1ζ,ζ;r,s)=P−,−(−,1)(−q−1ζ,ζ;1/r,−s/r) we have the following identity of multiple integrals of the elliptic theta function:(4.58)rzq2(q2;q2)∞4(q4;q4)∞2(1+r/z)(1−1/rz)(1−z/r)1−z2Θq4(z2)Θq4(q6r2)Θq2(q2rz)Θq2(q2r/z)+2+(1−z/r)∑k=1∞(−1)k(z−z−1)q2k/r+q2k+q−2k−(z+z−1)/r(1−q2kz/r)(1−q2k/rz)=q2(q2;q2)∞8(q4;q4)∞4Θq4(z2)1−z2(−q2∫∫∫C0(+,0)+∫∫∫C1(+,0))∏a=13dwa2π−1w1w2w35(1−rz)(1−rw3/q3)∏a=12(1−q2/zwa)∏a=12(1−rwa/q2)(1−q2w1/w2)(1−q4/w1w2)Θq2(w1w2)Θq2(w2/w1)Θq2(zw3/q)Θq2(qw3/z)∏a=13Θq4(wa2/q2)∏a=12Θq2(waz)Θq2(wa/z)Θq2(w3wa/q)Θq2(wa/qw3). Here we have set w3=v. The integration contour C0(+,0) is a simple closed curve such that the w1 encircles q2+2sz, q4+2s/z, q−1+2sw3, q5+2s/w3, the w2 encircles q4+2sz, q4+2s/z, q−1+2sw3, q5+2s/w3, and the w3 encircles q3+2swa, q5+2s/wa (a=1,2), for s=0,1,2,… . The integration contour C1(+,1) is a simple closed curve such that the w1 encircles q2+2sz, q4+2s/z, q3+2sw3, q5+2s/w3, the w2 encircles q4+2sz, q4+2s/z, q3+2sw3, q5+2s/w3, and w3 encircles q−1+2swa, q5+2s/wa (a=1,2), for s=0,1,2,… .Secondly, from(4.59)P+,+(+,1)(−q−1ζ,ζ;r,s)=P−,−(−,0)(−q−1ζ,ζ;1/r,−s/r) we have the following identity:(4.60)2+1−z/rz∑k=1∞(−q2)k(z−z−1)−(1+q4k)/r+(z+z−1)q2k(1−q2kz/r)(1−q2k/rz)=q2(q2;q2)∞8(q4;q4)∞4Θq4(z2)1−z2(q2∫∫∫C0(+,1)−∫∫∫C1(+,1))∏a=13dwa2π−1(1−1/rz)(1−q/rw3)∏a=12(1−q2/zwa)w22w33(1−q2w1/w2)(1−q4/w1w2)∏a=1,2(1−q2/rwa)Θq2(w1w2)Θq2(w2/w1)Θq2(zw3/q)Θq2(qw3/z)∏a=13Θq4(wa2/q2)∏a=12Θq2(waw3/q2)Θq2(wa/qw3)Θq2(waz)Θq2(wa/z). The integration contour Cl(+,1) is a simple closed curve such that wa (a=1,2) encircles q2/r in addition the same points as the integration contour Cl(+,0) does. The integration contour Cl(+,0) 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.5ConclusionIn 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 σ1± 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.Although our results provide the first examples of correlation functions computed in the case of non-diagonal boundary conditions, one of the most interesting open problem is to extend the present analysis to the case of general integrable non-diagonal boundary condition. Indeed, for a finite size and generic boundary conditions, due to the absence of reference states the algebraic Bethe ansatz cannot be applied. In this case, alternative approaches are necessary. At least, in the thermodynamic limit the above mentioned arguments about the structure of the eigenvectors in relation with the q-Onsager algebra representation theory hold. In this case, the explicit construction of monomials in terms of the q-Onsager generators can be also considered, opening the