Table Of ContentOchanomizuUniversity
YukawaInstituteKyoto
YITP-98-5
cond-mat./9801193
January 1998
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Lax pair for SU(n) Hubbard model
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e Ruihong Yuea1 and Ryu Sasakib
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t a Department of Physics, Ochanomizu University, Tokyo 112, Japan
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bYukawa Institute for Theoretical Physics, Kyoto University,
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d Kyoto 606-01, Japan
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1 Abstract
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0 For one dimensional SU(n) Hubbardmodel, a pair of Lax operators are derived, which
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9 give a set of fundamental equations for the quantum inverse scattering method under
/ both periodic and open boundary conditions. This provides another proof of the inte-
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grability of the model under periodic boundary condition.
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PACS Numbers: 46.10.+z, 05.40.+j, 05.60.+w
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1e-mail: [email protected]
Integrable strongly correlated electron systems have been an important research subject
in condensed matter physics and mathematics. One of the significant models is the 1-D
Hubbard. The exact solution was given by Lieb and Wu [1]. However, the integrability was
shown twenty years later by Shastry, Olmedilla and Wadati [2, 3]. The integrability and the
exact solution of the system under the open boundary condition were discussed by several
authors [4, 5]. The Lax pair was first given by Wadati, Olmedilla and Akutsu [6]. Recently,
Maassarani and Mathieu have constructed the hamiltonian SU(n) XX model and proved
its integrability [7]. Considering two coupled SU(n) XX models, Maassarani succeeded in
generalizing Shastry’s method to SU(n) Hubbard model [8]. Further, he solved the Yang-
Baxter equation to prove the integrability of one dimensional SU(n) Hubbard model [9].
( It is also proved by Martins for n = 3,4 [10].) In this paper, we apply the quantum
inverse scattering method to 1-dimensional SU(n) Hubbard model and derive the explicit
form of the Lax pair, which gives another proof of the integrability. It is worthy to note
that straightforward application of the reflection matrix method to study the problem of the
integrable open boundary in the present SU(n) Hubbard model would encounter difficulties
due to the fact that the R matrix in reference [9] does not satisfy the crossing symmetry
condition and the lack of invertibility of a certain matrix.
TheLaxpairformalismwillgiveaneffectivemethodforsuchasystem. The1-dimensional
SU(n) model in the Schr¨odinger picture is given by
N n−1 Un2 N
H = EnαEαn +EαnEnα +EnαEαn +EαnEnα + C C , (1)
σ,k σ,k+1 σ,k σ,k+1 τ,k τ,k+1 τ,k τ,k+1 4 σ,k τ,k
kX=1αX=1(cid:16) (cid:17) kX=1
αβ
where E (a = σ,τ) is a matrix with zeros everywhere except for a 1 at the intersection of
a,k
row α and column β:
(Eαβ) = δαδβ. (2)
lm l m
The subscripts a,k stand for two different E operators at site k (k = 1,...,N). The
n×n diagonal matrix C is defined by C = Eαα −Enn. The Hamiltonian enjoys the
α<n
(su(n−1)⊕u(1)) ⊕(su(n−1)⊕u(1)) syPmmetry. The generators are
σ τ
N N
Jαβ = Eαβ, and K = C , α,β = 1,...,n−1, a = σ, τ.
a a,k a a,k
k=1 k=1
X X
In the rest of this letter we discuss various operators in the Heisenberg picture. When we
deal with the operators corresponding to the matrices E and C in the Schr¨odinger picture,
they are denoted by adding a hat (ˆ) to the corresponding matrix:
Qˆ(t) = eiHtQe−iHt, Q = E or C.
2
In the following, we do not indicate the time dependence of the operators. Applying this
method to the Hamiltonian (1), we find
dEˆnn
a,k = i EˆβnEˆnβ −EˆnβEˆβn +EˆβnEˆnβ −EˆnβEˆβn ,
dt a,k a,k+1 a,k a,k+1 a,k a,k−1 a,k a,k−1
βX<n(cid:16) (cid:17)
dEˆnα Un2
a,k = i EˆβαEˆnβ +EˆβαEˆnβ −i EˆnnEˆnα +EˆnnEˆnα −i EˆnαCˆ ,
dt a,k a,k+1 a,k a,k−1 a,k a,k+1 a,k a,k−1 2 a,k a¯,k
βX<n(cid:16) (cid:17) (cid:16) (cid:17)
dEˆαn Un2
a,k = i EˆnnEˆαn +EˆnnEˆαn −i EˆαβEˆβn +EˆαβEˆβn +i EˆαnCˆ ,
dt a,k a,k+1 a,k a,k−1 a,k a,k+1 a,k a,k−1 2 a,k a¯,k
(cid:16) (cid:17) βX<n(cid:16) (cid:17)
dEˆαβ
a,k = i EˆnβEˆαn −EˆαnEˆnβ +EˆnβEˆαn −EˆαnEˆnβ , (3)
dt a,k a,k+1 a,k a,k+1 a,k a,k−1 a,k a,k−1
(cid:16) (cid:17)
where (a = σ,τ) and (a¯ = τ,σ). For an infinite system, it is not necessary to specify
the boundary condition. However, one should understand Eˆ is equal to Eˆ under the
a,0 a,N
periodic boundary condition. In an open boundary system, Eˆ and Eˆ in the r.h.s.
a,0 a,N+1
must be regarded as vanishing.
