Nested off-diagonal Bethe ansatz and exact solutions of the su(n) spin chain with generic integrable boundaries

Abstract

The nested off-diagonal Bethe ansatz method is proposed to diagonalize multi-component integrable models with generic integrable boundaries. As an example, the exact solutions of the su(n)-invariant spin chain model with both periodic and non-diagonal boundaries are derived by constructing the nested T - Q relations based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices.

Published for SISSA by Springer

Received: December 22, 2013 Revised: March 14, 2014 Accepted: March 29, 2014

Published: April 23, 2014

Nested o -diagonal Bethe ansatz and exact solutions of the su(n) spin chain with generic integrable boundaries

JHEP04(2014)143

Junpeng Cao,a,b Wen-Li Yang,c,d Kangjie Shic and Yupeng Wanga,b

aBeijing National Laboratory for Condensed Matter Physics,

Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China

bCollaborative Innovation Center of Quantum Matter,

Beijing, China

cInstitute of Modern Physics, Northwest University,

Xian 710069, China

dBeijing Center for Mathematics and Information Interdisciplinary Sciences, Beijing 100048, China

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: The nested o -diagonal Bethe ansatz method is proposed to diagonalize multi-component integrable models with generic integrable boundaries. As an example, the exact solutions of the su(n)-invariant spin chain model with both periodic and non-diagonal boundaries are derived by constructing the nested T Q relations based on the operator product identities among the fused transfer matrices and the asymptotic behavior of the transfer matrices.

Keywords: Bethe Ansatz, Lattice Integrable Models

ArXiv ePrint: 1312.4770

Open Access, c

The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP04(2014)143

Web End =10.1007/JHEP04(2014)143

Contents

1 Introduction 1

2 su(n)-invariant spin chain with periodic boundary conditions 32.1 Transfer matrix 32.2 Operator product identities 42.3 Nested T-Q relation 5

3 su(n)-invariant spin chain with general open boundary conditions 63.1 Transfer matrix 63.2 Operator product identities 83.3 Asymptotic behaviors of the transfer matrices 11

4 su(3)-invariant spin chain with non-diagonal boundary term 13

5 Exact solution of su(n)-invariant spin chain with general open boundaries 16

6 Conclusions 20

A Proofs of the operator identities 21

B Higher rank analogs of (4.8)(4.13) 22

1 Introduction

The appearance of integrability in planar AdS/CFT [1] is a rather unexpected occurrence and has led to many remarkable results [2] (see also references therein) and even ultimately to the exact solution of planar N = 4 supersymmetric Yang-Mills (SYM) theory. The anomalous dimensions of single-trace operators of N = 4 SYM are given by the eigenvalues of certain integrable closed spin chain Hamiltonians [2, 3]. Then it was shown [4, 5] that the computing of the anomalous dimensions of determinant-like operators of N = 4 SYM can be mapped to the eigenvalue problem of certain integrable open spin chain ( spin chain with boundary condition specied by reection K-matrices or boundary scattering matrices) Hamiltonians [2, 6, 7], while by AdS/CFT the K-matrices of the open chain correspond to open strings attached to maximal giant gravitons [5, 8]. Therefore spin chain model has played an important role in understanding the physical contents of planar N = 4 SYM theory and planar AdS/CFT. Moreover, it has already provided valuable insight into the important universality class of boundary quantum physical systems in condensed matter physics [9]. Motivated by the above great applications, in this paper,

1

JHEP04(2014)143

we develop the nested o -diagonal Bethe ansatz method, a generalization of the method proposed in [1013], to solve the eigenvalue problem of multi-component spin chains with the most general integrable boundary terms.

So far, there have been several well-known methods for deriving the Bethe ansatz (BA) solutions of quantum integrable models: the coordinate BA [1416], the T-Q approach [17 21], the algebraic BA [2227], the analytic BA [28], the functional BA [29, 30] or the separation of variables method [3134] and many others [3555]. However, there exists a quite unusual class of integrable models which do not possess the U(1) symmetry (whose transfer matrices contain not only the diagonal elements but also some o -diagonal elements of the monodromy matrix and the usual U(1) symmetry is broken, i.e., the total spin is no longer conserved). Normally, most of the conventional methods do not work for these models even though their integrability has been proven for many years [25].

Recently, a systematic method [1013] for dealing with such kind of models associated with su(2) algebra was proposed by the present authors, which had been shown successfully to construct the exact solutions of the open Heisenberg spin chain with unparallel boundary elds, the XXZ spin torus, the closed XYZ chain with odd site number and other models with general boundary terms [56, 57]. With the help of the Hirota equation, Nepomechie [58] generalized the results of [1013] to the arbitrary spin XXX open chain with general boundary terms. An expression for the corresponding eigenvectors was also proposed recently in [59].

The central idea of the method in [1013] is to construct a proper T Q ansatz with an extra o -diagonal term (comparing with the ordinary ones [20]) based on the functional relations between the transfer matrix (the trace of the monodromy matrix) and the quantum determinant q(u), at some special points of the spectral parameter u = j, i.e.,1

t( j)t( j ) q( j). (1.1)

In this paper, we generalize the o -diagonal Bethe ansatz method to the multi-component integrable models (integrable spin chains associated with higher rank algebras). This generalization allows us to construct the nested T Q relations based on the recursive operator product identities and the asymptotic behavior of the transfer matrices for the systems with both periodic and arbitrary integrable open boundary conditions. We elucidate our method with the su(n) spin chain (both periodic and open) model as an example. Our method might be used to the integrable systems associated with Bn, Cn and Dn algebras.

The paper is organized as follows. Section 2 serves as an introduction of our notations and some basic ingredients. We briey describe the inhomogeneous su(n)-invariant spin chain with periodic boundary condition. Based on some operator product relations for the antisymmetric fused transfer matrices and their asymptotic behaviors, the nested T Q ansatz of their eigenvalues and the corresponding Bethe ansatz equations (BAEs) are constructed. In section 3, we study the su(n)-invariant open spin chains with general

1These kind relations for the eigenvalues of the transfer matrices were previously obtained by the separation of variables method for the open XXX spin chain [31], for the XXZ spin chain with antiperiodic boundary condition and were used to determine the eigenvalues [32, 33]. Then this idea was generalized to the open XXZ chain with one general non-diagonal and one diagonal or triangular boundary K-matrices [34].

2

JHEP04(2014)143

open boundary integrable conditions. Based on some properties of the R-matrix and K-matrices, we obtain the important operator product identities among the fused transfer matrices of the open chains and their asymptotic behaviors when u . In section 4, we focus on the su(3)-invariant open spin chain with the most general non-diagonal boundary terms. The nested Bethe ansatz solution for the eigenvalues of the transfer matrix and the corresponding Bethe ansatz equations (BAEs) are given in detail based on the operator product identities of the transfer matrix and their asymptotic behaviors and values of the transfer matrices at some special points. The results for the su(n)-invariant case is given in section 5. We summarize our results and give some discussions in section6. Some detailed technical proof is given in appendices A and B.

2 su(n)-invariant spin chain with periodic boundary conditions

2.1 Transfer matrix

Let V denote an n-dimensional linear space. The Hamiltonian of su(n)-invariant quantum spin system with periodic boundary condition is given by [60, 61]

H =

N

JHEP04(2014)143

Xj=1Pj,j+1, (2.1)

where N is the number of sites, Pj,j+1 is permutation operator, P bdac = ad bc with a, b, c, d = 1, , n. The integrability of the system (2.1) is guaranteed by the su(n)-invariant R-matrix R(u) End(V V) [62, 63]

R12(u) = u + P1,2, (2.2)

where u is the spectral parameter and is the crossing parameter. The R-matrix satises the quantum Yang-Baxter equation (QYBE)

R12(u1 u2)R13(u1 u3)R23(u2 u3) = R23(u2 u3)R13(u1 u3)R12(u1 u2), (2.3)

and possesses the following properties:

Initial condition: R12(0) = P1,2, (2.4)

Unitarity: R12(u)R21(u) = 1(u) id, 1(u) = (u+ )(u ), (2.5)

Crossing-unitarity: Rt112(u)Rt121(un ) = 2(u) id, 2(u) = u(u + n ), (2.6) Fusion conditions: R12( ) = 2 P ()1,2, R12( ) = 2 P (+)1,2. (2.7)

Here R21(u) = P1,2R12(u)P1,2, P ()1,2 = 12{1 P1,2} is anti-symmetric (symmetric) project operator in the tensor product space V V, and ti denotes the transposition in the i-th space. Here and below we adopt the standard notation: for any matrix A End(V), Aj is

an embedding operator in the tensor space VV , which acts as A on the j-th space and as an identity on the other factor spaces; Rij(u) is an embedding operator of R-matrix in the tensor space, which acts as an identity on the factor spaces except for the i-th and j-th ones.

3

Let us introduce the row-to-row (or one-row ) monodromy matrix T (u), which is an n n matrix with operator-valued elements acting on VN,

T0(u) = R0N(u N)R0N1(u N1) R01(u 1). (2.8)

Here { j|j = 1, , N} are arbitrary free complex parameters which are usually called as inhomogeneous parameters.

The transfer matrix t(p)(u) of the spin chain with periodic boundary condition (or closed chain) is given by [24]

t(p)(u) = tr0T0(u). (2.9)

The QYBE implies that one-row monodromy matrix T (u) satises the following relation

R00 (u v)T0(u)T0 (v) = T0 (v)T0(u)R00 (u v). (2.10)

The above equation leads to the fact that the transfer matrices with di erent spectral parameters commute with each other: [t(p)(u), t(p)(v)] = 0. Then t(p)(u) serves as the generating functional of the conserved quantities, which ensures the integrability of the closed spin chain. The Hamiltonian (2.1) can be obtained from the transfer matrix as following

H = @ ln t(u)

@u |u=0, j=0. (2.11)

2.2 Operator product identities

Our main tool is the so-called fusion technique [6569]. We shall only consider the antisymmetric fusion procedure which leads to the desired operator identities to determine the spectrum of the transfer matrix t(p)(u) given by (2.9).

For this purpose, let us introduce the anti-symmetric projectors which are determined by the following induction relations

P ()1,2,,m+1 = 1

m + 1 (1 P1,2 P1,3 . . . P1,m+1) P ()2,3,,m+1, m = 1, . . . , n 1.