way for the computation of correlation functions within the q-vertex operator approach. Also, we would like to draw the reader's attention to other interesting open problems. It would be interesting to extend the analysis to models with higher symmetry – for instance Uq(slˆ(M|N)) – or the ABF model governed by the elliptic quantum group Bq,λ(sl2ˆ) [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–Zamolodchikov equations [43] and Koornwinder polynomials [44,45] (see also [46]).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.AcknowledgementsThe authors would like to thank S. Belliard, V. Fateev, K. Kozlowski, J.M. Maillet, G. Niccoli, and Y.-Z. Zhang. P.B. thanks J. Stokman for pointing out Ref. [40], and P. Zinn-Justin for discussions. T.K. thanks S. Tsujimoto for discussions. T.K. would like to thank Laboratoire de Mathématiques et Physique Théorique, Université de Tours, for kind invitation and warm hospitality during his stay in March 2013. This work is supported by the Grant-in-Aid for Scientific Research C (21540228) from JSPS and Visiting professorship from CNRS. Appendix AQuantum group Uq(sl2ˆ)In this appendix we recall the definition of Uq(sl2ˆ) [47,48] and fix the notation that are used in the main text. Let −1<q<0. Consider a free Abelian group on the letters Λ0,Λ1,δ. We call P=ZΛ0⊕ZΛ1⊕Zδ the weight lattice, Λi the fundamental weights and δ the null root. Define the simple roots αi (i=0,1) and the element ρ by α0+α1=δ, Λ1=Λ0+α12, ρ=Λ0+Λ1. Let (h0,h1,d) be an ordered basis of P⁎=Hom(P,Z) dual to (Λ0,Λ1,δ). We define a symmetric bilinear form (,):P×P→12Z by(A.1)(Λ0,Λ0)=0,(Λ0,α1)=0,(Λ0,δ)=1,(α1,α1)=2,(α1,δ)=0,(δ,δ)=0. Regarding P⁎⊂P via this bilinear form we have the identification(A.2)h0=α0,h1=α1,d=Λ0. The quantum group Uq(sl2ˆ) is a q-analogue of the universal enveloping algebra U(sl2ˆ) generated by Chevalley generators ej, fj (j=0,1) and qh (h∈P⁎) and through the defining relations:(A.3)q0=1,qhqh′=qh+h′,(A.4)qhejq−h=q(h,αj)ej,qhfjq−h=q−(h,αj)fj,(A.5)[ei,fj]=δi,jqhi−q−hiq−q−1,(A.6)ei3ej−[3]qei2ejei+[3]qeiejei2−ejei3=0(i≠j),(A.7)fi3fj−[3]qfi2fjfi+[3]qfifjfi2−fjfi3=0(i≠j). Here we denote:(A.8)[n]q=qn−q−nq−q−1. The quantum group Uq(sl2ˆ) has the coproduct Δ structure:(A.9)Δ(qh)=qh⊗qh,Δ(ej)=ej⊗1+qhj⊗ej,Δ(fj)=fj⊗q−hj+1⊗fj. The coproduct Δ satisfies an algebra automorphism Δ(XY)=Δ(X)Δ(Y).The quantum group Uq(sl2ˆ) has another realization called the Drinfeld's second realization. The generators of the Drinfeld's realizations are(A.10)am(m∈Z≠0),xm±(m∈Z),γ12,K,qd. In order to write down the defining relations, it is convenient to