- Email: [email protected]
Integrable eight-state supersymmetric U model with boundary terms and its Bethe ansatz solution
9 February 1998
PHYSICS
ELSEVIER
Physics Letters A 238 (1998)
LETTERS
A
309-314
Integrable eight-state supersymmetric U model with boundary terms and its Bethe ansatz solution Xiang-Yu Ge ‘, Mark D. Gould, Yao-Zhong Zhang 2, Huan-Qiang Department
of Mathematics,
Univemity of Queensland.
Zhou ’
Brisbane. Queenslund 4072. Amwulirr
Received 30 September 1997; accepted for publication 9 December
1997
Communicated by A.R. Bishop
Abstract A class of integrable boundary terms for the eight-state supersymmetric U model are presented by solving the graded reflection equations. The boundary model is solved by using the coordinate Bethe ansatz method and the Bethe ansatz equations are obtained. @ 1998 Elsevier Science B.V. PAC’S: 7S.lO.Jm; 75.1O.L~
Integrable systems with boundary interactions are a recent achievement that we think deserves careful investigations. Here we are only concerned with one-dimensional open-boundary lattice integrable models of strongly correlated electrons. Such lattice models can be treated by Sklyanin’s boundary quantum inverse scattering method (QISM) [ I] or its generalizations [ 2-51. More specifically, we present integrable boundary terms for the eight-state version of the supersymmetric U model recently introduced in Ref. [6]. The bulk Hamiltonian describes a supersymmetric electron model with correlated single-particle and pair hoppings as well as uncorrelated triple-particle hopping. So the model foran open lattice, which we consider here, involves many physically interesting processes with boundary interactions. The boundary model is solved by means of the coordinate Bethe ansatz method and the Bethe ansatz equations are derived. Let c,:.~ (c.;., ) denote a fermionic creation (annihilation) operator which creates (annihilates) an electron of species LY(= +, 0, -) at site j. They satisfy the anti-commutation where i.,j = 1,2.. . , L and CY,p = f, 0, -. We consider the following
relations given by {c/,~~, ci,~} = 6,,S,,, Hamiltonian with boundary terms,
’ E-mail:[email protected]. ’ Queen Elizabeth ‘On
II Fellow. E-mail: [email protected].
leave of absence from Department of Physics, Chongqing University. Chongqing 630044,
03759601/98/$19.00 P/I SO375960
@
1998 Elsevier Science B.V. All rights reserved
I (97)00959-6
China. E-mail: [email protected].
X.-Y. Ge et al./Physics
310
where H.&j:, is the Hamiltonian
-
2(g + I>
nZP+y
c
A 238 (1998) 309-311
density of the eight-state
H.K1 (g) = - C(cLc.t+I.n +h*c.)w * 1
Letten
-
U model [ 61,
supersymmetric
5C
(n,j,p
+fZj+l,P)
+
5
+
h.c.) exp
(c~,+c~,~c~,-Cj+l.-Cj+~,oc.~+~,+
+
’
(nj,pn,i.y
fz~,+l,~n,i+,,y~)
+
P#Y( +a)
PC+a)
(C,~,C,~~C.;+I,BC.~+I.~
C
(
- $‘l.iJ
+
ni+l,r)
>
(g
+
I:(,+
2)
h.c.)
+
C(“j,a n
+
~,j+l,~)
2
1 c (n.Mr.i,P + 1) azp
2(g+
nj+l.anj+l.p)
+
(rz.~~+f~.Pi.- +
l)Cg+2)
(gf
(2)
ni+l,+n,+l~onj+l._)
with g gs- 1’
77=-lnand Hpndary,
5 = ln(g t 1) - i lng(g + 21,
H);pundary are boundary
HL boundary
_
boundary _
H,
---
&?
