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Lax pair and boundary K-matrices for the one-dimensional Hubbard model
5 I C..=.=B
ELSEVIER
Nuclear Physics B 485 [FSI (1997) 685-693
Lax pair and boundary K-matrices for the one-dimensional Hubbard model Xi-Wen Guan a,b, Mei-Shan Wang c, Shan-De Yang b a CCAST (World Laboratory), PO. Box 8730, Beijing 100080, PR China b Center for Theoretical Physics and Department of Physics, Jilin University, Changchun 130023, PR China t c Department of Physics, Shandong Normal University, Jinan 250014, PR China
Received 11 June 1996; revised 26 September 1996; accepted 1 November 1996
Abstract The Lax pair for the one-dimensional (1D) Hubbard open chain is explicitly constructed. Our construction provides an alternative and direct demonstration for the quantum integrability of the model. As a further result, we obtain its two kinds of boundary K-matrices compatible with the integrability of the model. It seems to contribute to the different boundary terms in the Hamiltonian of the model. PACS: 75.10H; 75.10J Keywords: Lax pair; Hubbard open chain; Boundary K-matrix; Reflection equation
1. Introduction Recently there has been considerable interest in the study of completely integrable quantum spin open chains [ 1-5]. Sklyanin [ 1] showed that there is a variant of the usual formalism of the quantum inverse scattering method (QISM) [6,7] which may be used to describe systems on a finite interval with independent boundary conditions (BC) on each end. Central to his approach is the introduction of a new algebraic structure called the reflection equations (RE), which guarantee the integrability of the systems with open BC. However, it is assumed that the quantum R-matrix possesses both P and T symmetry. Mezincescu and Nepomechie [ 8 ] extended the Sklyanin formalism to the 1Mailing address. 0550-3213/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved. PII S0550-3213(96)00630-X
686
X.-W. Guan et aL /Nuclear Physics B 485 [FS] (1997) 685-693
case of the PT-invariant R-matrix. Thus one can show that quantum spin open chains -(j) and D~1~ in the fundamental representation associated with,A(1~ "1 ,"A(2~ ~2n ' --(2~. /-X2n--l' Bn~l~,{'~n are integrable and have the quantum algebra symmetry Uq(go) [9]. Lately, Zhou [10] proposed an extended Sklyanin formalism to treat the BC for the 1D Hubbard model. On the other hand, the traditional basis for applying the QISM to a completely integrable system is to represent the equations of motion of the system into the Lax form. Izergin and Korepin [ 11 ] proved that for systems with periodic BC, the existence of the quantum R-matrix allows one to express the original equations of motion in the Lax form. Meanwhile the explicit forms of Lax pairs for some physically important models have been given by many authors [ 12-14]. A natural problem is that there must exist one kind of revised form of the ordinary Lax formulation which can be used to describe quantum completely integrable lattice spin open chains [ 15]. The aim of this paper is to present an explicit construction of the Lax pair for the 1D Hubbard open chain. The construction provides an alternative and direct demonstration for the quantum integrability of the model. As a further result, we obtain its two kinds of boundary Kmatrices compatible with the integrability of the model. It seems to contribute to the different boundary terms in the Hamiltonian of the model. Indeed, the boundary Kmatrices thus constructed are consistent with those obtained by Zhou [ 10] through solving the RE. The outline of the present paper is as follows. In Section 2 the Lax pair formulation of the QISM for spin open chains is given. In Section 3 the 1D Hubbard model and its equivalent spin open chain are introduced; the Lax pair for the open chain is explicitly constructed and the corresponding boundary K-matrices are obtained. Section 4 is devoted to the conclusion.
2. Lax pair formulation The Lax pair formulation of the QISM for completely integrable lattice spin open chains may be described as follows. We consider an operator version of an auxiliary linear problem, a~i+l = L j ( u ) a S j ,
j = 1,2 . . . . . N,
d ~¢,j = m~(u)~j,
j = 2 , 3 . . . . . N,
d ~-~N+I =SN(U)~N+1,
d ~ ' ~ 1 = 81(U)~1,
(1)
where Lj (u), M r (u), 8N (u) and 81 (u) are matrices depending on the spectral parameter u, which does not depend on time t, and dynamical variables. Evidently, the consistency conditions for Eq. (1) yield the following Lax equations:
X.-W. Guan et al./Nuclear Physics B 485 [FS] (1997) 685-693
d -d~Lj(u)=Mj+l(u)Lj(u)-Lj(u)Mj(u),
687
j=2,3 ..... N-l,
d --~LN(U) =t~N(U)LN(U) -- LN(U)MN(U), d -~LI (u) = M2(u)L1 (u) - LI (U)~l (u).
(2)
If the equations of motion of the system can be expressed in the form of Eq. (2), provided the boundary K-matrices exist as the solutions of Eq. (5) and (6) below, then we call the lattice spin open chain completely integrable. In fact, it is readily shown from Eq. (2) that the transfer matrix [ 1]
r(u) = T r ( K + ( u ) T ( u ) K _ ( u ) T -l ( - u ) )
(3)
does not depend on time. Here T(u) is the monodromy matrix
T(u) = L N ( U ) . . . Ll (u).
(4)
This only requires that the boundary K-matrices satisfy the following constraint conditions: K_ (u) ~1 ( - - U ) = ~1 (U) K_ (u)
(5)
Tr( K+(U)6N(U)AN(U) ) = Tr( K+(u)AN(U)6N(-U) ),
(6)
and
where A N ( U ) = L N ( U ) . . . LI ( u ) K - ( u ) L I I ( - u )
...