introduce the generating function(A.11)X±(z)=∑m∈Zxm±z−m−1,(A.12)ψ+(z)=Kexp((q−q−1)∑n=1∞anz−n),(A.13)ψ−(z)=K−1exp(−(q−q−1)∑n=1∞a−nzn). Defining relations are given by(A.14)[γ12,Uq(sl2ˆ)]=0,(A.15)[am,an]=δm+n,0[2m]qmγm−γ−mq−q−1,(A.16)KamK−1=am,KX±(z)K−1=q±2X±(z),(A.17)[am,X±(z)]=±[2m]qmγ∓|m|/2zmX±(z),(A.18)(z−q±2w)X±(z)X±(w)=(q±2z−w)X±(w)X±(z),(A.19)[X+(z),X−(w)]=1(q−q−1)zw(ψ+(γ12w)δ(z/γw)−ψ−(γ−12w)δ(γz/w)). Here δ(z)=∑n∈Zzn is a formal power series. Other defining relations are given by(A.20)qdγ12q−d=γ12,qdKq−d=K,qdxm±q−d=qmxm±,qdamq−d=qmam. The Chevalley generators are related to the Drinfeld generators as follows:(A.21)K=qh1,x0+=e1,x0−=f1,γK−1=qh0,x1−=e0qh1,x−1+=q−h1f0. The Drinfeld realization is convenient to study bosonizations.Appendix BBosonizationIn this appendix we review the bosonizations of the quantum group Uq(sl2ˆ) and the q-vertex operators for level k=1 [3,9]. The center γ12 of Uq(sl2ˆ) satisfies (γ12)2=qh0+h1=qk on the level k representation. Hence γ=q on the level k=1 representation. For the level k=1 case, the defining relation of am (m∈Z≠0) (A.15) becomes(B.1)[am,an]=δm+n,0[2m]q[m]qm(m,n≠0). We introduce the zero-mode operator ∂,α by [∂,α]=2 and the normal ordering symbol : :(B.2):aman:={aman(m<0),anam(m>0),:α∂:=:∂α:=α∂. The bosonization of the irreducible highest representation V(Λi) with fundamental weights Λi are given by (3.4). On this space the actions of am, eα, ∂ are given by(B.3)am(f⊗eβ)={amf⊗eβ(m<0),[am,f]⊗eβ(m>0),(B.4)eα(f⊗eβ)=f⊗eα+β,∂(f⊗eβ)=(α,β)f⊗eβ, where f∈C[a−1,a−2,…], and we have set [∂,Λ0]=0 and Λ1=Λ0+α2.The action of other Drinfeld generators is given by(B.5)K=q∂,γ=q,(B.6)X±(z)=exp(R±(z))exp(S±(z))e±αz±∂,(B.7)qd(1⊗eΛi+nα)=q−(β,β)/2+i/4(1⊗eΛi+nα), where(B.8)R±(z)=±∑n=1∞a−n[n]qq∓n/2zn,S±(z)=∓∑n=1∞an[n]qq∓n/2z−n. We have the following bosonizations of the q-vertex operators.(B.9)Φ−(1−i,i)(ζ)=exp(P(ζ2))exp(Q(ζ2))eα/2(−q3ζ2)(∂+i)/2ζ−i,(B.10)Φ+(1−i,i)(ζ)=∮C˜1dw2π−1(1−q2)wζq(w−q2ζ2)(w−q4ζ2):Φ−(1−i,i)(ζ)X−(w):,(B.11)Ψ−⁎(1−i,i)(ξ)=exp(−P(q−1ξ2))exp(−Q(qξ2))e−α/2(−q3ξ2)(−∂+i)/2ξ1−i,(B.12)Ψ+⁎(1−i,i)(ξ)=∮C˜2dw2π−1q2(1−q2)ξ(w−q2ξ2)(w−q4ξ2):Ψ−⁎(1−i,i)(ξ)X+(w):, where(B.13)P(z)=∑n=1∞a−n[2n]qq7n/2zn,Q(z)=−∑n=1∞an[2n]qq−5n/2z−n. Here the integration contour C˜1 is a simple closed curve such that the w encircles q4ζ2 inside but not q2ζ2. The integration contour C˜2 is a simple closed curve such that the w encircles q2ζ2 inside but not q4ζ2.In what follows we summarize normal orderings.