The algebraic Bethe ansatz for the Izergin–Korepin model with open boundary conditions

Nuclear Physics B 670 [FS] (2003) 401–438 www.elsevier.com/locate/npe

The algebraic Bethe ansatz for the Izergin–Korepin model with open boundary conditions Guang-Liang Li Department of Applied Physics, Xi’an Jiaotong University, Xi’an 710049, China

Kang-Jie Shi, Rui-Hong Yue Institute of Modern Physics, Northwest University, Xi’an 710069, China Received 25 March 2003; received in revised form 2 June 2003; accepted 1 August 2003

Abstract We present the procedure of exactly solving the Izergin–Korepin model with open boundary conditions by using the algebraic Bethe ansatz, which include constructing the multi-particle state and achieving the eigenvalue of the transfer matrix and corresponding Bethe equations. We give a proof about our conclusions on the multi-particle state based on an assumption. When the model is Uq (su(2)) quantum invariant, our results agree with that obtained by analytic Bethe ansatz method.  2003 Elsevier B.V. All rights reserved. PACS: 75.10.J; 05.20; 05.30 Keywords: Izergin–Korepin model; Algebraic Bethe ansatz; Open boundary

1. Introduction The Bethe ansatz solutions to a integrable model will make it possible to study the thermodynamic properties of the model, such as correlation functions [1], specific heat, magnetic susceptibility [2,3], and finite size effects [4–6]. At same time, in the procedure of solving the models, it will also help us to better understand their underlying mathematical structures. E-mail address: [email protected] (G.-L. Li). 0550-3213/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2003.08.001

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G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

There are several ways to solve a model. One of the powerful mathematical tool is the quantum inverse scattering method (QISM) [7–11], namely, the algebraic Bethe ansatz method. Besides describing the spectra of quantum integrable systems, the Bethe ansatz is also used to construct exact and manageable expressions for correlation functions [1]. The QISM was first developed for the system with periodic boundary conditions. It was later generalized to systems with open boundary conditions [11]. Since Sklyanin’s work [11], many works have been done on the integrable models with open boundary conditions [12–22]. Solving a model with open boundary conditions will enable us to investigate the boundary effect of the system. It may also be helpful for us to study the impurities problems which have attracted considerable interest recently [23–34]. The algebraic Bethe ansatz have been applied to the six-vertex type model successfully. It is also have been used in other type models with periodic boundary conditions [35– 40] or with open boundary conditions [41,42], such as Izergin–Korepin (IK) model [43]. The R matrix of the IK model is the simplest example of an R matrix of the twisted type and the model with open boundary conditions has the important physical applications. For example, it can be related to the loop models [21] and self-avoiding walks at a boundary [44]. The IK model was solved by using other methods under the periodic boundary [40,45] or open boundary conditions [21,46–48]. Employing the algebraic Bethe ansatz method, for the IK model with periodic boundary conditions, Tarasov proposed the construction of two-particle state for the model and argued that his conclusion can be generalized to any n-particle state [35]. Relying on the previous work by Tarasov and Martins [35,36], Fan solved the IK model with open boundary conditions [41]. In Fan’s work, he give the expression of two-particle state and a conjecture for the nparticle state. He also checked the Bethe equation at the case n = 1. For the IK model, it is interesting to find the explicit expression of n-particle state and verify the Bethe equations in the case n
2, which will also help us to learn about the feasibility of applying the algebraic Bethe ansatz method to the model. In this paper, our aim is to do this work. The paper is organized as following. In Section 2 we introduce Izergin–Korepin model and the diagonal K± matrices of reflection equation which determines the nontrivial boundary terms in the Hamiltonian. In Section 3, by direct calculation, we present our results for the one-particle state, two-particle state and three-particle state and generalize our results to the case of n-particle state. We then prove our conclusions on n-particle state based on an assumption. When the model is Uq (su(2)) quantum invariant, our conclusions recover that obtained by analytic Bethe ansatz method [46]. For the nonquantum-group invariant cases, there are a little difference between our results and that proposed by Yung and Batchelor in Ref. [21]. The summary and some discussions of our main results are included in Section 4. In the appendix, some necessary relations and proofs are provided.

2. The vertex model and integrable boundary conditions The R matrix for the Izergin–Korepin model [43] is

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438



c 0 0 b  0 0   0 e¯  R(u) =  0 0  0 0  0 0  0 0 0 0

0 0 d 0 g¯ 0 f¯ 0 0

0 e 0 b 0 0 0 0 0

0 0 g 0 a 0 g¯ 0 0

0 0 0 0 0 b 0 e¯ 0

0 0 f 0 g 0 d 0 0

0 0 0 0 0 e 0 b 0

 0 0  0  0  0  0  0  0 c

403

(1)

with a(u) = sinh(u − 3q) − sinh(5q) + sinh(3q) + sinh(q), b(u) = sinh(u − 3q) + sinh(3q), c(u) = sinh(u − 5q) + sinh(q), d(u) = sinh(u − q) + sinh(q), e(u) = −2e−u/2 sinh(2q) cosh(u/2 − 3q), e(u) ¯ = −2eu/2 sinh(2q) cosh(u/2 − 3q), f (u) = −2e−u+2q sinh(q) sinh(2q) − e−q sinh(4q), f¯(u) = 2eu−2q sinh(q) sinh(2q) − eq sinh(4q), g(u) = 2e−u/2+2q sinh(u/2) sinh(2q), g(u) ¯ = −2eu/2−2q sinh(u/2) sinh(2q).

(2)

The R matrix satisfies the following properties regularity: unitarity: PT-symmetry:

R12 (0) = ρ(0)1/2P12 , t t

1 2 (−u) = ρ(u), R12 (u)R12

t1 t2 P12 R12 (u)P12 = R12 (u), 1

crossing-symmetry:

1

t2 R12 (u) = V R12 (−u − η)V −1 .

(3)

Here P is the exchange operator defined by P(x ⊗ y) = y ⊗ x, ti denotes transposition 1

2

in the space i, V = V ⊗ 1, V = 1 ⊗ V , η is the crossing parameter and V determines √ the crossing matrix M ≡ V t V = M t with η = −6q − −1π and M = diag(e2q , 1, e−2q ), ρ(u) = ((sinh(q) − sinh(5q + u))(sinh(q) − sinh(5q − u))). The R matrix also fulfil Yang–Baxter equation (YBE) [49] R12 (u − v)R13 (u)R23 (v) = R23 (v)R13 (u)R12 (u − v),

(4)

where R12 (u), R13 (u) and R23 (u) act on C 3 ⊗C 3 ⊗C 3 , with R12 (u) = R(u)⊗1, R23 (u) = 1 ⊗ R(u), etc.

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For an N × N square lattice, if we can find K± (u) which satisfy the following reflection equations [11,50] 1

2

t1 t2 (u + v)K − (v) R12 (u − v)K − (u)R12 2

1

t t

1 2 = K − (v)R12 (u + v)K − (u)R12 (u − v),

1

1

(5) 1 2

t1 t2 (−u − v − 2η)M K t+2 (v) R12 (−u + v)K t+1 (u)M −1 R12 2

1

1

1

t1 t2 (−u + v), = K t+2 (v)MR12 (−u − v − 2η)M −1 K t+1 (u)R12

(6)

where Eq. (5) is called reflection equation and Eq. (6) is called dual reflection equation, 1

2

K ± (u) = K± (u) ⊗ 1, K ± (u) = 1 ⊗ K± (u), then the transfer matrix t (u) defined as t (u) = tr K+ (u)U (u)

(7)

can constitute a one-parameter commutative family [t (u), t (v)] = 0. Here U (u) = T (u)K− (u)T −1 (−u),

(8)

T (u) = R01 (u)R02 · · · R0N (u),

(9)

the space V0 is usually called the auxiliary space, the space V1 ⊗ V2 · · · ⊗ VN is called the quantum space. The corresponding integrable open chain Hamiltonian takes the form H=

N−1  k=1

0

11 tr K + (0)HN,0 , Hk,k+1 + K− (0) + 2 tr K+ (0)

(10)

 where Hk,k+1 = Pk,k+1 Rkk+1 (u)|u=0 . From Eqs. (5) and (6), we can see that, given a solution K− (u) of Eq. (5), the matrix t (−u − η)M K+ (u) = K−

(11)

satisfies Eq. (6). The general solution to Eq. (5) of IK model have been obtained in Ref. [51], here we will choose the diagonal ones. Denote  K− (u) = diag K− (u)1 , K− (u)2 , K− (u)3 ,  K+ (u) = diag K+ (u)1 , K+ (u)2 , K+ (u)3 , by Eq. (11), we have (i): K− (u) = 1, (ii):

K+ (u) = M,

 K− (u) = diag e−u c cosh(q) + sinh(u − 2q) , c cosh(q + u) − sinh(2q),   eu c cosh(q) + sinh(u − 2q) ,

(12)

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405

 K+ (u) = diag eu−4q c cosh(q) + sinh(u − 4q) , c cosh(u − 7q) + sinh(2q),   (13) e−u+4q c cosh(q) + sinh(u − 4q) with c2 = c 2 = −1. In case (ii), we omit a factor −1 in K+ (u) for simplicity.

3. The algebraic Bethe ansatz 3.1. The vacuum state and commutation relations Firstly, we write the double-monodromy matrix as

A(u) B1 (u) F (u) U (u) = C1 (u) D1 (u) B2 (u) . G(u) C2 (u) D2 (u)

(14)

With the help of Eqs. (4), (5), we can prove that (14) also satisfy the reflecting equation (5) 1

2

2

1

t1 t2 t1 t2 R12 (u − v)U (u)R12 (u + v)U (v) = U (v)R12 (u + v)U (u)R12 (u − v), 1

(15)

2

where U (u) = U (u) ⊗ 1, U (u) = 1 ⊗ U (u). Now we introduce the vacuum state, |0 =

⊗N 

(1, 0, 0)t ,

(16)

where t denotes the transposition. Acting the double-row monodromy matrix (14) on the vacuum state, we can find Ci (u)|0 = 0,

Bi (u)|0 = 0 (i = 1, 2)

G(u)|0 = 0,

F (u)|0 = 0.

(17)

Considering the definition of U (u) Eq. (8), we have A(u)|0 = T (u)11 K− (u)1 T −1 (−u)11 |0 + T (u)12 K− (u)2 T −1 (−u)21 |0 + T (u)13 K − (u)3 T −1 (−u)31 |0,

(18)

D1 (u)|0 = T (u)21 K− (u)1 T −1 (−u)12 |0 + T (u)22 K− (u)2 T −1 (−u)22 |0 + T (u)23 K − (u)2 T −1 (−u)32 |0, D2 (u)|0 = T (u)31 K− (u)1 T −1 (−u)13 |0 + T (u)32 K− (u)2 T −1 (−u)23 |0 + T (u)33 K − (u)3 T −1 (−u)33 |0.

(19)

(20)

In above equations, the first term of Eq. (19) and the previous two terms of Eq. (20) cannot be calculated directly but it can be worked out by using the following method. Taking v = −u in the Yang–Baxter equation, we can get T2−1 (−u)R12 (2u)T1 (u) = T1 (u)R12 (2u)T2−1 (−u).

(21)

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Taking special indices in this relation and applying both sides of this relation to the vacuum state, we find: T (u)21 T −1 (−u)12 |0 = T (u)32 T −1 (−u)23 |0 =

 −1 e(2u) ¯ T (−u)11 T (u)11 − T (u)22 T −1 (−u)22 |0, c(2u) e(2u)c(2u) ¯ − f¯(2u)e(2u)

a(2u)c(2u) − e(2u)e(2u) ¯  × T −1 (−u)22 T (u)22 − T (u)33 T −1 (−u)33 |0,  ¯ f (2u) −1 T (−u)11 T (u)11 T (u)31 T −1 (−u)13 |0 = c(2u) a(2u)f¯(2u) − e(2u) ¯ e(2u) ¯ T (u)33 T −1 (−u)33 − a(2u)c(2u) − e(2u)e(2u) ¯  e(2u)c(2u) ¯ − f¯(2u)e(2u) −1 e(2u) ¯ 2 2 ( )T (−u)2 T (u)2 |0. − c(2u) a(2u)c(2u) − e(2u)e(2u) ¯ (22) After defining two new operators 1 (u) = D1 (u) − f1 (u)A(u), D 2 (u) = D2 (u) − f2 (u)A(u) − f3 (u)D 1 (u), D

(23) (24)

and substituting Eq. (22) to Eqs. (18)–(20), we obtain A(u)|0 = K− (u)1 c(u)2N ρ(u)−N |0 = ω1 (u)|0,  1 (u)|0 = K− (u)2 − f1 (u)K− (u)1 b(u)2N ρ(u)−N |0 = ω2 (u)|0, D  2 (u)|0 = K− (u)3 − f3 (u)K− (u)2 − f4 (u)K− (u)1 d(u)2N ρ(u)−N |0 D = ω3 (u)|0,

(25) (26) (27)

where f1 (u) =

e¯(2u) , c(2u)

f2 (u) =

f¯(2u) , c(2u)

e¯(2u)c(2u) − f¯(2u)e(2u) , a(2u)c(2u) − e¯(2u)e(2u) a(2u)f¯(2u) − e(2u) ¯ e(2u) ¯ f4 (u) = . a(2u)c(2u) − e(2u)e(2u) ¯ f3 (u) =