We introduce further the fused one-row monodromy matrices Th1,...,mi(u) (cf. (2.8))

Th1,...,mi(u) = P ()1,2,...,m T1(u)T2(u ) . . . Tm(u (m 1) ) P ()1,2,...,m, (2.12)

and the associated fused transfer matrices t(p)m(u)

t(p)m(u) = tr12m{Th1,...,mi(u)}, m = 1, , n, (2.13)

which includes the fundamental transfer matrix t(p)(u) given by (2.9) as the rst one, i.e., t(p)(u) = t(p)1(u). It follows from the fusion of the R-matrix [6569] that the fused transfer matrices constitute commutative families

[t(p)i(u), t(p)j(v)] = 0, i, j = 1, . . . , n. (2.14)

4

JHEP04(2014)143

We note that t(p)n(u) is the quantum determinant (proportional to the identity operator for generic u and { j}),

t(p)n(u) = (p)q(u) id =

Let us evaluate the product of the fundamental transfer matrix and the fused ones at some special points j and j . According to the denition (2.13), we thus have the following functional relations among the transfer matrices

t(p)( j)t(p)m( j ) = t(p)m+1( j), m = 1, . . . , n 1, j = 1, , N. (2.16)

The initial condition (2.4), the properties (2.7) of the R-matrix (e.g, Rij( )Rij( ) = 0)

and the properties (A.7) (see below) imply that the fused monodromy matrices Th1,...,mi(u) given by (2.12) vanishes at some special points,

Th1,...,mi( j + k ) = 0, for k = 1, . . . m 1, j = 1, . . . , N. (2.17)

This fact allows us to introduce some commutative operators {(p)m(u)} associated with the fused transfer matrices {t(p)m(u)}

t(p)m(u) =

N

2.3 Nested T-Q relation

The explicit expression (2.15) of the quantum determinant, the asymptotic behaviors (2.19) and the functional relations (2.20) allow one to determine the eigenvalues of all the operators {(p)m(u)} and consequently those of {t(p)m(u)} completely with the help of the relation (2.18) as follows. The commutativity of the transfer matrices with di erent spectral parameters implies that they have common eigenstates. Let | i be a common eigenstate of {t(p)m(u)}, which does not depend upon u, with the eigenvalue (p)m(u), i.e.,

t(p)m(u)| i = (p)m(u)| i, m = 1, . . . n.

5

N

Yl=1(u l + )

N

Yj=1

n1

Yk=1(u j k ) id. (2.15)

JHEP04(2014)143

N

Yl=1

m1

Yk=1(u l k )(p)m(u), [(p)l(u), (p)m(v)] = 0, l, m = 1, . . . , n. (2.18)

We use the convention: (p)(u) = (p)1(u). From the above equations and the denitions (2.13) of the fused transfer matrices, we conclude that the operators {(p)m(u)}, as

functions of u, are polynomials of degree N with the following asymptotic behaviors

(p)m(u) = n!

m!(n m)! uN + . . . , u . (2.19)

The operator identities (2.16) implies that these operators satisfy the following functional relations

(p)( j)(p)m( j ) =

Yl=1( j l )(p)m+1( j), j = 1, . . . , N, m = 1, . . . , n 1. (2.20)

The analyticity of the R-matrix implies that the eigenvalues (p)m(u) are polynomials of u with a degree of mN. The relations (2.18)(2.20) give rise to some similar relations of { (p)m(u)} which allow us to determine { (p)m(u)} completely. Here we give the nal result. The proof can be obtained by simple checking the solution satisfying the resulting relations.

Let us introduce n functions {z(l)p(u)|l = 1, . . . , n},

z(l)p(u) = Q(0)p(u)Q(l1)p(u + )Q(l)p(u )Q(l1)p(u)Q(l)p(u)

, l = 1, . . . n, (2.21)

where the functions Q(l)p(u) are given by

Q(0)p(u) =

N

Yj=1(u j), (2.22)

Q(r)p(u) =

Lr

Yl=1(u (r)l), r = 1, . . . , n 1, (2.23)

Q(n)p(u) = 1, (2.24)

where {Lr|r = 1, . . . n 1} are some non-negative integers and the parameters { (r)l|l = 1, . . . Lr, r = 1, . . . n 1} will be determined by the Bethe ansatz equations (2.26) (see below). The eigenvalues (p)m(u) of the m-th fused transfer matrix t(p)m(u) is then given by

(p)m(u) =

X1i1<i2<...<imnz(i1)p(u)z(i2)p(u ) . . . z(im)p(u(m1) ), m = 1, . . . , n. (2.25)

The regular property of (p)(u) implies that the residues of (p)(u) at each apparent simple pole (r)l have to vanish. This leads to the associated BAEs,

Lr

JHEP04(2014)143

Ym=1 (r)l (r+1)m (r)l (r+1)m, (2.26)

l = 1, . . . Lr, r = 1, 2, . . . , n 1, L0 = N, LN = 0, (0)l = l.

By taking the limit j = 0, the above BAEs are readily reduced to those previously obtained by other Bethe ansatz methods [6264].

3 su(n)-invariant spin chain with general open boundary conditions

3.1 Transfer matrix

Integrable open chain can be constructed as follows [15, 25]. Let us introduce a pair of K-matrices K(u) and K+(u). The former satises the reection equation (RE)

R12(u1 u2)K1(u1)R21(u1 + u2)K2(u2)

= K2(u2)R12(u1 + u2)K1(u1)R21(u1 u2), (3.1)

6

Lr+1

Yj=1,6=l (r)l (r)j (r)l (r)j + =

Lr1

Yk=1 (r)l (r1)k (r)l (r1)k +

and the latter satises the dual RE

R12(u2 u1)K+1(u1)R21(u1 u2 n )K+2(u2)

= K+2(u2)R12(u1 u2 n )K+1(u1)R21(u2 u1). (3.2)

For open spin-chains, instead of the standard row-to-row monodromy matrix T (u) (2.8), one needs to consider the double-row monodromy matrix J (u)

J0(u) = T0(u)K0(u) T0(u), (3.3) T0(u) = R01(u + 1)R02(u + 2) . . . R0N(u + N). (3.4)

Then the double-row transfer matrix t(u) of the open spin chain is given by

t(u) = tr0{K+0(u)J0(u)}. (3.5)

From the QYBE and the (dual) RE, one may check that the transfer matrices with di erent spectral parameters commute with each other: [t(u), t(v)] = 0. Thus t(u) serves as the generating functional of the conserved quantities, which ensures the integrability of the system.

In this paper, we consider a generic solution K(u) to the RE associated with the R-matrix (2.2) [7074]

K(u) = + uM, M2 = 1, (3.6)

where is a boundary parameter and M is an n n constant matrix (only depends on boundary parameters). Besides the RE, the K-matrix satises the following properties

K(0) = , K(u) = u M + . . . , u . (3.7)

Since the second power of M becomes the nn identity matrix, the eigenvalues of M must be 1. Suppose that there are p positive eigenvalues and q negative eigenvalues, then we have p + q = n and trM = p q. At the same time, we introduce the corresponding dual K-matrix K+(u) which is a generic solution of the dual RE (3.2)

K+(u) =

u + n2

where

is a boundary parameter and M is an nn boundary parameter dependent matrix, whose eigenvalues are 1. Again, we suppose that there are p positive eigenvalues and q negative eigenvalues, then we have p + q = n and tr M = p q. Besides the dual RE, the K-matrix also satises the following properties

K+

, K+(u) = u M + . . . , u . (3.9)

The Hamiltonian of the open spin chain specied by the K-matrices K(u) (3.6)

and (3.8) can be expressed in terms of the transfer matrix (3.5) as

H = @ ln t(u)

@u |u=0, j=0

= 2

N1

Xj=1

Pj,j+1 + tr0K+0(0)tr0K+0(0)+ 2tr0K+0(0)P0Ntr0K+0(0)+ 1 K1(0). (3.10)

7

JHEP04(2014)143

M, M2 = 1, (3.8)

n

2

=

3.2 Operator product identities

Similar to the closed spin chain case in the previous section, we apply the fusion technique to study the open spin chain. In this case, we need to use the fusion techniques both for R-matrices [6569] and for K-matrices [75, 76]. We only consider the antisymmetric fusion procedure which leads to the desired operator identities to determine the spectrum of the transfer matrix t(u) given by (3.5).

Following [75, 76], let us introduce the fused K-matrices and double-row monodromy matrices by the following recursive relations

K+1,...,m(u) = K+h2,...,mi(u )R1m(2u n + (m 1) ) . . .

R12(2u n + )K+1(u), (3.11)

K+h1,...,mi(u) = P ()1,...,mK+1,...,m(u)P ()1,...,m, (3.12)

K1,...,m(u) = K1(u)R21(2u ) . . . Rm1(2u (m 1) )Kh2,...,mi(u ), (3.13) Kh1,...,mi(u) = P ()1,...,m K1,...,m(u) P ()1,...,m, (3.14)

J1,...,m(u) = J1(u)R21(2u ) . . . Rm1(2u (m 1) )Jh2,...,mi(u ), (3.15) Jh1,...,mi(u) = P ()1,...,mJ1,...,m(u)P ()1,...,m = Th1,...,mi(u)Kh1,...,mi(u) Th1,...,mi(u), (3.16)

where the fused one-row monodromy matrix Th1,...,mi(u) is given by (2.12) and

Th1,...,mi(u) = P ()1,2,...,m T1(u) T2(u ) . . . Tm(u (m 1) ) P ()1,2,...,m. (3.17)

For the open spin chain, the m-th fused transfer matrix tm(u) constructed by the antisymmetric fusion procedure is given by

tm(u) = tr1,...,m{K+h1,...,mi(u)Jh1,...,mi(u)}, m = 1, . . . , n, (3.18)

which includes the fundamental transfer matrix t(u) given by (3.5) as the rst one, i.e., t(u) = t1(u). The relation (3.16) allows us to rewrite the transfer matric tm(u) in terms of the fused K-matrices and one-row monodromy matrices

tm(u) = tr1,...,m

nK+h1,...,mi(u) Th1,...,mi(u) Kh1,...,mi(u) Th1,...,mi(u)o. (3.19)

It follows from the fusion of the R-matrix [6569] and that of the K-matrices [75, 76] that the fused transfer matrices constitute commutative families, namely,

[ti(u), tj(v)] = 0, i, j = 1, . . . , n. (3.20)

Moreover, we remark that tn(u) is the so-called quantum determinant and that for generic u and { j} it is proportional to the identity operator, namely,

tn(u) = q(u) id, (3.21)

q(u) = q{T (u)} q{ T(u)} q{K+(u)} q{K(u)}

=

8

JHEP04(2014)143

N

Yl=1

n1

Yk=1(u l k )(u + l k )

N

Yl=1(u l + )(u + l + )

i

Yj=1(2u (i + j) )(2u + (n 2 i j) )

(1)q+q

q1

Yk=0

n1

Yi=1

u + n 2

2

k

p1

Yk=0

u + n 2

2 +

k

q1

Yk=0(u k )

p1

Yk=0(u + k ). (3.22)

(2.3) and (3.2) allow us to rewrite the fused K-matrix in another form

K+h1,...,mi(u) = P ()1,...,mK+m(u (m 1) )Rmm1(2u n + (2m 3) ) . . .