Let us first consider the degenerate case U = 0 (the SU(n) XX model [7]). In this case,
the equations of motion (3) decouple into two identical sets of equations for σ and τ. Let us
introduce the L-operator for each of them:
(+) (+) (+)
L (λ) = cos(λ)S +sin(λ)T +U , (4)
a,k a,k a,k a,k
(+) (+) (+)
in which the blocks S , T and U are defined by
a,k a,k a,k
S(+) = ( EˆαβEβα )+EˆnnEnn ,
a,k a,k a,au a,k a,au
α,β<n
X
T(+) = EˆnnEαα +EˆααEnn ,
a,k a,k a,au a,k a,au
α<n
X (cid:16) (cid:17)
U(+) = EˆnαEαn +EˆαnEnα . (5)
a,k a,k a,au a,k a,au
α<n
X (cid:16) (cid:17)
They satisfy various identities inherited from the definition in terms of Eαβ’s (2):
(+) (+) (+) (+) (+) (+) (+) (+) (+) (+) (+) (+) (+)
S T = T S = S U = U S = 0, T U = U T = U ,...,. (6)
a,k a,k a,k a,k a,k a,k a,k a,k a,k a,k a,k a,k a,k
We also introduce the M-operator for each species:
M (λ) = A EˆβnEˆnβ +A EˆnβEˆβn ) Enn
a,k 1 a,k a,k−1 2 a,k a,k−1 a,au
βX<nn o
+ B (Eˆnβ +Eˆnβ )Eβn +(Eˆβn +Eˆβn )Enβ
a,k a,k−1 a,au a,k a,k−1 a,au
βX<n n o
+ D EˆnαEˆαn +D EˆαnEˆnα + D (EˆβnEˆnβ +EˆnβEˆβn ) Eαα
1 a,k a,k−1 2 a,k a,k−1 3 a,k a,k−1 a,k a,k−1 a,au
αX<n β6=Xα<n
+ F EˆnβEˆαn −EˆαnEˆnβ Eβα , (7)
a,k a,k−1 a,k a,k−1 a,au
αX<nβ6=Xα<n n o
3
whereA,B,D,F areasyetundeterminedfunctionsofa(spectral)parameterλ. Thematrices
E are the constant matrices with the same definition as E ,k = 1,···,N. The subscript
a,au a,k
au stands for the auxiliary space instead of the quantum space. Thus L and M are n×n
a,k a,k
matrices in the auxiliary space. We want to rewrite equations (3) in a matrix Lax pair form
dL (λ)
a,k
= M (λ)L (λ)−L (λ)M (λ). (8)
a,k+1 a,k a,k a,k
dt
Substituting equations (4) and (7) into the above Lax-pair form (8), we find the solution
A = D = i+itan(λ), A = D = i−itan(λ),
1 1 2 2
(9)
D = i, B = −i/cos(λ), F = itan(λ).
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From now on we do not denote the λ-dependence of L and M for brevity. The transfer
a,k a,k
matrix for SU(n) XX model with N sites can be defined by
T = L ···L ,
a,XX a,N a,1
which satisfies
dT
a,XX
= M T −T M .
a,N+1 a,XX a,XX a,1
dt
Fromthis it is standard to show that the traceofT is independent of timeunder periodic
a,XX
boundary condition. So the SU(n) XX model is integrable. This Lax-pair is very important
for the construction of the Lax-pair of SU(n) Hubbard model. Notice that its integrability
was first given by Maassarani and Mathieu in the framework of Yang-Baxter relation [7].
Now, let us consider U 6= 0 case. In terms of the above L and M , we can rewrite
a,k a,k
equations (3) as
dL Un2
a,j = M L −L M +i [L ,C Cˆ ], (10)
a,j+1 a,j a,j a,j a,j a,au a¯,j
dt 4
in which the last term in the r.h.s. manifests the coupling between the two species σ and τ.