Rewriting Eq. (15) in component form R12 (u− )ac11ca22 U (u)cd11 R21 (u+ )db11 cd22 U (v)db22 = U (v)ac22 R12 (u+ )ac11dc22 U (u)cd11 R21 (u− )db11 db22 ,

(28)

where the repeated indices sum over 1 to 3, u− = u − v, u+ = u + v, we can obtain the following fundamental commutation relations

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

407

g(u− )b(2v) g(u+ ) 1 (v) F (u)A(v) + F (u)D d(u− )c(2v) b(u+ )   cosh( u2− + q) g(u+ ) g(−u− )b(2u)  F (v)A(u) + F (v) D = (v)B (u) − (u) , B 1 1 1 d(−u− )c(2u) b(u+ ) cosh( u2− − q) (29) 1 1 1  A(u)B1 (v) = a1 (u, v)B1 (v)A(u) + a2 (u, v)B1 (u)A(v) + a3 (u, v)B1 (u)D1 (v)

B1 (u)B1 (v) −

+ a41 (u, v)F (u)C1 (v) + a51 (u, v)F (u)C2 (v) + a61 (u, v)F (v)C1 (u), (30) 1 (u)B1 (v) = a12 (u, v)B1 (v)D 1 (u) + a22 (u, v)B1 (u)A(v) + a32 (u, v)B1 (u)D 1 (v) D 1 (v) + a62 (u, v)F (u)C1 (v) + a42 (u, v)B2 (u)A(v) + a52 (u, v)B2 (u)D + a72 (u, v)F (u)C2 (v) + a82 (u, v)F (v)C1 (u) + a92 (u, v)F (v)C2 (u), (31) 2 (u) + a23 (u, v)B1 (u)A(v) + a33 (u, v)B1 (u)D 1 (v) 2 (u)B1 (v) = a13 (u, v)B1 (v)D D 1 (v) + a 3 (u, v)F (u)C1 (v) + a 3 (u, v)B2 (u)A(v) + a 3 (u, v)B2 (u)D 4

5

6

+ a73 (u, v)F (u)C2 (v) + a83 (u, v)F (v)C1 (u) + a93 (u, v)F (v)C2 (u), (32) 1 1 1  A(u)F (v) = b1 (u, v)F (v)A(u) + b2 (u, v)F (u)A(v) + b3 (u, v)F (u)D1 (v) 2 (v) + b51 (u, v)B1 (u)B1 (v) + b61 (u, v)B1 (u)B2 (v), + b41 (u, v)F (u)D (33) 1 (u)F (v) = b12 (u, v)F (v)D 1 (u) + b22(u, v)F (u)A(v) + b32 (u, v)F (u)D 1 (v) D 2 (v) + b52(u, v)B1 (u)B1 (v) + b62 (u, v)B1 (u)B2 (v) + b42 (u, v)F (u)D + b72 (u, v)B2 (u)B1 (v) + b82 (u, v)B2 (u)B2 (v),

(34)

2 (u)F (v) = b3 (u, v)F (v)D 2 (u) + b3(u, v)F (u)A(v) + b3 (u, v)F (u)D 1 (v) D 1 2 3 3 3 3 2 (v) + b (u, v)B1 (u)B1 (v) + b (u, v)B1 (u)B2 (v) + b4 (u, v)F (u)D 5 6 3 (35) + b7 (u, v)B2 (u)B1 (v) + b83 (u, v)B2 (u)B2 (v), C1 (u)B1 (v) = c11 (u, v)B1 (v)C1 (u) + c21 (u, v)B1 (v)C2 (u) + c31 (u, v)B1 (u)C2 (u) + c41 (u, v)B2 (u)C2 (v) + c51 (u, v)F (v)G(u) + c61 (u, v)A(v)A(u) 1 (u) + c81 (u, v)A(u)A(v) + c91 (u, v)A(u)D 1 (v) + c71 (u, v)A(v)D 1 1 1 (u)A(v) + c11 1 (u)D 1 (v), + c10 (u, v)D (u, v)D

(36)

C2 (u)B1 (v) = c12 (u, v)B1 (v)C1 (u) + c22 (u, v)B1 (v)C2 (u) + c32 (u, v)B1 (u)C2 (u) + c42 (u, v)B2 (u)C2 (v) + c52 (u, v)F (v)G(u) + c62 (u, v)A(v)A(u) 1 (u) + c82 (u, v)A(v)D 2 (u) + c92 (u, v)A(u)A(v) + c72 (u, v)A(v)D 2 2 1 (v) + c11 1 (u)A(v) + c10 (u, v)A(u)D (u, v)D 2 2 1 (u)D 1 (v) + c13 (u, v)D 2 (u)A(v) + c12 (u, v)D 2 2 (u)D 1 (v), + c14 (u, v)D

(37)

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B1 (u)B2 (v) = d11 (u, v)B2 (v)B1 (u) + d21 (u, v)B1 (v)B1 (u) + d31 (u, v)F (v)A(u) 1 (u) + d51 (u, v)F (u)A(v) + d61 (u, v)F (u)D 1 (v) + d41 (u, v)F (v)D 2 (v), + d71 (u, v)F (u)D

(38)

F (u)B1 (v) = e11 (u, v)B1 (v)F (u) + e21 (u, v)F (v)B1 (u) + e31 (u, v)F (v)B2 (u), B1 (u)F (v) = e12 (u, v)F (v)B1 (u) + e22 (u, v)F (v)B2 (u) + e32 (u, v)B1 (v)F (u) + e42 (u, v)B2 (v)F (u), F (u)B2 (v) = e13 (u, v)B2 (v)F (u) + e23 (u, v)B1 (v)F (u) + e33 (u, v)F (v)B1 (u) + e43 (u, v)F (v)B2 (u), B2 (u)F (v) = e14 (u, v)F (v)B2 (u) + e24 (u, v)B1 (v)F (u) + e34 (u, v)B2 (v)F (u), F (u)F (v) = F (v)F (u),

(39) (40) (41) (42) (43)

where the coefficients of Eqs. (30)–(42) are defined in Appendix A. 3.2. The one-particle state Considering Eqs. (23), (24), the transfer matrix (7) can be rewritten as 1 (u) + w3 (u)D 2 (u), t (u) = w1 (u)A(u) + w2 (u)D

(44)

with w1 (u) = K+ (u)1 + f1 (u)K+ (u)2 + f2 (u)K+ (u)3 , w2 (u) = K+ (u)2 + f3 (u)K+ (u)3 ,

w3 (u) = K+ (u)3 .

(45)

The one-particle state can be constructed as |Φ1 (v1 ) = B1 (v1 )|0.

(46)

Using the commutation relations (30)–(32), we can find A(u)|Φ1 (v1 ) = ω1 (u)a11 (u, v1 )|Φ1 (v1 ) + ω1 (v1 )a21 (u, v1 )B1 (u)|0 + ω2 (v1 )a31 (u, v1 )B1 (u)|0, 1 (u)|Φ1 (v1 ) = ω2 (u)a12 (u, v1 )|Φ1 (v1 ) + ω1 (v1 )a22 (u, v1 )B1 (u)|0 D + ω2 (v1 )a32 (u, v1 )B1 (u)|0 + ω1 (v1 )a42 (u, v1 )B2 (u)|0 + ω2 (v1 )a52 (u, v1 )B2 (u)|0, 2 (u)|Φ1 (v1 ) = ω3 (u)a 3 (u, v1 )|Φ1 (v1 ) + ω1 (v1 )a 3 (u, v1 )B1 (u)|0 D 1 2 + ω2 (v1 )a33 (u, v1 )B1 (u)|0 + ω1 (v1 )a43 (u, v1 )B2 (u)|0 + ω2 (v1 )a53 (u, v1 )B2 (u)|0. Then

 t (u)|Φ1 (v1 ) = w1 (u)ω1 (u)a11 (u, v1 ) + w2 (u)ω2 (u)a12 (u, v1 )  + w3 (u)ω3 (u)a13 (u, v1 ) |Φ1 (v1 )

(47)

(48)

(49)

(50)

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

409

and all the unwanted terms vanish if the rapidity satisfy the Bethe ansatz equation ω1 (v1 ) = β(u, v1 ), ω2 (v1 )

(51)

where β(u, v) =

e−2q sinh(v − 2q) sinh(v)

(52)

corresponding to Eq. (12) and

  e−v sinh(v − 2q) cosh( u+v − 3q) − c sinh( u−v − 2q) 2 2   β(u, v) = u+v  sinh(v) cosh( u−v 2 − q) − c sinh( 2 − 4q) √ for Eq. (13) with c = ± −1.

(53)

3.3. The two-particle state Considering the commutation relation (29), we define 2 2 1 (v2 )gd,2 Φ2 (v1 , v2 ) = B1 (v1 )B1 (v2 ) + F (v1 )A(v2 )ga,2 (v1 , v2 ) + F (v1 )D (v1 , v2 ) (54)

with α(v1 , v2 ) =

2 cosh( v1 −v 2 + q)

2 cosh( v1 −v 2 − q) g(v1 + v2 ) 2 . gd,2 (v1 , v2 ) = b(v1 + v2 )

,

2 ga,2 (v1 , v2 ) = −

g(v1 − v2 )b(2v2 ) , d(v1 − v2 )c(2v2 ) (55)

Then, the two particles state can be constructed as |Φ2 (v1 , v2 ) = Φ2 (v1 , v2 )|0

(56)

1 (u), and it has the property |Φ2 (v1 , v2 ) = α(v1 , v2 )|Φ2 (v2 , v1 ). Acting operators A(u), D 2 (u) on it, respectively, and using the corresponding commutation relations Eqs. (29)– D (38), we finally obtain the following simplified results A(u)|Φ2 (v1 , v2 ) = ω1 (u)

2 

a11 (u, vi )|Φ2 (v1 , v2 )

i=1

  + B1 (u)|Φ1 (v2 ) ω1 (v1 )a21(u, v1 )a11 (v1 , v2 ) + ω2 (v1 )a31 (u, v1 )a12 (v1 , v2 )  + α(v1 , v2 )B1 (u)|Φ1 (v1 ) ω1 (v2 )a21 (u, v2 )a11 (v2 , v1 )  + ω2 (v2 )a31(u, v2 )a12 (v2 , v1 )  + F (u)|0 ω1 (v1 )ω1 (v2 )H1A (u, v1 , v2 ) + ω2 (v1 )ω1 (v2 )H2A (u, v1 , v2 )  + ω1 (v1 )ω2 (v2 )H3A (u, v1 , v2 ) + ω2 (v1 )ω2 (v2 )H4A (u, v1 , v2 ) , (57)

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1 (u)|Φ2 (v1 , v2 ) D = ω2 (u)

2 

a12 (u, vi )|Φ2 (v1 , v2 )

i=1

  + B1 (u)|Φ1 (v2 ) ω1 (v1 )a22(u, v1 )a11 (v1 , v2 ) + ω2 (v1 )a32 (u, v1 )a12 (v1 , v2 )  + α(v1 , v2 )B1 (u)|Φ1 (v1 ) ω1 (v2 )a22 (u, v2 )a11 (v2 , v1 )  + ω2 (v2 )a32(u, v2 )a12 (v2 , v1 )   + B2 (u)|Φ1 (v2 ) ω1 (v1 )a42(u, v1 )a11 (v1 , v2 ) + ω2 (v1 )a52 (u, v1 )a12 (v1 , v2 )  +α(v1 , v2 )B2 (u)|Φ1 (v1 ) ω1 (v2 )a42 (u, v2 )a11 (v2 , v1 )  + ω2 (v2 )a52 (u, v2 )a12 (v2 , v1 )  + F (u)|0 ω1 (v1 )ω1 (v2 )H1D1 (u, v1 , v2 ) + ω2 (v1 )ω1 (v2 )H2D1 (u, v1 , v2 )

 + ω1 (v1 )ω2 (v2 )H3D1 (u, v1 , v2 ) + ω2 (v1 )ω2 (v2 )H4D1 (u, v1 , v2 ) ,

2 (u)|Φ2 (v1 , v2 ) D = ω3 (u)

2 

(58)

a13 (u, vi )|Φ2 (v1 , v2 )

i=1

  + B1 (u)|Φ1 (v2 ) ω1 (v1 )a23(u, v1 )a11 (v1 , v2 ) + ω2 (v1 )a33 (u, v1 )a12 (v1 , v2 )  + α(v1 , v2 )B1 (u)|Φ1 (v1 ) ω1 (v2 )a23 (u, v2 )a11 (v2 , v1 )  + ω2 (v2 )a33(u, v2 )a12 (v2 , v1 )   + B2 (u)|Φ1 (v2 ) ω1 (v1 )a43(u, v1 )a11 (v1 , v2 ) + ω2 (v1 )a53 (u, v1 )a12 (v1 , v2 )  + α(v1 , v2 )B2 (u)|Φ1 (v1 ) ω1 (v2 )a43 (u, v2 )a11 (v2 , v1 )  + ω2 (v2 )a53(u, v2 )a12 (v2 , v1 )  + F (u)|0 ω1 (v1 )ω1 (v2 )H1D2 (u, v1 , v2 ) + ω2 (v1 )ω1 (v2 )H2D2 (u, v1 , v2 )