Rm1(2u n + (m 1) )K+h1,...,m1i(u)P ()1,...,m. (3.23)

The above equation, (3.13)(3.14), (2.5)(2.7) and the degenerate properties of the R-matrix and the K-matrices:

R12(0) = P1,2, K(0) = , K+

n

2

JHEP04(2014)143

=

, (3.24)

implies that at the following 2m special points

0, 2, . . . ,

m1

2 , and

n

2 +(m1) ,

n

2 +(m1)

2, . . . ,

n

2 +

m1

2 , (3.25)

the transfer matrix tm(u) may be written in terms of {tl(u)|l = m1, . . . , 0} (we have used the convention t0(u) = id), for examples (4.8)(4.13) for the su(3)-case and (B.1)(B.12) for the su(4)-case. The commutativity of the transfer matrices with di erent spectral parameters implies that they have common eigenstates. Let | i be a common eigenstate of {tm(u)}, which does not depend upon u, with the eigenvalue m(u), i.e.,

tm(u)| i = m(u)| i, m = 1, . . . n. (3.26)

Now let us evaluate the product of the fundamental transfer matrix and the fused ones at some special points

t( j)tm( j ) = tr1...m+1

J t11( j)K+1( j)t1

Jh2,...,m+1i( j )K+h2,...m+1i( j )[bracerightBig]

(2.6)

=

12(2 j k ) tr1...m+1 [braceleftbig]J t11( j)K+1( j)t1

Rt112(2 j + n ) . . . Rt11m+1(2 j + m n ) Rt11m+1(2 j m ) . . . Rt112(2 j )

Jh2,...,m+1i( j )K+h2,...m+1i( j )[bracerightBig]

=

m

[productdisplay]

k=1

m

[productdisplay]

k=1

12(2 j k ) tr1...m+1 {

R1 m+1(2 j + m n ) . . . R12(2 j + n )K+1( j)

J1,...,m+1( j)K+h2,...,m+1i( j )[bracerightBig]

9

12(2 j k ) tr1...m+1 [braceleftBig]K+h2,...,m+1i( j )

R1 m+1(2 j + m n ) . . . R12(2 j + n )K+1( j) J1,...,m+1( j)}

=

=

m

[productdisplay]

k=1

m

[productdisplay]

k=1

12(2 j k ) tr1...m+1 [braceleftbig]K+1

,...,m+1( j)J1,...,m+1( j)

[bracerightbig]

(A.2)

=

m

[productdisplay]

k=1

12(2 j k ) tr1...m+1 [braceleftBig]K+1,...,m+1( j)P ()1,...,m+1J1,...,m+1( j)[bracerightBig]

(A.11)

=

m

[productdisplay]

k=1

12(2 j k ) tr1...m+1 [braceleftBig]K+h1,...,m+1i( j)J1,...,m+1( j)P ()1,...,m+1[bracerightBig]

(A.10)

=

JHEP04(2014)143

m

[productdisplay]

k=1

12(2 j k ) tr1...m+1 [braceleftBig]K+h1,...,m+1i( j)Jh1,...,m+1i( j)[bracerightBig] .

According to the denition (3.18), we thus have the following functional relations among the transfer matrices

t( j)tm( j ) = tm+1( j)

m

Yk=112(2 j k ), (3.27)

j = 1, . . . , N; m = 1, . . . , n 1.

In terms of the corresponding eigenvalues, the above relations become

( j) m( j ) = m+1( j)

m

Yk=112(2 j k ), (3.28)

j = 1, . . . , N; m = 1, . . . , n 1.

Using the similar method that we have derived the zero points (2.17) of the fused monodromy matrix Th1,...,mi(u), we can gure out the zero points of the fused monodromy matrix Th1,...,mi(u) and those of the fused K-matrices Kh1,...,mi(u) respectively. Thanks to the alternative expression (3.19) of the fused transfer matrix tm(u), we know that these zero points all together constitute the zero points of the transfer matrix, which allows us to rewrite the transfer matrix as

tm(u) =

m1

Yi=1

i

Yj=1(2u i j )(2u + (2m 2 n) i j )

m1

Yk=1(u l k )(u + l k ) m(u). (3.29)

Since the operator m(u) is proportional to the transfer matrix tm(u) by c-number coe -cient, the corresponding eigenvalue

m(u) has the following relation with m(u)

m(u) =

m1

Yi=1

N

Yl=1

i

Yj=1(2u i j )(2u + (2m 2 n) i j )

m1

Yk=1(u l k )(u + l k ) m(u). (3.30)

10

N

Yl=1

It follows from the denitions of the fused transfer matrices (3.18) that the eigenvalue

m(u) of the resulting commutative operator m(u), as a function of u, is a polynomial of degree 2N + 2m. The functional relations (3.27) give rise to that the eigenvalue

m(u) of

m(u) satises the following relations

( j)

m( j ) =

m+1( j)

m

Yk=112(2 j k )0( j), (3.31)

m = 1, . . . , n 1, j = 1, . . . , N,

where the function 0(u) is given by

0(u) =

N

Yl=1(u l )(u + l )

m+1

JHEP04(2014)143

Yk=2(2u k )(2u k + (n 2) ).

Then n(u) is proportional to identity operator with a known coe cient

n(u)

n(u)=

N

Yl=1(u l + )(u + l + )

q1

Yk=0

u + n 2

2

k

(1)q+q

p1

Yk=0

u+ n2

2 +

k

q1

Yk=0 (uk )

p1

Yk=0(u+k ). (3.32)

3.3 Asymptotic behaviors of the transfer matrices

The denitions (3.11)(3.18) of the fused K-matrices, the fused monodromy matrices and the fused transfer matrices and the asymptotic behaviors (3.7) and (3.9) imply that the asymptotic behaviors of the operators {m(u)} given by (3.29) is completely xed by the eigenvalues of the product matrix MM (see (3.36) below). Firstly let us give some properties of the eigenvalues of MM. Suppose { l|l = 1, . . . , n} be the eigenvalues. The fact that M2 = M2 = 1 allows one to derive the following relations among the eigenvalues,

n

Xl=1 kl = tr{( MM)k} = tr{(M M)k} = tr{( MM)k} =

n

Xl=1 kl, k. (3.33)

Meanwhile we know that

Det| MM| = 1 . . . n = (1)q+q. (3.34)

This implies that the eigenvalues of M M should take the following form

{ 1, . . . , n} = {1, . . . , 1, 1, . . . , 1, ei#1, ei#1, . . . , ei#r, ei#r}, (3.35)

where #j are some continuous free parameters which are related to boundary interaction terms (e.g., the boundary magnetic elds). The maximum number of the continuous parameters is n/2 if n is even and is (n 1)/2 if n is odd.

Some remarks are in order. When M and M commute with each other and thus can be diagonalized simultaneously by some gauge transformation, the corresponding open spin

11

chain can be diagonalized by the algebraic Bethe ansatz method after a global gauge transformation [71]. In case of the boundary parameters (which are related to the matrices M and M) have some constraints so that a proper local vacuum state exists, the generalized algebraic Bethe ansatz method [41, 77, 78] can be used to obtain the Bethe ansatz solutions of the associated open spin chains [7981]. However, the results in [1013] strongly suggest that for generic M and M such a simple local vacuum state do not exist even for the su(2) case.

The asymptotic behaviors (3.7) and (3.9) enable us to derive that the eigenvalue

m(u)

of the operators {m(u)} given by (3.29) have the following asymptotic behaviors

m(u) = (1)m m u2N+2m + . . . , m = 1, . . . , n, u , (3.36)

where m =

X1i1<i2...<imn i1 . . . im, m = 1, . . . , n.

Keeping the fact that

n(u) has been already xed (3.32) in the mind, we need to determine the eigenvalues of the other n 1 transfer matrices {m(u)|m = 1, . . . , n 1}. It is

also known from (3.29) that

m(u) , as a function of u, is a polynomial of degree 2N + 2m.

Thanks to the very functional relations (3.31) and the asymptotic behaviors (3.36), one can completely determine the eigenvalues of the transfer matrix and the other higher fused transfer matrices by providing some other values of the eigenvalue functions at

JHEP04(2014)143

Pn1 m=1 2m special points (3.25) (e.g. see (4.8)(4.13) for the su(3)-case and (B.1)(B.12) for the su(4)-case). The method has been proven in [1013] to be successful in solving the open spin chains related to su(2) algebra. In the following section, we shall apply the method to solve the open spin chains associated with su(n) algebra.

For this purpose, let us rst factorize out the contributions of K-matrices which are relevant to the quantum determinant

n(u) (3.32) by introducing n functions {K(l)(u)|l = 1, . . . , n} which are polynomials of u with a degree 2. The functions depend only on the boundary parameters and

and satisfy the following relations

n

Yl=1K(l)(u (l 1) ) = (1)q+q

q1

Yk=0

u + n 2

2

k

p1

Yk=0

u + n 2

2 +

k

q1

Yk=0(u k )

p1

Yk=0(u + k ), (3.37)

K(l)(u)K(l)(u l ) = K(l+1)(u)K(l+1)(u l ), l = 1, , n 1. (3.38)

From the solution to the above equations, one can construct a nested T-Q ansatz for the eigenvalues m(u). It is remarked that there are some di erent solutions to the above equations. However, it was shown in [1013, 58] that for the su(2) open spin chain any choice of the above equation leads to a complete set of solutions of the the corresponding model. It is believed that di erent choices of the solution might only give rise to di erent parameterizations of the eigenvalues.

12

Before closing this section, let us give a summary of the set of properties which characterize the eigenvalue of the transfer matrix tm(u):

explicit expression of tn(u) or the quantum determinant (3.22).

Analytical property and asymptotical behaviors (3.36) of the transfer matrices.

Functional relations (3.27) for the fused transfer matrices.

The values of the transfer matrices at the special points (3.25) (for examples (4.8) (4.13) for the su(3)-case and (B.1)(B.12) for the su(4)-case).

The above condition are believed to determine the eigenvalues of the transfer matrices tm(u).

4 su(3)-invariant spin chain with non-diagonal boundary term

In this section, we use the method outlined in the previous section to give the Bethe ansatz solution of the su(3)-invariant spin chain with generic boundary terms. Without loss of generality, we take the corresponding M and M with p = p = 1 and the eigenvalues of MM being

( 1, 2, 3) = (1, ei#, ei#), (4.1)

as an example to demonstrate our method in detail.Let us introduce 3 functions {K(l)|l = 1, 2, 3} as follows

K(1)(u) =

+ 12 u ( + u), (4.2)

K(2)(u) =

JHEP04(2014)143

+ 32 + u

( u ), (4.3)

( u ), (4.4)

which satisfy (3.37) and (3.38) for n = 3. From the denitions (3.18) of the fused transfer matrices tm(u) and the asymptotic behaviors of the K-matrices K(u), we have that the eigenvalues of the transfer matrices have the following asymptotic behaviors

(u)|u = tr( MM)u2N+2 + . . . =

3

K(3)(u) =

+ 32 + u

Xi=1 iu2N+2 + . . .

= (1 + 2 cos #)u2N+2 + . . . ., (4.5)

2(u)|u = tr12

nP ()1,2( MM)1( MM)2P ()1,2

o

u2N+4 + . . .

X1i1<i23 i1 i2u2N+4 + . . .