L~vl(-u).
This implies that the system possesses an infinite number of conserved quantities.
3.
1D Hubbard open chain
Let us consider the 1D Hubbard open chain determined by the Hamiltonian N--l j=l
N S
j=l
+p+(2nlT -- 1) + p_(2nj t -- l) + q+(2nNT -- 1) + q-(2nNl -- 1).
(7)
Here p+ and q+ are the constants describing the boundary effect. The boundary terms in the Hamiltonian equation (7) are specially chosen, and only related to the diagonal boundary K-matrices. U is the coupling constant describing the Coulomb interaction and a~s and ajs are creation and annihilation operators with spins (s =T or ~.) at site j and njs = a+ajs is the density operator. They satisfy the anti-commutation relations
X.-W. Guan et al./Nuclear Physics B 485 [FS] (1997) 685-693
688
{ajs, aj, s , } = {ajs,~,s, + + } = O,
(8)
{ ajs, a;s, } = t~jj, t~ss, .
Applying the Jordan-Wigner transformation, which relates fermion operators and spin operators,
ajT =exp
ilr E o'~-o-~ o-7, I--I
ilrEo'-[o f
a+T=ex p
\
ilr E o'+o"c
ajt=ex p \
a;=exp \
o-+,
l=1
ilr E r~-,~- r~,
exp
l=l
l=l
iTrEo-~-of 1=1
ilr E r[,~-
exp \
t=l
r +, /
where o- and r are two species of Pauli matrices (they commute with each other) we obtain the Hamiltonian of a spin model which is equivalent to the Hubbard model (7): N--l H = E (o'f+l o"; j=l
U N
+h.c.) + (~r ~ r) + 7 E ~ f f r ff j=l
+p+~ + p_r~ + q+o~zN+ q_zZN.
(9)
In the case U = 0, this Hamiltonian reduces to a pair of uncoupled XY open chains. Thus, the problem reduces to the study of two identical Heisenberg XY open chains coupled to each other. We note that the Hamiltonian (9) is related to the transfer matrix r(u) (3) in the following way:
r(u) = CIU + C2u2 -I- C3( H -l- const.)u 3 + . . . Here Ci (i = 1,2 . . . . ) are some scalar functions of the boundary constants p+ and q+. This is similar to the situation that occurs in the 1D Heisenberg XY open chain [ 1 ]. Note that those K-matrices determine the BC and contribute non-trivial terms to the Hamiltonian. For our purpose, let us now recall some basic results for the 1D Hubbard chain with periodic BC [ 16,17]. It is well known that the transfer matrix of the six-vertex free-fermion model commutes with the Hamiltonian of the XY model. In Ref. [ 16], Shastry showed that the Hamiltonian of the 1D Hubbard periodic chain commutes with the transfer matrix of an equivalent coupled spin chain, which is the trace of the monodromy matrix T(u) Eq. (4), i.e.
r(u) = TroLNo(U)LN-lO(U) ... Lm(u),
(10)
X.-W, Guan et al./Nuclear Physics B 485 [FS] (1997) 685-693
689
where
Lj(u)=loL~a)(u)®L~r)(U)Io,
j=l ..... N
(11)
with z oz sinh 2h ' Io = cosh ~h + O-o~-
(
and mJ
cosu+sinu +
(u) =
2
U sin2u, sinh2h = ~-
j--sin z "2
0")
0";
o+
cosu+si.o
2
(12) ~
•
j
Note that L~~) (u) has the same form as LI ") (u) with ~-'s replacing o-'s. The parameter h controls the strength of the interlayer interactions. The local monodromy matrix Lj(u) acts in the tensor product of the physical space Wj and the auxiliary space l,b (= C 2 ® C2), and the trace as well as the matrix products are carried out in the auxiliary space V0. Moreover, it is also shown that the elements of the matrix T(u) are the generators of an associative algebra T defined by the relation 1
2
2
1
R12(Ul,U2) T(ul)T(uz)=T(u2)T(ul)
R12(Ul,UZ),
(13)
where 1
2
X=-X®idv2,
X-=idv~®X
for any matrix X E End(V). The explicit form of the R-matrix Rl2(ul, u2) can be found in Refs. [16-18]. It has been shown [12] that in the case of periodic BC, the L-matrix Lj(u) (11) and the M-matrix Mj(u), Bj(I,1)
mj(u) =
ot(e-t'r++ehr;_l ) a(e_ho.++eho.+_l) 0
~(eh'rT+e--hrT_l)
Bj(2,2)
0 h + --h o '+j _ 1 ) o:(e o'j +e
ot(eh o'j-- +e--/to'7_l ) 0 B./(3,3) h + a(e
0
)
ot(e-"o-7 +eho-~_1)
ot(e-h~ +ehrT_l)
~'j+e 1"3_l ) --h +
'
B./(4,4) (14)
where Bj(1,
l)=Iz+ip(o'+o';_l
+ T+7"}-_,)+ iK(o-~-o-+_l + ~-f~-+_.),
B j ( 2 , 2 ) = - t z + ip(~rj-aT_ , + r ; ~ _ , )
+ iK(g7~rJ-_ ~ + r}rT_,),
Bj(3, 3) = - / ~ + ip(~rT~r+_ l + ~f r;_,) + iK(o-+o-;_, + r;r+_,), B j ( 4 , 4 ) =/~ + ip(o'-~o-+_, + T;TJ-_I ) "Jr-iK(o-+o-~-_,+ z + , f _ , ) , with /~----
U(2 - cos2u),
p = 1 -tanu,
K= 1 +tanu,
o~-
COSU
,
X.-W. Guan et al./Nuclear Physics B 485 [FS] (1997) 685-693
690
may be presented into the Lax equation. In order to construct the matrices ~l (u) and 6N(U), one needs the following equations of motion: U.
q_ z
d ° " + = _io.+o.~ + 2 '°'l rl + 2p+io'~, U.
