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Yangian symmetry and quantum inverse scattering method for the one-dimensional Hubbard model
I7 March 1997
PHYSICS ELSEVIER
LETTERS
A
Physics Letters A 227 ( 1997) 2 16-226
Yangian symmetry and quantum inverse scattering method for the one-dimensional Hubbard model Shuichi Murakami a,1,Frank Gijhmann b*2 a Department o/Applied Physics. Fuculty ofEngineering. University ofTokyo. Hongo 7-3-l. Bunkyo-ku, Tokyo 113, Japun b Department ofPhysics, Faculty of Science, University ofTokyo, Hongo 7-3-l. Bunkyo-ku, Tokyo 113. Japan ’ Received 3 October
1996; accepted for publication 13 December Communicated by A.R. Bishop
1996
Abstract We develop the quantum inverse scattering method for the one-dimensional Hubbard model on the infinite interval at zero density. The R-matrix and monodromy matrix are obtained as limits from their known counterparts on the finite interval. The R-matrix greatly simplifies in the considered limit. The new R-matrix contains a submatrix which turns into the rational R-matrix of the XXX-chain by an appropriate reparametrization. The corresponding submatrix of the monodromy matrix thus provides a representation of the Y(su(2)) Yangian. From its quantum determinant we obtain an infinite series of mutually commuting Yangian invariant operators which includes the Hamiltonian. Keywords:
Hubbard
model; Quantum
inverse scattering;
Yangian
1. Introduction Among the exactly solvable Id quantum systems the Hubbard model has probably the most interesting applications in solid state physics. Its elementary excitations, their dispersion relations and S-matrix at half filling [I] have been calculated exactly by use of the coordinate Bethe ansatz in conjunction with the SO(4) symmetry of the model [2,3]. The story of this development is long and originates in the seminal paper [4] of Lieb and Wu. A recent overview is offered tiy the reprint volume [S]. From the point of view of the coordinate Bethe ansatz the one-dimensional Hubbard model appears similar to the fermionic nonlinear Schrijdinger model. In fact, Lieb and Wu in their article used this analogy to obtain the Bethe ansatz equations from Yang’s earlier result [6,7]. Algebraically, however, the model seems to be more complicated. Nearly 20 years passed before the basic tools of quantum inverse scattering method (QISM), R-matrix and L-matrix were derived by Shastry [S;9] and by Olmedilla et al. [ 10-121, and it was shown only
’ E-mail: [email protected]. 2 E-mail: [email protected]. 3 Address from October 1996: Physikalisches 0375-9601/97/$17.00 Copyright PII SO375-9601(96)00953-X
lnstitut der UniversitZt Bayreuth,
TPI, 95440 Bayreuth,
0 1997 Elsevier Science B.V. All rights reserved
Germany.
S. Murukumi, F. Giihmutm/Physics
Letters A 227 (1997) 216-226
217
recently that the R-matrix satisfies the Yang-Baxter equation (YBE) 1131. The R-matrix and monodromy matrix of the Hubbard model have unusual features. The monodromy matrix is 4 X 4 rather than 3 X 3, as one might have guessed naively from the fact that there are two levels of Bethe ansatz equations or from the analogy with the fermionic nonlinear Schrijdinger model. It seems to be impossible to find a parametrization of the R-matrix, such that it becomes a function of the difference of the spectral parameters. For these reasons an algebraic Bethe ansatz is difficult and was performed only recently by Ramos and Martins [ 141. There is another algebraic structure related to the Hubbard model on the infinite line. As was discovered by Uglov and Korepin [15] the Hubbard Hamiltonian commutes with two independent and mutually commuting representations of the Yangian Y(su(2)). Yangians are the quantum groups connected to rational solutions of the YBE [ 16-181. The Yangian invariance of the Hubbard Hamiltonian became likely after the observation that the S-matrix of elementary excitations at half filling is essentially a direct sum of two rational solutions of the YBE, each corresponding to the XXX spin chain [19]. There is some hope that the Yangian symmetry might be used to obtain excitation spectrum and n-point correlators of the Hubbard model in a way similar to the calculation of these quantities for the XXZ-chain by usage of its II,@) symmetry [20]. Such a kind of approach might also be applicable to an extended class of non-nearest-neighbour Hubbard models [21], which have recently been shown to be Yangian symmetric, too [22], and for which a QISM approach is unlikely to exist. The main concern of this Letter is to show how QISM and Yangian symmetry of the Hubbard model are connected. We benefit from the experience of one of the authors with the fermionic nonlinear Schrodinger model [23,24]. It turns out that the situation in case of the Hubbard model is to a large extent analogous. The Yangian symmetry is shown, when the model is considered on the infinite interval. Below the R-matrix and monodromy matrix are obtained as limits from their known counterparts on the finite interval. The R-matrix greatly simplifies in the considered limit. The new R-matrix contains a submatrix which turns into the rational R-matrix of the XXX-chain by an appropriate reparametrization. The corresponding submatrix of the monodromy matrix thus provides a representation of the Y(su(2)) Yangian. This representation is identified as the Yangian representation constructed earlier by Uglov and Korepin using ad hoc methods. From the quantum determinant of the considered submatrix of the monodromy matrix we obtain an infinite series of mutually commuting Yangian invariant operators which includes the Hamiltonian.