= (2 cos # + 1)u2N+4 + . . . . (4.6)

13

=

The explicit expressions of the K-matrices (3.6) and (3.8) imply the following identities

K(u) K(u) = (2 u2), K+

u + n2

2 u2). (4.7)

The above relations and some degenerated properties (3.24) of the R-matrix and the K-matrices allow us to derive that the fused transfer matrices satisfy the following properties at some special points (3.25), namely, 0, 32 ; 0, 2, 2, for the su(3)-case:

t(0) = (1)N

N

K+

u + n

2

= (

Yl=1( l + )( l )tr{K+(0)} id, (4.8)

t

JHEP04(2014)143

3

2

= (1)N

N

Yl=1

l + 32 l 3 2

tr

K 32 id, (4.9)

t2

2

= tr12

P 12K+2

2

R12(3 )K+1

2

P 12

2

4 2

N

Yl=1

l + 32 l 32 l + 2 l 2 id, (4.10)

t2( ) = tr12

P 12K1( )R21(3 )K2(2 )P 12

2

4

Yl=1( l + )( l )( l + 2 )( l 2 ) id, (4.11)

t2(0) = (1)N2 2

N

N

Yl=1( l + )( l )tr{K+(0)} t( ), (4.12)

t2

K 32 t 2 , (4.13)

These relations allow us to derive similar relations of the eigenvalues {

m(u)}. Then

the resulting relations (total number of the conditions is equal to 2 + 4 = 6), the very relations (3.31) for n = 3 and the asymptotic behaviors (4.5)(4.6) allow us to determine the eigenvalues

m(u) (also m(u) via the relations (3.30)).

Let us dene the corresponding Q(r)(u) for the open spin chains

Q(0)(u) =

N

Yj=1(u j)(u + j), (4.14)

Q(r)(u) =

Lr

2

= (1)N2

2

N

Yl=1

l + 32 l 32 tr

Yl=1(u (r)l)(u + (r)l + r ), r = 1, . . . , n 1, (4.15)

Q(n)(u) = 1, (4.16)

where {Lr|r = 1, . . . , n 1} are some non-negative integers. In the following part of the paper, we adopt the convention

a(u) = Q(0)(u + ), d(u) = Q(0)(u). (4.17)

14

In order to construct the solution of the open su(3) spin chain, we introduce three functions

1(u) = z1(u) + x1(u),2(u) = z2(u),3(u) = z3(u). (4.18)

Here zm(u) is given by the following relations

zm(u) = u(u + 32 )(u + (m1)2 )(u + m2 )

K(m)(u)d(u)Q(m1)(u + )Q(m)(u ) Q(m1)(u)Q(m)(u) ,

m = 1, 2, 3, (4.19)

with {K(m)(u)|m = 1, 2, 3} are given by (4.2)(4.4) (here we have assumed Q(3)(u) = 1 since that the su(3)-case is considered) and x1(u) is dened as

x1(u) = u

u + 3 2

a(u)d(u) F1(u)

Q(1)(u). (4.20)

The nested functional T-Q ansatz is expressed as

(u) =

3

Xi1=1i1(u) =

3

Xi1=1zi1(u) + u

u + 3 2

a(u)d(u) F1(u)

Q(1)(u), (4.21)

2(u) = 2(2u )

X1i1<i23i1(u)i2(u ) x1(u)z2(u )

JHEP04(2014)143

. (4.22)

We remark that the extra term x1(u) in (4.18) given by (4.20) does not violate the very functional relation (3.28) with n = 3 due to the fact a( j ) = d( j) = 0, but it

does change the form of the resulting BAEs (see (4.29) below). The function x1(u) can be

determined by regularity of (u) and 2(u) given by (4.21) and (4.22) and their asymptotic behaviors as follows. The vanishing of the residues of (u) at (1)j and (1)j requires

F1(u) = f1(u)Q(2)(u ), (4.23)

with

f1(u) = f1(u ). (4.24)

In order not to violate the relations (4.8)(4.13), let all terms with x1(u) in m(u) be zero at all the degenerate points considered in (4.8)(4.13), then the function f1(u) is given by

f1(u) = c u

u + 1 2

2(u + ). (4.25)

The asymptotic behaviors of (u) and 2(u) then x the constant c,

c = 2(cos # 1), (4.26)

where # is specied by the eigenvalues of the matrix MM (4.1). It is remarked that F1(u)

is a polynomial of degree 2L1 2N. Then the above relations lead to the constraint among the non-negative integers L1 and L2

L1 = N + L2 + 2. (4.27)

15

(4.23)(4.27) and the relations (3.30) between

m(u) and m(u) lead to that the asymptotic behaviors (4.5)(4.6) of the eigenvalues

m(u) are automatically satised. Keeping the

following identities in mind,

a( j ) = d( j) = 0, j = 1, . . . , N, (4.28)

with the help of (3.22), by putting u = j, j in (4.21) and (4.22) we can easily show that the ansatz (4.21)(4.22) make the very functional relations (3.28) with n = 3 fullled. The regular property of (u) and 2(u) leads to the associated Bethe ansatz equations,

1 + (1)l (1)l +

K(2)( (1)l)d( (1)l)

K(1)( (1)l)a( (1)l)

Q(1)( (1)l + )Q(2)( (1)l )

Q(1)( (1)l )Q(2)( (1)l)

= c ( (1)l)2( (1)l + 12 )3( (1) + )d( (1)l)Q(2)( (1)l )K(1)( (1)l)Q(1)( (1)l )

, l = 1, . . . , L1, (4.29)

JHEP04(2014)143

(2)l + 32

(2)l + 12

K(2)( (2)l)

K(3)( (2)l)

Q(1)( (2)l + )Q(2)( (2)l )

Q(1)( (2)l)Q(2)( (2)l + )

= 1, l = 1, . . . L2. (4.30)

The eigenvalue of the Hamiltonian (3.10) in the case of n = 3 is given by

E =

L1

Xl=12 2 (1)l( (1)l + )+ 2(N 1) +

+ 32 p

(

+ 32 p )

3, (4.31)

where the parameters { (1)l} are the roots of the BAEs (4.29)(4.30) in the homogeneous limit j = 0.

5 Exact solution of su(n)-invariant spin chain with general open boundaries

The analogs of (4.8)(4.13) for arbitrary n at the special points listed in (3.25) can also be constructed with the properties of (3.24) and (4.7). To show the procedure clearly, we construct those relations for n = 4 in appendix B. In fact, those relations are ensured for the diagonal case as already demonstrated by the algebraic Bethe ansatz. For the non-diagonal case, since we put xi(u) to be zero for u at those degenerate points, the relations must also hold no matter how their exact forms are. By following the same procedure as the previous section, we may derive the solutions of the su(n)-invariant quantum spin chain with general open boundary conditions. Here we present the nal result. The functions zm(u) now read

zm(u) = 2u(2u + n )

(2u + (m 1) )(2u + m )K(m)(u)Q(0)(u)

+ 2

Q(m1)(u + )Q(m)(u )

Q(m1)(u)Q(m)(u) , (5.1)

m = 1, . . . , n,

where {K(l)(u)|l = 1, . . . , n} satisfy (3.37)(3.38) and {Q(m)(u)|m = 0, 1, . . . , n} are given by (4.14)(4.16). In principle, K(l)(u) could be any decomposition of (3.37). For simplicity,

16

we parameterize them satisfying the following relations

K(l)(u) = K(l+1)(u l ), l = 1, , n 1. (5.2)

Similarly as (4.18), let us introduce the functions {i(u)|i = 1, . . . , n} by

i(u) = zi(u) + xi(u), i = 1, . . . , n, (5.3)

where the functions xi(u) are

(x2l1(u) = u(u + n2 )a(u)d(u) F2l1(u)Q(2l1)(u) x2l(u) = 0

, (5.4)

and l = 1, 2, . . . , n2 if n is even, l = 1, 2, . . . , n12. The functions {F2l1(u)} are given by

F1(u) = f1(u)Q(2)(u ), (5.5)

F2l1(u) = f2l1(u)Q(2l2)(u (2l 1) )Q(2l)(u (2l 1) )a(u (2l 1) ), (5.6)

where l = 2, . . . , n2 if n is even and l = 2, . . . , n12 if n is odd, and

f2l1(u) = c2l1

n1

Yk=1

u + k2 u + (2l 1) k2 , l = 1, 2, . (5.7)

The functions f2l1(u) has the following crossing symmetry relation

f2l1(u) = f2l1(u (2l 1) ). (5.8)

Here the parameters {c2l1} are determined, with helps of the asymptotic behaviors of the eigenvalues of the transfer matrices, by the following relations

X1i1<i2<...<imni1i2 . . .im +

JHEP04(2014)143

m1

Xk=1

m2

Xl=k

X1i1<i2<...<i2k22l2i1i2 . . .i2k22l1

X2l+1i2k+1<i2k+2<...<imni2k+1i2k+2 . . .im +

m3

Xk=2

m4

Xl=k

X1 i1<i2<...<i2k32l2 i1

i2 . . .i2k32l1

X2l+1i2k<i2k+1<...<imni2ki2k+1 . . .im

= (1)m

X1i1<i2<...<imn i1 i2 . . . im, (5.9)

where

i =

1 + c2l1, i = 2l 1,

1, i = 2l, 1, i = n,

and

(i). m1 = m3 = m

2 , m2 = m4 =

n m 2k

2 , if m is even and n is even; (5.10)

17

(ii). m1 = m 1

2 , m2 = m4 =

n m 2k 12 , m3 =

m + 1 2 ,

if m is odd and n is even, (5.11)

(iii). m1 = m3 = m2 , m2 = m4 =

n m 2k 1

2 ,if m is even and n is odd, (5.12)

(iv). m1 = m 12 , m2 =

n m 2k

2 , m3 =

m + 1

2 , m4 =

n m 2k 2 2 ,

if m is odd and n is odd. (5.13)

The constants { i|i = 1, . . . , n} are the eigenvalues of the matrix MM given by (3.35), while the matrices M and M are related to the K-matrices K(u). Then the nested T-Q ansatz of the eigenvalues (u) of the transfer matrix t(u) is

(u) =

n

Xi=1i(u), (5.14)

where the functions {i(u)} are given by (5.3). The other eigenvalues m(u) of the fused transfer matrix tm(u) are given by

m(u) =

m1

Yl=1

l

Yk=12(2u k l + )

JHEP04(2014)143

X1i1<i2<...<imni1(u)i2(u ) . . .im(u (m 1) )

m1

Xk=1

m2

Xl=k

X1i1<i2<...<i2k22l2i1(u)i2(u ) . . .i2k2(u (2k 3) )

f2l1(u (2k 2) )2l(u (2k 1) )

X2 l+1i2k+1<i2k+2<...<imn

i2k+1(u 2k )i2k+2(u (2k + 1) ) . . .im(u (m 1) )

m3

Xk=2

m4

Xl=k

X1i1<i2<...<i2k32l2i1(u)i2(u ) . . .i2k3(u (2k 4) )

f2l1(u (2k 3) )2l(u (2k 2) )

X2 l+1i2k<i2k+1<...<imn

i2k(u (2k 1) )i2k+1(u 2k ) . . .im(u (m 1) )

), (5.15)

where the m1, m2, m3 and m4 are the same as those in the equations (5.10)(5.13). The parameters { (r)l} satisfy the associated Bethe ansatz equations

K(1)( (1)j)a( (1)j)Q(1)( (1)j ) +

(1)j

(1)j +

K(2)( (1)j)d( (1)j)Q(1)( (1)j + )

Q(2)( (1)j )

Q(2)( (1)j)

+ (1)j

(1)j + 2 a( (1)j)d( (1)j)F1( (1)j) = 0, j = 1, . . . , L1. (5.16)

18

2 (2l)k+(2l+1)

2 (2l)k+(2l1)

K(2l)( (2l)k)

K(2l+1)( (2l)k)

Q(2l1)( (2l)k+ )Q(2l+1)( (2l)k)

Q(2l1)( (2l)k)Q(2l+1)( (2l)k )

=Q(2l)( (2l)k+ ) Q(2l)( (2l)k )

,

k = 1, . . . , L2l, (5.17)

K(2s+1)( (2s+1)j)Q(2s+1)( (2s+1)j ) +

(2s+1)j + s

(2s+1)j + (s + 1)

K(2s+2)( (2s+1)j)

Q(2s+1)( (2s+1)j + )

Q(2s)( (2s+1)j)Q(2s+2)( (2s+1)j )

Q(2s)( (2s+1)j + )Q(2s)( (2s+1)j)

+ ( (2s+1)j + s )

JHEP04(2014)143

(2s+1)j + 2s + 12

a( (2s+1)j)

Q(2s)( (2s+1)j)

Q(2s)( (2s+1)j + )

F2s+1( (2s+1)j) = 0, (5.18)

j = 1, . . . , L2s+1,

where l = s = 1, . . . , n2 1 if n is even, l = 1, . . . , n12 and s = 1, . . . , n12 1 if n is odd.