+
z
d °- N + = --IO'NO'N-1 • z + + _~tO.Nr N + 2q+io.+, -'~ d _ U. z "-~o" l = io'~ o'~ - "~to'-~ T 1 - 2p+io" 1, d U --~o';¢ = io'ZuO'N_l -- -~ io'~ rZN -- 2q +io'~ , d z1 = 2i(o.+o.~- _ o.+o-2) ~O" d Nz ~'~O"
= _ 2 i ( o . + o . ~v_ 1 -- ° ' N ° ' + - l )
(15)
d d --~7"= --~o'(o" --, r , p + ~ p _ , q + ~ q _ ) ,
(16)
and
yielded by the Hamiltonian (9). From Eq. (2), it follows that { DI(1,1)+/zCOSU 2 ~I(U)
= |
\
M_v~-
M+o'?
0
N_v~
DI (2,2) -/zcosu 2
0
A+o"1
N+°"+
0
Ol(3,3)--I~cosu z
a_v;-
B+o"+
B-v +
o
"~
/]
(17)
DI (4,4) +/zcosu 2 ]
with
Dl(1,1)=-2i(p_~'Fz++p+o'~o'+),
D1 ( 2 , 2 ) = -2i(p+o-~-o -+ - p_~-+~-~-),
D 1 ( 3 , 3 ) = 2i(p+o'~-o-~- -- p_7"~-~-~-),
D 1 ( 4 , 4 ) = 2i(p+o-+o-~- + p _ z + V l ) ,
M+=--i(ehcosu+e-h2p+sinu),
N+ = - - i ( e - h c o s u - - e h 2 p + s i n u ) ,
A+ = - - i ( e - h c o s u
B± = - - i ( e h c o s u -- e - h 2 p + sinu)
+ eh2p+ sin u),
and DN(I,I)+IXCOSU /
(~N(U) =
+
N'_v;
N'+o-~
0 t
M_v N,+
DN(2,2)--UCOSU 2
0
g+o-~,
M+ o-N
0
Du (3,3) --/zoos u2
B _ ru-
0
A+o"N
t
+
I
+
A_vjv
(18)
DN(4,4)+pCOSU 2
with
D N ( 1, 1) = - - 2 i ( q _ r ~ v r + + q+o-~to'+), D N ( 3 , 3) = 2i(q+o'+~r~v -- q_r~vz+),
O N ( 2 , 2) = --2i(q+o-~o -+ -- q - r + r ~ ) , D N ( 4 , 4) = 2i(q+cr+o'~, + q_r+r~v),
691
X.-W. Guan et al./Nuclear Physics B 485 [FS] (1997) 685-693 r
t
M ± = - i ( e h cosu + e - h 2 q + sin u), I
A± = --i(e-hcosu
N+ = - - i ( e - h c o s u
-- eh2q~ sin u),
t
+ eh2q± sin u),
B± = - - i ( e h c o s u -- e - h 2 q ± sin u).
Thus we have obtained the Lax pair for 1D Hubbard spin open chain. This also gives a straightforward proof of the integrability of the model. Now let us study the constraint conditions (5) and (6) for finding the boundary K-matrix K+. Let
K+(u) =
i
0
0
K±(2,2) 0
0 K ± ( 3 , 3)
0
0
.
(19)
K+(4,4)
Substituting (17) and (19) into Eq. (5), one obtains sixteen equations. By tedious algebraic calculations, we may get only two kinds of solutions as follows: Case I e - h cos u - e h 2 ( - sin u K_ ( 1, 1 ), K_ ( 2 , 2 ) - e h c o s u + e_h2~: - sinu e - h c o s u -- eh2~_ s i n u K _ (1 ' 1), K _ ( 3 , 3) - eh cos u + e_h2~: - sinu
( e h cos u - e - h 2 ( - sin u) ( e - h cos u -- e h 2 ~ - sin u) .. K_(4,4)--(-~-_~osu~_e-~_sinu)(ehcosu+e_hZ~_si--~u) t~_(1,1)
(20)
with p+ = p _ = ~:_, Case II e - h COSu -- eh2~_ sin u K_ ( 1, 1 ), K_ (2, 2) = eh COSU -q- e - h 2 s c_ sin u e - h COSU + eh2~_ sin u K_ (3, 3) = e h c o s u _ e _ h 2 ~ _ sin u K _ ( 1 , 1 ) , K _ ( 4 , 4 ) = K _ ( 1 , 1) with p+ = - p _ = ( _ . Determined up to an unimportant constant, K _ ( u ) forms:
(21)
may be written in the following
Case I K _ ( 1, 1 ) = A_ ( e - h cos u q- e h 2 ( _ sin u) ( e h cos u + e - h 2 ( _ sin u),
K_ (2, 2) = A_ ( e - h cos u -- eh2~_ sin u) ( e - h cos u -q- e h 2 ( _ sin u), K_ (3, 3 ) = A_ ( e - h cos u -- eh2~_ sin u) ( e - h cos u + e h 2 ( _ sin u), K_ (4, 4) = a _ (e h cos u - e - h 2 sc_ sin u) (e - h cos u - eh2~:_ sin u),
(22)
692
X.-W. Guan et al./Nuclear Physics B 485 [FS] (1997) 685-693
Case II K _ ( 1 , 1) = A_ ( eh cos u + e - h 2 ~ _ sin u ) ( eh cos u -- e - h 2 ~ _ sin u), K_ ( 2, 2 ) = ,~_ ( e - h COSU -- e h 2~:_ sin u ) ( e h cos u -- e - h2so_ sin u ), K_ (3, 3) = A_ ( e - h cos u q- e h 2 ( - sin u) ( e h cos u q- e - h 2 ~ _ sin u), K_ ( 4 , 4 ) = A _ ( e h c o s u + e - h 2 ~ _ s i n u ) ( e h c o s u -- e - h 2 ~ _ s i n u ) ,
(23)
respectively. Here ~_ and A_ are arbitrary constants describing the boundary effect. In order to determine K + ( u ) , one notes that A N ( u ) = L N ( u ) A l v - 1 (u)LF¢ 1 ( - u ) .
