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Open-boundary conditions for new integrable nineteen-vertex models
__ _If3
29 April 19%
CmJ
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 2 13 (I 996) 273-278
Open-boundary conditions for new integrable nineteen-vertex models Xi-Wen Guan a*b,Shan-De Yang b a CCAST (World Laboratory). P.O. Box 8730, Beijing 100080, Chino b Center for Theoretical Physics and Department of Physics, Jilin University, Changchun 130023, Chinu
’
Received 12 January 19%; accepted 29 January 1996 Communicated by A.R. Bishop
Abstract Using Sklyanin’s formalism, we prove that only two of four new integrable nineteen-vertex models exist with open boundary conditions (BC) compatible with integrability. By solving the reflection equation (RE), the diagonal boundary K matrices for the two models are obtained and the explicit forms of the Hamiltonians are given.
1. Introduction Recently there has been a considerable interest in the study of the lattice integrable models with open BC. This was initiated by Cherednik and Sklyanin [1,2]. In this case, the Yang-Baxter equation (YBE) is not a sufficient condition of integrability. To describe the BC compatible with integrability, Sklyanin has introduced a new algebraic structure called the reflection equation (RE),
uI +u2)~_(uz)
- 4~-(WM
R,,(u, R,,( -u,
+ ~~:I(+,z(-u,
2 =
KxU2)R12(
-
=y@x,
PIIn
SO3759601(96)00109-O
(1)
(2)
in the ith space; 1~ x 8 idV2, 2 = id,, @ x for any matrix
x E End(v), and the R
(3)
operator in vi 8 vj i.e. x, y E v,
(4)
I Mailing address. 0375.9601/%/$12.00
-u2).
- 4 -27&W
=R,2,
where Pij is the permutation P(x@y)
+u&‘_(u,)R,,(u,
u,-u2-2&‘(u,)R,,(-u,+u,).
Here ti denotes transposition matrix must have (i) P symmetry, P,,R,,P,,
=k_(u&(ul
Copyright 0 1996 Published
by Elsevier
Science
B.V. All rights reserved
X.-W. Gum, S.-D. Yung / Physics Letters A 213 (I 996) 273-278
274
and T symmetry R’,$( u) = R;$( u) ;
(5)
(ii) unitarity, R,z( “)R,z(
-u)
= P(U);
(6)
(iii) crossing unitarity, R;b( u)R’,i( --u - 271) = fi( u),
(7)
where p(u) and c(u) are some scalar functions and n is a constant characterizing the R matrix. For other kinds of R matrices. the generalized RE appear in Refs. [3] and [4]. It follows that the transfer matrices r(u) defined by T(U) = tr K+(u)T(u)K_(u)T-‘(
-u)
(8)
commute with each other for different values of the spectral parameter the generators T,,(u) ((-u, /3= 1,2,... ,dim W,, W, is the auxiliary (YBA) is R(u,
-u,)i-(u,)+(u,)
=&)+(u,)R(u,
u. Here the algebra T(u) is defined by space) and the Yang-Baxter algebra
-zQ).
(9)
Therefore, one may consider T(U) as a generating function of the integrals of motion for quantum systems. Now much attention has been paid to the solutions of the RE which present boundary K matrices compatible with integrability [5-71. In a recent paper 181, a complete list of exactly solvable nineteen-vertex models was presented. The list in Ref. [8] comprises several well-known models, e.g. the Fateev-Zamolodchikov [9] model, but also four non-trivial new ones. The four new models are solved by Klliper etc. in a recent paper [lo]. A natural problem is: do there exist boundary conditions compatible with integrability for these models? This is just the subject of the present paper. Using Skylanin’s formalism, we show that two of them exist obeying the BC. By solving the RE, the diagonal boundary K matrices are obtained, and the explicit forms of the Hamiltonians are given.