The rule for constructing {xi(u)|i = 1, . . . , n} is the following: (1)they must vanish for u at all special points appeared in the operator identities (3.28) and in (3.25) and therefore their existence does not a ect the operator identities and the analogs of (4.8)(4.13); (2)the denominators must not generate new poles and therefore should be Qi(u) or part of them;

(3)the functions Fi(u) must satisfy the corresponding crossing symmetry properties to keep the self-consistency of the BAEs. The explicit expressions (5.4)(5.7) of the functions {xi(u)|i = 1, . . . , n} and the nested T-Q ansatz (5.14)(5.15) of the eigenvalues i(u) imply that the nested T-Q ansatz does satisfy the very function relations (3.28) due to the fact a( j ) = d( j) = 0. Since that these functions {xi(u)|i = 1, . . . , n} also vanish at the degenerated points (3.25), hence they do not violate the relations (for examples (4.8)(4.13) for the su(3)-case and (B.1)(B.12) for the su(4)-case) of the transfer matrices at these degenerated points. Moreover, the special choice of these functions (5.4)(5.7) also assures that the regularity of the all eigenvalues { i(u)|i = 1, . . . , n} of the transfer matrices can be guaranteed consistently by the resulting BAEs (5.16)(5.18). The asymptotic behaviors (3.36) of the eigenvalues of the transfer matrices lead to the equations (5.9)(5.13), which xes the constants {ci} in (5.7).

Some remarks about the existence of solutions of (5.9) are in order. Due to the construction rule (5.15) for m(u), the asymptotic behaviors of m(u) and nm(u) give the same condition in (5.9), namely, the total number of the independent equations in (5.9) is n/2 if n is even and is (n 1)/2 if n is odd. Thus, the equations (5.9)(5.13) will x the constants {ci} in (5.7). We have checked that for n = 3 there exists an unique solution to (5.9)(5.13) (e.g. (4.26)) and that for n > 4 there are more solutions (actually there are two solutions for n = 4) to (5.9)(5.13). For an example, the independent equations for n = 4 and p = p = 2 are

(4 + c1 + c3 = 2 cos #1 2 cos #24 2c1 2c3 + c1c3 = 4 cos #1 cos #2 . (5.19)

19

However, the di erent solutions only give di erent parameterizations of i(u) in the T

Q-type. It has been shown in [1013] that even for the open chains related to the su(2) case there indeed exist di erent T-Q ansatz for the eigenvalue of the transfer matrix. The numerical check [1013, 58] for the small sites of lattice shows that any of them gives the complete set of eigenvalues.

The eigenvalue of the Hamiltonian (3.10) is

E =

L1

Xl=12 2 (1)l( (1)l + )+ 2(N 1) + [K(1)(u)]

K(1)(u) |u0 +2n, (5.20)

where the parameters { (1)l} are the roots of the BAEs (5.16)(5.18) with j = 0.

When two K-matrices K+(u) and K(u) are both diagonal matrices, or they can be diagonalized simultaneously by some gauge transformation, all the parameters c2l1 vanish, leading to F2l1(u) = 0. The eigenvalue (u) of the transfer matrix t(u) and the BAEs recover those obtained by the other Bethe ansatz methods [77, 78, 8288].

6 Conclusions

In this paper, we propose the nested o -diagonal Bethe ansatz method for solving the multi-component integrable models with generic integrable boundaries, a generalization of the method proposed in [1013] (related to su(2) algebra) for integrable models associated with higher rank algebras. In the method some functional relations (for the su(n) case such as (2.16) for the closed chain or (3.27) for the open chain) among the antisymmetric fused transfer matrices play a very important role. Taking the su(n)-invariant spin chain model with both periodic and non-diagonal boundaries as examples, we elucidate how the method works for constructing the Bethe ansatz solutions of the model. For the su(n)-invariant closed chain, we re-derive the results obtained previously by other methods [6264], but with a simplied process. For the open boundary case specied by the most general K-matrices (3.6) and (3.8), the very functional relations (3.27) are derived only via some properties of the R-matrix and K-matrices. Based on these relations, the asymptotic behaviors (3.36) and the values of the eigenvalue functions (for examples, (4.8)(4.13) for the su(3)-case and (B.1)(B.12) for the su(4)-case) at

JHEP04(2014)143

Pn1m=1 2m special points (3.25), we obtain the eigenvalues of the transfer matrix. When the K-matrices are both diagonal ones, our results can be reduced to those obtained by the conventional Bethe ansatz methods. Therefore, our method provides an unied procedure for approaching the integrable models both with and without U(1) symmetry. We remark that this method might also be applied to other quantum integrable models dened in other algebras.

Acknowledgments

The nancial support from the National Natural Science Foundation of China (Grant Nos. 11174335, 11031005, 11375141, 11374334), the National Program for Basic Research of MOST (973 project under grant No.2011CB921700), the State Education Ministry of

20

China (Grant No. 20116101110017) and BCMIIS are gratefully acknowledged. We would like to thanks for the anonymous referees for their valuable suggestions for revising the manuscript. Two of the authors (W. -L. Yang and K. Shi) would like to thank IoP, CAS for the hospitality and they enjoyed during their visit there. W.-L. Yang also would like to thank KITPC and MCM for their hospitalities where some parts of work have been done during his visiting.

A Proofs of the operator identities

In this appendix, we give the detailed proof of the following identities which are crucial to obtain the functional relations (2.16) and (3.27):

T1( j)Th2,3,...,mi( j ) = P ()1,2,...,mT1( j)T2( j ) . . . Tm( j (m 1) ) P ()2,...,m, (A.1)

J1,...,m( j) = P ()1,2,...,mJ1,...,m( j). (A.2)

The exchange relation (2.10) of the one row monodromy matrix T (u) implies

R21( )T2(u )T1(u) = T1(u)T2(u )R21( ).

The above relation and the fusion condition (2.7) allow one to derive the following identity

P ()1,2 T1(u)T2(u )R12( ) = T1(u)T2(u )R12( ). (A.3)

Let us evaluate the product of the operators T1( j) and T2( j )

T1( j)T2( j ) = R1N ( j N) . . . R1 j+1( j j+1)R1j(0)R1 j1( j j1) . . . R1 1( j 1)R2N ( j N ) . . . R2 j+1( j j+1 )R2j( )

R2 j1( j j1 ) . . . R2 1( j 1 )

=Rj j1( j j1) . . . Rj 1( j 1)

R1N ( j N) . . . R1j+1( j j+1)

R2N ( j N ) . . . R2j+1( j j+1 )R21( )

R1j(0)R2 j1( j j1 ) . . . R2 1( j 1 )

(A.3)

= Rj j1( j j1) . . . Rj 1( j 1) P ()1,2

R1N ( j N) . . . R1j+1( j j+1)

R2N ( j N ) . . . R2j+1( j j+1 )R21( )

R1j(0)R2 j1( j j1 ) . . . R2 1( j 1 ) =P ()1,2 T1( j)T2( j ),

namely, we have

T1( j)T2( j ) = P ()1,2 T1( j)T2( j ), j = 1, . . . , N. (A.4)

Similarly, we have

T1( j) T2( j ) = P ()1,2 T1( j) T2( j ), j = 1, . . . , N. (A.5)

21

JHEP04(2014)143

Due to the fact that R12( ) is proportional to the antisymmetric projector (2.7), the relation (A.3) also implies

Th1,2i(u) = P ()1,2T1(u)T2(u )P ()1,2 = T1(u)T2(u )P ()1,2. (A.6)

Using similar method to derive the above relation and following the procedure [6569], we can derive the following relations

Th1,2,...,mi(u) = P ()1,2,...,mT1(u)T2(u ) . . . Tm(u (m 1) )P ()1,2,...,m

= T1(u)T2(u ) . . . Tm(u (m 1) )P ()1,2,...,m. (A.7)

Combining the above relation with (A.4), we can show that

Pl,l+1 T1( j)Th2,3,...,mi( j ) = T1( j)Th2,3,...,mi( j ), l = 1, . . . , m 1. (A.8)

Then we can conclude that T1( j)Th2,3,...,mi( j ) satisfy the relation (A.1).

With the similar method used to prove (A.4) and the reection equation (3.1), we can obtain the following relations:

J1( j)R21(2 j )J2( j ) = P ()1,2J1( j)R21(2 j )J2( j ),

j = 1, . . . , N. (A.9) Jh1,...,mi(u) = J1,...,m(u) P ()1,...,m, m = 1, . . . n, (A.10)

K+h1,...,mi(u) = K+1,...,m(u) P ()1,...,m, m = 1, . . . n. (A.11)

Using the relations (A.9) and (A.10), we can derive that

Pl,l+1 J1,...,m( j) = J1,...,m( j), l = 1, . . . , m 1. (A.12)

(A.2) is a consequence of the above relations. Hence we complete the proof of (A.2).