(24)
Obviously, the matrix elements of AN-1 are independent of each other and commute with the ones of L N ( u ) . Therefore one can assume that the corresponding coefficients of the matrix elements of AN-1 (u) vanish respectively in Eq. (6). It follows that Case I K+( 1, 1) = A+(e -h sinu + eh2(+ COSu) (e h sinu + e - h 2 ~ + COSU), K+ (2, 2) = ~+ ( e - h sin u -- e h2~+ COSu) ( e - h sin u + e h 2 ( + COSU),
K+ (3, 3) = ~+ ( e - h sin u - e h 2 ( + COSu) ( e - h sin u + eh2~ + COSU), K+ (4, 4) = a+ ( e h sin u - e - h 2 ( + COSu) ( e - h sin u -- eh2sc+ cos u),
(25)
Case II K + ( 1 , 1) = A+(e h sinu + e - h 2 ~ + c o s u ) ( e h sinu -- e - h 2 ( + cos u), K+ (2, 2) = A+ ( e - h sin u
-
K+ ( 3, 3 ) = A+ ( e - h sin u +
eh
2~:+ COSU) (e h sin u - e-h 2(+ cos u),
eh
2~+ COSU) ( e h sin u + e-hsC+ COSu),
K+ ( 4 , 4 ) = A + ( e h s i n u + e - h 2 ~ + c o s u ) ( e h sinu - e-h2~:+ cos u).
(26)
Here A+ and ~:+ are also arbitrary constants describing the boundary effect. We have checked that the K+ (u) satisfy the generalized RE [ 10] for the 1D Hubbard open chain. It is pointed out that in case I, for K+, if so+ ~ tan~+ and A+ ~ ~ , the K + ( u ) (22) and (25) are the same as those obtained by Zhou [ 10] through solving the RE. It seems non-trivial to derive the Bethe equations. It is shown that the K + ( u ) indeed describe the BC compatible with the integrability of the model.
4. Conclusion
We have presented the Lax pair formulation for the ID Hubbard spin open chain. The explicit form of the Lax pair for the model has been obtained and two kinds of boundary K-matrices have been given. It seems non-trivial to derive the Bethe ansatz equations from either the algebraic Bethe ansatz or the analytical Bethe ansatz approach.
x.-w. Guan et aL/Nuclear Physics B 485 [FS] (1997) 685-693
693
It shows that the Lax pair formulation coincides with Sklyanin's argument for finding the boundary K-matrix. Therefore, the Lax pair formulation exposes another scheme for dealing with lattice spin open chains. We may conjecture that the combination of the two formulations can be used to study open chains associated with matrices R ( u ) which do not have PT symmetry and crossing symmetry or with the matrix R ( u , v) which does not depend only on the difference u - v. How to diagonalize the Hamiltonian (9) and obtain its Bethe equations are very interesting questions. Another interesting question is to give a Lax formulation for the lattice fermion open chain. In this case the monodromy matrix is a supermatrix and the relations (5) and (6) should be understood in the graded sense. This question will be considered elsewhere.
Acknowledgements X.-W.G. would like to thank Prof. Zhou for his helpful discussions. This work is supported in part by the National Science Foundation of China.