2. Models The three-state vertex models considered in Ref. [8] are defined on a square lattice where the spin variables are placed on the bonds. Each spin may take three values, say 0, + 1. The corresponding local Boltzmann weights are denoted by R,$T for a spin configuration as P P
+
V a
We impose the ice-rule a + E_C = /3 + v which leads to nineteen while the further symmetries R;L$=R-P-“=R;&=R”P -v-o
b=RI;;,
i.e. non-zero
Boltzmann
p = RO’ 019
d=R&
Pa
reduce them to only seven independent a=Rfl,
allowed,
c=R-”
weights: I-I,
e=RE,
g=R’-’00 3
weights,
X.-W. Guan. S.-D. Yang/Physics
and the corresponding I
a
0
Letters A 213 (1996) 273-278
275
R matrix is given by 0
0
0
0
0
0
0
OpOeOOOOO
OObOgOcOO OeOpOOOOO
OOgOdOgOO
R=
(10)
OOOOOpOeO
OOcOgObOO OOOOOeOpO
0
\
0
0
0
0
0
0
0
a
In this paper we shall consider the four new models, in Ref. [s] labeled as #2, #3, #8 and #9 which we label as I to IV, respectively. Model I a=c=d=cosh
u+cosh
71 sinh U,
b=g=O,
e= 1,
p=
sinh 7) sinh u;
Model II a=c=cosh
u+cosh
77 sinh U,
d=cosh
u-cash
7) sinh u,
e=
1,
p =
sinh v sinh u,
b=g=O; Model III 4-u4 a=31/2,
2(U4 - 1) b=
3u2
u4+2 )
a e=--,
c=3u2’
Ub S=F’
u
p=O,
d= 1;
Model IV cJ2(U4a=
CY’)
1 - u4 ’
+r+u‘y
b=
U2(a+U4)
U(-afU4) e=
a2((Y+U4)
U’( 1 - (Y’) ’
c=
a2(a+U4)
’
1 -u4 g=Jzu‘lti
’
U(a+P)
’
p=o,
d= 1,
where U = e’. For the given R matrices, one can find that all four R matrices satisfy P symmetry, (5), and the following unitarity condition, R,A U)R,,(
-u>
Eq. (3), T symmetry,
Eq.
= P(U)*
P(U) = 1 - sinh*r] si&u = $(4 - e4”)(4-
for model I and model II, e-4U)
(e4@ - a2)(e-4U
- a2)
for model III, for model IV
= a4(,+e4”)(a+e-4U) But the R matrices of model I and model II do not satisfy the crossing unitarity, Eq. (7). This means that there do not exist boundary conditions compatible with integrability for the two models in Sklyanin’s formalism. The R matrices of model III and model IV satisfy the following crossing unitarity condition, R’@)R’+u+2h)
=fi(u).
X.-W. Gum. S.-D. Yang/Physics
276
Letters A 213 (1996) 273-278
For model III h=ln
fi( U) = _!(eJll-
fi,
1)(e-4M”‘-
1),
and for model IV
Now we can look for solutions needs a slight change, i.e.
=
to Eq. (1) and Eq. (2) in order to find the open BC. But in our case, Eq. (2)
~~(U*)R,*(-U, -u,+2*)kJ(u,)R,*(-u,
(11)
fu,)
for ensuring integrability of the models. Eqs. (1) and (1 I> are equivalent, which maps K_ into K,,
because there exists an automorphism
4: K_(u)+K+(u)=K!(-u+h). We now look for the diagonal
K,(u)
Inserting
solutions
0
K2(U)
0
0
of the RE for model III and model IV in the form 0
0
1
K_(u)=
(12)
0
.
(‘3)
K3(U) I
the R matrix (10) (with p = 01 and K_(u) (13) into (I), one may obtain three equations,
i.e.
bc’K,K;+gg’K2K;+cb’K3K;=cb’K,K;+gg’K2K;+bc’K3K;,
(14)
bg’K, K; + gd’K, K; + cg’K, K; = gb’K, K; + dg’K, K; + gc’K, K;,
(15)
cg’K, K; i- gd’K, K; + bg’K, K; = gc’K, K; + dg’K, K; -I-gb’K, K;;
(16)
here X’ = x(u, + u,), x = X(U, - u2), K = K(u,), K’ = K(u,). Substituting each related Boltzmann weight in model III and model IV into Eqs. (14)-(161, respectively, and by analyzing the algebraic structures of the above equations and an algebraic calculation, we may obtain the boundary K_(u) matrices as follows: for model III,
l A- (ee4’ K_(u)
= \
t-)
0
0
0
A_e-2U( e4” + t_)
0
0
0
+
A_ (eM4’ +
\ ,
(17)
c-) ,
and for model IV
’ A_ ( aew4’ K_(u)
= \
- e4’ +
cr&_)
0
0
A_ee2”( ae4u - eC4’ + at_)
0
0
0 0
\ 9
A_ ( aem4’ - e4’ + a&) (18)
X.-W. &an.