(B.14)Φ−(i,1−i)(ζ1)Φ−(1−i,i)(ζ2)=(−q3ζ12)12(q2ζ22/ζ12;q4)∞(q4ζ22/ζ12;q4)∞:Φ−(i,1−i)(ζ1)Φ−(1−i,i)(ζ2):,(B.15)Ψ−⁎(i,1−i)(ξ1)Ψ−⁎(1−i,i)(ξ2)=(−q3ξ12)12(ξ22/ξ12;q4)∞(q2ξ22/ξ12;q4)∞:Ψ−⁎(1−i,i)(ξ1)Ψ−⁎(1−i,i)(ξ2):,(B.16)Φ−(i,1−i)(ζ)Ψ−⁎(1−i,i)(ξ)=(−q3ζ2)−12(q3ξ2/ζ2;q4)∞(qξ2/ζ2;q4)∞:Φ−(i,1−i)(ζ)Ψ−⁎(1−i,i)(ξ):,(B.17)Ψ−⁎(i,1−i)(ξ)Φ−(1−i,i)(ζ)=(−q3ξ2)−12(q3ζ2/ξ2;q4)∞(qζ2/ξ2;q4)∞:Ψ−⁎(i,1−i)(ξ)Φ−(1−i,i)(ζ):,(B.18)X+(w1)X+(w2)=w12(1−w2/w1)(1−w2/q2w1):X+(w1)X+(w2):,(B.19)X−(w1)X−(w2)=w12(1−w2/w1)(1−q2w2/w1):X−(w1)X−(w2):,(B.20)X+(w1)X−(w2)=1w12(1−qw2/w1)(1−w2/qw1):X+(w1)X−(w2):,(B.21)X−(w1)X+(w2)=1w12(1−qw2/w1)(1−w2/qw1):X−(w1)X+(w2):,(B.22)Φ−(i,1−i)(ζ)X+(w)=(−q3ζ2)(1−w/q3ζ2):Φ−(i,1−i)(ζ)X+(w):,(B.23)X+(w)Φ−(i,1−i)(ζ)=w(1−q3ζ2/w):X+(w)Φ−(i,1−i)(ζ):,(B.24)Φ−(i,1−i)(ζ)X−(w)=−1q3ζ2(1−w/q2ζ2):Φ−(i,1−i)(ζ)X−(w):,(B.25)X−(w)Φ−(i,1−i)(ζ)=1w(1−q4ζ2/w):X−(w)Φ−(i,1−i)(ζ):,(B.26)Ψ−⁎(i,1−i)(ξ)X+(w)=−1q3ξ2(1−w/q4ξ2):Ψ−⁎(i,1−i)(ξ)X+(w):,(B.27)X+(w)Ψ−⁎(i,1−i)(ξ)=1w(1−q2ξ2/w):X+(w)Ψ−⁎(i,1−i)(ξ):,(B.28)Ψ−⁎(i,1−i)(ξ)X−(w)=(−q3ξ2)(1−w/q3ξ2):Ψ−⁎(i,1−i)(ξ)X−(w):,(B.29)X−(w)Ψ−⁎(i,1−i)(ξ)=w(1−q3ξ2/w):X−(w)Ψ−⁎(i,1−i)(ξ):.Appendix CConvenient formulaeIn this appendix we summarize convenient formulae for calculations of vacuum expectation values.(C.1)exp(∑n=1∞1(q−q−1)q(c+2)nn[2n]qyn)=(qc+4y;q4)∞,(C.2)exp(∑n=1∞1(q−q−1)[n]qq(c+2)nn[2n]q2yn)=(qc+5y;q4,q4)∞(qc+7y;q4,q4)∞,(C.3)exp(∑n=1∞1(q−q−1)q(c+2)nn[n]qyn)=(qc+3y;q4)∞,(C.4)exp(∑n>0[n]qn[2n]qqcn1−αnγnyn)=(qc+3y;q4,q4)∞(qc+1y;q4,q4)∞,(C.5)exp(∑n>0[2n]qn[n]qqcn1−αnγnyn)=1(qc−1y;q2)∞,(C.6)exp(∑n=1∞[n]qn(1−αnγn)δn(i)qcn/2yn)=((qc+3y2;q8,q8)∞(qc+1y2;q8,q8)∞(qc−1y2;q8,q8)∞(qc+5y2;q8,q8)∞)1/2((q(c−3)/2+2ir1−2iy;q4,q4)∞(q(c+1)/2+2ir1−2iy;q4,q4)∞)1−2i,(C.7)exp(∑n=1∞[2n]qn(1−αnγn)δn(i)qcn/2yn)=((qc−1y2;q8)∞(qc−3y2;q8)∞)1/2(q(c−5)/2+2ir1−2iy;q4)∞1−2i,(C.8)exp(∑n=1∞[n]qn(1−αnγn)βn(i)qcn/2yn)=((qc+7y2;q8,q8)∞(qc+13y2;q8,q8)∞(qc+9y2;q8,q8)∞(qc+11y2;q8,q8)∞)1/2((q(c+9)/2−2ir1−2iy;q4,q4)∞(q(c+13)/2−2ir1−2iy;q4,q4)∞)1−2i,(C.9)exp(∑n=1∞[2n]qn(1−αnγn)βn(i)qcn/2yn)=((qc+5y2;q8)∞(qc+7y2;q8)∞)1/2(q(c+7)/2−2ir1−2iy;q4)∞1−2i.References[1]H.A.BetheZur theorie der metalle I. Eigenfunktionen der linearen atomketteZ. Phys.711931205226[2]R.J.BaxterExactly Solved Models in Statistical Mechanics1982Academic PressLondon[3]B.DaviesO.FodaM.JimboT.MiwaA.NakayashikiDiagonalization of the XXZ Hamiltonian by vertex operatorCommun. Math. Phys.151199389153M.JimboK.MikiT.MiwaA.NakayashikiCorrelation functions of the XXZ model for Δ<−1Phys. Lett. A1681992256263M.JimboT.MiwaAlgebraic Analysis of Solvable Lattice ModelCBMS Regional Conference Series in Mathematicsvol. 