 + ω1 (v1 )ω2 (v2 )H3D2 (u, v1 , v2 ) + ω2 (v1 )ω2 (v2 )H4D2 (u, v1 , v2 ) . (59) x The coefficients Hi (u, v1 , v2 ), i = 1, 2, 3, 4, x = A, D1 , D2 are defined as  H1A (u, v1 , v2 ) = a41 (u, v1 ) c61 (v1 , v2 ) + c81 (v1 , v2 )  + a51 (u, v1 ) c62 (v1 , v2 ) + c92 (v1 , v2 ) 2 2 + b21 (u, v1 )ga,2 (v1 , v2 ) + α(v1 , u)a11 (u, v1 )a21 (u, v2 )ga,2 (u, v1 ),  A 1 1 1 H2 (u, v1 , v2 ) = a4 (u, v1 ) c7 (v1 , v2 ) + c10 (v1 , v2 )  2 (v1 , v2 ) + a51 (u, v1 ) c72 (v1 , v2 ) + c11 2 2 + b31 (u, v1 )ga,2 (v1 , v2 ) + α(v1 , u)a11 (u, v1 )a21 (u, v2 )gd,2 (u, v1 ), 2 2 H3A (u, v1 , v2 ) = a41 (u, v1 )c91 (v1 , v2 ) + a51 (u, v1 )c10 (v1 , v2 ) + b21 (u, v1 )gd,2 (v1 , v2 )

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

411

2 + α(v1 , u)a11 (u, v1 )a31 (u, v2 )ga,2 (u, v1 ), 1 2 2 H4A (u, v1 , v2 ) = a41 (u, v1 )c11 (v1 , v2 ) + a51 (u, v1 )c12 (v1 , v2 ) + b31 (u, v1 )gd,2 (v1 , v2 ) 2 + α(v1 , u)a11 (u, v1 )a31 (u, v2 )gd,2 (u, v1 ),

(60)

 H1D1 (u, v1 , v2 ) = a62 (u, v1 ) c61 (v1 , v2 ) + c81 (v1 , v2 )  + a72 (u, v1 ) c62 (v1 , v2 ) + c92 (v1 , v2 ) 2 2 (v1 , v2 ) + α(v1 , u)a12 (u, v1 )a22 (u, v2 )ga,2 (u, v1 ) + b22 (u, v1 )ga,2

+ a12 (u, v1 )a42 (u, v2 )d31 (v1 , u),  1 H2D1 (u, v1 , v2 ) = a62 (u, v1 ) c71 (v1 , v2 ) + c10 (v1 , v2 )  2 (v1 , v2 ) + a72 (u, v1 ) c72 (v1 , v2 ) + c11 2 2 (v1 , v2 ) + α(v1 , u)a12 (u, v1 )a22 (u, v2 )gd,2 (u, v1 ) + b32 (u, v1 )ga,2

+ a12 (u, v1 )a42 (u, v2 )d41 (v1 , u), D

2 2 H3 1 (u, v1 , v2 ) = a62 (u, v1 )c91 (v1 , v2 ) + a72 (u, v1 )c10 (v1 , v2 ) + b22 (u, v1 )gd,2 (v1 , v2 ) 2 (u, v1 ) + α(v1 , u)a12(u, v1 )a32 (u, v2 )ga,2

+ a12 (u, v1 )a52 (u, v2 )d31 (v1 , u), D

1 2 2 (v1 , v2 ) + a72 (u, v1 )c12 (v1 , v2 ) + b32 (u, v1 )gd,2 (v1 , v2 ) H4 1 (u, v1 , v2 ) = a62 (u, v1 )c11 2 (u, v1 ) + α(v1 , u)a12(u, v1 )a32 (u, v2 )gd,2

+ a12 (u, v1 )a52 (u, v2 )d41 (v1 , u),

(61)

 H1D2 (u, v1 , v2 ) = a63 (u, v1 ) c61 (v1 , v2 ) + c81 (v1 , v2 ) + a73 (u, v1 )(c62 (v1 , v2 ) + c92 (v1 , v2 )) 2 2 (v1 , v2 ) + α(v1 , u)a13 (u, v1 )a23 (u, v2 )ga,2 (u, v1 ) + b23 (u, v1 )ga,2

+ a13 (u, v1 )a43 (u, v2 )d31 (v1 , u),  1 H2D2 (u, v1 , v2 ) = a63 (u, v1 ) c71 (v1 , v2 ) + c10 (v1 , v2 )  2 (v1 , v2 ) + a73 (u, v1 ) c72 (v1 , v2 ) + c11 2 2 (v1 , v2 ) + α(v1 , u)a13 (u, v1 )a23 (u, v2 )gd,2 (u, v1 ) + b33 (u, v1 )ga,2

+ a13 (u, v1 )a43 (u, v2 )d41 (v1 , u), D

2 (v1 , v2 ) H3 2 (u, v1 , v2 ) = a63 (u, v1 )c91 (v1 , v2 ) + a73 (u, v1 )c10 2 + b23 (u, v1 )gd,2 (v1 , v2 ) 2 + α(v1 , u)a13(u, v1 )a33 (u, v2 )ga,2 (u, v1 )

+ a13 (u, v1 )a52 (u, v2 )d31 (v1 , u), 1 2 2 H4D2 (u, v1 , v2 ) = a63 (u, v1 )c11 (v1 , v2 ) + a73 (u, v1 )c12 (v1 , v2 ) + b33 (u, v1 )gd,2 (v1 , v2 )

412

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438 2 + α(v1 , u)a13(u, v1 )a33 (u, v2 )gd,2 (u, v1 )

+ a13 (u, v1 )a53 (u, v2 )d41 (v1 , u) and we can find that

Hix (u, v1 , v2 )

(62)

have the following properties

H1x (u, v1 , v2 ) = α(v1 , v2 )H1x (u, v2 , v1 ),

(63)

H2x (u, v1 , v2 ) = α(v1 , v2 )H3x (u, v2 , v1 ), H4x (u, v1 , v2 ) = α(v1 , v2 )H4x (u, v2 , v1 ).

(64) (65)

Acting t (u) (44) on |Φ2 (v1 , v2 ) with the help of Eqs. (57)–(59), we get 2 2   t (u)|Φ2 (v1 , v2 ) = w1 (u)ω1 (u) a11 (u, vi ) + w2 (u)ω2 (u) a12 (u, vi ) i=1

+ w3 (u)ω3 (u)

2 

i=1

a13 (u, vi ) |Φ2 (v1 , v2 )

(66)

i=1

and all the unwanted terms vanish by the Bethe equations 2 2 ω1 (vi ) j =1,=i a1 (vi , vj ) = β(u, vi ) 2 (i = 1, 2). 1 ω2 (vi ) j =1,=i a1 (vi , vj )

(67)

3.4. The three-particle state Let Φ3 (v1 , v2 , v3 ) = B1 (v1 )Φ2 (v2 , v3 )  3 3 1 (v3 )gd,3 + F (v1 )Φ1 (v2 ) A(v3 )ga,3 (v2 , v3 ) + D (v2 , v3 )  3 3 1 (v2 )gd,2 + F (v1 )Φ1 (v3 ) A(v2 )ga,2 (v2 , v3 ) + D (v2 , v3 ) .

(68)

Then the three-particle state can be defined as |Φ3 (v1 , v2 , v3 ) = Φ3 (v1 , v2 , v3 )|0.

(69)

Requiring the exchange symmetry |Φ3 (v1 , v2 , v3 ) = α(v2 , v3 )|Φ3 (v1 , v3 , v2 ),

(70)

|Φ3 (v1 , v2 , v3 ) = α(v1 , v2 )|Φ3 (v2 , v1 , v3 )

(71)

and using Eqs. (38), (30), (31), (39), (40), (41), we can get 3 2 ga,2 (v1 , v2 , v3 ) = a11 (v2 , v3 )ga,2 (v1 , v2 ), 3 2 ga,3 (v1 , v2 , v3 ) = α(v2 , v3 )a11 (v3 , v2 )ga,2 (v1 , v3 ), 3 2 gd,2 (v1 , v2 , v3 ) = a12 (v2 , v3 )gd,2 (v1 , v2 ), 3 2 gd,3 (v1 , v2 , v3 ) = α(v2 , v3 )a12 (v3 , v2 )gd,2 (v1 , v3 ).

(72)

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

413

Before tackling the eigenvalue problem of three-particle state, we have to know the results when the operators C1 (u) and C2 (u) act on the two-particle state, which will need more extra commutation relations as listed in Appendix B. Using the commutation relations (30)–(37), (B.1)–(B.5), and after a tedious calculation, we obtain the following simplified forms (u, v1 , v2 ) + B1 (u)ψ˜ c(2) (u, v1 , v2 ) + B2 (u)ψ˜ c(3) (u, v1 , v2 ), Cs (u)|Φ2 (v1 , v2 ) = ψ˜ c(1) s s s (73) (1) (2) where s = 1, 2, ψ˜ cs (u, v1 , v2 ) is a linear combination of |Φ1 ’s, ψ˜ cs (u, v1 , v2 ) and ˜ (l) ˜ (l) ψ˜ c(3) s (u, v1 , v2 ) are coefficients. The ψcs (u, v1 , v2 ) (l = 1, 2, 3) satisfy ψcs (u, v1 , v2 ) = (l) (l) α(v1 , v2 )ψ˜ cs (u, v2 , v1 ). The explicit expressions of ψ˜ cs (u, v1 , v2 ) are rather long and we omit them here for the sake of simplicity. With the aid of Eqs. (57)–(58), (73) and commutation relations (30)–(37), (B.1)–(B.5), we finally obtain

A(u)|Φ3 (v1 , v2 , v3 ) = ω1 (u)

3 

a11 (u, vi )|Φ3 (v1 , v2 , v3 )

i=1

+

3  i−1 

α(vj , vi )B1 (u)|Φ2 (v1 , vˇi , v3 )

i=1 j =1

 ×

ω1 (vi )a21 (u, vi )

3  j =1,=i

+

2 

3  i−1 

 a12 (vi , vj )

j =1,=i j −1

α(vk , vi )

i=1 j =i+1 k=1

3 

a11 (vi , vj ) + ω2 (vi )a31 (u, vi )

α(vl , vj )F (u)|Φ1 (vˇi , v2 , vˇj )

l=1=i

 × ω1 (vi )ω1 (vj )a11 (vi , vm )a11 (vj , vm )H1A (u, vi , vj ) + ω2 (vi )ω1 (vj )a12 (vi , vm )a11 (vj , vm )H2A (u, vi , vj ) + ω1 (vi )ω2 (vj )a11 (vi , vm )a12 (vj , vm )H3A (u, vi , vj )

 + ω2 (vi )ω2 (vj )a12 (vi , vm )a12 (vj , vm )H4A (u, vi , vj ) ,

(74)

1 (u)|Φ3 (v1 , v2 , v3 ) D = ω2 (u)

3 

a12 (u, vi )|Φ3 (v1 , v2 , v3 )

i=1

+

3  i−1 

α(vj , vi )B1 (u)|Φ2 (v1 , vˇi , v3 )

i=1 j =1

 ×

ω1 (vi )a22 (u, vi )

3  j =1,=i

a11 (vi , vj ) + ω2 (vi )a32 (u, vi )

3  j =1,=i

 a12 (vi , vj )

414

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

+

3  i−1 

α(vj , vi )B2 (u)|Φ2 (v1 , vˇi , v3 )

i=1 j =1

 ×

ω1 (vi )a42 (u, vi )

3  j =1,=i

+

2  3  i−1 

 a12 (vi , vj )

j =1,=i j −1

α(vk , vi )

i=1 j =i+1 k=1

3 

a11 (vi , vj ) + ω2 (vi )a52 (u, vi )

α(vl , vj )F (u)|Φ1 (vˇi , v2 , vˇj )

l=1=i

 × ω1 (vi )ω1 (vj )a11 (vi , vm )a11 (vj , vm )H1D1 (u, vi , vj ) + ω2 (vi )ω1 (vj )a12 (vi , vm )a11 (vj , vm )H2D1 (u, vi , vj ) + ω1 (vi )ω2 (vj )a11 (vi , vm )a12 (vj , vm )H3D1 (u, vi , vj )

 D + ω2 (vi )ω2 (vj )a12 (vi , vm )a12 (vj , vm )H4 1 (u, vi , vj ) ,

(75)

2 (u)|Φ3 (v1 , v2 , v3 ) D = ω3 (u)

3 

a13 (u, vi )|Φ3 (v1 , v2 , v3 )

i=1

+

3  i−1 

α(vj , vi )B1 (u)|Φ2 (v1 , vˇi , v3 )

i=1 j =1

 ×

ω1 (vi )a23 (u, vi )

3  j =1,=i

+

3  i−1 

3 

a11 (vi , vj ) + ω2 (vi )a33 (u, vi )

 a12 (vi , vj )

j =1,=i

α(vj , vi )B2 (u)|Φ2 (v1 , vˇi , v3 )

i=1 j =1

 ×

ω1 (vi )a43 (u, vi )

3  j =1,=i

+

2  3  i−1  i=1 j =i+1 k=1

α(vk , vi )

3 

a11 (vi , vj ) + ω2 (vi )a53 (u, vi )

 a12 (vi , vj )

j =1,=i j −1

α(vl , vj )F (u)|Φ1 (vˇi , v2 , vˇj )

l=1=i

 × ω1 (vi )ω1 (vj )a11 (vi , vm )a11 (vj , vm )H1D2 (u, vi , vj ) + ω2 (vi )ω1 (vj )a12 (vi , vm )a11 (vj , vm )H2D2 (u, vi , vj ) + ω1 (vi )ω2 (vj )a11 (vi , vm )a12 (vj , vm )H3D2 (u, vi , vj )

 D + ω2 (vi )ω2 (vj )a12 (vi , vm )a12 (vj , vm )H4 2 (u, vi , vj ) .