S.-D. Yang/Physics
Letters A 213 (1996) 273-278
277
respectively. From Eq. (121, K, (u> for the two models are given by \
/ K+(u)
A+ (e4” + 6,)
0
0
0
A+e2”‘(eC4”’ + 6,)
0
0
0
= \
A + (e41r’+ {+ ) ,
and
0
A+e2u’( ae-4rr’ - e4’ +
0
a[+)
0
A+( ae4’ - ew4” + a[+)
0
(20) respectively, where u’ = u - A. We note that the diagonal boundary K matrices for the two models depend on two continuous parameters.
3. Hamiltonians We now consider the Hamiltonians of systems with boundary conditions. We assume that the L matrix L(U) coincides with the R matrix R(u) in the space W, Q W,, kl( u) = R&C u) 9 and R(u) and K_(u) satisfy R,,,(O) = Pm, and K_(O) = 1, respectively. It follows that there exists a relation between the transfer matrix T(U) and the Hamiltonian for the lattice spin open chain: r’(0) = 2H tr K+(O) + tr K’(O),
(21)
where N-l
H =
Hj,j+, +;
c
tr P,,K’(O)P,’
+
tr K+ (0) L’No(o)pid
j= I
tr K+(O)
(22)
with Hj.j+
I =
'j,j+
ILj,j+
I(O)*
prime denotes the derivative with respect to the spectral parameter u. By straightforward calculation, we may obtain the following explicit forms of the Hamiltonians, Here
he
N-l
H =
c j=
Hj,j,,
+ b.t.
1
for model III 3Hj.j. 1= 4(2 - ~)SjSj+ +4(2+1/2)&S;+, -5[(S;)*1,+,
! + 4~(
SjSj+ I)* - 4(2 - ~)(
+SySy,,)‘+Z,(S;+,)*]
Sj”Sj”i1+ ‘i”~+ 1)
4(2 - fi)S;S;+,
- 161,
b.t. = ~[-(3-~-)(s:)*+(l-b)z,]-~[~3-5+~~~~~*+acI+45,~1,1. +
+ 4(4+ fi)(sjLS;+,)*
X.-W. Guctn, S.-D.
278
Yang/Physics
Letters
A 213 (1996)
273-278
and for model IV Hj.j+
I = 4( Jl+cy
- 4(iE
- “)S,S,+ , +4JiTz(SjSj+,)*-4(~-cr)(S;S:, + ff)(s;s;+,
+ sj’s/!+ J2 + (3cu + 2 - 4Jl+cy)S;S;+,
+(3a+4_4~~)(~~S,;,,)‘+(5_6~~)[(SIL~~~j+i
3 b.t. = 5--2
](
+s;sjyt,)
+‘j(‘ltl)‘]
-‘rr’,
-8-5u+~_)(Sf)2+(4+3a-5-)1,]
(1 f2u)A+ +
2cu-3+(1
+2ar)5+
[(-8-5a+St)(S~)*+2j4-2a)(2+5+)1,].
respectively, where S’ (i = x, y, z) are spin-l generators of ~$2).
4, Conclusion We have studied the boundary conditions compatible with integrability for the four new nineteen-vertex models and found the diagonal solutions of the RE for two of the models, and obtained the Hamiltonians with nontrivial boundary terms. We note that the boundary terms of the Hamiltonians include quadratic terms (S’j2. This relates to the relations between the elements of the K-matrix and this is different from the six-vertex model [S]. For finding the BC compatibly with integrability for mode1 I and model II, maybe we must use the generalized RE [3,41 or another generalized RE. However, how to diagonalize the Hamiltonians is a problem to be solved. One way may be to use Sklyanin’s formalism and a fusion procedure [ 11,121.
Acknowledgement
X.-W. Guan would like to thank Professor H.-Q. Zhou for useful discussions.