References [1] E.K. Sklyanin, J. Phys. A 21 (1988) 2375. [2] H.J. de Vega and A. Gonzfilez-Ruiz,J. Phys. A 27 (1994) 6129; A 26 (1993) L519. [3] B.Y. Hou and R.H. Yue, Phys. Lett. A 183 (1993) 167. [4] L. Mezincescu,R.I. Nepomechie and U. Rittenberg, Phys. Lett. A 147 (1990) 70. [5] X.W. Guan and S.D. Yang, Phys. Lett. A 213 (1996) 273. [6] L.D. Faddeev, Les Houches 1982, ed. J.B. Zuber and R. Stora (North-Holland,Amsterdam, 1984). [7] P.P. Kulish and E.K. Sklyanin,in Lecture Notes in Physics, Vol. 151 (Springer,Berlin, 1982) p. 61. [8] L. Mezincescuand R.I. Nepomechie, J. Phys. A 24 (1991) L17. [9] L. Mezincescuand R.I. Nepomechie, Int. J. Mod. Phys. A 6 (1991) 52231. [ 10] H.Q. Zhou, Quantum Integrabilityfor the One-DimensionalHubbardOpen Chain, CQU-TH-95-8. [111 A.G. Izergin and V.E. Korepin,Sov. J. Part. Nucl. 13 (1982) 207. [12] K. Sogo and M. Wadati, Prog. Theor. Phys. 68 (1982) 85. [13] M. Wadati, E. Olmedillaand Y. Akutu, J. Phys. Soc. Jpn. 56 (1987) 1340. [14] H.Q. Zhou and L.J. Jiang, Phys. Lett. A 134 (1989) 469. [ 15] H.Q. Zhou, Acta Phys. Sinica, in press. [16] B.S. Shastry, Phys. Rev. Lett. 56 (1986) 1529, 2453. [17] E. Oimedilla, M. Wadati and Y. Akutsu, J. Phys. Soc. Jpn. 56 (1987) 2298; E. Olmedilla and M. Wadati, Phys. Rev. Lett. 60 (1988) 1595. [18] H.Q. Zhou and J.G. Tang, Phys. Rev. B 8 (1988) 11915.
ELSEVIER
Nuclear Physics B 485 [FSI (1997) 685-693
Lax pair and boundary K-matrices for the one-dimensional Hubbard model Xi-Wen Guan a,b, Mei-Shan Wang c, Shan-De Yang b a CCAST (World Laboratory), PO. Box 8730, Beijing 100080, PR China b Center for Theoretical Physics and Department of Physics, Jilin University, Changchun 130023, PR China t c Department of Physics, Shandong Normal University, Jinan 250014, PR China
Received 11 June 1996; revised 26 September 1996; accepted 1 November 1996
Abstract The Lax pair for the one-dimensional (1D) Hubbard open chain is explicitly constructed. Our construction provides an alternative and direct demonstration for the quantum integrability of the model. As a further result, we obtain its two kinds of boundary K-matrices compatible with the integrability of the model. It seems to contribute to the different boundary terms in the Hamiltonian of the model. PACS: 75.10H; 75.10J Keywords: Lax pair; Hubbard open chain; Boundary K-matrix; Reflection equation
1. Introduction Recently there has been considerable interest in the study of completely integrable quantum spin open chains [ 1-5]. Sklyanin [ 1] showed that there is a variant of the usual formalism of the quantum inverse scattering method (QISM) [6,7] which may be used to describe systems on a finite interval with independent boundary conditions (BC) on each end. Central to his approach is the introduction of a new algebraic structure called the reflection equations (RE), which guarantee the integrability of the systems with open BC. However, it is assumed that the quantum R-matrix possesses both P and T symmetry. Mezincescu and Nepomechie [ 8 ] extended the Sklyanin formalism to the 1Mailing address. 0550-3213/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved. PII S0550-3213(96)00630-X
686
X.-W. Guan et aL /Nuclear Physics B 485 [FS] (1997) 685-693
case of the PT-invariant R-matrix. Thus one can show that quantum spin open chains -(j) and D~1~ in the fundamental representation associated with,A(1~ "1 ,"A(2~ ~2n ' --(2~. /-X2n--l' Bn~l~,{'~n are integrable and have the quantum algebra symmetry Uq(go) [9]. Lately, Zhou [10] proposed an extended Sklyanin formalism to treat the BC for the 1D Hubbard model. On the other hand, the traditional basis for applying the QISM to a completely integrable system is to represent the equations of motion of the system into the Lax form. Izergin and Korepin [ 11 ] proved that for systems with periodic BC, the existence of the quantum R-matrix allows one to express the original equations of motion in the Lax form. Meanwhile the explicit forms of Lax pairs for some physically important models have been given by many authors [ 12-14]. A natural problem is that there must exist one kind of revised form of the ordinary Lax formulation which can be used to describe quantum completely integrable lattice spin open chains [ 15]. The aim of this paper is to present an explicit construction of the Lax pair for the 1D Hubbard open chain. The construction provides an alternative and direct demonstration for the quantum integrability of the model. As a further result, we obtain its two kinds of boundary Kmatrices compatible with the integrability of the model. It seems to contribute to the different boundary terms in the Hamiltonian of the model. Indeed, the boundary Kmatrices thus constructed are consistent with those obtained by Zhou [ 10] through solving the RE. The outline of the present paper is as follows. In Section 2 the Lax pair formulation of the QISM for spin open chains is given. In Section 3 the 1D Hubbard model and its equivalent spin open chain are introduced; the Lax pair for the open chain is explicitly constructed and the corresponding boundary K-matrices are obtained. Section 4 is devoted to the conclusion.
2. Lax pair formulation The Lax pair formulation of the QISM for completely integrable lattice spin open chains may be described as follows. We consider an operator version of an auxiliary linear problem, a~i+l = L j ( u ) a S j ,
j = 1,2 . . . . . N,
d ~¢,j = m~(u)~j,
j = 2 , 3 . . . . . N,
d ~-~N+I =SN(U)~N+1,
d ~ ' ~ 1 = 81(U)~1,
(1)
where Lj (u), M r (u), 8N (u) and 81 (u) are matrices depending on the spectral parameter u, which does not depend on time t, and dynamical variables. Evidently, the consistency conditions for Eq. (1) yield the following Lax equations:
X.-W. Guan et al./Nuclear Physics B 485 [FS] (1997) 685-693
d -d~Lj(u)=Mj+l(u)Lj(u)-Lj(u)Mj(u),
687
j=2,3 ..... N-l,
d --~LN(U) =t~N(U)LN(U) -- LN(U)MN(U), d -~LI (u) = M2(u)L1 (u) - LI (U)~l (u).