(76)

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

415

Where m = 3 if i = 1, j = 2; m = 2 if i = 1, j = 3; m = 1 if i = 2, j = 3,  Φ2 (v2 , v3 ), i = 1, Φ2 (v1 , vˇi , v3 ) = Φ2 (v1 , v3 ), i = 2, Φ2 (v1 , v2 ), i = 3,  Φ (v ), i = 1, j = 2, 1 3 Φ1 (vˇi , v2 , vˇj ) = Φ1 (v2 ), i = 1, j = 3, Φ1 (v1 ), i = 2, j = 3.

(77)

Acting t (u) Eq. (44) on |Φ3 (v1 , v2 , v3 ), we get  2 2   a11 (u, vi ) + w2 (u)ω2 (u) a12 (u, vi ) t (u)|Φ3 (v1 , v2 , v3 ) = w1 (u)ω1 (u) i=1

+ w3 (u)ω3 (u)

2 



i=1

a13 (u, vi ) |Φ3 (v1 , v2 , v3 )

(78)

i=1

and all the unwanted terms vanish by the Bethe equations 3 2 ω1 (vi ) j =1,=i a1 (vi , vj ) = β(u, vi ) 2 (i = 1, 2, 3). 1 ω2 (vi ) j =1,=i a1 (vi , vj )

(79)

3.5. The n-particle state Let Φn (v1 , . . . , vn ) = B1 (v1 )Φn−1 (v2 , . . . , vn ) n  n Φn−2 (v2 , . . . , vˇi , . . . , vn )A(vi )ga,i (v1 , . . . , vn ) + F (v1 ) i=2

+ F (v1 )

n 

n 1 (vi )gd,i Φn−2 (v2 , . . . , vˇi , . . . , vn )D (v1 , . . . , vn ),

i=2

(80)

then the general n-particle state is constructed as |Φn (v1 , . . . , vn ) = Φn (v1 , . . . , vn )|0.

(81)

n n (v1 , . . . , vn ) and gd,i (v1 , . . . , vn ) are given by (see We can prove that if the coefficients ga,i Appendix C) n ga,i (v1 , . . . , vn ) =

i−1 

2 α(vj , vi )Λn−2 1 (vi ; v2 , . . . , vˇ i , . . . , vn )ga,2 (v1 , vi ),

(82)

2 α(vj , vi )Λn−2 2 (vi ; v2 , . . . , vˇi , . . . , vn )gd,2 (v1 , vi ),

(83)

j =2 n gd,i (v1 , . . . , vn ) =

i−1  j =2

416

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

the |Φn (v1 , . . . , vn ) own the property as below |Φn (v1 , . . . , vi , vi+1 , . . . , vn ) = α(vi , vi+1 )|Φn (v1 , . . . , vi+1 , vi , . . . , vn ).

(84)

Where Λnl (u; v1 , v2 , . . . , vn ) =

n 

(l = 1, 2, 3),

a1l (u, vi )

(85)

i=1

Λn−1 (v; vi , {vm }) = Λn−1 (v; v1 , . . . , vˇi , . . . , vn ) l l

(l = 1, 2),

Λn−2 (v; vi , vj , {vm }) = Λn−2 (v; v1 , . . . , vˇi , . . . , vˇj , . . . , vn ) l l

(86) (l = 1, 2).

(87)

α(vj , vi ) B1 (u)|Φn−1 (v1 , . . . , vˇi , . . . , vn ),

(88)

Denote  (1)  Ψ (u, vi ; {vm }) = n−1

 i−1 



j =1

 (2)  Ψ (u, vi ; {vm }) = n−1

 i−1 

 α(vj , vi ) B2 (u)|Φn−1 (v1 , . . . , vˇi , . . . , vn ),

j =1

 (3)  Ψ (u, vi , vj ; {vm }) = n−2

 i−1 

α(vk , vi )

k=1

j −1

(89)

 α(vl , vj ) F (u)

l=1=i

× |Φn−2 (v1 , . . . , vˇi , . . . , vˇj , . . . , vn ).

(90)

Relying on the previous results of directly calculating the one, two and three-particle state, for the n-particle state |Φn (v1 , . . . , vn ), we can infer that A(u)|Φn (v1 , . . . , vn ) = ω1 (u)Λn1 (u; v1 , . . . , vn )|Φn (v1 , . . . , vn ) +

n   (1)  Ψ (u, vi ; {vm }) n−1 i=1

  n−1 1 × ω1 (vi )a21 (u, vi )Λn−1 1 (vi ; vi , {vm }) + ω2 (vi )a3 (u, vi )Λ2 (vi ; vi , {vm }) +

n−1  n   (3)  Ψ (u, vi , vj ; {vm }) n−2 i=1 j =i+1



n−2 × ω1 (vi )ω1 (vj )H1A (u, vi , vj )Λn−2 1 (vi ; vi , vj , {vm })Λ1 (vj ; vi , vj , {vm }) n−2 + ω2 (vi )ω1 (vj )H2A (u, vi , vj )Λn−2 2 (vi ; vi , vj , {vm })Λ1 (vj ; vi , vj , {vm }) n−2 + ω1 (vi )ω2 (vj )H3A (u, vi , vj )Λn−2 1 (vi ; vi , vj , {vm })Λ2 (vj ; vi , vj , {vm }) n−2 + ω2 (vi )ω2 (vj )H4A (u, vi , vj )Λn−2 2 (vi ; vi , vj , {vm })Λ2 (vj ; vi , vj , {vm })



(91)

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

417

1 (u)|Φn (v1 , . . . , vn ) D = ω2 (u)Λn2 (u; v1 , . . . , vn )|Φn (v1 , . . . , vn ) +

n   (1)  Ψ (u, vi ; {vm }) n−1 i=1

  n−1 2 × ω1 (vi )a22 (u, vi )Λn−1 1 (vi ; vi , {vm }) + ω2 (vi )a3 (u, vi )Λ2 (vi ; vi , {vm }) +

n   (2)  Ψ (u, vi ; {vm }) n−1 i=1

  n−1 2 × ω1 (vi )a42 (u, vi )Λn−1 1 (vi ; vi , {vm }) + ω2 (vi )a5 (u, vi )Λ2 (vi ; vi , {vm }) +

n n−1    (3)  Ψ (u, vi , vj ; {vm }) n−2 i=1 j =i+1



n−2 × ω1 (vi )ω1 (vj )H1D1 (u, vi , vj )Λn−2 1 (vi ; vi , vj , {vm })Λ1 (vj ; vi , vj , {vm }) n−2 + ω2 (vi )ω1 (vj )H2D1 (u, vi , vj )Λn−2 2 (vi ; vi , vj , {vm })Λ1 (vj ; vi , vj , {vm }) n−2 + ω1 (vi )ω2 (vj )H3D1 (u, vi , vj )Λn−2 1 (vi ; vi , vj , {vm })Λ2 (vj ; vi , vj , {vm }) n−2 + ω2 (vi )ω2 (vj )H4D1 (u, vi , vj )Λn−2 2 (vi ; vi , vj , {vm })Λ2 (vj ; vi , vj , {vm })

2 (u)|Φn (v1 , . . . , vn ) D



(92)

= ω3 (u)Λn3 (u; v1 , . . . , vn )|Φn (v1 , . . . , vn ) +

n   (1)  Ψ (u, vi ; {vm }) n−1 i=1

  n−1 3 × ω1 (vi )a23 (u, vi )Λn−1 1 (vi ; vi , {vm }) + ω2 (vi )a3 (u, vi )Λ2 (vi ; vi , {vm }) +

n   (2)  Ψ (u, vi ; {vm }) n−1 i=1

  n−1 3 × ω1 (vi )a43 (u, vi )Λn−1 1 (vi ; vi , {vm }) + ω2 (vi )a5 (u, vi )Λ2 (vi ; vi , {vm }) +

n−1  n   (3)  Ψ (u, vi , vj ; {vm }) n−2 i=1 j =i+1



n−2 × ω1 (vi )ω1 (vj )H1D2 (u, vi , vj )Λn−2 1 (vi ; vi , vj , {vm })Λ1 (vj ; vi , vj , {vm }) n−2 + ω2 (vi )ω1 (vj )H2D2 (u, vi , vj )Λn−2 2 (vi ; vi , vj , {vm })Λ1 (vj ; vi , vj , {vm }) n−2 + ω1 (vi )ω2 (vj )H3D2 (u, vi , vj )Λn−2 1 (vi ; vi , vj , {vm })Λ2 (vj ; vi , vj , {vm }) n−2 + ω2 (vi )ω2 (vj )H4D2 (u, vi , vj )Λn−2 2 (vi ; vi , vj , {vm })Λ2 (vj ; vi , vj , {vm })



(93)

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We can prove that our conclusions Eqs. (91)–(93) are still hold at the case n + 1 (see Appendix D) by using the following assumption. Assumption. For n variables s1 , s2 , . . . , sn , if (2)

(3)

(4)

(1) ϕm + B1 (s1 )ϕm−1 + B2 (s1 )ϕm−1 + F (s1 )ϕm−2 = 0,

(94)

then (1) = 0, ϕm

(2)

ϕm−1 = 0,

(3)

ϕm−1 = 0,

(4)

ϕm−2 = 0,

(95)

(1) where ϕm is a linear combination of B1 (sj1 )|Φm−1 (sj2 , sj3 , . . . , sjm )’s, B2 (sj1 ) × |Φm−1 (sj2 , sj3 , . . . , sjm )’s, and F (sj1 )|Φm−2 (sj2 , sj3 , . . . , sjm−1 )’s, with m  n − 1, the spectrum parameters sj1 , sj2 , . . . , sjm−1 , sjm ∈ {s2 , s3 , . . . , sn } and j1 < j2 < · · · < jm . (2) (3) (4) (1) (l) The structure of ϕm−1 , ϕm−1 and ϕm−2 are similar to that of ϕm . In ϕm (l = 1, 2, 3, 4), the variable s1 do not appear in the spectrum parameter of operators B1 , B2 , F and Φk (k < m), but it can appear in their coefficients. Eqs. (94) and (95) can be viewed as a linear (1) (2) (3) (4) independence assumption of ϕm , B1 (s1 )ϕm−1 , B2 (s1 )ϕm−1 , and F (s1 )ϕm−2 .

Using Eqs. (91)–(93) and (44), we get t (u)|Φn (v1 , . . . , vn )  = w1 (u)ω1 (u)Λn1 (u; v1 , . . . , vn ) + w2 (u)ω2 (u)Λn2 (u; v1 , . . . , vn )  + w3 (u)ω3 (u)Λn3 (u; v1 , . . . , vn ) |Φn (v1 , . . . , vn ) + u.t.,

(96)

where u.t. denotes the unwanted terms. We can easily check as we have done on the twoparticle state that if the rapidities satisfy the Bethe equations Λn−1 (vi ; v1 , . . . , vˇi , . . . , vn ) ω1 (vi ) = β(u, vi ) 2n−1 ω2 (vi ) Λ1 (vi ; v1 , . . . , vˇi , . . . , vn )

(i = 1, . . . , n)

(97)

all the unwanted terms vanish. The explicit expression for Eq. (96) is t (u)|Φn (v1 , . . . , vn ) = w1 (u)w1 (u)c(u)2N ρ(u)−N ×

n  sinh( u+2v˜i + q) sinh( u−2v˜i + q) i=1

sinh( u+2v˜i − q) sinh( u−2v˜i − q)

+ w2 (u)w2 (u)b(u)2N ρ(u)−N ×

n  i=1

sinh( u+2v˜i − 3q) sinh( u−2v˜i − 3q) cosh( u+2v˜i ) cosh( u−2v˜i )

sinh( u+2v˜i − q) sinh( u−2v˜i − q) cosh( u+2v˜i − 2q) cosh( u−2v˜i − 2q)

+ w3 (u)w3 (u)d(u)2N ρ(u)−N

n  sinh(cosh( u+2v˜i − 4q) cosh( u−2v˜i − 4q) i=1

cosh( u+2v˜i − 2q) cosh( u−2v˜i − 2q)

, (98)