References [I] I.V. Chetednik, Theor. Math. Phys. 61 (1984) 911. (21 E.K. Sklyanin, J. Phys. A 21 (1988) 2375. [3] L. Mezincescu and R.I. Nepomechi, J. Phys. A 24 (1991) L17. [4] R.H. Yue and Y.X. Chen, J. Phys. A 26 (1993) 2989. [5] H.I. de Vega and A. Gonzalez-Ruiz, J. Phys. A 27 (1994) 6129; 26 (1993) LS19. (61 B.Y. Hou and R.H. Yue Phys. Lett. A 183 (1993) 167. 171 C.M. Yung and M.T. Batchelor, Phys. Lett. A 198 (1995) 395. (81 M. Idzumi, T. Tokihiro and M. Arai, J. Phys. I (Paris) 4 (1994) 1151. [9] A.B. Zamolodchikov and V.A. Fateev, Sov. J. Nucl. Phys. 32 (1980) 298. IO] A. Kliiper, 81. Matveenko and J. Zittartz, Z. Phys. B 96 (1995) 401. Ill P.P. Kulish and E.K. Sklyanin, in: Lecture notes in physics, Vol. 151 (Springer, Berlin, 121 L. Mezincescu. R.I. Nepomechie and U. Rittenberg, Phys. Lett. A 147 (1990) 70.
1982) p. 61,
29 April 19%
CmJ
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 2 13 (I 996) 273-278
Open-boundary conditions for new integrable nineteen-vertex models Xi-Wen Guan a*b,Shan-De Yang b a CCAST (World Laboratory). P.O. Box 8730, Beijing 100080, Chino b Center for Theoretical Physics and Department of Physics, Jilin University, Changchun 130023, Chinu
’
Received 12 January 19%; accepted 29 January 1996 Communicated by A.R. Bishop
Abstract Using Sklyanin’s formalism, we prove that only two of four new integrable nineteen-vertex models exist with open boundary conditions (BC) compatible with integrability. By solving the reflection equation (RE), the diagonal boundary K matrices for the two models are obtained and the explicit forms of the Hamiltonians are given.
1. Introduction Recently there has been a considerable interest in the study of the lattice integrable models with open BC. This was initiated by Cherednik and Sklyanin [1,2]. In this case, the Yang-Baxter equation (YBE) is not a sufficient condition of integrability. To describe the BC compatible with integrability, Sklyanin has introduced a new algebraic structure called the reflection equation (RE),
uI +u2)~_(uz)
- 4~-(WM
R,,(u, R,,( -u,
+ ~~:I(+,z(-u,
2 =
KxU2)R12(
-
=y@x,
PIIn
SO3759601(96)00109-O
(1)
(2)
in the ith space; 1~ x 8 idV2, 2 = id,, @ x for any matrix
x E End(v), and the R
(3)
operator in vi 8 vj i.e. x, y E v,
(4)
I Mailing address. 0375.9601/%/$12.00
-u2).
- 4 -27&W
=R,2,
where Pij is the permutation P(x@y)
+u&‘_(u,)R,,(u,
u,-u2-2&‘(u,)R,,(-u,+u,).
Here ti denotes transposition matrix must have (i) P symmetry, P,,R,,P,,
=k_(u&(ul
Copyright 0 1996 Published
by Elsevier
Science
B.V. All rights reserved
X.-W. Gum, S.-D. Yung / Physics Letters A 213 (I 996) 273-278
274
and T symmetry R’,$( u) = R;$( u) ;
(5)
(ii) unitarity, R,z( “)R,z(
-u)
= P(U);
(6)
(iii) crossing unitarity, R;b( u)R’,i( --u - 271) = fi( u),
(7)
where p(u) and c(u) are some scalar functions and n is a constant characterizing the R matrix. For other kinds of R matrices. the generalized RE appear in Refs. [3] and [4]. It follows that the transfer matrices r(u) defined by T(U) = tr K+(u)T(u)K_(u)T-‘(
-u)
(8)
commute with each other for different values of the spectral parameter the generators T,,(u) ((-u, /3= 1,2,... ,dim W,, W, is the auxiliary (YBA) is R(u,
-u,)i-(u,)+(u,)
=&)+(u,)R(u,
u. Here the algebra T(u) is defined by space) and the Yang-Baxter algebra
-zQ).
(9)
Therefore, one may consider T(U) as a generating function of the integrals of motion for quantum systems. Now much attention has been paid to the solutions of the RE which present boundary K matrices compatible with integrability [5-71. In a recent paper 181, a complete list of exactly solvable nineteen-vertex models was presented. The list in Ref. [8] comprises several well-known models, e.g. the Fateev-Zamolodchikov [9] model, but also four non-trivial new ones. The four new models are solved by Klliper etc. in a recent paper [lo]. A natural problem is: do there exist boundary conditions compatible with integrability for these models? This is just the subject of the present paper. Using Skylanin’s formalism, we show that two of them exist obeying the BC. By solving the RE, the diagonal boundary K matrices are obtained, and the explicit forms of the Hamiltonians are given.