2 - 5+
-
g+2’
terms 8
2 - -_(~zl+nlo 5-
-- 2 -2&?[-
&=--lnL
z5+
+
111012~_ +
?zl+n,__)
+
5_(2+~_)~ll+~z10n1-
8 (m.+nL0 + nL0nL- + nL+nL- ) + 5+ (2 + 5+)
>
nL+aOnL-
1
.
t In the above equations, njU is the number density operator nja = cjacj,, n., = n,+ -I n.jo + n,j-; c* are some parameters describing the boundary effects. As was shown in Ref. [ 61, the supersymmetry algebra underlying the bulk model is g1( 311). The boundary terms may spoil this g1( 311) supersymmetry, leaving the boundary model with a smaller symmetry algebra. If one projects out any one species, then the projected Hamiltonian is nothing but that of the supersymmetric I/ model with boundary terms [7] with the following identification of the parameters in the two models: u = f2/(g+ 1). We will establish the quantum integrability of the Hamiltonian for the boundary eight-state supersymmetric I/ model ( 1) by showing that it can be derived from the (graded) boundary quantum inverse scattering method. Let us first of all recall that the Hamiltonian of the eight-state supersymmetric lJ model with the periodic boundary conditions commutes with the bulk transfer matrix, which is the supertrace of the monodromy matrix T(u), T(u)
=RoL(u)...RoI(u),
(5)
where the subscript 0 denotes the auxiliary superspace V = C 4,4. It should be noted that the supertrace out for the auxiliary superspace V. The R-matrix R(u) s P&(U), where P is the graded permutation is given by
(U+2gHrc+2g+2)P
u + 2g p2+ u - 2g
ii(u) = P, - -
(u - 2g)(u
- 2g - 2)
_ (tz+2g)(u+2g+2)(u+2g+4) s
(u-2g)(u-2g-2)(u-2g-4P
is carried operator,
”
(6)
where 4, k = 1,2,3,4, are four projection operators whose explicit formulae can be found in Ref. [6]. The elements of the supermatrix T(u) are the generators of an associative superalgebra A defined by the relations RI~(UI
-
~2)
+
(UI)
+
(~2)
=
+
(~2)
+
(UI)RI~(UI
-
u2),
(7)
X.-Y
Ge et al./Physics
Letters
A 238
(19981
309-314
31
I
2
where ;( G X 8 1, X = 1 @ X for any supermatrix X E End(V). For later use, we list some useful properties enjoyed by the R-matrix: (i) Unitarity: Rr2( 14)RZI(-u) = 1 and (ii) crossing-unitarity: Rs: ( --II + 4) h’yj (u) = b(u) with p(u) being a scalar function. In order to construct integrable electronic models with open boundary conditions, we introduce the following graded reflection equations (REs) that the so-called boundary K-(super) matrices K* (II)satisfy,
where the supertransposition st, (/J = I, 2) is only carried out in the ,uth factor superspace of V ;QV. whereas ist, denotes the inverse operation of st,. Following Sklyanin’s arguments [ I], one may show that the quantity 7-(/l) given by I_(r,)
=T(u)K_(u)T-'(-u)
satisfies the same relation
as K_
( IO)
(24):
R,?(u,- ~12) ;- (zI,)R~~(u~ +Uz) +- (uz)=;- (&)Rl2(ti1 +U?) +. Thus if one defines the boundary ~(11) = str(K+(li)I_(u))
transfer matrix r(u)
as
=str(K+(u)T(u)K_(u)T-‘(-u)).
then it can be shown [5] that ]7(NI
),T(U?)l = 0.
We now solve (8) and (9) for K+(u) and K_(u).Let us restrict ourselves one may check that the matrix K_ (10 given by
K_(u) =
1 5-(2-5_)(2f5_>
'A_(u) 0 0 0 0 0 0 \ 0
0 B_(U)
0 0 0 0 0 0
0
0
0
0
to the diagonal
solutions.
Then.
0 0 0 0 0 0 0 0 B_(u) 0 0 0 0 0 0 B_(u) 0 0 0 0 0 0 C_(M) 0 0 0 0 0 0 C_(u) 0 0 0 0 0 0 C_(u) 0 0 0 0 0 0 D_(u) ( 14)
where
A_(u) =(-[_ +t1)(2-5_
+u)(-2-5-
+B),
-u)(-2-e_
+u),
-u)(-2-l_
-10,
B_(U)=(-&_+U)(2-~_-U)(-2-~-+II), C_(u)
=(-t_
-u)(2-5_
D_(l4)=(-[_ -u)(2-5_
(15)
312
X.-r
Ge et d/Physics
Lettm
A 238 (1998) 309-314
satisfies (8). The matrix K+(u) can be obtained from the isomorphism K_(u) of (8), then K+(u) defined by K:‘(u)
of the two REs. Indeed, given a solution
= K_(-uf2)
( 16)
is a solution of (9). The proof follows from some algebraic computations upon substituting ( 16) into (9) and making use of the properties of the R-matrix. Therefore, one may choose the boundary matrix K+(u) as
A+(u) 0 0 0 0 0 0 0
K+(u) =
0 B+(u) 0 0 0 0 0 0
0
0 0
0
B+(u )O B+(u) 0 0
0 0 0 0 0
0 0 0 0 C+(u) 0
0 0 0 0 0 C+(u)
0 0
0 0
0 0
0 0 0 0 0 0
0
0 0 0 0 0 C+(u) 0
( 17)
0 D+(u),
where A+(u)=(-2g+2+5+-u)(-2g+5+-u)(-2g-2+~+-u), B+(il)=(-2g-2+!$++u)(-2g+<+-u)(-2g-2+5+-u). C+(u)
= (-2g-2+~++u)(-2g-4+~++u)(-2g-2+~+
-u),
D+(u)=(-2g-2+t++u)(-2g-4+~++u)(-2g-6+~++u).