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

419

where w1 (u) = 1, sinh(u) , sinh(u − 2q) sinh(u) cosh(u − 5q) w3 (u) = e2q , sinh(u − 4q) cosh(u − 3q) sinh(u − 6q) cosh(u − q) , w1 (u) = sinh(u − 2q) cosh(u − 3q) sinh(u − 6q) , w2 (u) = e−2q sinh(u − 4q)

w2 (u) = e2q

w3 (u) = e−2q

(99)

for the case (i) Eq. (12) and w1 (u) = e−u [c cosh(q) + sinh(u − 2q)], sinh(u) cosh(u − q) w2 (u) = c, sinh(u − 2q) sinh(u) cosh(u − q) w3 (u) = eu−4q [c cosh(q) + sinh(u − 4q)], sinh(u − 4q) cosh(u − 3q) sinh(u − 6q) cosh(u − 5q)  [c cosh(q) + sinh(u − 2q)], w1 (u) = eu sinh(u − 2q) cosh(u − 3q) sinh(u − 6q) cosh(u − 5q)  c, w2 (u) = sinh(u − 4q) w3 (u) = e4q−u [c cosh(q) + sinh(u − 4q)]

(100)

for the case Eq. (13). The Bethe equations (97) are sinh( v˜2i − q)2N

sinh( v˜2i + q)2N =

v˜ +v˜ v˜ −v˜ v˜ +v˜ v˜ −v˜ n  sinh( i 2 j − 2q) sinh( i 2 j − 2q) cosh( i 2 j + q) cosh( i 2 j + q) j =1=i

sinh(

v˜ i +v˜ j 2

+ 2q) sinh(

v˜ i −v˜ j 2

+ 2q) cosh(

v˜ i +v˜ j 2

− q) cosh(

v˜ i −v˜ j 2

− q)

(i = 1, . . . , n)

(101)

and sinh( v˜2i − q)2N cosh( u−2v˜i − 2q) − c sinh( u+2v˜i − 3q) cosh(q) − c sinh(v˜i ) cosh(v˜i + q) sinh( v˜ i + q)2N cosh( u+v˜i − 2q) − c sinh( u−v˜i − 3q) 2

=

2

2

v˜ +v˜ v˜ −v˜ v˜ +v˜ v˜ −v˜ n  sinh( i 2 j − 2q) sinh( i 2 j − 2q) cosh( i 2 j + q) cosh( i 2 j + q) j =1=i

sinh(

v˜ i +v˜ j 2

+ 2q) sinh(

v˜ i −v˜ j 2

+ 2q) cosh(

v˜ i +v˜ j 2

− q) cosh(

(i = 1, . . . , n) for Eqs. (12) and (13), respectively. Here vi = v˜i

v˜ i −v˜ j 2

− q) (102)

+ 2q, c2

= c 2

= −1.

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4. Conclusions

In this paper, we solve Izergin–Korepin model with open boundary conditions by using the algebraic Bethe ansatz method. Based on the direct calculation for the oneparticle state, two-particle state and three-particle state, we give the explicit expression of general n-particle state, the eigenvalue of the transfer matrix and the corresponding Bethe equations. To the Uq (su(2)) quantum invariant case, our results coincide with that obtained by analytic Bethe ansatz method [46] and [21]. For the non-quantum-group invariant cases, there are a little difference between our results and that proposed in Ref. [21] by means of analytic Bethe ansatz method. For example, the spectrum parameter u exists in our Bethe equations (102), however there is no u in Bethe equations of Ref. [21]. The difference may be caused by the characters of the two different Bethe ansatz methods. In the frame of QISM, we construct a multi-particle state with some exchange symmetry and act the transfer matrix on it to achieve the eigenvalue and Bethe equations. The construction of the multi-particle state is not unique. Different multi-particle state may lead to different results. The analytic Bethe ansatz method use the symmetry property and the asymptotic behavior of the transfer matrix to obtain the eigenvalue and Bethe equations. In Ref. [52], Nepomechie argued the symmetry property of the transfer matrix of IK model. He proved the transfer matrix corresponding to Eq. (13) also have Uq (o(3)) symmetry, but with a nonstandard coproduct, which may lead to the difference between our conclusions and that in Ref. [21] when the analytic Bethe ansatz method is used. We propose an assumption by which we show our conclusions for the general nparticle state with mathematical induction method. In the procedure of proof, the exchange symmetry property of n-particle state Eq. (84) plays a key role, which ensure some unreasonable terms do not appear so that we can successfully use the algebraic Bethe ansatz method. We may apply the above discussion to other similar models, such as all the nineteen-vertex models or Hubbard-like models. Our assumption hold for n  3 by directly calculating the two and three-particle state and we have also examined some conclusions obtained by the assumption for the general n-particle state. It is an open problem for us to prove the assumption for the general n-particle state. Using the Bethe ansatz equations and energy spectrum, we can study the boundary contributions to the thermodynamic quantities and the surface critical behavior of twodimensional polymers. We also can use the n-particle state to calculate the corresponding form factor and correlation functions which have rather important applications in condensed matter physics.

Acknowledgement

This work is supported by the National Natural Science Foundation of China under Grant No. 10175050.

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

421

Appendix A b(u+ )c(−u− ) , c(u+ )b(−u− ) b(2v)e(−u ¯ −) a21 (u, v) = − , c(2v)b(−u− ) e(u+ ) a31 (u, v) = − , c(u+ ) g(u+ )e(−u ¯ −) a41 (u, v) = − , c(u+ )b(−u− ) f (u+ ) a51 (u, v) = − , c(u+ ) g(u+ )c(−u− ) a61 (u, v) = , c(u+ )b(−u− ) a11 (u, v) =

(A.1)



  ¯ +) ¯ −) a(u+ ) e(u+ )e(u a(u− ) g(u− )g(u − − , b(u+ ) b(u+ )c(u+ ) b(u− ) b(u− )d(u− ) ¯ − )g(u− ) ¯ −) g(u ¯ + )e(u a(u+ )e(u − f1 (v) a22 (u, v) = b(u+ )b(u− )d(u− ) b(u+ )b(u− )

a12 (u, v) =

+ X1 (u, v)a11 (v, u) − X2 (u, v)a21 (u, v), ¯ −) a(u+ )e(u − X2 (u, v)a31 (u, v), a32 (u, v) = − b(u+ )b(u− ) d(u+ )g(u− ) g(u+ ) − f1 (v) , a42 (u, v) = b(u+ )d(u− ) b(u+ ) g(u+ ) a52 (u, v) = − , b(u+ ) ¯ −) g(u− )e(u a62 (u, v) = + X1 (u, v)a61 (v, u) − X2 (u, v)a41 (u, v), b(u− )d(u− ) ¯ −) e(u+ )e(u − X2 (u, v)a51 (u, v), a72 (u, v) = − b(u+ )b(u− ) g(u ¯ − ) f¯(u− )g(u− ) a82 (u, v) = − + X1 (u, v)a41 (v, u) − X2 (u, v)a61 (u, v), b(u− ) b(u− )d(u− )   ¯ −) e(u+ ) a(u− ) g(u− )g(u a92 (u, v) = − + X1 (u, v)a51 (v, u), b(u+ ) b(u− ) b(u− )d(u− ) a13 (u, v) = a23 (u, v) =

¯ + )g(u+ )b(u− ) b(u+ )b(u− ) g(u − , d(u+ )d(u− ) b(u+ )d(u+ )d(u− )

¯ + )e(u ¯ − )b(u− ) g(u ¯ + )g(u e(u ¯ + )f¯(u− ) − f1 (v) + X3 (u, v)a11 (v, u) b(u+ )d(u+ )d(u− )d(u− ) d(u+ )d(u− )

(A.2)

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− X4 (u, v)a21 (u, v) − X5 (u, v)a22 (u, v), e(u ¯ + )f¯(u− ) a33 (u, v) = − − X4 (u, v)a31 (u, v) − X5 (u, v)a32 (u, v), d(u+ )d(u− ) g(u ¯ + )b(u− )b(u− ) b(u+ )g(u ¯ −) − f1 (v) − X5 (u, v)a42 (u, v), a43 (u, v) = b(u+ )d(u− )d(u− ) d(u+ )d(u− ) b(u+ )g(u ¯ −) a53 (u, v) = − − X5 (u, v)a52 (u, v), d(u+ )d(u− ) ¯ − )b(u− ) g(u ¯ + )e(u + X3 (u, v)a61 (v, u) a63 (u, v) = d(u+ )d(u− )d(u− ) − X4 (u, v)a41 (u, v) − X5 (u, v)a62 (u, v), c(u+ )f¯(u− ) − X4 (u, v)a51 (u, v) − X5 (u, v)a72 (u, v), a73 (u, v) = − d(u+ )d(u− ) g(u ¯ + )f¯(u− )b(u− ) + X3 (u, v)a41 (v, u) a83 (u, v) = − d(u+ )d(u− )d(u− ) − X4 (u, v)a61 (u, v) − X5 (u, v)a82 (u, v), g(u ¯ + )e(u+ )g(u ¯ −) ¯ − )b(u− ) c(u+ )e(u − a93 (u, v) = d(u+ )d(u− ) b(u+ )d(u+ )d(u− )d(u− ) + X3 (u, v)a51 (v, u) − X5 (u, v)a92 (u, v), d(u+ )c(−u− ) , c(u+ )d(−u− ) ¯ b(u+ )g(−u −) b51 (u, v) = − , c(u+ )d(−u− ) e(u+ ) b61 (u, v) = − , c(u+ ) d(u+ )f¯(−u− ) g(u+ )g(−u f (u+ ) ¯ −) b21 (u, v) = − − f1 (v) − f2 (v) , c(u+ )d(−u− ) c(u+ )d(−u− ) c(u+ ) g(u+ )g(−u f (u+ ) ¯ −) b31 (u, v) = − − f3 (v) , c(u+ )d(−u− ) c(u+ ) f (u+ ) , b41 (u, v) = − c(u+ )

(A.3)

b11 (u, v) =

   ¯ +) ¯ −) g(u+ )g(u e(u− )e(u b12 (u, v) = 1 − 1− , b(u+ )b(u+ ) b(u− )b(u− )   e(u− )e(u ¯ −) g(u ¯ +) 2 1− α(v, u)ga,2 (u, v) − X6 (u, v)b21 (u, v) b22 (u, v) = b(u+ ) b(u− )b(u− ) ¯ −) ¯ −) e(u− )e(u e(u+ )e(u − f2 (v) , + f1 (v) b(u− )b(u− ) b(u+ )b(u− )

(A.4)

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

g(u ¯ +) e(u− )e(u ¯ −) 2 (u, v) − X6 (u, v)b31 (u, v) (1 − )α(v, u)gd,2 b(u+ ) b(u− )b(u− ) e(u− )e(u ¯ −) ¯ −) e(u+ )e(u + − f3 (v) , b(u− )b(u− ) b(u+ )b(u− ) ¯ −) e(u+ )e(u − X6 (u, v)b41 (u, v), b42 (u, v) = − b(u+ )b(u− )   ¯ −) g(u ¯ +) e(u− )e(u b52 (u, v) = 1− α(v, u) b(u+ ) b(u− )b(u− ) ¯ −) g(u ¯ + ) e(u− )e(u − X6 (u, v)b51 (u, v), + b(u+ ) b(u− )b(u− ) ¯ −) a(u+ )e(u − X6 (u, v)b61 (u, v), b62 (u, v) = − b(u+ )b(u− ) d(u+ )e(u− ) , b72 (u, v) = b(u+ )b(u− ) g(u+ ) b82 (u, v) = − , b(u+ )

423

b32 (u, v) =

b13 (u, v) =

b23 (u, v) =

(A.5)

 c(u− ) c(u+ ) g(u+ )f¯(u+ ) − d(u− ) d(u+ ) c(u+ )d(u+ )   e(u ¯ + )f (u+ ) e(u ¯ + ) e(u+ ) − e−4q − , d(u+ ) h(u+ ) h(u+ )c(u+ )  c(u− ) f¯(u+ )c(u− ) d(u− ) c(u+ )d(u− )   e(u ¯ + ) e(u+ )d(u+ )c(u− ) e(u ¯ − )e(u+ ) − f2 (v) − e−4q d(u+ ) h(u+ )c(u+ )d(u− ) e(u− )h(u+ ) ¯ c(u+ )f (u− ) − X7 (u, v)b21 (u, v) − f2 (v) d(u+ )d(u− ) 2 − X8 (u, v)b22 (u, v) − X9 (u, v)α(v, u)ga,2 (u, v),

b33 (u, v) = e−4q

¯ − )e(u+ ) e¯(u+ )c(u− ) e(u c(u+ )f¯(u− ) f3 (v) − f3 (v) d(u+ )d(u− ) e(u− )h(u+ ) d(u+ )d(u− )

2 − X7 (u, v)b31 (u, v) − X8 (u, v)b32 (u, v) − X9 (u, v)α(v, u)gd,2 (u, v), ¯ − )e(u+ ) c(u+ )f¯(u− ) e¯(u+ )c(u− ) e(u − b43 (u, v) = e−4q d(u+ )d(u− ) e(u− )h(u+ ) d(u+ )d(u− )