2. Quantum
inverse scattering
The Hamiltonian
method on the infinite interval
of the one-dimensional
Hubbard
model is
(2.1) where c-
and cl’ (r are annihilation and creation operators of electrons of spin CT at site j of a Id lattice, and is the particle number operator. Since we want to study finite excitations over the zero density vacuum IO> of the infinite interval, we normalized the Hamiltonian such that fi IO> = 0. Starting point for the QISM for the Hubbard model is the exchange relation [ 111 njc = c;L :jg
(2.2) where a,, denotes the Grassmann
direct product (2.3)
218
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Letters A 227 (1997) 216-226
with grading P(1) = P(4) = 0, P(2) = P(3) = 1. We adopt the expressions for the matrices 9 and 2 in terms of two parametrizing functions a(A), -y(A) from Ref. [I 1I. For later convenience, however, we shift the arguments of &A) and -y(A) by i T, such that we simply have (Y(A)= cos A, y(A) = sin A. The L-matrix is I
-eh(AYmtfml
-if,,
-%AA)=
+
‘mT _
,-h(A,,+
f ml mt
=-
sin 2A
c
mf
icmtcmLeh(*)
eeh(*)cmT
CnT
ckl
I
icmTg,,
(2.4)
7 eeh(*)gm
ml
t f,
.L
ct ml
= ~(h)(l - n,,)
sinh 2/z(A)
icm,fml
ewh(*)fmT g,,
CL1
ieh( h)cf
\
where f,,(A)
-fmtcml
1
g mt
‘m1
L ig,,
gm1
+ i&A)nmV, g,,(A)
ckl
--ET,,
&Iieh(*)
= o(A)(l - nmo) - iy(A)n,,,
and h(A) is defined as
U (2.5)
4’
Due to space limitations we do not reproduce the R-matrix here. It is 16 X 16 and contains 36 nonvanishing entries, only ten of which are different modulo signs. The ten different entries are denoted by pi, i = 1,. . . , 10, in Ref. [ 1 I]. They are rational functions of a(A), y(A) and e ‘(*) . We provide a list and some basic formulae which have been used in our calculations in Appendix A. The (m - n)-site Hubbard model (m > n) is characterized by the monodromy matrix ym,(A> =%-,(A)%,(A)
. . .-%(A).
(2.6)
It has been shown in Refs. [8,11] that the logarithmic derivative of the graded trace of Y,,,(A)
at A = 0 reproduces the Hamiltonian (2.1) under periodic boundary conditions. Like -E”,(A) the monodromy matrix Ym,( A) satisfies (2.2). In contrast to the classical case [25] a formulation of QISM on the infinite interval [26,27] has so far only been possible for zero density of elementary particles. This is due to the complicated structure of the finite density (finite band filling) vacua formed by an infinite number of interacting particles. For the Hubbard model, there are four simple vacua, the empty band, the completely filled band and the half filled band with all spins up or all spins down. In the following we will consider the empty band 10) (c,, 10) = 0) as reference state. Zero particle density means considering only states with a finite number of particles in the empty band. Expectation values of the L-matrix with respect to this space have a finite limit for ]m ( + x, which formally can be obtained by setting normal ordered products of operators equal to zero [26,27],
_Y’(A) + V(A) = diag( - y( A)*eh(*), CI( A)y( A)emh’*‘, a( A)y( A)eChCA), -CI( A)2eh(“‘). V(A) can be used to split off the asymptotics Fm,( A) = V(A) -?7-,,(
(2.7)
of Ym,,(A>. We expect the matrix
A)V( A)”
(2.8)
to have a finite limit for m --* x, n + --tc. This limit, y(
A) = m !iF+Ey_J
A).