(18)
Now it can be shown that Hamiltonian (1) is related to the second derivative r(u) (up to an unimportant additive constant)
of the boundary
transfer matrix
H = 2g HR, HR =
r”(0) 4(V+2W) L-l
=c
H.tj+l
./=I
x [strc (i+
+$tr-to) + (0)Gr.a)
2(v
+2stro
t
2w)
(““-‘+ (O)@a)
fstro
(g+
(0) (Hfc)2)]3
(19)
where V = straKi(O),
W=stra
(k+
(O)@a)
7
H,!j = Pi,jRI,i(O),
G,,,j = Pi,,jR:lj(O).
(20)
Here Pi.j denotes the graded permutation operator acting on the ith and jth quantum spaces. ( 19) implies that the boundary eight-state supersymmetric ZJ model admits an infinite number of conserved currents which are in involution with each other, thus assuring its integrability. It should be emphasized that the Hamiltonian ( 1) appears as the second derivative of the boundary transfer matrix r(u) with respect to the spectral parameter u at u = 0. This is due to the fact that the supertrace of K+(O) equals zero. The reason for the zero supertrace of K+(O) is related to the fact that the quantum space is the eight-dimensional typical irreducible representation of gl( 3 11) . A similar situation has already appeared in many spin chain models [4,7,5].
X.-Y Ge et d/Physics
Letters A 238 (1998) 309-314
313
Having established the quantum integrability of the boundary model, we now solve it by using the coordinate space Bethe ansatz method. The whole procedure is similar to that for other models [8,7]. The Bethe ansatz equations are
c ““2’L-‘)~(kl;p,)~(k,;pL)
=
nM’8.-A~‘)+ic/28i+Al”+ic/2 .’
,,=, 8.j - Aa’ - ic/2 0, + AjJ ) - k/C! ’
Al;’’ - Oi + ic/2 A,$” + 0.i + ic/2
2
A:;’ - AL’) + ic A!:’ + AL” + ic
,=, A”’ (r - Bi - ic/2 Al;” + Bi - ic/2 = - p=, Aj;” -- AL” - ic Al:’ + AL” ~ ic x
5 :i: -
Ar'
-
ic/2 AL’) + AY) - ic/2,
CT= l,...,
/>=, (1 - Ab2) + ic/2 Aa’ + A;" + k/2 MI A’” - AL” + ic/2 A:” + A;‘) + ic/2 Y n/,=, A;” - A”’ P - ic/2 Ab2’ + Ai” - ic/2 -
Mu
A:"
_ Ab"
M,.
+ ic AY)
+ Ab2) + ic
A?) - Ah’) - ic At” + Ah’) _ ic’
y=
l.....M’,
(21) where
mP)=‘-pe-i”p,=_,_&, 1 - p elk ’
and c = eq -
I; the charge
rapidities
pL=_,
8i E O( k.i) are related
(22)
2zgt.; .t
to the single-particle
quasi-momenta
k, by
O(k) = $ tan(k/2). The energy eigenvalue E of the model is given by E = -2 Es, cos k, (modular an unimportant additive constant coming from the chemical potential term). In conclusion, we have studied integrable open-boundary conditions for the eight-state supersymmetric CI model. The quantum integrability of the system follows from the fact that the Hamiltonian may be embedded into a one-parameter family of commuting transfer matrices. Moreover, the Bethe ansatz equations are derived by use of the coordinate space Bethe ansatz approach. This provides us with a basis for computing the finite size corrections (see, e.g., Refs. [ 8,9] ) to the low-lying energies in the system, which in turn allow us to use the boundary conformal field theory technique to study the critical properties of the boundary. Details will lx treated in a separate publication. This work is supported by the Australian Research Council, University of Queensland New Staff Research Grant and Enabling Research Grant. H.-Q.Z. would like to thank the Department of Mathematics of the University of Queensland for their kind hospitality. He also thanks the National Natural Science Foundation ot China and the Sichuan Young Investigators Science and Technology Fund for support.