B Higher rank analogs of (4.8)(4.13)

The explicit expressions of the K-matrices (3.6) and (3.8) imply some identities (4.7) among them. These identities and some degenerated properties (3.24) of the R-matrix and the K-matrices allow one to derive that the fused transfer matrices satisfy certain relations at some special points (3.25). For the su(3)-case they are given by (4.8)(4.13). Here we present their analogs for the su(4)-case:

t(0) = (1)N

N

Yl=1

t2(0) = 3(1)N 2

JHEP04(2014)143

N

Yl=1( l + )( l )tr{K+(0)} id, (B.1)

t(2 ) = (1)N

l + 32 ( l 2 )tr{K(2 )} id, (B.2)

N

Yl=1( l + )( l )tr{K+(0)} t1( ), (B.3)

22

t2

2

= tr12

K+h12i

2

24 2

N

Yl=1

l 2 l + 2 l 32 l + 32 id, (B.4)

t2

3

2

=

24 2

N

Yl=1

l 5 2

l + 52

l 32

l + 32

tr12

Kh12i

3

2

id, (B.5)

JHEP04(2014)143

t2( ) = 3(1)N

2

N

Yl=1( l + 2 )( l 2 )tr{K(2 )} t1( ), (B.6)

and

t3(0) = 12(1)N 4

N

Yl=1( l + )( l )tr{K+(0)} t2( ), (B.7)

t3(0) = 12(1)N

4

N

Yl=1( l + 2 )( l 2 )tr{K(2 )} t2(0), (B.8)

t3

2

= 12tr12

K+h12i

2

5

24 2 t1 3 2

N

Yl=1

l 2 l + 2 l 3 2

l + 32

, (B.9)

t3

2

= 12 5

24 2

tr23

Kh23i

3

2

t1

2

N

Yl=1

l 52 l + 52 l 3 2

l + 32

, (B.10)

@@ut3(u) |u= = 4 2(2 2)(1)Ntr123{K+h123i( )}

N

Yl=1 2l( l )( l + )( l 2 )( l + 2 ) id, (B.11)

@@ut3(u) |u= = 4

2( 2

2)(1)Ntr123{Kh123i( )}

N

Yl=1( l )( l + )( l 2 )( l +2 )( l 3 )( l +3 ) id. (B.12)

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

23

References

[1] J.M. Maldacena, The large-N limit of superconformal eld theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [http://dx.doi.org/10.1023/A:1026654312961

Web End =Int. J. Theor. Phys. 38 (1999) 1113 ] [http://arxiv.org/abs/hep-th/9711200

Web End =hep-th/9711200 ] [http://inspirehep.net/search?p=find+J+Adv.Theor.Math.Phys.,2,231

Web End =INSPIRE ].

[2] N. Beisert et al., Review of AdS/CFT integrability: an overview,

http://dx.doi.org/10.1007/s11005-011-0529-2

Web End =Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.3982

Web End =INSPIRE ].

[3] J.A. Minahan and K. Zarembo, The Bethe ansatz for N = 4 super Yang-Mills, http://dx.doi.org/10.1088/1126-6708/2003/03/013

Web End =JHEP 03 (2003) 013 [http://arxiv.org/abs/hep-th/0212208

Web End =hep-th/0212208 ] [http://inspirehep.net/search?p=find+J+JHEP,0303,013

Web End =INSPIRE ].

[4] D. Berenstein and S.E. Vazquez, Integrable open spin chains from giant gravitons, http://dx.doi.org/10.1088/1126-6708/2005/06/059

Web End =JHEP 06 (2005) 059 [http://arxiv.org/abs/hep-th/0501078

Web End =hep-th/0501078 ] [http://inspirehep.net/search?p=find+J+JHEP,0506,059

Web End =INSPIRE ].

[5] D.M. Hofman and J.M. Maldacena, Reecting magnons, http://dx.doi.org/10.1088/1126-6708/2007/11/063

Web End =JHEP 11 (2007) 063 [arXiv:0708.2272] [http://inspirehep.net/search?p=find+J+JHEP,0711,063

Web End =INSPIRE ].

[6] R. Murgan and R.I. Nepomechie, Open-chain transfer matrices for AdS/CFT, http://dx.doi.org/10.1088/1126-6708/2008/09/085

Web End =JHEP 09 (2008) 085 [arXiv:0808.2629] [http://inspirehep.net/search?p=find+J+JHEP,0809,085

Web End =INSPIRE ].

[7] R.I. Nepomechie, Revisiting the Y = 0 open spin chain at one loop, http://dx.doi.org/10.1007/JHEP11(2011)069

Web End =JHEP 11 (2011) 069 [arXiv:1109.4366] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1109.4366

Web End =INSPIRE ].

[8] J. McGreevy, L. Susskind and N. Toumbas, Invasion of the giant gravitons from anti-de Sitter space, http://dx.doi.org/10.1088/1126-6708/2000/06/008

Web End =JHEP 06 (2000) 008 [http://arxiv.org/abs/hep-th/0003075

Web End =hep-th/0003075 ] [http://inspirehep.net/search?p=find+J+JHEP,0006,008

Web End =INSPIRE ].

[9] J. Dukelsky, S. Pittel and G. Sierra, Colloquium: exactly solvable Richardson-Gaudin models for many-body quantum systems, http://dx.doi.org/10.1103/RevModPhys.76.643

Web End =Rev. Mod. Phys. 76 (2004) 643 [http://arxiv.org/abs/nucl-th/0405011

Web End =nucl-th/0405011 ] [http://inspirehep.net/search?p=find+J+Rev.Mod.Phys.,76,643

Web End =INSPIRE ].

[10] J. Cao, W. Yang, K. Shi and Y. Wang, O -diagonal Bethe ansatz and exact solution of a topological spin ring, http://dx.doi.org/10.1103/PhysRevLett.111.137201

Web End =Phys. Rev. Lett. 111 (2013) 137201 [arXiv:1305.7328] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1305.7328

Web End =INSPIRE ].

[11] J. Cao, W.-L. Yang, K. Shi and Y. Wang, O -diagonal Bethe ansatz solution of the XXX spin-chain with arbitrary boundary conditions, http://dx.doi.org/10.1016/j.nuclphysb.2013.06.022

Web End =Nucl. Phys. B 875 (2013) 152 [arXiv:1306.1742] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.1742

Web End =INSPIRE ].

[12] J. Cao, W.-L. Yang, K.-J. Shi and Y. Wang, Spin-1/2 XYZ model revisit: general solutions via o -diagonal Bethe ansatz, arXiv:1307.0280 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1307.0280

Web End =INSPIRE ].

[13] J. Cao, W.-L. Yang, K. Shi and Y. Wang, O -diagonal Bethe ansatz solutions of the anisotropic spin-1/2 chains with arbitrary boundary elds, http://dx.doi.org/10.1016/j.nuclphysb.2013.10.001

Web End =Nucl. Phys. B 877 (2013) 152 [arXiv:1307.2023] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1307.2023

Web End =INSPIRE ].

[14] H. Bethe, On the theory of metals. 1. Eigenvalues and eigenfunctions for the linear atomic chain, http://dx.doi.org/10.1007/BF01341708

Web End =Z. Phys. 71 (1931) 205 [http://inspirehep.net/search?p=find+J+Z.Physik,71,205

Web End =INSPIRE ].

[15] F.C. Alcaraz, M.N. Barber, M.T. Batchelor, R.J. Baxter and G.R.W. Quispel, Surface exponents of the quantum XXZ, Ashkin-Teller and Potts models, http://dx.doi.org/10.1088/0305-4470/20/18/038

Web End =J. Phys. A 20 (1987) 6397 [ http://inspirehep.net/search?p=find+J+J.Phys.,A20,6397

Web End =INSPIRE ].

[16] N. Cramp and. Ragoucy, Generalized coordinate Bethe ansatz for non diagonal boundaries, http://dx.doi.org/10.1016/j.nuclphysb.2012.01.020

Web End =Nucl. Phys. B 858 (2012) 502 [arXiv:1105.0338] [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B858,502

Web End =INSPIRE ].

[17] R.J. Baxter, Eight-vertex model in lattice statistics, http://dx.doi.org/10.1103/PhysRevLett.26.832

Web End =Phys. Rev. Lett. 26 (1971) 832 [http://inspirehep.net/search?p=find+J+Phys.Rev.Lett.,26,832

Web End =INSPIRE ].

[18] R.J. Baxter, One-dimensional anisotropic Heisenberg chain, http://dx.doi.org/10.1103/PhysRevLett.26.834

Web End =Phys. Rev. Lett. 26 (1971) 834 [ http://inspirehep.net/search?p=find+J+Phys.Rev.Lett.,26,834

Web End =INSPIRE ].

24

JHEP04(2014)143

[19] R.J. Baxter, One-dimensional anisotropic Heisenberg chain, http://dx.doi.org/10.1016/0003-4916(72)90270-9

Web End =Annals Phys. 70 (1972) 323 [ http://inspirehep.net/search?p=find+J+Ann.Phys.,70,323

Web End =INSPIRE ].

[20] R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press, U.S.A. (1982).

[21] W.-L. Yang, R.I. Nepomechie and Y.-Z. Zhang, Q-operator and T-Q relation from the fusion hierarchy, http://dx.doi.org/10.1016/j.physletb.2005.12.022

Web End =Phys. Lett. B 633 (2006) 664 [http://arxiv.org/abs/hep-th/0511134

Web End =hep-th/0511134 ] [http://inspirehep.net/search?p=find+J+Phys.Lett.,B633,664

Web End =INSPIRE ].

[22] E.K. Sklyanin and L.D. Faddeev, Quantum mechanical approach to completely integrable eld theory models, Sov. Phys. Dokl. 23 (1978) 902 [Dokl. Akad. Nauk Ser. Fiz. 243 (1978) 1430] [http://inspirehep.net/search?p=find+J+Sov.Phys.Dokl.,23,902

Web End =INSPIRE ].

[23] L.A. Takhtajan and L.D. Faddeev, The quantum method of the inverse problem and the Heisenberg XYZ model, http://dx.doi.org/10.1070/RM1979v034n05ABEH003909

Web End =Russ. Math. Surveys 34 (1979) 11 [Usp. Mat. Nauk 34 (1979) 13] [http://inspirehep.net/search?p=find+J+Russ.Math.Surv.,34,11

Web End =INSPIRE ].

[24] V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation function, Cambridge Univ. Press, Cambridge U.K. (1993).

[25] E.K. Sklyanin, Boundary conditions for integrable quantum systems,http://dx.doi.org/10.1088/0305-4470/21/10/015

Web End =J. Phys. A 21 (1988) 2375 [http://inspirehep.net/search?p=find+J+J.Phys.,A21,2375

Web End =INSPIRE ].

[26] H. Fan, B.-Y. Hou, K.-J. Shi and Z.-X. Yang, Algebraic Bethe ansatz for eight vertex model with general open boundary conditions, http://dx.doi.org/10.1016/0550-3213(96)00398-7

Web End =Nucl. Phys. B 478 (1996) 723 [http://arxiv.org/abs/hep-th/9604016

Web End =hep-th/9604016 ] [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B478,723

Web End =INSPIRE ].

[27] S. Belliard, N. Crampe and E. Ragoucy, Algebraic Bethe ansatz for open XXX model with triangular boundary matrices, http://dx.doi.org/10.1007/s11005-012-0601-6

Web End =Lett. Math. Phys. 103 (2013) 493 [arXiv:1209.4269] [arXiv:1209.4269].

[28] N.Y. Reshetikhin, The functional equation method in the theory of exactly soluble quantum systems, Sov. Phys. JETP 57 (1983) 691.

[29] E.K. Sklyanin, The quantum Toda chain, http://dx.doi.org/10.1007/3-540-15213-X_80

Web End =Lect. Notes Phys. 226 (1985) 196 [http://inspirehep.net/search?p=find+J+LNPHA,226,196

Web End =INSPIRE ].