(2)
If the equations of motion of the system can be expressed in the form of Eq. (2), provided the boundary K-matrices exist as the solutions of Eq. (5) and (6) below, then we call the lattice spin open chain completely integrable. In fact, it is readily shown from Eq. (2) that the transfer matrix [ 1]
r(u) = T r ( K + ( u ) T ( u ) K _ ( u ) T -l ( - u ) )
(3)
does not depend on time. Here T(u) is the monodromy matrix
T(u) = L N ( U ) . . . Ll (u).
(4)
This only requires that the boundary K-matrices satisfy the following constraint conditions: K_ (u) ~1 ( - - U ) = ~1 (U) K_ (u)
(5)
Tr( K+(U)6N(U)AN(U) ) = Tr( K+(u)AN(U)6N(-U) ),
(6)
and
where A N ( U ) = L N ( U ) . . . LI ( u ) K - ( u ) L I I ( - u )
...
L~vl(-u).
This implies that the system possesses an infinite number of conserved quantities.
3.
1D Hubbard open chain
Let us consider the 1D Hubbard open chain determined by the Hamiltonian N--l j=l
N S
j=l
+p+(2nlT -- 1) + p_(2nj t -- l) + q+(2nNT -- 1) + q-(2nNl -- 1).
(7)
Here p+ and q+ are the constants describing the boundary effect. The boundary terms in the Hamiltonian equation (7) are specially chosen, and only related to the diagonal boundary K-matrices. U is the coupling constant describing the Coulomb interaction and a~s and ajs are creation and annihilation operators with spins (s =T or ~.) at site j and njs = a+ajs is the density operator. They satisfy the anti-commutation relations
X.-W. Guan et al./Nuclear Physics B 485 [FS] (1997) 685-693
688
{ajs, aj, s , } = {ajs,~,s, + + } = O,
(8)
{ ajs, a;s, } = t~jj, t~ss, .
Applying the Jordan-Wigner transformation, which relates fermion operators and spin operators,
ajT =exp
ilr E o'~-o-~ o-7, I--I
ilrEo'-[o f
a+T=ex p
\
ilr E o'+o"c
ajt=ex p \
a;=exp \
o-+,
l=1
ilr E r~-,~- r~,
exp
l=l
l=l
iTrEo-~-of 1=1
ilr E r[,~-
exp \
t=l
r +, /
where o- and r are two species of Pauli matrices (they commute with each other) we obtain the Hamiltonian of a spin model which is equivalent to the Hubbard model (7): N--l H = E (o'f+l o"; j=l
U N
+h.c.) + (~r ~ r) + 7 E ~ f f r ff j=l
+p+~ + p_r~ + q+o~zN+ q_zZN.
(9)
In the case U = 0, this Hamiltonian reduces to a pair of uncoupled XY open chains. Thus, the problem reduces to the study of two identical Heisenberg XY open chains coupled to each other. We note that the Hamiltonian (9) is related to the transfer matrix r(u) (3) in the following way:
r(u) = CIU + C2u2 -I- C3( H -l- const.)u 3 + . . . Here Ci (i = 1,2 . . . . ) are some scalar functions of the boundary constants p+ and q+. This is similar to the situation that occurs in the 1D Heisenberg XY open chain [ 1 ]. Note that those K-matrices determine the BC and contribute non-trivial terms to the Hamiltonian. For our purpose, let us now recall some basic results for the 1D Hubbard chain with periodic BC [ 16,17]. It is well known that the transfer matrix of the six-vertex free-fermion model commutes with the Hamiltonian of the XY model. In Ref. [ 16], Shastry showed that the Hamiltonian of the 1D Hubbard periodic chain commutes with the transfer matrix of an equivalent coupled spin chain, which is the trace of the monodromy matrix T(u) Eq. (4), i.e.