(2.9)
will be the monodromy matrix on the infinite asymptotics W( A, p) of _T$(A) @,Zm( p) for operators. Then W(A, p)-“(S,,(A) BsYmn( just the tensor product V(A) @,vV( p). Due to W( A,
~112.21
W(k
PL)~~.~~=W(A~
W(
=
+L
I-&,~,
=
-iy(Ah(
P)uAx=~~(A)@(PCL)~
A, ~u),~,~,= -eh(h)*h(CI).
interval. To derive an exchange relation for &A), consider the large 1m ) again by omitting all normal ordered products of Fermi p))W(A, /.L)” is expected to have a finite limit. W(A, /_L)is not normal ordering there appear additional off-diagonal elements, CL),
WA, W(A,
~u)14.23= ~cL)23,41
-W(k =
-W(k
P)w~= PL)s~A,
-WA)a(pu), =
-ia(A)y(p),
S. Murakami,
It follows from the exchange 9(
relation
F. Giihmann/Physics
for the monodromy
um( P, A)-%A, = [%,(
relation
~u)u,(A.
(2.10)
and the definition
#m,(A)
P) @.,%(A)]Cr,(
matrix that
A, CL)*
A, P)W( A, P) = W( P> A)S(
Using once more the exchange
219
Letters A 227 (1997) 216-226
(2.8) of ym,,(A) we obtain
@G%,(P)]
~3 A)-‘s(A,
P)U,(A,
(2.11)
CL),
U~(A.~)=W(A.CL)-~[V(A)~~,V(~)~]. Postponing the discussion by use of the definitions u,(A,
of convergence
CL) = m!“,,Lim(A. _
we arrive at the exchange $+‘(A,
p)[y(A)
for a while we formally
5@‘)(
P),
A, CL) = U,(
relation for the monodromy S,$=(
(2.12)
c()] = [y(p)
take the limits
I-L, A)-‘9’(
matrix y(h)
e$-(A)]R”‘-‘(A,
The matrices (/+(A, pu), their inverses and the R-matrices &-(“(A, pi’s by utilizing the formulae provided in Appendix A. The non-zero
&(A. ~)ap.up= 1,
u&L
m, - n + x in (2.11). Then
A, P)&(
on the infinite
interval, (2.14)
CL).
CL) can be calculated as functions of the matrix elements of CT,(A, p) are
P)IW = Ps
-
P4
’
-ii3 u,(
A,
lu)m
=
U&L
PL),w,
(2.13)
A. /J).
=
-
PI0
ik
’
u,(
A,
CL),~.B
=
-
u,
( A,
CL)WU
=
___
P3_PI
*
where pi = pi( A, p). The corresponding matrices &” * ) are given in (2.15) (Scheme I>. The reader is urged to compare (2.15) with the R-matrix on the finite interval [ 111. Instead of the 36 nonvanishing elements of the original R-matrix we have only 18 nonvanishing elements here, which brings about simpler commutation relations between elements of the monodromy matrix. From our experience with the finite interval case we know that the monodromy matrix is of the following block form, /&(A)
y(A) =
C,,(A)
C,*(A)
012(A)
\
B,,(A) A,,(A) An(A) ButA) &I( A> fbd A) A**(A) &2(A) ’
,&,(A)
C,,(A)
C&A)
Q,(A)
I
(2.16)
where on the finite interval A(A) corresponds to the su(2) Lie algebra of rotations, D(A) corresponds to the q-pairing su(2) Lie algebra and the blocks B(A) and C(A) are connected to each other by particle-hole and gauge transformations [28]. Using the explicit form of the matrices &( * ) the exchange relation (2.16) implies
PA A, P) P,O(A, CL) z
P?( A,
CL) P~(A,
PL) -
PZ( A,
[A,&% A,,( P)] =A+( P>A~G(A)-Av~t+%d
E.L). (2.17)
~1
i.e. the commutation relations between the matrix elements of A(A) are decoupled from the rest of the algebra. This fact will be crucial for the derivation of the Y(su(2)) Yangian representation below.
220
S. Murakumi,
@“)(A,
p)=si+‘(h.
0
0
0
0
0
0
Giihmann/Physics
Letters
A 227
(1997)
216-226
CL)
0
0
31
F.