Using the relation [L ,C ] = −[L ,Cˆ ], we obtain
a,j a,au a,j a,j
dL Un2
a,j = M L −L M −i [L ,Cˆ Cˆ ]. (11)
a,j+1 a,j a,j a,j a,j a,j a¯,j
dt 4
We define the following operators for the coupled system
L = L L , M = M +M , (12)
j σ,j τ,j j σ,j τ,j
e f
then equation (11) can be written as
dL Un2
j = M L −L M −i [L ,Cˆ Cˆ ]. (13)
j+1 j j j j σ,j τ,j
dt 4
e
f e e f e
4
Now, we want to rewrite the last term in the above equation so that we could obtain the
Lax-pair formfor the coupled system in a similar form to (8). Following the method given by
Wadati et al for SU(2) Hubbard model, we introduce a “rotation” matrix by the u(1)⊗u(1)
charge:
h
I = cosh(h/2)+sinh(h/2)C C = exp{ C C }, (14)
au σ,au τ,au σ,au τ,au
2
where h is a free parameter to be determined later and the “rotated” operators are
L = I L I , M = I−1M I . (15)
j au j au j au j au
e f
(+) (+) (+)
By this the fundamental blocks S , T and U are mapped to their ‘anti-symmetric’
a,k a,k a,k
(−) (−) (−)
counterparts S , T and U :
a,k a,k a,k
S(−) = C S(+) = S(+)C = ( EˆαβEβα )−EˆnnEnn ,
a,k a,au a,k a,k a,au a,k a,au a,k a,au
α,β<n
X
T(−) = C T(+) = T(+)C = EˆnnEαα −EˆααEnn ,
a,k a,au a,k a,k a,au a,k a,au a,k a,au
α<n
X (cid:16) (cid:17)
U(−) = C U(+) = −U(+)C = EˆnαEαn −EˆαnEnα , (16)
a,k a,au a,k a,k a,au a,k a,au a,k a,au
α<n
X (cid:16) (cid:17)
which also satisfy identities similar to those given in (6). Then equation (13) becomes
dL Un2
j = I2 M I−2L −L M −i [L ,Cˆ Cˆ ]. (17)
dt au j+1 au j j j 4 j σ,j τ,j
Using the definition of I and M, we obtain
au
I2 M I−2 = M +Q +Q , (18)
au j+1 au j+1 j+1 j
2i
(−) (−)
Q = − sinh(h) U C +U C , (19)
j cos(λ) σ,j τ,au τ,j σ,au
(cid:16) (cid:17)
and
dL
j
= (M +Q )L −L (M +Q )
j+1 j+1 j j j j
dt
(20)
Un2
+L Q +Q L −i [L ,Cˆ Cˆ ].
j j j j j σ,j τ,j
4
Detailed calculation shows the last line in the above equation to be
Un2
ˆ ˆ
Q L +L Q −i [L ,C C ]
j j j j j σ,j τ,j
4
i 1 Un2 a+b
= b− sinh(2h)+i [L ,C C ] (21)
j σ,au τ,au
(a (cid:18) a−b(cid:19) 4 a−b)
i a Un2 2ab
(−) (−) (−) (−) (−) (−)
− 2 sinh(2h)−i I U (S +T )+U (S +T ) I ,
(aa−b 4 a−b) au σ,j τ,j τ,j τ,j σ,j σ,j au
(cid:16) (cid:17)
5
where a = cos(λ), b = sin(λ). In deriving equation (21), use has been made of the following
identities
ˆ ˆ
(a−b)[L ,C C ]+(a+b)[L ,C C ]
j σ,j τ,j j σ,au τ,au
(−) (−) (−) (−) (−) (−)
= −4abI U (S +T )+U (S +T ) I , (22)
au σ,j τ,j τ,j τ,j σ,j σ,j au
i (cid:16) 2i (cid:17)
I Q I−1 = − sinh(2h) U(−)C +U(−)C − sinh2(h) U(+) +U(+) ,
au j au a σ,j τ,au τ,j σ,au a σ,j τ,j
(cid:16) (cid:17) (cid:16) (cid:17)
which are straightforward consequences of the identities (6). In expression (21) let us choose
the parameter h by
Un2 Un2
sinh(2h) = 2ab = sin(2λ), (23)
4 4
so that the second line in the r.h.s. of (21) vanish. Then the Lax-pair can be written as
dL
j
= B L −L B , (24)
j+1 j j j
dt
Un2 1 a+b
B = M +Q +i 2b b− + C C . (25)
j j j σ,au τ,au
4 ( (cid:18) a−b(cid:19) a−b)
Here, M ,Q are defined by equations (15) and (19), respectively. It is clear that the
j j
condition (23) guarantees the parametrizability for SU(n) Hubbard model, which was first
introduced in [9] through the Yang-Baxter equation. As we know, the standard method to
construct the open system is to study the reflection equations [11]. The R matrix must enjoy
unitarity and crossing symmetry. For SU(n) Hubbard model, the R matrix in [9] does not
have the crossing symmetry and the invertibility of a related matrix is lacking. In the Lax
pair formalism, however, the detailed properties of the R matrix do not come in. This is
another way to study the integrability of the open boundary systems. Therefore, the Lax-
pair derived in the present letter would be a useful tool for analyzing open systems. We will
consider its applications elsewhere.
ACKNOWLEDGMENTS
This work was supported by Grant–in–Aid of Ministry of Education, Science and Culture of
Japan. R.Y. thanks the Japan Society for the Promotion of Science.
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