References I I 1 E.K. Sklyanin, J. Phys. A 21 (1988)
121L. Mezincescu, R. Nepomechie, 13I H.J. de Vega, A. GonGlez-Ruiz, A. Gonziilez-Ruiz.
2375.
J. Phys. A 24 (1991)
Ll7.
J. Phys. A 26 ( 1993) LS19;
Nucl. Phys. B 424 ( 1994) 553.
141 H.-Q. Zhou, Phys. Rev. B 54 ( 1996) 41; 53 ( 1996) 5089.
15I
A.J. Bracken. X.Y. Ge, Y.-Z. Zhang, H.-Q. Zhou, An open-boundary integrable model of three coupled XY spin chains, and Integrable open-boundary preprints.
conditions for the y-deformed
supersymmetric
Cr model of strongly correlated electrons, University
of Queenslnnd
314
X.-Y Ge et (11./Ph.ysics Letter-s A 238 (1998) 309-314
161 M.D. Gould, Y.-Z. Zhang, H.-Q. Zhou, preprint cond-mat/9709120. 17 ) Y.-Z. Zhang, H.-Q. Zhou, preprint cond-m&9707263. 18 1 H. Asakawa, M. Suzuki. J. Phys. A 29 (1996) 225; M. Shiroishi, M. Wadati. J. Phys. Sot. Jpn. 66 i 1977) I. 191 CM. Yung, M.T. Batchelor, Nucl. Phys. B 435 ( 1995) 430; F.H. Essler, J. Phys. A 29 (1996) 6183: G. Bediirftig, EH. Essler. H. Frahm. Phys. Rev. Lett. 77 (1996) 5098
PHYSICS
ELSEVIER
Physics Letters A 238 (1998)
LETTERS
A
309-314
Integrable eight-state supersymmetric U model with boundary terms and its Bethe ansatz solution Xiang-Yu Ge ‘, Mark D. Gould, Yao-Zhong Zhang 2, Huan-Qiang Department
of Mathematics,
Univemity of Queensland.
Zhou ’
Brisbane. Queenslund 4072. Amwulirr
Received 30 September 1997; accepted for publication 9 December
1997
Communicated by A.R. Bishop
Abstract A class of integrable boundary terms for the eight-state supersymmetric U model are presented by solving the graded reflection equations. The boundary model is solved by using the coordinate Bethe ansatz method and the Bethe ansatz equations are obtained. @ 1998 Elsevier Science B.V. PAC’S: 7S.lO.Jm; 75.1O.L~
Integrable systems with boundary interactions are a recent achievement that we think deserves careful investigations. Here we are only concerned with one-dimensional open-boundary lattice integrable models of strongly correlated electrons. Such lattice models can be treated by Sklyanin’s boundary quantum inverse scattering method (QISM) [ I] or its generalizations [ 2-51. More specifically, we present integrable boundary terms for the eight-state version of the supersymmetric U model recently introduced in Ref. [6]. The bulk Hamiltonian describes a supersymmetric electron model with correlated single-particle and pair hoppings as well as uncorrelated triple-particle hopping. So the model foran open lattice, which we consider here, involves many physically interesting processes with boundary interactions. The boundary model is solved by means of the coordinate Bethe ansatz method and the Bethe ansatz equations are derived. Let c,:.~ (c.;., ) denote a fermionic creation (annihilation) operator which creates (annihilates) an electron of species LY(= +, 0, -) at site j. They satisfy the anti-commutation where i.,j = 1,2.. . , L and CY,p = f, 0, -. We consider the following
relations given by {c/,~~, ci,~} = 6,,S,,, Hamiltonian with boundary terms,
’ E-mail:[email protected]. ’ Queen Elizabeth ‘On
II Fellow. E-mail: [email protected].
leave of absence from Department of Physics, Chongqing University. Chongqing 630044,
03759601/98/$19.00 P/I SO375960
@
1998 Elsevier Science B.V. All rights reserved
I (97)00959-6
China. E-mail: [email protected].