0 PIP4
0
iPI
0
0
0
0
0
o
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
PI P4
0
0
0
ip,,
0
0
0
0
0
-ipI
0
0
0
0
0
0
0
0
0
P4
0
0
0
0
0
0
0
0
0
0
0
-ipI
0
0
0
0
0
oo-
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
o-o
0 7 0
P3_PI
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
P9PlO
o
o
0
0
0
0
0
0
0
0
P3 -PI 0
0
0
0
ip,
p4 P9
0
0
0
P3P4_PI
0
0
0
P4
P5 - P4
0
0
0
0
0 -PI
0
0
0
P9PlO
o
0
0
0
P3 P4 - P: -0 P3_PI
P3_PI
0
0
0
0
0
0 0
P4 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ip, p4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ip9
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ip,
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
PI
PI-P,
0
0
0
PO
Scheme 1. Eq. (2.15)
So far we avoided to comment on the analytic structure of the exchange relation (2.14). This is indeed a delicate point. Along with the calculation of the limits U*(A, p) we obtain convergence conditions. The convergence conditions for U+( A, ~1 and U_ (A, ~1 are complementary. This situation is typical for the QISM on the infinite interval. Thus the first equation in (2.15) is only formal for the present time. We conjecture, however, that analytic continuation in A and p respects the exchange relation (2.14) with the possible exception of A = p (modulo periods), where singular terms (like 6(A - ~1) may destroy the first equality in (2.15). We know from our experience with the fermionic nonlinear Schradinger model [23,24] that such singular terms are irrelevant for the derivation of a Yangian representation from the exchange relation (2.14). 3. Yangian symmetry The Y(su(2)) Yangian the following relations,
16-181 algebra is generated
by six generators
Qt (n = 0, 1; a = 1, 2, 3), satisfying
Q,h] =fu”=Q;
(3.1)
[Q& Q;] =fu”QL
(3.2)
[Q;,
[[Qr,
Q;],
[Q;,
Q;]]
+ [[Q;.
Qf],
[Q& Q:]]
= ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Qd, Q;}, (3.3)
where
K
is a nonzero
constant,
cr ’ (a = 1, 2, 3) are he Pauli matrices,
fuhc = igobc
is the antisymmetric
S. Murukami. F. Giihmann/Phy.sics
221
Letrers A 227 (1997) 216-226
tensor of structure constants of su(2), and Aahcdef = fodkfbe’fcfmfk’m. Here and in the following we are using implicit summation over doubly occurring indices. The bracket { ) denotes the symmetrized product (3.4) Now we will show how to obtain a representation matrix (2.16). Introducing the reparametrization
of Y(su(2))
from the submatrix
A(A) of the monodromy
u( A) = - 2i cot 2 A cash 2 h( A),
(3.5)
the prefactor on the 1.h.s. of (2.17) becomes
&At CL)P,O(A, CL) 4 A) - 4 CL) icI ’ p3(A, P)P~(A, P) - idA P)*=
(3.6)
and we can write (2.17) in matrix form as (3.7)
(iu+(~(A)-u(E*-))~)[A(A)~A(~)1=[A(~)~A(A)l(i~+~o(A)-u(~))~).
Here LP is a 4 X 4 permutation matrix. (icI + {u(A) - u( ~)}LP) is the rational R-matrix of the XXX spin chain. (3.7) implies that A,,( A) is a generating function of the Y(su(2)) Yangian. The Yangian generators Q,” and Q; are the first coefficients in the asymptotic expansion [29,24] 1 A,p(A)=l+iUC nzO
u(A)‘+’
i 0=l
Q:k$
+
(3.8)
Qk,,
In order to obtain compact expression for Q,” and Qf we introduced the abbreviations 4 5 ’ = - u2, G2 = a’, (73 = (T 3. There are several possibilities to perform the limit u( A> --) =. However, we found that only one of these yields finite results for Q: and Qf. We have to take Im(A1 -+ 3~ and further have to choose the proper branch of solution in Eq. (2.5) which determines h(A) as a function of A. Some of the details of the calculation are given in Appendix B. The final result is (3.9)
(3.10) In our conventions the constant K in (3.3) is equal to iU. Comparing Korepin [ 151, we find complete equivalence,
(3.9), (3.10) with the result of Uglov and
E,=Qh+iQ,‘,
F,=Qh-iQ6,
H,=pQ;,
(3.11)
E, =Ql
F, =Qi - iQy,
H, = 2Q:.
(3.12)
+iQf,
where the expressions on the 1.h.s. are taken from the paper of Uglov and Korepin. Associated with the exchange relation (3.7) we can consider the quantum determinant
[30,31],
Det, A( A) = A,,( u(A)) A,,( u(A) - ic/) - A,,( u(A)) A2,( LJ(A) - iv), which is in the center of the Yangian [Det,A(A),
A,p( P)]
=O,
and provides a generating [Det, A( A), Det, A( p)]
function = 0.
of mutually
(3.13) commuting
operators, (3.14)
’ The reader should not worry about this notation. II results from our choice of the L-matrix, which we took from Ref. [I I ] to facilitate comparison with earlier work. It is easy to introduce a slight change of the L-matrix, compatible with the exchange relation, such that 6” is replaced by 17” in (3.8).