[30] E.K. Sklyanin, Separation of variables new trends,

http://dx.doi.org/10.1143/PTPS.118.35

Web End =Prog. Theor. Phys. Suppl. 118 (1995) 35 [http://arxiv.org/abs/solv-int/9504001

Web End =solv-int/9504001 ] [http://inspirehep.net/search?p=find+J+Prog.Theor.Phys.Suppl.,118,35

Web End =INSPIRE ].

[31] H. Frahm, A. Seel and T. Wirth, Separation of variables in the open XXX chain, http://dx.doi.org/10.1016/j.nuclphysb.2008.04.008

Web End =Nucl. Phys. B 802 (2008) 351 [arXiv:0803.1776] [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B802,351

Web End =INSPIRE ].

[32] G. Niccoli, Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: complete spectrum and form factors, http://dx.doi.org/10.1016/j.nuclphysb.2013.01.017

Web End =Nucl. Phys. B 870 (2013) 397 [arXiv:1205.4537] [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B870,397

Web End =INSPIRE ].

[33] G. Niccoli, Antiperiodic dynamical 6-vertex and periodic 8-vertex models I: complete spectrum by SOV and matrix elements of the identity on separate states,http://dx.doi.org/10.1088/1751-8113/46/7/075003

Web End =J. Phys. A 46 (2013) 075003 [arXiv:1207.1928] [http://inspirehep.net/search?p=find+J+J.Phys.,A46,075003

Web End =INSPIRE ].

[34] G. Niccoli, Non-diagonal open spin-1/2 XXZ quantum chains by separation of variables: complete spectrum and matrix elements of some quasi-local operators,http://dx.doi.org/10.1088/1742-5468/2012/10/P10025

Web End =J. Stat. Mech. 10 (2012) P10025 [arXiv:1206.0646] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1206.0646

Web End =INSPIRE ].

[35] G.E. Andrews, R.J. Baxter and P.J. Forrester, Eight vertex SOS model and generalized Rogers-Ramanujan type identities, http://dx.doi.org/10.1007/BF01014383

Web End =J. Stat. Phys. 35 (1984) 193 [http://inspirehep.net/search?p=find+J+J.Stat.Phys.,35,193

Web End =INSPIRE ].

[36] V.V. Bazhanov and N.Y. Reshetikhin, Critical RSOS models and conformal eld theory, http://dx.doi.org/10.1142/S0217751X89000042

Web End =Int. J. Mod. Phys. A 4 (1989) 115 [http://inspirehep.net/search?p=find+J+Int.J.Mod.Phys.,A4,115

Web End =INSPIRE ].

[37] R.I. Nepomechie, Bethe ansatz solution of the open XX spin chain with nondiagonal boundary terms, http://dx.doi.org/10.1088/0305-4470/34/47/306

Web End =J. Phys. A 34 (2001) 9993 [http://arxiv.org/abs/hep-th/0110081

Web End =hep-th/0110081 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0110081

Web End =INSPIRE ].

25

JHEP04(2014)143

[38] R.I. Nepomechie, Solving the open XXZ spin chain with nondiagonal boundary terms at roots of unity, http://dx.doi.org/10.1016/S0550-3213(01)00585-5

Web End =Nucl. Phys. B 622 (2002) 615 [http://arxiv.org/abs/hep-th/0110116

Web End =hep-th/0110116 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0110116

Web End =INSPIRE ].

[39] R.I. Nepomechie, Functional relations and Bethe ansatz for the XXZ chain,http://dx.doi.org/10.1023/A:1023016602955

Web End =J. Stat. Phys. 111 (2003) 1363 [http://arxiv.org/abs/hep-th/0211001

Web End =hep-th/0211001 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0211001

Web End =INSPIRE ].

[40] R.I. Nepomechie, Bethe ansatz solution of the open XXZ chain with nondiagonal boundary terms, http://dx.doi.org/10.1088/0305-4470/37/2/012

Web End =J. Phys. A 37 (2004) 433 [http://arxiv.org/abs/hep-th/0304092

Web End =hep-th/0304092 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0304092

Web End =INSPIRE ].

[41] J. Cao, H.-Q. Lin, K.-J. Shi and Y. Wang, Exact solutions and elementary excitations in the XXZ spin chain with unparallel boundary elds, http://dx.doi.org/10.1016/S0550-3213(03)00372-9

Web End =Nucl. Phys. B 663 (2003) 487 [http://arxiv.org/abs/cond-mat/0212163

Web End =cond-mat/0212163 ] [http://arxiv.org/abs/cond-mat/0212163

Web End =cond-mat/0212163 ].

[42] W.-L. Yang, Y.-Z. Zhang and M.D. Gould, Exact solution of the XXZ Gaudin model with generic open boundaries, http://dx.doi.org/10.1016/j.nuclphysb.2004.07.029

Web End =Nucl. Phys. B 698 (2004) 503 [http://arxiv.org/abs/hep-th/0411048

Web End =hep-th/0411048 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0411048

Web End =INSPIRE ].

[43] J. de Gier and P. Pyatov, Bethe ansatz for the Temperley-Lieb loop model with open boundaries, http://dx.doi.org/10.1088/1742-5468/2004/03/P002

Web End =J. Stat. Mech. 03 (2004) P03002 [http://arxiv.org/abs/hep-th/0312235

Web End =hep-th/0312235 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0312235

Web End =INSPIRE ].

[44] A. Nichols, V. Rittenberg and J. de Gier, One-boundary Temperley-Lieb algebras in the XXZ and loop models, http://dx.doi.org/10.1088/1742-5468/2005/03/P03003

Web End =J. Stat. Mech. 03 (2005) P03003 [http://arxiv.org/abs/cond-mat/0411512

Web End =cond-mat/0411512 ] [http://inspirehep.net/search?p=find+EPRINT+cond-mat/0411512

Web End =INSPIRE ].

[45] J. de Gier, A. Nichols, P. Pyatov and V. Rittenberg, Magic in the spectra of the XXZ quantum chain with boundaries at = 0 and = 1/2, http://dx.doi.org/10.1016/j.nuclphysb.2005.09.005

Web End =Nucl. Phys. B 729 (2005) 387 [http://arxiv.org/abs/hep-th/0505062

Web End =hep-th/0505062 ] [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B729,387

Web End =INSPIRE ].

[46] J. de Gier and F.H.L. Essler, Bethe ansatz solution of the asymmetric exclusion process with open boundaries, http://dx.doi.org/10.1103/PhysRevLett.95.240601

Web End =Phys. Rev. Lett. 95 (2005) 240601 [http://arxiv.org/abs/cond-mat/0508707

Web End =cond-mat/0508707 ].

[47] J. de Gier and F.H.L. Essler, Exact spectral gaps of the asymmetric exclusion process with open boundaries, http://dx.doi.org/10.1088/1742-5468/2006/12/P12011

Web End =J. Stat. Mech. 12 (2006) 11 [http://arxiv.org/abs/cond-mat/0609645

Web End =cond-mat/0609645 ].

[48] A. Doikou and P.P. Martin, On quantum group symmetry and Bethe ansatz for the asymmetric twin spin chain with integrable boundary, http://dx.doi.org/10.1088/1742-5468/2006/06/P06004

Web End =J. Stat. Mech. 06 (2006) P06004 [http://arxiv.org/abs/hep-th/0503019

Web End =hep-th/0503019 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0503019

Web End =INSPIRE ].

[49] A. Doikou, The open XXZ and associated models at q root of unity,http://dx.doi.org/10.1088/1742-5468/2006/05/P05010

Web End =J. Stat. Mech. 05 (2006) P05010 [http://arxiv.org/abs/hep-th/0603112

Web End =hep-th/0603112 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0603112

Web End =INSPIRE ].

[50] Z. Bajnok, Equivalences between spin models induced by defects,http://dx.doi.org/10.1088/1742-5468/2006/06/P06010

Web End =J. Stat. Mech. 06 (2006) P06010 [http://arxiv.org/abs/hep-th/0601107

Web End =hep-th/0601107 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0601107

Web End =INSPIRE ].

[51] P. Baseilhac and K. Koizumi, Exact spectrum of the XXZ open spin chain from the q-Onsager algebra representation theory, http://dx.doi.org/10.1088/1742-5468/2007/09/P09006

Web End =J. Stat. Mech. 09 (2007) P09006 [http://arxiv.org/abs/hep-th/0703106

Web End =hep-th/0703106 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0703106

Web End =INSPIRE ].

[52] W. Galleas, Functional relations from the Yang-Baxter algebra: eigenvalues of the XXZ model with non-diagonal twisted and open boundary conditions,http://dx.doi.org/10.1016/j.nuclphysb.2007.09.011

Web End =Nucl. Phys. B 790 (2008) 524 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B790,524

Web End =INSPIRE ].

[53] H. Frahm, J.H. Grelik, A. Seel and T. Wirth, Functional Bethe ansatz methods for the open XXX chain, http://dx.doi.org/10.1088/1751-8113/44/1/015001

Web End =J. Phys. A 44 (2011) 015001 [arXiv:1009.1081] [http://inspirehep.net/search?p=find+J+J.Phys.,A44,015001

Web End =INSPIRE ].

[54] S. Niekamp, T. Wirth and H. Frahm, The XXZ model with anti-periodic twisted boundary conditions, http://dx.doi.org/10.1088/1751-8113/42/19/195008

Web End =J. Phys. A 42 (2009) 195008 [arXiv:0902.1079] [http://inspirehep.net/search?p=find+J+J.Phys.,A42,195008

Web End =INSPIRE ].

[55] A.M. Grabinski and H. Frahm, Non-diagonal boundary conditions for gl(1|1) super spin chains, http://dx.doi.org/10.1088/1751-8113/43/4/045207

Web End =J. Phys. A 43 (2010) 045207 [arXiv:0910.4029].

26

JHEP04(2014)143

[56] Y.-Y. Li, J. Cao, W.-L. Yang, K. Shi and Y. Wang, Exact solution of the one-dimensional Hubbard model with arbitrary boundary magnetic elds, http://dx.doi.org/10.1016/j.nuclphysb.2013.12.004

Web End =Nucl. Phys. B 879 (2014) 98 [arXiv:1311.0432] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1311.0432

Web End =INSPIRE ].

[57] X. Zhang, J. Cao, W.-L. Yang, K. Shi and Y. Wang, Exact solution of the one-dimensional super-symmetric t-J model with unparallel boundary elds, arXiv:1312.0376 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1312.0376

Web End =INSPIRE ].

[58] R.I. Nepomechie, An inhomogeneous T-Q equation for the open XXX chain with general boundary terms: completeness and arbitrary spin, http://dx.doi.org/10.1088/1751-8113/46/44/442002

Web End =J. Phys. A 46 (2013) 442002 [arXiv:1307.5049] [http://inspirehep.net/search?p=find+J+J.Phys.,A46,442002

Web End =INSPIRE ].

[59] S. Belliard and N. Cramp, Heisenberg XXX model with general boundaries: eigenvectors from algebraic Bethe ansatz, http://dx.doi.org/10.3842/SIGMA.2013.072

Web End =SIGMA 9 (2013) 072 [arXiv:1309.6165] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1309.6165

Web End =INSPIRE ].