851994AMS[4]V.E.KorepinN.M.BogoliubovA.G.IzerginQuantum Inverse Scattering Method and Correlation FunctionsCambridge Monographs on Mathematical Physics1993Cambridge University PressCambridgeA.G.IzerginV.E.KorepinThe quantum inverse scattering method approach to correlation functionsCommun. Math. Phys.9419846792A.G.IzerginV.E.KorepinCorrelation functions for the Heisenberg XXZ-antiferromagnetCommun. Math. Phys.991985271302[5]N.KitanineJ.M.MailletV.TerrasForm factors of the XXZ Heisenberg spin-12 finite chainNucl. Phys. B5541999647678N.KitanineJ.M.MailletV.TerrasCorrelation functions of the XXZ Heisenberg spin-12 chain in a magnetic fieldNucl. Phys. B5672000554582J.M.MailletV.TerrasOn the quantum inverse scattering problemNucl. Phys. B5752000627644N.KitanineJ.M.MailletN.A.SlavnovV.TerrasSpin–spin correlation functions of the XXZ −12 Heisenberg chain in a magnetic fieldNucl. Phys. B6412002487518[6]M.JimboR.KedemT.KojimaH.KonnoT.MiwaXXZ chain with a boundaryNucl. Phys. B4411995437470[7]N.ReshetikhinCalculation of the norm of Bethe vectors in models with SU(3)-symmetryJ. Sov. Math.46198916941706M.GaudinLa fonction d'onde de Bethe1983MassonParisV.E.KorepinCalculation of norms of Bethe wave functionsCommun. Math. Phys.861982391418[8]M.WheelerMultiple integral formulae for the scalar product of on-shell and off-shell Bethe vectors in SU(3)-invariant modelsNucl. Phys. B8752013186212S.BelliardS.PakuliakE.RagoucyN.A.SlavnovBethe vectors of quantum integrable models with GL(3) trigonometric R-matrixpreprintarXiv:1304.76022013[9]I.B.FrenkelN.JingVertex representations of quantum affine algebrasProc. Natl. Acad. Sci. USA85198893739377[10]Y.KoyamaStaggered polarization of vertex models with Uq(sl(n)ˆ)-symmetryCommun. Math. Phys.1641994277291[11]N.JingS.KangY.KoyamaVertex operators of quantum affine Lie algebras Uq(Dn(1))Commun. Math. Phys.1741995367392[12]D.BernardVertex operator representations of the quantum affine algebra Uq(Br(1))Lett. Math. Phys.171989239245[13]K.KimuraJ.ShiraishiJ.UchiyamaA level-one representation of the quantum affine superalgebra Uq(slˆ(M+1|N+1))Commun. Math. Phys.1881997367378Y.-Z.ZhangLevel-one representations and vertex operators of quantum affine superalgebra Uq(glˆ(N|N))J. Math. Phys.40199961106124[14]E.K.SklyaninBoundary conditions for integrable quantum systemsJ. Phys. A21198823752389[15]R.NepomechieBethe ansatz solution of the open XXZ chain with nondiagonal boundary termsSpecial issue on recent advances in the theory of quantum integrable systemsJ. Phys. A372004433440[16]J.CaoH.-Q.LinK.-J.ShiY.WangExact solution of XXZ spin chain with unparallel boundary fieldsNucl. Phys. B6632003487519W.-L.YangY.-Z.ZhangOn the second reference state and complete eigenstates of the open XXZ chainJ. High Energy Phys.0704200744(11 pp.)