− X7 (u, v)b41 (u, v) − X8 (u, v)b42 (u, v), b53 (u, v) = −X7 (u, v)b51 (u, v) − X8 (u, v)b52 (u, v) − X9 (u, v)α(v, u), ¯ + )f¯(u− ) e¯(u+ )c(u− ) e¯(u− )a(u+ ) e(u − b63 (u, v) = e−4q d(u+ )d(u− ) e(u− )h(u+ ) d(u+ )d(u− ) − X7 (u, v)b61 (u, v) − X8 (u, v)b62 (u, v),

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b73 (u, v) = −X8 (u, v)b72 (u, v), e¯(u+ )c(u− ) b(u− )g(u+ ) b(u+ )g(u ¯ −) − − X8 (u, v)b82 (u, v), b83 (u, v) = e−4q d(u+ )d(u− ) e(u− )h(u+ ) d(u+ )d(u− ) (A.6) a(u+ ) , c(u+ ) g(u+ )e(u− ) c21 (u, v) = , c(u+ )b(u− ) ¯ −) g(u+ )e(u c31 (u, v) = − , c(u+ )b(u− ) f (u+ ) c41 (u, v) = − , c(u+ ) e(u+ ) , c51 (u, v) = c(u+ ) e¯(u+ )c(u− ) b(2u) c61 (u, v) = , c(u+ )b(u− ) c(2u) b(u+ )e(u− ) c71 (u, v) = , c(u+ )b(u− ) e(u ¯ − )b(2u) c81 (u, v) = −f1 (v) , b(u− )c(2u) e(u ¯ − )b(2u) , c91 (u, v) = − b(u− )c(2u) e(u+ ) 1 c10 , (u, v) = −f1 (v) c(u+ ) e(u+ ) 1 c11 , (u, v) = − c(u+ ) c11 (u, v) =

c12 (u, v) = c22 (u, v) = c32 (u, v) = c42 (u, v) = c52 (u, v) = c62 (u, v) =

  e(u ¯ + )g(u a(u+ ) ¯ −) 1− , b(u+ )d(u− ) c(u+ ) a(u− ) e(u ¯ + )g(u ¯ − )g(u+ )e(u− ) − , d(u− ) b(u+ )d(u− )c(u+ )b(u− ) ¯ − )g(u+ )e(u ¯ − ) f¯(u− ) e(u ¯ + )g(u − , b(u+ )d(u− )c(u+ )b(u− ) d(u− )   g(u ¯ − ) e(u ¯ + )g(u+ ) e(u+ ) − , d(u− ) b(u+ )c(u+ ) b(u+ )   ¯ + )e(u+ ) g(u ¯ − ) c(u+ ) e(u − , d(u− ) b(u+ ) b(u+ )c(u+ ) g(2u) ¯ e(u ¯ + )c(u− ) , c(2u)c(u+ )b(u− )

(A.7)

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

c72 (u, v) =

g(u ¯ + )a(u− ) e¯(u+ )g(u d(u+ )g(u− ) ¯ − )e(u− ) − + f3 (u) , b(u+ )d(u− ) b(u+ )d(u− )b(u− ) b(u+ )d(u− )

c82 (u, v) =

d(u+ )g(u− ) , b(u+ )d(u− )

c92 (u, v) = −f1 (v)

425

g(2u) ¯ e(u ¯ −) , c(2u)c(u− )

g(2u) ¯ e(u ¯ −) , c(2u)c(u− )   ¯ − )a(u+ ) e¯(u+ )g(u ¯ − )e(u+ ) g(u+ ) g(u 2 c11 + − (u, v) = −f1 (v) f3 (u) , b(u+ ) b(u+ )d(u− ) b(u+ )d(u− )c(u+ )   ¯ − )a(u+ ) e(u ¯ + )g(u g(u+ ) g(u ¯ − )e(u+ ) 2 + − c12 (u, v) = − f3 (u) , b(u+ ) b(u+ )d(u− ) b(u+ )d(u− )c(u+ ) 2 c10 (u, v) = −

2 (u, v) = − c13

g(u+ ) , b(u+ )

2 c14 (u, v) = −f1 (v)

g(u+ ) , b(u+ )

(A.8)

c(u+ ) , h(u+ )   ¯ + )e(−u ¯ c(u+ ) g(u −) − X10 (u, v)α(u, v) , d21 (u, v) = h(u+ ) d(u+ )b(−u− )  e(u ¯ + )c(−u− ) ¯ c(u+ ) b(u+ )e(−u −) 1 f1 (u) − d3 (u, v) = h(u+ ) d(u+ )b(−u− ) c(u+ )b(−u− )  2 − X10 (u, v)α(u, v)ga,2 (v, u) ,

d11 (u, v) =

  ¯ c(u+ ) b(u+ )e(−u −) 2 − X10 (u, v)α(u, v)gd,2 (v, u) , h(u+ ) d(u+ )b(−u− )  e(u ¯ +) 1 ¯ c(u+ ) b(u+ )e(−u e(u+ ) −) 1 f1 (v) + f2 (v) + b (u, v) d5 (u, v) = − h(u+ ) d(u+ )b(−u− ) d(u+ ) d(u+ ) 2  2 + X10 (u, v)ga,2 (u, v) ,

d41 (u, v) = −

d61 (u, v) = −

 e(u ¯ +) 1 ¯ c(u+ ) b(u+ )e(−u e(u+ ) −) + f3 (v) + b (u, v) h(u+ ) d(u+ )b(−u− ) d(u+ ) d(u+ ) 3  2 + X10 (u, v)gd,2 (u, v) ,

c(u+ ) d71 (u, v) = − h(u+ )



 e(u ¯ +) 1 e(u+ ) + b (u, v) , d(u+ ) d(u+ ) 4

(A.9)

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where Xi (u, v) (i = 1, . . . , 10) and h(u) are defined as below   ¯ −) e(u ¯ + ) a(u− ) g(u− )g(u − X1 (u, v) = , b(u+ ) b(u− ) b(u− )d(u− ) ¯ −) e(u ¯ + )e(u X2 (u, v) = f1 (u) + , b(u+ )b(u− ) ¯ + )e(u ¯ − ) g(u ¯ + )g(u ¯ − )b(u− ) f¯(u+ )e(u − , X3 (u, v) = d(u+ )d(u− ) b(u+ )d(u+ )d(u− )d(u− ) g(u ¯ + )g(u ¯ −) f¯(u+ )f¯(u− ) X4 (u, v) = + f1 (u) + f2 (u), d(u+ )d(u− ) d(u+ )d(u− ) ¯ −) g(u ¯ + )g(u + f3 (u), X5 (u, v) = d(u+ )d(u− ) ¯ −) e(u ¯ + )e(u + f1 (u), X6 (u, v) = b(u+ )b(u− ) f¯(u+ )f¯(u− ) X7 (u, v) = d(u+ )d(u− )   b(u+ )b(u− ) ¯ + )c(u− ) e¯(u+ )e(u ¯ −) −4q e(u −e + f1 (u) d(u+ )d(u− ) h(u+ )e(u− ) h(u+ )e(u− ) ¯ −) g(u ¯ + )g(u + f2 (u), + f1 (u) d(u+ )d(u− ) ¯ −) e(u ¯ + )c(u− ) b(u+ )b(u− ) g(u ¯ + )g(u + f3 (u) − e−4q , X8 (u, v) = d(u+ )d(u− ) d(u+ )d(u− ) h(u+ )e(u− ) ¯ −) f¯(u+ )c(u− ) b(u+ )g(u X9 (u, v) = d(u+ )d(u− ) c(u+ )d(u− )   ¯ + )c(u− ) g(u ¯ −) ¯ + )b(u− ) e¯(u+ ) b(u+ )g(u −4q e(u − +e , d(u+ )d(u− ) h(u+ )e(u− ) h(u+ ) c(u+ )d(u− ) e(u ¯ +) 1 ¯ g(u ¯ + )e(−u −) − b (u, v), X10 (u, v) = d(u+ )b(−u− ) d(u+ ) 5     u u − 4q cosh − 3q , h(u) = 2 sinh 2 2 b(u+ )b(u− ) , c(u− )d(u+ ) g(u+ )b(u− ) , e21 (u, v) = c(u− )d(u+ ) e(u ¯ −) e31 (u, v) = , c(u− ) b(u+ )b(u− ) e12 (u, v) = , c(u− )k(u+ ) e11 (u, v) =

(A.10)

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

  ¯ g(u+ )b(u+ ) e(u− ) e(−u −) − , d(u+ )k(u+ ) c(u− ) c(−u− ) ¯ + )e(−u ¯ b(u+ )b(u+ )e(u− ) g(u+ )g(u −) − , e32 (u, v) = d(u+ )k(u+ )c(u− ) d(u+ )k(u+ )c(−u− ) g(u+ )b(−u− ) , e42 (u, v) = − c(−u− )k(u+ ) ¯ +) 2 d(u+ )b(u− ) g(u − e (v, u), e13 (v, u) = c(u− )b(u+ ) b(u+ ) 4 ¯ +) 2 g(u ¯ + )e(u ¯ − ) g(u − e (v, u), e23 (v, u) = c(u− )b(u+ ) b(u+ ) 3 g(u ¯ +) 2 e (v, u), e33 (v, u) = − b(u+ ) 1 ¯ +) 2 e(u ¯ − ) g(u − e (v, u), e43 (v, u) = c(u− ) b(u+ ) 2 b(u+ )b(u− ) , e14 (v, u) = c(u− )d(u+ ) g(u ¯ + )b(u− ) , e24 (v, u) = c(u− )d(u+ ) e(u− ) e34 (v, u) = , c(u− )     u u k(u) = 2 sinh − 5q . cosh 2 2

427

e22 (u, v) =

(A.11)

Appendix B

A(u)B2 (v) = X11 (u, v)B2 (v)A(u) + X21 (u, v)B1 (v)A(u) + X31 (u, v)B1 (u)A(v) 1 (v) + X51 (u, v)B1 (u)D 2 (v) + X61 (u, v)F (u)C1 (v) + X41 (u, v)B1 (u)D + X71 (u, v)F (u)C2 (v) + X81 (u, v)F (v)C1 (u),

(B.1)

1 (u)B2 (v) D 1 (u) + X22 (u, v)B1 (v)D 1 (u) + X32 (u, v)B1 (u)A(v) = X12 (u, v)B2 (v)D 1 (v) + X2 (u, v)B1 (u)D 2 (v) + X2 (u, v)B2 (u)A(v) + X42 (u, v)B1 (u)D 5 6 1 (v) + X82 (u, v)B2 (u)D 2 (v) + X92 (u, v)F (u)C1 (v) + X72 (u, v)B2 (u)D 2 2 2 + X10 (u, v)F (u)C2 (v) + X11 (u, v)F (v)C1 (u) + X12 (u, v)F (v)C2 (u), (B.2)

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2 (u)B2 (v) D 2 (u) + X3 (u, v)B1 (v)D 2 (u) + X3 (u, v)B1 (u)A(v) = X13 (u, v)B2 (v)D 2 3 3 3   + X (u, v)B1 (u)D1 (v) + X (u, v)B1 (u)D2 (v) + X3 (u, v)B2 (u)A(v)

4 5 6 3 3   + X7 (u, v)B2 (u)D1 (v) + X8 (u, v)B2 (u)D2 (v) + X93 (u, v)F (u)C1 (v) 3 3 3 + X10 (u, v)F (u)C2 (v) + X11 (u, v)F (v)C1 (u) + X12 (u, v)F (v)C2 (u),

(B.3)

C1 (u)F (v) = Y11 (u, v)F (v)C1 (u) + Y21 (u, v)F (v)C2 (u) + Y31 (u, v)F (u)C1 (v) 1 (u) + Y41 (u, v)F (u)C2 (v) + Y51 (u, v)B1 (v)A(u) + Y61 (u, v)B1 (v)D 1 (u) + Y91 (u, v)B1 (u)A(v) + Y71 (u, v)B2 (v)A(u) + Y81 (u, v)B2 (v)D 1 1 1 1 (v) + Y11 2 (v) + Y12 (u, v)B1 (u)D (u, v)B1 (u)D (u, v)B2 (u)A(v) + Y10 1 1 1 (v) + Y14 (u, v)B2 (u)D 2 (v), + Y13 (u, v)B2 (u)D (B.4) C2 (u)F (v) = Y12 (u, v)F (v)C2 (u) + Y22 (u, v)F (v)C1 (u) + Y32 (u, v)F (u)C1 (v) 1 (u) + Y42 (u, v)F (u)C2 (v) + Y52 (u, v)B1 (v)A(u) + Y62 (u, v)B1 (v)D 2 (u) + Y82 (u, v)B2 (v)A(u) + Y92 (u, v)B2 (v)D 1 (u) + Y72 (u, v)B1 (v)D 2 2 2 2 (u) + Y11 1 (v) (u, v)B2 (v)D (u, v)B1 (u)A(v) + Y12 (u, v)B1 (u)D + Y10 2 2 2 2 (v) + Y14 1 (v) (u, v)B1 (u)D (u, v)B2 (u)A(v) + Y15 (u, v)B2 (u)D + Y13 2 2 (v). (u, v)B2 (u)D + Y16 (B.5)

Here the explicit form of all the coefficients of Eqs. (B.1)–(B.5) are not presented for their long and tedious expressions.