r(u) = TroLNo(U)LN-lO(U) ... Lm(u),
(10)
X.-W, Guan et al./Nuclear Physics B 485 [FS] (1997) 685-693
689
where
Lj(u)=loL~a)(u)®L~r)(U)Io,
j=l ..... N
(11)
with z oz sinh 2h ' Io = cosh ~h + O-o~-
(
and mJ
cosu+sinu +
(u) =
2
U sin2u, sinh2h = ~-
j--sin z "2
0")
0";
o+
cosu+si.o
2
(12) ~
•
j
Note that L~~) (u) has the same form as LI ") (u) with ~-'s replacing o-'s. The parameter h controls the strength of the interlayer interactions. The local monodromy matrix Lj(u) acts in the tensor product of the physical space Wj and the auxiliary space l,b (= C 2 ® C2), and the trace as well as the matrix products are carried out in the auxiliary space V0. Moreover, it is also shown that the elements of the matrix T(u) are the generators of an associative algebra T defined by the relation 1
2
2
1
R12(Ul,U2) T(ul)T(uz)=T(u2)T(ul)
R12(Ul,UZ),
(13)
where 1
2
X=-X®idv2,
X-=idv~®X
for any matrix X E End(V). The explicit form of the R-matrix Rl2(ul, u2) can be found in Refs. [16-18]. It has been shown [12] that in the case of periodic BC, the L-matrix Lj(u) (11) and the M-matrix Mj(u), Bj(I,1)
mj(u) =
ot(e-t'r++ehr;_l ) a(e_ho.++eho.+_l) 0
~(eh'rT+e--hrT_l)
Bj(2,2)
0 h + --h o '+j _ 1 ) o:(e o'j +e
ot(eh o'j-- +e--/to'7_l ) 0 B./(3,3) h + a(e
0
)
ot(e-"o-7 +eho-~_1)
ot(e-h~ +ehrT_l)
~'j+e 1"3_l ) --h +
'
B./(4,4) (14)
where Bj(1,
l)=Iz+ip(o'+o';_l
+ T+7"}-_,)+ iK(o-~-o-+_l + ~-f~-+_.),
B j ( 2 , 2 ) = - t z + ip(~rj-aT_ , + r ; ~ _ , )
+ iK(g7~rJ-_ ~ + r}rT_,),
Bj(3, 3) = - / ~ + ip(~rT~r+_ l + ~f r;_,) + iK(o-+o-;_, + r;r+_,), B j ( 4 , 4 ) =/~ + ip(o'-~o-+_, + T;TJ-_I ) "Jr-iK(o-+o-~-_,+ z + , f _ , ) , with /~----
U(2 - cos2u),
p = 1 -tanu,
K= 1 +tanu,
o~-
COSU
,
X.-W. Guan et al./Nuclear Physics B 485 [FS] (1997) 685-693
690
may be presented into the Lax equation. In order to construct the matrices ~l (u) and 6N(U), one needs the following equations of motion: U.
q_ z
d ° " + = _io.+o.~ + 2 '°'l rl + 2p+io'~, U.
+
z
d °- N + = --IO'NO'N-1 • z + + _~tO.Nr N + 2q+io.+, -'~ d _ U. z "-~o" l = io'~ o'~ - "~to'-~ T 1 - 2p+io" 1, d U --~o';¢ = io'ZuO'N_l -- -~ io'~ rZN -- 2q +io'~ , d z1 = 2i(o.+o.~- _ o.+o-2) ~O" d Nz ~'~O"
= _ 2 i ( o . + o . ~v_ 1 -- ° ' N ° ' + - l )
(15)
d d --~7"= --~o'(o" --, r , p + ~ p _ , q + ~ q _ ) ,
(16)
and
yielded by the Hamiltonian (9). From Eq. (2), it follows that { DI(1,1)+/zCOSU 2 ~I(U)
= |
\
M_v~-
M+o'?
0
N_v~
DI (2,2) -/zcosu 2
0
A+o"1
N+°"+
0
Ol(3,3)--I~cosu z
a_v;-
B+o"+
B-v +
o
"~
/]
(17)
DI (4,4) +/zcosu 2 ]
with
Dl(1,1)=-2i(p_~'Fz++p+o'~o'+),
D1 ( 2 , 2 ) = -2i(p+o-~-o -+ - p_~-+~-~-),
D 1 ( 3 , 3 ) = 2i(p+o'~-o-~- -- p_7"~-~-~-),
D 1 ( 4 , 4 ) = 2i(p+o-+o-~- + p _ z + V l ) ,
M+=--i(ehcosu+e-h2p+sinu),
N+ = - - i ( e - h c o s u - - e h 2 p + s i n u ) ,
A+ = - - i ( e - h c o s u
B± = - - i ( e h c o s u -- e - h 2 p + sinu)
+ eh2p+ sin u),
and DN(I,I)+IXCOSU /
(~N(U) =
+
N'_v;
N'+o-~
0 t
M_v N,+
DN(2,2)--UCOSU 2
0
g+o-~,
M+ o-N
0
Du (3,3) --/zoos u2
B _ ru-
0
A+o"N
t
+
I
+
A_vjv
(18)
DN(4,4)+pCOSU 2
with
D N ( 1, 1) = - - 2 i ( q _ r ~ v r + + q+o-~to'+), D N ( 3 , 3) = 2i(q+o'+~r~v -- q_r~vz+),
O N ( 2 , 2) = --2i(q+o-~o -+ -- q - r + r ~ ) , D N ( 4 , 4) = 2i(q+cr+o'~, + q_r+r~v),
691
X.-W. Guan et al./Nuclear Physics B 485 [FS] (1997) 685-693 r
t
M ± = - i ( e h cosu + e - h 2 q + sin u), I
A± = --i(e-hcosu
N+ = - - i ( e - h c o s u
-- eh2q~ sin u),
t
+ eh2q± sin u),
B± = - - i ( e h c o s u -- e - h 2 q ± sin u).
Thus we have obtained the Lax pair for 1D Hubbard spin open chain. This also gives a straightforward proof of the integrability of the model. Now let us study the constraint conditions (5) and (6) for finding the boundary K-matrix K+. Let
K+(u) =
i
0
0
K±(2,2) 0
0 K ± ( 3 , 3)
0
0
.