[17]R.MurganR.I.NepomechieC.ShiBoundary energy of the open XXZ chain from new exact solutionsAnn. Henri Poincaré7200614291448[18]J.CaoW.-L.YangK.ShiY.WangOff-diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary fieldsNucl. Phys. B8772013152175arXiv:1307.2023[19]S.BelliardN.Crampé HeisenbergXXX model with general boundaries: Eigenvectors from Algebraic Bethe ansatzSIGMA92013072arXiv:1309.6165[20]N.KitanineK.K.KozlowskiJ.M.MailletG.NiccoliN.A.SlavnovV.TerrasCorrelation functions of the open XXZ chain. IJ. Stat. Mech.2007P10009(37 pp.)N.KitanineK.K.KozlowskiJ.M.MailletG.NiccoliN.A.SlavnovV.TerrasCorrelation functions of the open XXZ chain. IIJ. Stat. Mech.2008P07010(33 pp.)[21]H.FurutsuT.KojimaThe Uq(slˆn) analogue of the XXZ chain with a boundaryJ. Math. Phys.41200044134436[22]W.-L.YangY.-Z.ZhangIzergin–Korepin model with a boundaryNucl. Phys. B5962001495512T.KojimaA remark on ground state of boundary Izergin–Korepin modelInt. J. Mod. Phys. A26201119731989[23]T.KojimaDiagonalization of transfer matrix of supersymmetry Uq(slˆ(M+1|N+1)) chain with a boundaryJ. Math. Phys.542013043507(40 pp.)[24]S.BelliardN.CrampéE.RagoucyAlgebraic Bethe ansatz for open XXX model with triangular boundary matricesLett. Math. Phys.1032013493506[25]R.A.PimentaA.Lima-SantosAlgebraic Bethe ansatz for the six vertex model with upper triangular K-matricesJ. Phys. A462013455002arXiv:1308.4446[26]P.BaseilhacK.KoizumiA new (in)finite-dimensional algebra for quantum integrable modelsNucl. Phys. B7202005325347P.BaseilhacK.KoizumiExact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theoryJ. Stat. Mech.2007P09006(27 pp.)P.BaseilhacThe q-deformed analogue of the Onsager algebra: beyond the Bethe ansatz approachNucl. Phys. B7542006309328P.BaseilhacK.ShigechiA new current algebra and the reflection equationLett. Math. Phys.9220104765[27]P.BaseilhacS.BelliardThe half-infinite XXZ chain in Onsager's approachNucl. Phys. B8732013550583[28]P.BaseilhacS.BelliardCentral extension of the reflection equations and an analog of Miki's formulaJ. Phys. A442011415205arXiv:1104.1591[29]H.J.de VegaA.Gonza'lez-RuizBoundary K-matrices for the XYZ, XXZ and XXX spin chainsJ. Phys. A27199461296137[30]S.GhoshalAl.ZamolodchikovBoundary S matrix and boundary state in two-dimensional integrable quantum field theoryInt. J. Mod. Phys. A9199438413885[31]T.KojimaThe massless XXZ chain with a boundaryInt. J. Mod. Phys. A162001409424[32]W.GalleasFunctional relations from the Yang–Baxter algebra: Eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditionsNucl. Phys. B7902008524542arXiv:1002.1623[33]G.NiccoliAntiperiodic spin-12 XXZ quantum chains by separation of variables: complete spectrum and form factorsNucl. Phys. B8702013397420G.NiccoliNon-diagonal open spin-12-XXZ quantum chains by separation of variables: complete spectrum and matrix elements of some quasi-local operatorsJ. Stat. Mech.2012P10025(42 pp.)