X.-Y. Ge et al./Physics
310
where H.&j:, is the Hamiltonian
-
2(g + I>
nZP+y
c
A 238 (1998) 309-311
density of the eight-state
H.K1 (g) = - C(cLc.t+I.n +h*c.)w * 1
Letten
-
U model [ 61,
supersymmetric
5C
(n,j,p
+fZj+l,P)
+
5
+
h.c.) exp
(c~,+c~,~c~,-Cj+l.-Cj+~,oc.~+~,+
+
’
(nj,pn,i.y
fz~,+l,~n,i+,,y~)
+
P#Y( +a)
PC+a)
(C,~,C,~~C.;+I,BC.~+I.~
C
(
- $‘l.iJ
+
ni+l,r)
>
(g
+
I:(,+
2)
h.c.)
+
C(“j,a n
+
~,j+l,~)
2
1 c (n.Mr.i,P + 1) azp
2(g+
nj+l.anj+l.p)
+
(rz.~~+f~.Pi.- +
l)Cg+2)
(gf
(2)
ni+l,+n,+l~onj+l._)
with g gs- 1’
77=-lnand Hpndary,
5 = ln(g t 1) - i lng(g + 21,
H);pundary are boundary
HL boundary
_
boundary _
H,
---
&?
2 - 5+
-
g+2’
terms 8
2 - -_(~zl+nlo 5-
-- 2 -2&?[-
&=--lnL
z5+
+
111012~_ +
?zl+n,__)
+
5_(2+~_)~ll+~z10n1-
8 (m.+nL0 + nL0nL- + nL+nL- ) + 5+ (2 + 5+)
>
nL+aOnL-
1
.
t In the above equations, njU is the number density operator nja = cjacj,, n., = n,+ -I n.jo + n,j-; c* are some parameters describing the boundary effects. As was shown in Ref. [ 61, the supersymmetry algebra underlying the bulk model is g1( 311). The boundary terms may spoil this g1( 311) supersymmetry, leaving the boundary model with a smaller symmetry algebra. If one projects out any one species, then the projected Hamiltonian is nothing but that of the supersymmetric I/ model with boundary terms [7] with the following identification of the parameters in the two models: u = f2/(g+ 1). We will establish the quantum integrability of the Hamiltonian for the boundary eight-state supersymmetric I/ model ( 1) by showing that it can be derived from the (graded) boundary quantum inverse scattering method. Let us first of all recall that the Hamiltonian of the eight-state supersymmetric lJ model with the periodic boundary conditions commutes with the bulk transfer matrix, which is the supertrace of the monodromy matrix T(u), T(u)
=RoL(u)...RoI(u),
(5)
where the subscript 0 denotes the auxiliary superspace V = C 4,4. It should be noted that the supertrace out for the auxiliary superspace V. The R-matrix R(u) s P&(U), where P is the graded permutation is given by
(U+2gHrc+2g+2)P
u + 2g p2+ u - 2g
ii(u) = P, - -
(u - 2g)(u
- 2g - 2)
_ (tz+2g)(u+2g+2)(u+2g+4) s
(u-2g)(u-2g-2)(u-2g-4P
is carried operator,
”
(6)
where 4, k = 1,2,3,4, are four projection operators whose explicit formulae can be found in Ref. [6]. The elements of the supermatrix T(u) are the generators of an associative superalgebra A defined by the relations RI~(UI
-
~2)
+
(UI)
+
(~2)
=
+
(~2)
+
(UI)RI~(UI
-
u2),
(7)
X.-Y
Ge et al./Physics
Letters
A 238
(19981
309-314
31
I
2
where ;( G X 8 1, X = 1 @ X for any supermatrix X E End(V). For later use, we list some useful properties enjoyed by the R-matrix: (i) Unitarity: Rr2( 14)RZI(-u) = 1 and (ii) crossing-unitarity: Rs: ( --II + 4) h’yj (u) = b(u) with p(u) being a scalar function. In order to construct integrable electronic models with open boundary conditions, we introduce the following graded reflection equations (REs) that the so-called boundary K-(super) matrices K* (II)satisfy,
where the supertransposition st, (/J = I, 2) is only carried out in the ,uth factor superspace of V ;QV. whereas ist, denotes the inverse operation of st,. Following Sklyanin’s arguments [ I], one may show that the quantity 7-(/l) given by I_(r,)
=T(u)K_(u)T-'(-u)
satisfies the same relation
as K_
( IO)
(24):
R,?(u,- ~12) ;- (zI,)R~~(u~ +Uz) +- (uz)=;- (&)Rl2(ti1 +U?) +. Thus if one defines the boundary ~(11) = str(K+(li)I_(u))
transfer matrix r(u)
as
=str(K+(u)T(u)K_(u)T-‘(-u)).