(19)
K+(4,4)
Substituting (17) and (19) into Eq. (5), one obtains sixteen equations. By tedious algebraic calculations, we may get only two kinds of solutions as follows: Case I e - h cos u - e h 2 ( - sin u K_ ( 1, 1 ), K_ ( 2 , 2 ) - e h c o s u + e_h2~: - sinu e - h c o s u -- eh2~_ s i n u K _ (1 ' 1), K _ ( 3 , 3) - eh cos u + e_h2~: - sinu
( e h cos u - e - h 2 ( - sin u) ( e - h cos u -- e h 2 ~ - sin u) .. K_(4,4)--(-~-_~osu~_e-~_sinu)(ehcosu+e_hZ~_si--~u) t~_(1,1)
(20)
with p+ = p _ = ~:_, Case II e - h COSu -- eh2~_ sin u K_ ( 1, 1 ), K_ (2, 2) = eh COSU -q- e - h 2 s c_ sin u e - h COSU + eh2~_ sin u K_ (3, 3) = e h c o s u _ e _ h 2 ~ _ sin u K _ ( 1 , 1 ) , K _ ( 4 , 4 ) = K _ ( 1 , 1) with p+ = - p _ = ( _ . Determined up to an unimportant constant, K _ ( u ) forms:
(21)
may be written in the following
Case I K _ ( 1, 1 ) = A_ ( e - h cos u q- e h 2 ( _ sin u) ( e h cos u + e - h 2 ( _ sin u),
K_ (2, 2) = A_ ( e - h cos u -- eh2~_ sin u) ( e - h cos u -q- e h 2 ( _ sin u), K_ (3, 3 ) = A_ ( e - h cos u -- eh2~_ sin u) ( e - h cos u + e h 2 ( _ sin u), K_ (4, 4) = a _ (e h cos u - e - h 2 sc_ sin u) (e - h cos u - eh2~:_ sin u),
(22)
692
X.-W. Guan et al./Nuclear Physics B 485 [FS] (1997) 685-693
Case II K _ ( 1 , 1) = A_ ( eh cos u + e - h 2 ~ _ sin u ) ( eh cos u -- e - h 2 ~ _ sin u), K_ ( 2, 2 ) = ,~_ ( e - h COSU -- e h 2~:_ sin u ) ( e h cos u -- e - h2so_ sin u ), K_ (3, 3) = A_ ( e - h cos u q- e h 2 ( - sin u) ( e h cos u q- e - h 2 ~ _ sin u), K_ ( 4 , 4 ) = A _ ( e h c o s u + e - h 2 ~ _ s i n u ) ( e h c o s u -- e - h 2 ~ _ s i n u ) ,
(23)
respectively. Here ~_ and A_ are arbitrary constants describing the boundary effect. In order to determine K + ( u ) , one notes that A N ( u ) = L N ( u ) A l v - 1 (u)LF¢ 1 ( - u ) .
(24)
Obviously, the matrix elements of AN-1 are independent of each other and commute with the ones of L N ( u ) . Therefore one can assume that the corresponding coefficients of the matrix elements of AN-1 (u) vanish respectively in Eq. (6). It follows that Case I K+( 1, 1) = A+(e -h sinu + eh2(+ COSu) (e h sinu + e - h 2 ~ + COSU), K+ (2, 2) = ~+ ( e - h sin u -- e h2~+ COSu) ( e - h sin u + e h 2 ( + COSU),
K+ (3, 3) = ~+ ( e - h sin u - e h 2 ( + COSu) ( e - h sin u + eh2~ + COSU), K+ (4, 4) = a+ ( e h sin u - e - h 2 ( + COSu) ( e - h sin u -- eh2sc+ cos u),
(25)
Case II K + ( 1 , 1) = A+(e h sinu + e - h 2 ~ + c o s u ) ( e h sinu -- e - h 2 ( + cos u), K+ (2, 2) = A+ ( e - h sin u
-
K+ ( 3, 3 ) = A+ ( e - h sin u +
eh
2~:+ COSU) (e h sin u - e-h 2(+ cos u),
eh
2~+ COSU) ( e h sin u + e-hsC+ COSu),
K+ ( 4 , 4 ) = A + ( e h s i n u + e - h 2 ~ + c o s u ) ( e h sinu - e-h2~:+ cos u).
(26)
Here A+ and ~:+ are also arbitrary constants describing the boundary effect. We have checked that the K+ (u) satisfy the generalized RE [ 10] for the 1D Hubbard open chain. It is pointed out that in case I, for K+, if so+ ~ tan~+ and A+ ~ ~ , the K + ( u ) (22) and (25) are the same as those obtained by Zhou [ 10] through solving the RE. It seems non-trivial to derive the Bethe equations. It is shown that the K + ( u ) indeed describe the BC compatible with the integrability of the model.
4. Conclusion
We have presented the Lax pair formulation for the ID Hubbard spin open chain. The explicit form of the Lax pair for the model has been obtained and two kinds of boundary K-matrices have been given. It seems non-trivial to derive the Bethe ansatz equations from either the algebraic Bethe ansatz or the analytical Bethe ansatz approach.
x.-w. Guan et aL/Nuclear Physics B 485 [FS] (1997) 685-693
693
It shows that the Lax pair formulation coincides with Sklyanin's argument for finding the boundary K-matrix. Therefore, the Lax pair formulation exposes another scheme for dealing with lattice spin open chains. We may conjecture that the combination of the two formulations can be used to study open chains associated with matrices R ( u ) which do not have PT symmetry and crossing symmetry or with the matrix R ( u , v) which does not depend only on the difference u - v. How to diagonalize the Hamiltonian (9) and obtain its Bethe equations are very interesting questions. Another interesting question is to give a Lax formulation for the lattice fermion open chain. In this case the monodromy matrix is a supermatrix and the relations (5) and (6) should be understood in the graded sense. This question will be considered elsewhere.
Acknowledgements X.-W.G. would like to thank Prof. Zhou for his helpful discussions. This work is supported in part by the National Science Foundation of China.
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