Appendix C Considering the definition of n-particle state Eqs. (81), (80), from |Φn (v1 , . . . , vi , vi+1 , . . . , vn ) = α(vi , vi+1 )|Φn (v1 , . . . , vi+1 , vi , . . . , vn )

(C.1)

with i = 1, we can easily obtain n n (v1 , . . . , vi , vi+1 , . . . , vn ) = α(vi , vi+1 )ga,i (v1 , . . . , vi+1 , vi , . . . , vn ), ga,i+1

(C.2)

n n (v1 , . . . , vi , vi+1 , . . . , vn ) = α(vi , vi+1 )gd,i (v1 , . . . , vi+1 , vi , . . . , vn ). gd,i+1

(C.3)

Let i = 1 in Eq. (C.1), we have |Φn (v1 , v2 , . . . , vn )  = α(v1 , v2 ) B1 (v2 )B1 (v1 )|Φn−2 (v3 , . . . , vn )

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

+ B1 (v2 )F (v1 )

n 

429

Φn−3 (v3 , . . . , vˇi , . . . , vn )

j =3

  n−1 n−1  × A(vj )ga,j −1 (v1 , v3 , . . . , vn ) + D1 (vj )gd,j −1 (v1 , v3 , . . . , vn ) |0  n + F (v2 )Φn−2 (v3 , . . . , vn ) A(v1 )ga,2 (v2 , v1 , . . . , vn )  n  (v2 , v1 , . . . , vn ) |0 + D1 (v1 )gd,2  n  + F (v2 ) B1 (v1 )Φn−3 (v3 , . . . , vˇi , . . . , vn ) j =3

+ F (v1 ) 



Φn−4 (v3 , . . . , vˇi , . . . , vˇj , . . . , vn )

j =3,j =i

n−2 ˇi , . . . , vn ) × A(vj )ga,j  (v1 , v3 , . . . , v



1 (vj )g n−2 (v1 , v3 , . . . , vˇi , . . . , vn ) +D d,j

   n n  × A(vi )ga,i (v2 , v1 , . . . , vn ) + D1 (vi )gd,i (v2 , v1 , . . . , vn ) |0 ,

(C.4)

where j  = j − 1 whenj < i and j  = j − 2 when j > i. Firstly, we substitute com1 (v2 ), mutation relation (29) to the term B1 (v2 )B1 (v1 ) and act the operators A(v2 ), D  A(v1 ), D1 (v1 ), respectively, on |Φn−2 (v3 , . . . , vn ) with the help of the conclusions (91), (92). Then we compare the coefficients of F (v1 )Φn−2 (v3 , . . . , vn )A(v2 )|0 and 1 (v2 )|0 in the l.h.s. of Eq. (C.4) with that in the r.h.s. of F (v1 )Φn−2 (v3 , . . . , vn )D Eq. (C.4), respectively, and we will get the following results immediately n 2 (v1 , . . . , vn ) = Λn−2 ga,2 1 (v2 ; v3 , . . . , vn )ga,2 (v1 , v2 ),

(C.5)

n 2 (v1 , . . . , vn ) = Λn−2 gd,2 2 (v2 ; v3 , . . . , vn )gd,2 (v1 , v2 ).

(C.6)

From Eqs. (C.2), (C.5) and (C.3), (C.6), we can arrive at n ga,i (v1 , . . . , vn ) =

i−1 

2 α(vj , vi )Λn−2 1 (vi ; v2 , . . . , vˇ i , . . . , vn )ga,2 (v1 , vi ),

(C.7)

2 α(vj , vi )Λn−2 2 (vi ; v2 , . . . , vˇi , . . . , vn )gd,2 (v1 , vi ).

(C.8)

j =2 n gd,i (v1 , . . . , vn ) =

i−1  j =2

Now we have to test Eqs. (C.7), (C.8). Putting Eqs. (C.7), (C.8) into Eq. (C.4) and applying relations (40), (39), (41), (43) to the terms B1 (v2 )F (v1 ), F (v2 )B1 (v1 ), F (v2 )B2 (v1 ), F (v2 )F (v1 ) in the r.h.s. of Eq. (C.4), respectively, (reminding that the fist step have been done), we will find that in the r.h.s. of Eq. (C.4), the terms F (v1 )B2 (v2 )|Φn−3 (v3 , . . . , vˇi , . . . , vn ) and B2 (v1 )F (v2 )|Φn−3 (v3 , . . . , vˇi , . . . , vn ) are easily cancelled out and the rest terms can be simplified into one term |Φn (v1 , v2 , . . . , vn ) which is just the l.h.s. of Eq. (C.4). We have done this work with the aid of computer.

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Appendix D 1 (u)|Φn+1 (v1 , . . . , vn+1 ) as an example, because the In this appendix, we only take D 1 (u)|Φn+1 (v1 , . . . , vn+1 ) can be applied to other similar procedure of calculating the D 1 (u) acting operators. We now carry on our induction. In order to calculate the operators D on the |Φn+1 (v1 , . . . , vn+1 ), besides Eqs. (91)–(93), we also need to know the results of Cs (u)|Φn (v1 , . . . , vn ) (s = 1, 2). See Eq. (D.5). Let us assume Cs (u)|Φn (v1 , . . . , vn ) = ψc(1) + B1 (u)ψc(2) + B2 (u)ψc(3) + F (u)ψc(4) , s s s s (1)

(2)

(D.1)

(3)

where ψcs is a linear combination of |Φn−1 ’s. ψcs , ψcs are respectively the linear combination of |Φn−2 ’s. ψc(4) are the linear combination of |Φn−3 ’s. All the ψc(i) s s , i = 1, 2, 3, 4 satisfy the following property (v1 , . . . , vj , vj +1 , . . . , vn ) = α(vj , vj +1 )ψc(i) (v1 , . . . , vj +1 , vj , . . . , vn ) ψc(i) s s

(D.2)

and the explicit expression of ψc(1) s are given by ψc(1) (v1 , v2 , . . . , vn ) 1 =

n  i−1 

α(vj , vi )|Φn−1 (v1 , . . . , vˇi , . . . , vn )

i=1 j =1

  × ω1 (u)ω1 (vi )Λn1 (u; vi , {vm })Λn1 (vi ; vi , {vm }) c61 (u, vi ) + c81 (u, vi ) + ω1 (u)ω2 (vi )Λn1 (u; vi , {vm })Λn2 (vi ; vi , {vm })c91 (u, vi )  1 + ω2 (u)ω1 (vi )Λn2 (u; vi , {vm })Λn1 (vi ; vi , {vm }) c71 (u, vi ) + c10 (u, vi ) 1 + ω2 (u)ω2 (vi )Λn2 (u; vi , {vm })Λn2 (vi ; vi , {vm })c11 (u, vi ) ,

(D.3)

ψc(1) (v1 , v2 , . . . , vn ) 2 =

n  i−1 

α(vj , vi )|Φn−1 (v1 , . . . , vˇi , . . . , vn )

i=1 j =1

  × ω1 (u)ω1 (vi )Λn1 (u; vi , {vm })Λn1 (vi ; vi , {vm }) c62 (u, vi ) + c92 (u, vi ) 2 (u, vi ) + ω1 (u)ω2 (vi )Λn1 (u; vi , {vm })Λn2 (vi ; vi , {vm })c10  2 n n 2 + ω2 (u)ω1 (vi )Λ2 (u; vi , {vm })Λ1 (vi ; vi , {vm }) c7 (u, vi ) + c11 (u, vi ) 2 + ω2 (u)ω2 (vi )Λn2 (u; vi , {vm })Λn2 (vi ; vi , {vm })c12 (u, vi )  n n 2 2 + ω3 (u)ω1 (vi )Λ3 (u; vi , {vm })Λ1 (vi ; vi , {vm }) c8 (u, vi ) + c13 (u, vi ) 2 + ω3 (u)ω2 (vi )Λn3 (u; vi , {vm })Λn2 (vi ; vi , {vm })c14 (u, vi ) .

(D.4)

We can prove that the assumption Eq. (D.1) hold for the case n + 1 by the same method 1 (u) act on the employed in the following induction procedure. Now let operator D

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

431

|Φn+1 (v1 , . . . , vn+1 ), we have 1 (u)|Φn+1 (v1 , . . . , vn+1 ) D  1 (u) + a22 (u, v1 )B1 (u)A(v1 ) = a12 (u, v1 )B1 (v1 )D 1 (v1 ) + a42 (u, v1 )B2 (u)A(v1 ) + a52 (u, v1 )B2 (u)D 1 (v1 ) + a32 (u, v1 )B1 (u)D + a62 (u, v1 )F (u)C1 (v1 ) + a72 (u, v1 )F (u)C2 (v1 ) + a82(u, v1 )F (v1 )C1 (u) + a92 (u, v1 )F (v1 )C2 (u) × |Φn (v2 , . . . , vn+1 )  1 (u) + b12(u, v1 )F (v1 )D 1 (v1 ) + b42 (u, v1 )F (u)D 2 (v1 ) + b22 (u, v1 )F (u)A(v1 ) + b32 (u, v1 )F (u)D + b52 (u, v1 )B1 (u)B1 (v1 ) + b62 (u, v1 )B1 (u)B2 (v1 ) + b72 (u, v1 )B2 (u)B1 (v1 ) + b82 (u, v1 )B2 (u)B2 (v1 ) ×

n+1 

|Φn−1 (v2 , . . . , vˇi , . . . , vn+1 )

i=2

  n+1 n+1 (v1 , . . . , vn+1 ) + ω2 (vi )gd,i (v1 , . . . , vn+1 ) . × ω1 (vi )ga,i

(D.5)

After a not very hard deducing by using Eqs. (91)–(93), (D.1), (38)–(43), we find that Eq. (D.5) can be written as 1 (u)|Φn+1 (v1 , . . . , vn+1 ) = ψ (1) + B1 (u)ψ (2) + B2 (u)ψ (3) + F (u)ψ (4) , D

(D.6)

where ψ (1) is a linear combination of B1 |Φn ’s and F |Φn−1 ’s, ψ (2) and ψ (3) , respectively, are the linear combination of B1 |Φn−1 ’s, B2 |Φn−1 ’s and F |Φn−2 ’s and ψ (4) is a linear combination of B1 |Φn−2 ’s, B2 |Φn−2 ’s and F |Φn−3 ’s. For instance, the expression of ψ (2) are ψ (2) (v1 , v2 , . . . , vn+1 ) = ω1 (v1 )a22 (u, v1 )Λn1 (v1 ; v1 , {vm })|Φn (v2 , . . . , vn+1 ) + ω2 (v1 )a32 (u, v1 )Λn2 (v1 ; v1 , {vm })|Φn (v2 , . . . , vn+1 ) + B1 (v1 )

n+1 

|Φn−1 (v2 , . . . , vˇi , . . . , vn+1 )Kib1 (v1 , vi , {vm })

i=2

+ B2 (v1 )

n+1 

|Φn−1 (v2 , . . . , vˇi , . . . , vn+1 )Kib2 (v1 , vi , {vm })

i=2

+ F (v1 )

n  n+1 

f

|Φn−2 (v2 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )Kij (v1 , vi , vj , {vm }), i=2 j =i+1 (D.7) f

where Kib1 , Kib2 , and Kij are coefficients. Then, by the assumption Eqs. (94), (95) and the property of Eq. (84), we find all the ψ (i) , i = 1, 2, 3, 4 satisfy

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G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

ψ (i) (v1 , . . . , vj , vj +1 , . . . , vn+1 ) = α(vj , vj +1 )ψ (i) (v1 , . . . , vj +1 , vj , . . . , vn+1 )

(D.8)

for the rapidities v1 , . . . , vn+1 . In the following procedure, we will determine the explicit expressions of ψ (i) with the aid of Eq. (D.8) and Eqs. (94), (95). Without losing of generality, we take ψ (2) as an example here. From Eqs. (D.7) and (D.8), we have ψ (2) (v1 , v2 , . . . , vn+1 ) = α(v1 , v2 )ψ (2) (v2 , v1 , . . . , vn+1 )  = α(v1 , v2 ) ω1 (v2 )a22 (u, v2 )Λn1 (v2 ; v2 , {vm })|Φn (v1 , v3 , . . . , vn+1 ) + ω2 (v2 )a32 (u, v2 )Λn2 (v2 ; v2 , {vm })|Φn (v1 , v3 , . . . , vn+1 ) + B1 (v2 )

n+1 

|Φn−1 (v1 , . . . , vˇi , . . . , vn+1 )Kib1 (v2 , vi , {vm })

i=1=2

+ B2 (v2 )

n+1 

|Φn−1 (v1 , . . . , vˇi , . . . , vn+1 )Kib2 (v2 , vi , {vm })

i=1=2

+ F (v2 )

n 

n+1 

|Φn−2 (v1 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )

i=1=2 j =i+1=2



f × Kij (v1 , vi , vj , {vm })

.