then it can be shown [5] that ]7(NI
),T(U?)l = 0.
We now solve (8) and (9) for K+(u) and K_(u).Let us restrict ourselves one may check that the matrix K_ (10 given by
K_(u) =
1 5-(2-5_)(2f5_>
'A_(u) 0 0 0 0 0 0 \ 0
0 B_(U)
0 0 0 0 0 0
0
0
0
0
to the diagonal
solutions.
Then.
0 0 0 0 0 0 0 0 B_(u) 0 0 0 0 0 0 B_(u) 0 0 0 0 0 0 C_(M) 0 0 0 0 0 0 C_(u) 0 0 0 0 0 0 C_(u) 0 0 0 0 0 0 D_(u) ( 14)
where
A_(u) =(-[_ +t1)(2-5_
+u)(-2-5-
+B),
-u)(-2-e_
+u),
-u)(-2-l_
-10,
B_(U)=(-&_+U)(2-~_-U)(-2-~-+II), C_(u)
=(-t_
-u)(2-5_
D_(l4)=(-[_ -u)(2-5_
(15)
312
X.-r
Ge et d/Physics
Lettm
A 238 (1998) 309-314
satisfies (8). The matrix K+(u) can be obtained from the isomorphism K_(u) of (8), then K+(u) defined by K:‘(u)
of the two REs. Indeed, given a solution
= K_(-uf2)
( 16)
is a solution of (9). The proof follows from some algebraic computations upon substituting ( 16) into (9) and making use of the properties of the R-matrix. Therefore, one may choose the boundary matrix K+(u) as
A+(u) 0 0 0 0 0 0 0
K+(u) =
0 B+(u) 0 0 0 0 0 0
0
0 0
0
B+(u )O B+(u) 0 0
0 0 0 0 0
0 0 0 0 C+(u) 0
0 0 0 0 0 C+(u)
0 0
0 0
0 0
0 0 0 0 0 0
0
0 0 0 0 0 C+(u) 0
( 17)
0 D+(u),
where A+(u)=(-2g+2+5+-u)(-2g+5+-u)(-2g-2+~+-u), B+(il)=(-2g-2+!$++u)(-2g+<+-u)(-2g-2+5+-u). C+(u)
= (-2g-2+~++u)(-2g-4+~++u)(-2g-2+~+
-u),
D+(u)=(-2g-2+t++u)(-2g-4+~++u)(-2g-6+~++u).
(18)
Now it can be shown that Hamiltonian (1) is related to the second derivative r(u) (up to an unimportant additive constant)
of the boundary
transfer matrix
H = 2g HR, HR =
r”(0) 4(V+2W) L-l
=c
H.tj+l
./=I
x [strc (i+
+$tr-to) + (0)Gr.a)
2(v
+2stro
t
2w)
(““-‘+ (O)@a)
fstro
(g+
(0) (Hfc)2)]3
(19)
where V = straKi(O),
W=stra
(k+
(O)@a)
7
H,!j = Pi,jRI,i(O),
G,,,j = Pi,,jR:lj(O).
(20)
Here Pi.j denotes the graded permutation operator acting on the ith and jth quantum spaces. ( 19) implies that the boundary eight-state supersymmetric ZJ model admits an infinite number of conserved currents which are in involution with each other, thus assuring its integrability. It should be emphasized that the Hamiltonian ( 1) appears as the second derivative of the boundary transfer matrix r(u) with respect to the spectral parameter u at u = 0. This is due to the fact that the supertrace of K+(O) equals zero. The reason for the zero supertrace of K+(O) is related to the fact that the quantum space is the eight-dimensional typical irreducible representation of gl( 3 11) . A similar situation has already appeared in many spin chain models [4,7,5].