[60] B. Sutherland, A general model for multicomponent quantum systems, http://dx.doi.org/10.1103/PhysRevB.12.3795

Web End =Phys. Rev. B 12 (1975) 3795 [http://inspirehep.net/search?p=find+J+Phys.Rev.,B12,3795

Web End =INSPIRE ].

[61] P. Schlottmann, Integrable narrow-band model with possible relevance to heavy-fermion systems, http://dx.doi.org/10.1103/PhysRevB.36.5177

Web End =Phys. Rev. B 36 (1987) 5177 [http://inspirehep.net/search?p=find+J+Phys.Rev.,B36,5177

Web End =INSPIRE ].

[62] H.J. de Vega and E. Lopes, Exact solution of the Perk-Schultz model, http://dx.doi.org/10.1103/PhysRevLett.67.489

Web End =Phys. Rev. Lett. 67 (1991) 489 [http://inspirehep.net/search?p=find+J+Phys.Rev.Lett.,67,489

Web End =INSPIRE ].

[63] E. Lopes, Exact solution of the multicomponent generalized six vertex model, http://dx.doi.org/10.1016/0550-3213(92)90426-C

Web End =Nucl. Phys. B 370 (1992) 636 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B370,636

Web End =INSPIRE ].

[64] I. Krichever, O. Lipan, P. Wiegmann and A. Zabrodin, Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations,http://dx.doi.org/10.1007/s002200050165

Web End =Commun. Math. Phys. 188 (1997) 267 [http://arxiv.org/abs/hep-th/9604080

Web End =hep-th/9604080 ] [http://inspirehep.net/search?p=find+J+Comm.Math.Phys.,188,267

Web End =INSPIRE ].

[65] M. Karowski, On the bound state problem in (1 + 1)-dimensional eld theories, http://dx.doi.org/10.1016/0550-3213(79)90600-X

Web End =Nucl. Phys. B 153 (1979) 244 [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B153,244

Web End =INSPIRE ].

[66] P.P. Kulish, N.Y. Reshetikhin and E.K. Sklyanin, Yang-Baxter equation and representation theory. 1, http://dx.doi.org/10.1007/BF02285311

Web End =Lett. Math. Phys. 5 (1981) 393 [http://inspirehep.net/search?p=find+J+Lett.Math.Phys.,5,393

Web End =INSPIRE ].

[67] P.P. Kulish and E.K. Sklyanin, Quantum spectral transform method: recent developments, http://dx.doi.org/10.1007/3-540-11190-5_8

Web End =Lect. Notes Phys. 151 (1982) 61 [http://inspirehep.net/search?p=find+J+LNPHA,151,61

Web End =INSPIRE ].

[68] A.N. Kirillov and N.Y. Reshetikhin, Exact solution of the Heisenberg XXZ model of spins,http://dx.doi.org/10.1007/BF01083768

Web End =J. Sov. Math. 35 (1986) 2627 .

[69] A.N. Kirillov and N.Y. Reshetikhin, Exact solution of the integrable XXZ Heisenberg model with arbitrary spin. I. The ground state and the excitation spectrum,http://dx.doi.org/10.1088/0305-4470/20/6/038

Web End =J. Phys. A 20 (1987) 1565 [http://inspirehep.net/search?p=find+J+J.Phys.,A20,1565

Web End =INSPIRE ].

[70] H. Frahm and N.A. Slavnov, New solutions to the reection equation and the projecting method, http://dx.doi.org/10.1088/0305-4470/32/9/005

Web End =J. Phys. A 32 (1999) 1547 [http://arxiv.org/abs/cond-mat/9810312

Web End =cond-mat/9810312 ].

[71] W. Galleas and M.J. Martins, Solution of the SU(N) vertex model with non-diagonal open boundaries, http://dx.doi.org/10.1016/j.physleta.2004.12.017

Web End =Phys. Lett. A 335 (2005) 167 [http://arxiv.org/abs/nlin/0407027

Web End =nlin/0407027 ] [http://inspirehep.net/search?p=find+EPRINT+NLIN/0407027

Web End =INSPIRE ].

[72] H.J. de Vega and A. Gonzlez-Ruiz, Boundary K matrices for the XYZ, XXZ and XXX spin chains, http://dx.doi.org/10.1088/0305-4470/27/18/021

Web End =J. Phys. A 27 (1994) 6129 [http://arxiv.org/abs/hep-th/9306089

Web End =hep-th/9306089 ] [http://inspirehep.net/search?p=find+J+J.Phys.,A27,6129

Web End =INSPIRE ].

[73] P.P. Kulish, Yang-Baxter equation and reection equations in integrable models, http://arxiv.org/abs/hep-th/9507070

Web End =hep-th/9507070 [http://inspirehep.net/search?p=find+EPRINT+hep-th/9507070

Web End =INSPIRE ].

[74] M. Mintchev,. Ragoucy and P. Sorba, Spontaneous symmetry breaking in the gl(N)-NLS hierarchy on the half line, http://dx.doi.org/10.1088/0305-4470/34/40/311

Web End =J. Phys. A 34 (2001) 8345 [http://arxiv.org/abs/hep-th/0104079

Web End =hep-th/0104079 ] [http://inspirehep.net/search?p=find+J+J.Phys.,A34,8345

Web End =INSPIRE ].

27

JHEP04(2014)143

[75] L. Mezincescu and R.I. Nepomechie, Fusion procedure for open chains,http://dx.doi.org/10.1088/0305-4470/25/9/024

Web End =J. Phys. A 25 (1992) 2533 [http://inspirehep.net/search?p=find+J+J.Phys.,A25,2533

Web End =INSPIRE ].

[76] Y.-K. Zhou, Row transfer matrix functional relations for Baxters eight vertex and six vertex models with open boundaries via more general reection matrices,http://dx.doi.org/10.1016/0550-3213(95)00553-6

Web End =Nucl. Phys. B 458 (1996) 504 [http://arxiv.org/abs/hep-th/9510095

Web End =hep-th/9510095 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9510095

Web End =INSPIRE ].

[77] W.L. Yang and R. Sasaki, Solution of the dual reection equation for A(1)n1 SOS model,

http://dx.doi.org/10.1063/1.1795972

Web End =J. Math. Phys. 45 (2004) 4301 [http://arxiv.org/abs/hep-th/0308118

Web End =hep-th/0308118 ] [http://inspirehep.net/search?p=find+J+J.Math.Phys.,45,4301

Web End =INSPIRE ].

[78] W.L. Yang and R. Sasaki, Exact solution of Zn Belavin model with open boundary condition, http://dx.doi.org/10.1016/j.nuclphysb.2003.11.039

Web End =Nucl. Phys. B 679 (2004) 495 [http://arxiv.org/abs/hep-th/0308127

Web End =hep-th/0308127 ] [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B679,495

Web End =INSPIRE ].

[79] W.-L. Yang and Y.-Z. Zhang, Non-diagonal solutions of the reection equation for the trigonometric A(1)n1 vertex model, http://dx.doi.org/10.1088/1126-6708/2004/12/019

Web End =JHEP 12 (2004) 019 [http://arxiv.org/abs/hep-th/0411160

Web End =hep-th/0411160 ] [http://inspirehep.net/search?p=find+J+JHEP,0412,019

Web End =INSPIRE ].

[80] W.-L. Yang and Y.-Z. Zhang, Exact solution of the A(1)n1 trigonometric vertex model with non-diagonal open boundaries, http://dx.doi.org/10.1088/1126-6708/2005/01/021

Web End =JHEP 01 (2005) 021 [http://arxiv.org/abs/hep-th/0411190

Web End =hep-th/0411190 ] [http://inspirehep.net/search?p=find+J+JHEP,0501,021

Web End =INSPIRE ].

[81] R.I. Nepomechie, Nested algebraic Bethe ansatz for open GL(N) spin chains with projected K-matrices, http://dx.doi.org/10.1016/j.nuclphysb.2010.01.006

Web End =Nucl. Phys. B 831 (2010) 429 [arXiv:0911.5494] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0911.5494

Web End =INSPIRE ].

[82] C.S. Melo, G.A.P. Ribeiro and M.J. Martins, Bethe ansatz for the XXX-S chain with non-diagonal open boundaries, http://dx.doi.org/10.1016/j.nuclphysb.2004.12.008

Web End =Nucl. Phys. B 711 (2005) 565 [http://arxiv.org/abs/nlin/0411038

Web End =nlin/0411038 ].

[83] A. Doikou, Fusion and analytical Bethe ansatz for the A(1)N1 open spin chain,http://dx.doi.org/10.1088/0305-4470/33/26/303

Web End =J. Phys. A 33 (2000) 4755 [http://arxiv.org/abs/hep-th/0006081

Web End =hep-th/0006081 ] [http://inspirehep.net/search?p=find+J+J.Phys.,A33,4755

Web End =INSPIRE ].

[84] A. Doikou and R.I. Nepomechie, Duality and quantum algebra symmetry of the A(1)N1 open spin chain with diagonal boundary elds, http://dx.doi.org/10.1016/S0550-3213(98)00567-7

Web End =Nucl. Phys. B 530 (1998) 641 [http://arxiv.org/abs/hep-th/9807065

Web End =hep-th/9807065 ]

[http://inspirehep.net/search?p=find+J+Nucl.Phys.,B530,641

Web End =INSPIRE ].

[85] A. Doikou and R.I. Nepomechie, Bulk and boundary S matrices for the SU(N) chain, http://dx.doi.org/10.1016/S0550-3213(98)00239-9

Web End =Nucl. Phys. B 521 (1998) 547 [http://arxiv.org/abs/hep-th/9803118

Web End =hep-th/9803118 ] [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B521,547

Web End =INSPIRE ].

[86] H.J. de Vega and A. Gonzlez-Ruiz, Exact solution of the SUq(n) invariant quantum spin chains, http://dx.doi.org/10.1016/0550-3213(94)90484-7

Web End =Nucl. Phys. B 417 (1994) 553 [http://arxiv.org/abs/hep-th/9309022

Web End =hep-th/9309022 ] [http://inspirehep.net/search?p=find+J+Nucl.Phys.,B417,553

Web End =INSPIRE ].

[87] H.J. de Vega and A. Gonzlez-Ruiz, Exact Bethe ansatz solution for An1 chains with non-SUq(n) invariant open boundary conditions, http://dx.doi.org/10.1142/S0217732394002069

Web End =Mod. Phys. Lett. A 09 (1994) 2207 [http://arxiv.org/abs/hep-th/9404141

Web End =hep-th/9404141 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/9404141

Web End =INSPIRE ].

[88] L. Mezincescu, R.I. Nepomechie and V. Rittenberg, Bethe ansatz solution of the Fateev-Zamolodchikov quantum spin chain with boundary terms, http://dx.doi.org/10.1016/0375-9601(90)90016-H

Web End =Phys. Lett. A 147 (1990) 70 [ http://inspirehep.net/search?p=find+J+Phys.Lett.,A147,70

Web End =INSPIRE ].

28

JHEP04(2014)143