[34]M.JimboT.MiwaQuantum KZ equation with |q|=1 and correlation functions of the XXZ model in the gapless regimeJ. Phys. A29199629232958M.JimboH.KonnoT.MiwaMassless XXZ model and degeneration of the elliptic algebra Aq,p(slˆ2)Deformation Theory and Symplectic GeometryAscona, 1996Math. Phys. Stud.vol. 201997KluwerDordrecht117138[35]R.J.BaxterVariational approximations for square lattice models in statistical mechanicsJ. Stat. Phys.191978461478[36]T.MiwaR.WestonBoundary ABF modelsNucl. Phys. B4861997517545T.KojimaDiagonalization of infinite transfer matrix of boundary Uq,p(AN−1(1)) face modelJ. Math. Phys.522011013501(30 pp.)[37]M.JimboT.MiwaA.NakayashikiDifference equations for the correlation functions of the eight-vertex modelJ. Phys. A26199321992209[38]O.FodaT.MiwaCorner transfer matrices and quantum affine algebrasInt. J. Mod. Phys. A7Suppl. 1A1992279302[39]V.TarasovA.VarchenkoIdentities between q-hypergeometric and hypergeometric integrals of different dimensionsAdv. Math.19120052945[40]E.M.RainsTransformations of elliptic hypergeometric integralsAnn. Math.1712010169243[41]V.P.SpiridonovTheta hypergeometric integralsAlgebra Anal.152003161215arXiv:math/0303205v2[42]P.TerwilligerA.N.KirillovA.TsuchiyaH.UmemuraProceedings of the Nagoya 1999 International Workshop on Physics and Combinatorics2001World ScientificRiver Edge, NJ377398arXiv:math.QA/0307016[43]M.JimboR.KedemH.KonnoT.MiwaR.WestonDifference equations in spin chains with a boundaryNucl. Phys. B4481995429456arXiv:hep-th/9502060[44]M.KasataniBoundary quantum Knizhnik–Zamolodchikov equationNew Trends in Quantum Integrable Systems2011World ScientificHackensack, NJ157171[45]N.ReshetikhinJ.StokmanB.VlaarReflection quantum Knizhnik–Zamolodchikov equations and Bethe vectorsarXiv:1305.1113J.StokmanB.VlaarKoornwinder polynomials and the XXZ spin chainarXiv:1310.5545[46]P.Di FrancescoP.Zinn-JustinQuantum Knizhnik–Zamolodchikov equation: reflecting boundary conditions and combinatoricsJ. Stat. Mech. Theory Exp.122007P12009(30 pp.)arXiv:0709.3410[47]M.JimboA q-difference analogue of U(g) and the Yang–Baxter equationLett. Math. Phys.1019856369[48]V.G.DrinfeldA new realization of Yangians and of quantum affine algebrasSov. Math. Dokl.361988212216