X.-Y Ge et d/Physics
Letters A 238 (1998) 309-314
313
Having established the quantum integrability of the boundary model, we now solve it by using the coordinate space Bethe ansatz method. The whole procedure is similar to that for other models [8,7]. The Bethe ansatz equations are
c ““2’L-‘)~(kl;p,)~(k,;pL)
=
nM’8.-A~‘)+ic/28i+Al”+ic/2 .’
,,=, 8.j - Aa’ - ic/2 0, + AjJ ) - k/C! ’
Al;’’ - Oi + ic/2 A,$” + 0.i + ic/2
2
A:;’ - AL’) + ic A!:’ + AL” + ic
,=, A”’ (r - Bi - ic/2 Al;” + Bi - ic/2 = - p=, Aj;” -- AL” - ic Al:’ + AL” ~ ic x
5 :i: -
Ar'
-
ic/2 AL’) + AY) - ic/2,
CT= l,...,
/>=, (1 - Ab2) + ic/2 Aa’ + A;" + k/2 MI A’” - AL” + ic/2 A:” + A;‘) + ic/2 Y n/,=, A;” - A”’ P - ic/2 Ab2’ + Ai” - ic/2 -
Mu
A:"
_ Ab"
M,.
+ ic AY)
+ Ab2) + ic
A?) - Ah’) - ic At” + Ah’) _ ic’
y=
l.....M’,
(21) where
mP)=‘-pe-i”p,=_,_&, 1 - p elk ’
and c = eq -
I; the charge
rapidities
pL=_,
8i E O( k.i) are related
(22)
2zgt.; .t
to the single-particle
quasi-momenta
k, by
O(k) = $ tan(k/2). The energy eigenvalue E of the model is given by E = -2 Es, cos k, (modular an unimportant additive constant coming from the chemical potential term). In conclusion, we have studied integrable open-boundary conditions for the eight-state supersymmetric CI model. The quantum integrability of the system follows from the fact that the Hamiltonian may be embedded into a one-parameter family of commuting transfer matrices. Moreover, the Bethe ansatz equations are derived by use of the coordinate space Bethe ansatz approach. This provides us with a basis for computing the finite size corrections (see, e.g., Refs. [ 8,9] ) to the low-lying energies in the system, which in turn allow us to use the boundary conformal field theory technique to study the critical properties of the boundary. Details will lx treated in a separate publication. This work is supported by the Australian Research Council, University of Queensland New Staff Research Grant and Enabling Research Grant. H.-Q.Z. would like to thank the Department of Mathematics of the University of Queensland for their kind hospitality. He also thanks the National Natural Science Foundation ot China and the Sichuan Young Investigators Science and Technology Fund for support.
References I I 1 E.K. Sklyanin, J. Phys. A 21 (1988)
121L. Mezincescu, R. Nepomechie, 13I H.J. de Vega, A. GonGlez-Ruiz, A. Gonziilez-Ruiz.
2375.
J. Phys. A 24 (1991)
Ll7.
J. Phys. A 26 ( 1993) LS19;
Nucl. Phys. B 424 ( 1994) 553.
141 H.-Q. Zhou, Phys. Rev. B 54 ( 1996) 41; 53 ( 1996) 5089.
15I
A.J. Bracken. X.Y. Ge, Y.-Z. Zhang, H.-Q. Zhou, An open-boundary integrable model of three coupled XY spin chains, and Integrable open-boundary preprints.
conditions for the y-deformed
supersymmetric
Cr model of strongly correlated electrons, University
of Queenslnnd
314
X.-Y Ge et (11./Ph.ysics Letter-s A 238 (1998) 309-314
161 M.D. Gould, Y.-Z. Zhang, H.-Q. Zhou, preprint cond-mat/9709120. 17 ) Y.-Z. Zhang, H.-Q. Zhou, preprint cond-m&9707263. 18 1 H. Asakawa, M. Suzuki. J. Phys. A 29 (1996) 225; M. Shiroishi, M. Wadati. J. Phys. Sot. Jpn. 66 i 1977) I. 191 CM. Yung, M.T. Batchelor, Nucl. Phys. B 435 ( 1995) 430; F.H. Essler, J. Phys. A 29 (1996) 6183: G. Bediirftig, EH. Essler. H. Frahm. Phys. Rev. Lett. 77 (1996) 5098