2. Models The three-state vertex models considered in Ref. [8] are defined on a square lattice where the spin variables are placed on the bonds. Each spin may take three values, say 0, + 1. The corresponding local Boltzmann weights are denoted by R,$T for a spin configuration as P P
+
V a
We impose the ice-rule a + E_C = /3 + v which leads to nineteen while the further symmetries R;L$=R-P-“=R;&=R”P -v-o
b=RI;;,
i.e. non-zero
Boltzmann
p = RO’ 019
d=R&
Pa
reduce them to only seven independent a=Rfl,
allowed,
c=R-”
weights: I-I,
e=RE,
g=R’-’00 3
weights,
X.-W. Guan. S.-D. Yang/Physics
and the corresponding I
a
0
Letters A 213 (1996) 273-278
275
R matrix is given by 0
0
0
0
0
0
0
OpOeOOOOO
OObOgOcOO OeOpOOOOO
OOgOdOgOO
R=
(10)
OOOOOpOeO
OOcOgObOO OOOOOeOpO
0
\
0
0
0
0
0
0
0
a
In this paper we shall consider the four new models, in Ref. [s] labeled as #2, #3, #8 and #9 which we label as I to IV, respectively. Model I a=c=d=cosh
u+cosh
71 sinh U,
b=g=O,
e= 1,
p=
sinh 7) sinh u;
Model II a=c=cosh
u+cosh
77 sinh U,
d=cosh
u-cash
7) sinh u,
e=
1,
p =
sinh v sinh u,
b=g=O; Model III 4-u4 a=31/2,
2(U4 - 1) b=
3u2
u4+2 )
a e=--,
c=3u2’
Ub S=F’
u
p=O,
d= 1;
Model IV cJ2(U4a=
CY’)
1 - u4 ’
+r+u‘y
b=
U2(a+U4)
U(-afU4) e=
a2((Y+U4)
U’( 1 - (Y’) ’
c=
a2(a+U4)
’
1 -u4 g=Jzu‘lti
’
U(a+P)
’
p=o,
d= 1,
where U = e’. For the given R matrices, one can find that all four R matrices satisfy P symmetry, (5), and the following unitarity condition, R,A U)R,,(
-u>
Eq. (3), T symmetry,
Eq.
= P(U)*
P(U) = 1 - sinh*r] si&u = $(4 - e4”)(4-
for model I and model II, e-4U)
(e4@ - a2)(e-4U
- a2)
for model III, for model IV
= a4(,+e4”)(a+e-4U) But the R matrices of model I and model II do not satisfy the crossing unitarity, Eq. (7). This means that there do not exist boundary conditions compatible with integrability for the two models in Sklyanin’s formalism. The R matrices of model III and model IV satisfy the following crossing unitarity condition, R’@)R’+u+2h)
=fi(u).
X.-W. Gum. S.-D. Yang/Physics
276
Letters A 213 (1996) 273-278
For model III h=ln
fi( U) = _!(eJll-
fi,
1)(e-4M”‘-
1),
and for model IV
Now we can look for solutions needs a slight change, i.e.
=
to Eq. (1) and Eq. (2) in order to find the open BC. But in our case, Eq. (2)
~~(U*)R,*(-U, -u,+2*)kJ(u,)R,*(-u,
(11)
fu,)
for ensuring integrability of the models. Eqs. (1) and (1 I> are equivalent, which maps K_ into K,,
because there exists an automorphism
4: K_(u)+K+(u)=K!(-u+h). We now look for the diagonal
K,(u)
Inserting
solutions
0
K2(U)
0
0
of the RE for model III and model IV in the form 0
0
1
K_(u)=
(12)
0
.
(‘3)
K3(U) I
the R matrix (10) (with p = 01 and K_(u) (13) into (I), one may obtain three equations,
i.e.
bc’K,K;+gg’K2K;+cb’K3K;=cb’K,K;+gg’K2K;+bc’K3K;,
(14)
bg’K, K; + gd’K, K; + cg’K, K; = gb’K, K; + dg’K, K; + gc’K, K;,
(15)
cg’K, K; i- gd’K, K; + bg’K, K; = gc’K, K; + dg’K, K; -I-gb’K, K;;
(16)
here X’ = x(u, + u,), x = X(U, - u2), K = K(u,), K’ = K(u,). Substituting each related Boltzmann weight in model III and model IV into Eqs. (14)-(161, respectively, and by analyzing the algebraic structures of the above equations and an algebraic calculation, we may obtain the boundary K_(u) matrices as follows: for model III,
l A- (ee4’ K_(u)
= \
t-)
0
0
0
A_e-2U( e4” + t_)
0
0
0
+
A_ (eM4’ +
\ ,
(17)
c-) ,
and for model IV
’ A_ ( aew4’ K_(u)
= \
- e4’ +
cr&_)
0
0
A_ee2”( ae4u - eC4’ + at_)
0
0
0 0
\ 9
A_ ( aem4’ - e4’ + a&) (18)
X.-W. &an.