(D.9)

From Eq. (D.9), we can find that in the l.h.s. of Eq. (D.9), there are such terms B2 (v1 )B1 (v3 )|Φn−2 (v4 , . . . , vn+1 )K2b2 (v1 , v2 , {vm }), B2 (v1 )B1 (v2 )

n+1 

|Φn−2 (v3 , . . . , vˇi , . . . , vn+1 )Kib2 (v1 , vi , {vm }).

i=3

However, in the r.h.s. of Eq. (D.9), we cannot obtain this terms above when we use the commutation relations Eqs. (29), (38)–(43) and Eqs. (91)–(93) to permute the spectrum parameters v1 and v2 . Subtracting the r.h.s. of Eq. (D.9) from l.h.s. of Eq. (D.9) and putting the spectrum parameters v1 to the left side of v2 by necessary commutation relations, we turn Eq. (D.9) into the form of Eq. (94). Using our assumption Eqs. (94), (95), we can easily get b

Ki 2 (v1 , vi , {vm }) = 0,

i ∈ [2, . . . , n + 1].

So ψ (2) can be written as (2)

(2)

ψ (2) (v1 , v2 , . . . , vn+1 ) = ψ1 (v1 , v2 , . . . , vn+1 ) + ψ2 (v1 , v2 , . . . , vn+1 )

(D.10)

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

433

with ψ1(2) (v1 , v2 , . . . , vn+1 ) =

n+1  i−1 

α(vj , vi )|Φn (v1 , . . . , vˇi , . . . , vn+1 )

i=1 j =1

! " × ω1 (vi )a22 (u, vi )Λn1 (vi ; vi , {vm }) + ω2 (vi )a32 (u, vi )Λn2 (vi ; vi , {vm }) , (D.11)

ψ2(2) (v1 , v2 , . . . , vn+1 ) = B1 (v1 )

n+1 

|Φn−1 (v2 , . . . , vˇi , . . . , vn+1 )Kib1 (v1 , vi , {vm })

i=2

+ F (v1 )

n  n+1 

f

|Φn−2 (v2 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )Kij (v1 , vi , vj , {vm })

i=2 j =i+1

−

i−1 n+1  

α(vj , vi )ω1 (vi )a22 (u, vi )Λn1 (vi ; vi , {vm })|Φn (v1 , . . . , vˇi , . . . , vn+1 )

i=2 j =1

−

n+1  i−1 

α(vj , vi )ω2 (vi )a32 (u, vi )Λn2 (vi ; vi , {vm })|Φn (v1 , . . . , vˇi , . . . , vn+1 )

i=2 j =1

= B1 (v1 )

n+1 

 1 (v1 , vi , {vm }) |Φn−1 (v2 , . . . , vˇi , . . . , vn+1 )K i b

i=2

+ F (v1 )

n  n+1 

f (v1 , vi , vj , {vm }), |Φn−2 (v2 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )K ij

i=2 j =i+1

(D.12)

b1 are new coefficients which include K b1 and the contributions from the terms of where K i i f . We can easily check that |Φn (v1 , . . . , vˇi , . . . , vn+1 ). So do K ij (2)

(2)

ψ1 (v1 , . . . , vj , vj +1 , . . . , vn+1 ) = α(vj , vj +1 )ψ1 (v1 , . . . , vj +1 , vj , . . . , vn+1 ), (D.13) therefore, from Eqs. (D.8) and (D.10) we should have ψ2(2) (v1 , . . . , vj , vj +1 , . . . , vn+1 ) = α(vj , vj +1 )ψ2(2) (v1 , . . . , vj +1 , vj , . . . , vn+1 ). (D.14)  in Eq. (D.12) with the aid of Eq. (D.14) in b1 and K We will determine the coefficients K i ij the following step. From f

(2)

(2)

ψ2 (v1 , v2 , . . . , vn+1 ) = α(v1 , v2 )ψ2 (v2 , v1 , . . . , vn+1 ),

(D.15)

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G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

we get b1 (v1 , v2 , {vm }) B1 (v1 )|Φn−1 (v3 , v4 , . . . , vn+1 )K 2 + B1 (v1 )

n+1 

b1 (v1 , vi , {vm }) |Φn−1 (v2 , . . . , vˇi , . . . , vn+1 )K i

i=3

+ F (v1 )

n  n+1 

f (v1 , vi , vj , {vm }) |Φn−2 (v2 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )K ij

i=2 j =i+1

b1 (v2 , v1 , {vm }) = α(v1 , v2 ){B1 (v2 )|Φn−1 (v3 , v4 , . . . , vn+1 )K 2 + B1 (v2 )

n+1 

b1 (v2 , vi , {vm }) |Φn−1 (v1 , v3 , . . . , vˇi , . . . , vn+1 )K i

i=3 n 

+ F (v2 )

n+1 

|Φn−2 (v1 , v3 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )

i=1=2 j =i+1=2

 (v2 , vi , vj , {vm })}. ×K ij f

(D.16)

It can be seen clearly that in the r.h.s. of Eq. (D.16) the first term will receive no any contributions from the rest terms in the procedure of permuting the spectrum parameters v1 and v2 with the help of the commutation relations Eqs. (29), (38)–(43). Using l.h.s. of Eq. (D.16) minus the r.h.s. of Eq. (D.16) and putting v1 to the left side of v2 by necessary commutation relations, we change Eq. (D.16) into the form of Eq. (94). With the help of the assumption Eqs. (94), (95), we easily get b1 = 0. K 2 So we can get the following result ψ2(2) (v1 , v2 , . . . , vn+1 ) = B1 (v1 )

n+1 

b1 (v1 , vi , {vm }) |Φn−1 (v2 , . . . , vˇi , . . . , vn+1 )K i

i=3

+ F (v1 )

n  n+1 

f (v1 , vi , vj , {vm }). |Φn−2 (v2 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )K ij

i=2 j =i+1

(D.17)

From (2)

(2)

ψ2 (v1 , v2 , v3 , . . . , vn+1 ) = α(v2 , v3 )ψ2 (v1 , v3 , v2 , . . . , vn+1 ), we get b1 (v1 , v3 , {vm }) B1 (v1 )|Φn−1 (v2 , v4 , . . . , vn+1 )K 3 + B1 (v1 )

n+1  i=4

b1 (v1 , vi , {vm }) |Φn−1 (v2 , v3 , . . . , vˇi , . . . , vn+1 )K i

(D.18)

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

+ F (v1 )

n  n+1 

435

 (v1 , vi , vj , {vm }) |Φn−2 (v2 , v3 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )K ij f

i=2 j =i+1

b1 (v1 , v2 , {vm }) = α(v2 , v3 ){B1 (v1 )|Φn−1 (v3 , v4 , . . . , vn+1 )K 3 + B1 (v1 )

n+1 

b1 (v1 , vi , {vm }) |Φn−1 (v3 , v2 , . . . , vˇi , . . . , vn+1 )K i

i=4

+ F (v1 )

n  n+1 

|Φn−2 (v3 , v2 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )

i=2 j =i+1

f (v1 , vi , vj , {vm })}. ×K ij

(D.19)

Changing Eq. (D.19) into the form of Eq. (94) by subtracting the r.h.s. of Eq. (D.19) from l.h.s. of Eq. (D.19), we then obtain b1 = 0 K 3 with the help of the assumption Eqs. (94), (95). Repeating the same procedure as above, we achieve b1 = K b1 = · · · = K b1 = 0. K 2 3 n+1 (2)

At this moment, ψ2 become (2)

ψ2 (v1 , . . . , vn+1 ) = F (v1 )

n  n+1 

f (v1 , vi , vj , {vm }). |Φn−2 (v2 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )K ij

i=2 j =i+1

(D.20)

From Eq. (D.15), we have F (v1 )

n+1 

 (v1 , vi , vj , {vm }) |Φn−2 (v3 , . . . , vˇj , . . . , vn+1 )K 2j f

j =3

+ F (v1 )

n+1 n  

f (v1 , vi , vj , {vm }) |Φn−2 (v2 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )K ij

i=3 j =i+1



= α(v1 , v2 ) F (v2 )

n+1 

 (v2 , vi , vj , {vm }) |Φn−2 (v3 , . . . , vˇj , . . . , vn+1 )K 2j f

j =3

+ F (v2 )

n  n+1 

|Φn−2 (v1 , v3 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )

i=3 j =i+1

 f  × Kij (v1 , vi , vj , {vm }) .

(D.21)

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G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

Using Eqs. (39), (43) to permute v1 and v2 in the r.h.s. of Eq. (D.21) and changing Eq. (D.21) into the form of Eq. (94) by l.h.s. of Eq. (D.21) minus the r.h.s. of Eq. (D.21), we can obtain f = 0, K 2j

j = 3, 4, . . . , n + 1

based on the assumption Eqs. (94), (95). Carrying on the similar steps as that we determine b1 , we finally get the coefficients K i f = 0, K ij

i = 2, 3, . . . , n, j = 3, 4, . . . , n + 1, j > i.

Thus, we have the conclusion (2)

ψ2 (v1 , v2 , . . . , vn+1 ) = 0

(D.22)

and (2)

ψ (2) (v1 , v2 , . . . , vn+1 ) = ψ1 (v1 , v2 , . . . , vn+1 ).

(D.23)

Now the ψ 2 is totally determined. From above deducing procedure we can see that ψ 2 is a specifically linear combination of specific |Φn ’s and such linear combination is unique. Carrying on the similar procedure as we analyse the structure of ψ 2 , we can prove that 1 ψ , ψ 3 , ψ 4 are only the linear combination of the |Φn+1 , |Φn ’s, |Φn−1 ’s, respectively. It is not very hard to deduce the explicit expressions of ψ 1 , ψ 3 , ψ 4 . They are ψ 1 (v1 , v2 , . . . , vn+1 ) = ω2 (u)Λn+1 2 (u; {vm })|Φn+1 (v1 , . . . , vn+1 ), ψ

(3)

=

(D.24)

(v1 , v2 , . . . , vn+1 ) n+1  i−1 

α(vj , vi )[ω1 (vi )a42 (u, vi )Λn1 (vi ; vi , {vm })

i=1 j =1

+ ω2 (vi )a52 (u, vi )Λn2 (vi ; vi , {vm })]|Φn (v1 , . . . , vˇi , . . . , vn+1 ), ψ

(4)

=

(D.25)

(v1 , v2 , . . . , vn+1 ) n  n+1  i−1  i=1 j =i+1 k=1

α(vk , vi )

j −1

α(vl , vj )|Φn−1 (v1 , . . . , vˇi , . . . , vˇj , . . . , vn+1 )

l=1=i

 × H1D1 (u, vi , vj )Λn−1 1 (vi ; vi , vj , {vm }) × Λn−1 1 (vj ; vi , vj , {vm })ω1 (vi )ω1 (vj ) + H2D1 (u, vi , vj )Λn−1 2 (vi ; vi , vj , {vm }) × Λn−1 1 (vj ; vi , vj , {vm })ω2 (vi )ω1 (vj ) D

+ H3 1 (u, vi , vj )Λn−1 1 (vi ; vi , vj , {vm }) × Λn−1 2 (vj ; vi , vj , {vm })ω1 (vi )ω2 (vj ) + H4D1 (u, vi , vj )Λn−1 2 (vi ; vi , vj , {vm }) × Λn−1 2 (vj ; vi , vj , {vm })ω2 (vi )ω2 (vj ) .

(D.26)

G.-L. Li et al. / Nuclear Physics B 670 [FS] (2003) 401–438

437

In Eq. (D.26), the procedure of obtaining the coefficients of |Φn−1 (v2 , . . . , vˇj , . . . , vn+1 ) is the same as that we have carried out in calculating the two-particle state. At this step, all the ψ (i) are determined. We can see that Eq. (D.6) can be obtained from Eq. (92) with n replaced by n + 1, which means that our conclusion Eq. (92) still hold at the case n + 1. Although we cannot prove our assumption Eqs. (94), (95) so far, we can examine the conclusions obtained by using the assumption. We verify that the coefficients Kib2 in f in Eq. (D.12) to be zeroes can be simplified b1 , K Eq. (D.9) are zeroes indeed. Verifying K i ij into the same work as we have done in directly calculating the three-particle state, in which (3) 1 we have used the explicit expressions of ψc(2) s and ψcs inferred from Eq. (73). The ψ can be easily verified by six kinds of simple identities. Such terms B2 (v1 )|Φn−1  in ψ 3 are cancelled out by the similar identities to Kib2 in Eq. (D.9) and such terms B2 (v1 )|Φn−2  in ψ 4 Eq. (D.26) are also cancelled out by the same identities used in the three-particle state. 2 (u) acting on the The whole procedure above can be applied to the operators A(u), D (n + 1)-particle state, respectively. Similarly, given G(u)|Φn (v1 , . . . , vn ) = ψg(1) + B1 (u)ψg(2) + B2 (u)ψg(3) + F (u)ψg(4) ,

(D.27)

where ψg(1) is the linear combination of |Φn−2 ’s, ψg(2) and ψg(3) are respectively the linear combination of |Φn−3 ’s, ψg(4) is a linear combination of |Φn−4 ’s, and all (i) (i) the ψg , i = 1, 2, 3, 4 satisfy the following property ψcs (v1 , . . . , vj , vj +1 , . . . , vn ) = α(vj , vj +1 )ψc(i) s (v1 , . . . , vj +1 , vj , . . . , vn ), we can prove that Eqs. (D.1), (D.27) still hold at the n + 1. Here we omit this steps for the sake of simplicity.

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