S.-D. Yang/Physics
Letters A 213 (1996) 273-278
277
respectively. From Eq. (121, K, (u> for the two models are given by \
/ K+(u)
A+ (e4” + 6,)
0
0
0
A+e2”‘(eC4”’ + 6,)
0
0
0
= \
A + (e41r’+ {+ ) ,
and
0
A+e2u’( ae-4rr’ - e4’ +
0
a[+)
0
A+( ae4’ - ew4” + a[+)
0
(20) respectively, where u’ = u - A. We note that the diagonal boundary K matrices for the two models depend on two continuous parameters.
3. Hamiltonians We now consider the Hamiltonians of systems with boundary conditions. We assume that the L matrix L(U) coincides with the R matrix R(u) in the space W, Q W,, kl( u) = R&C u) 9 and R(u) and K_(u) satisfy R,,,(O) = Pm, and K_(O) = 1, respectively. It follows that there exists a relation between the transfer matrix T(U) and the Hamiltonian for the lattice spin open chain: r’(0) = 2H tr K+(O) + tr K’(O),
(21)
where N-l
H =
Hj,j+, +;
c
tr P,,K’(O)P,’
+
tr K+ (0) L’No(o)pid
j= I
tr K+(O)
(22)
with Hj.j+
I =
'j,j+
ILj,j+
I(O)*
prime denotes the derivative with respect to the spectral parameter u. By straightforward calculation, we may obtain the following explicit forms of the Hamiltonians, Here
he
N-l
H =
c j=
Hj,j,,
+ b.t.
1
for model III 3Hj.j. 1= 4(2 - ~)SjSj+ +4(2+1/2)&S;+, -5[(S;)*1,+,
! + 4~(
SjSj+ I)* - 4(2 - ~)(
+SySy,,)‘+Z,(S;+,)*]
Sj”Sj”i1+ ‘i”~+ 1)
4(2 - fi)S;S;+,
- 161,
b.t. = ~[-(3-~-)(s:)*+(l-b)z,]-~[~3-5+~~~~~*+acI+45,~1,1. +
+ 4(4+ fi)(sjLS;+,)*
X.-W. Guctn, S.-D.
278
Yang/Physics
Letters
A 213 (1996)
273-278
and for model IV Hj.j+
I = 4( Jl+cy
- 4(iE
- “)S,S,+ , +4JiTz(SjSj+,)*-4(~-cr)(S;S:, + ff)(s;s;+,
+ sj’s/!+ J2 + (3cu + 2 - 4Jl+cy)S;S;+,
+(3a+4_4~~)(~~S,;,,)‘+(5_6~~)[(SIL~~~j+i
3 b.t. = 5--2
](
+s;sjyt,)
+‘j(‘ltl)‘]
-‘rr’,
-8-5u+~_)(Sf)2+(4+3a-5-)1,]
(1 f2u)A+ +
2cu-3+(1
+2ar)5+
[(-8-5a+St)(S~)*+2j4-2a)(2+5+)1,].
respectively, where S’ (i = x, y, z) are spin-l generators of ~$2).
4, Conclusion We have studied the boundary conditions compatible with integrability for the four new nineteen-vertex models and found the diagonal solutions of the RE for two of the models, and obtained the Hamiltonians with nontrivial boundary terms. We note that the boundary terms of the Hamiltonians include quadratic terms (S’j2. This relates to the relations between the elements of the K-matrix and this is different from the six-vertex model [S]. For finding the BC compatibly with integrability for mode1 I and model II, maybe we must use the generalized RE [3,41 or another generalized RE. However, how to diagonalize the Hamiltonians is a problem to be solved. One way may be to use Sklyanin’s formalism and a fusion procedure [ 11,121.
Acknowledgement
X.-W. Guan would like to thank Professor H.-Q. Zhou for useful discussions.
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