222
S. Murakumi, F. Giihmnnn/Physics
Letters A 227 (1997) 216-226
The second of these equations is of course a consequence of the first one. Performing again the asymptotic expression in terms of u(A), 1
Det,A(h)
= 1 +iU i n=O u(A)n+l rnT
(3.15)
we obtain I, = 0, I, = ifi, i.e. the Hamiltonian is among the commuting operators. All the conserved operators are Yangian invariant by construction. It will be interesting to investigate their relation to the formerly known conserved quantities [8,9,12,32], which were obtained for the finite periodic model. In closing this section we shall add a comment. The Hubbard Hamiltonian on the infinite interval is invariant under the transformation Cj,? --) ( I)+ J.t ’ u+ -lJ. (3.16) ‘i. 1 + c./,I ’ The Yangian generators Q,” and Qf, however, are transformed into a pair of generators Q,” and Qp of a second, independent representation of Y(su(2)) [ 151. These two representations mutually commute. Therefore they can be combined to a direct sum Y(su(2)) @ Y(su(2)). The reason why we get only one of these representations from our QISM approach is that, in order to perform the passage to the infinite interval, we refer to the zero density vacuum IO). This vacuum has lower symmetry than the Hamiltonian. It is invariant under the su(2) Lie algebra of rotations, but does not respect the q-pairing su(2) symmetry of the Hamiltonian. A fully su(2) @ ~$2) invariant vacuum would be the singlet ground state at half filling [19]. It seems to be yet a formidable task to formulate the QISM with respect to this state.
4. Concluding
remarks
and discussion
We have developed the QISM for the Hubbard model on the infinite interval with respect to the zero density vacuum. The R-matrix (2.15) thus obtained is greatly simplified in comparison with the R-matrix of the finite periodic model. Particularly, it reveals a hidden rational structure, which arises from a certain combination (3.6) of the functions pi. This structure was discovered earlier by Ramos and Martins [14] as part of the exchange relation for the Hubbard model on the finite interval. Note, however, that our reparametrization (3.6), which is essentially unique, differs from that given in Ref. [14]. A comparison is obstructed by the fact that the authors do not show their parameters cyj. Along with the simplified R-matrix we obtained the aymptotic expansion (3.8) of the submatrix A(A) of the monodromy matrix, which naturally provides a representation of Y(su(2)) and generates an infinite series of mutually commuting Yangian invariant operators including the Hamiltonian. There is a number of interesting open problems related to the QISM on the infinite interval. The analytic properties of the R-matrices 2 “( * ) (2.15) and of the monodromy matrix (2.9) deserve further investigations. Only the submatrix A(h) of the monodromy matrix has a limit for Im(h) -+ m. All other matrix elements diverge. It is therefore not clear at the present stage of investigation how to obtain creation operators of elementary excitations that are compatible with the Yangian generators Q,” and Qr. Creation operators are indispensable for the discussion of irreducible Yangian representations [23,24]. Another interesting task will be the construction of Dunk1 operators [33] associated with the Yangian representation discussed in this Letter. Dunk1 operators are building blocks of Yangian generators [33,35,23,24]. They are useful for the investigation of eigenstates. For the one-dimensional Hubbard model, although several attempts [34,35] have been made, no satisfactory Dunk1 operator is known.
Acknowledgement
We are grateful to Professor Miki Wadati for continuous encouragement and comments. We would also like to thank K. Hikami, V.E. Korepin and M. Shiroishi for fruitful discussions and comments. This work has been
S. Murukumi, F. Gijhmann/Physics
223
Letters A 227 (1997) 216-226
supported by the Japan Society for the Promotion of Science and the Ministry of Science, Culture and Education of Japan. Appendix A. Relations between the elements of the R-matrix In this appendix we collect functional relations among the elements of the R-matrix, which have been used in the calculation of the matrix U+(A, CL).We begin with the defining relations of the matrix elements, which are
Pi *9 P> =eb(*)a( tt) +e-‘r(*)y(p)9 PA** I*) p4(**Pu) =e’r(*)+Q +eW*Mh4 P*(*, I*> p9(*, P*) = -e'cu(A)y( p) +e-'r(A)a( CL), P*(A,FL) Plot*, P) =e’r(*)a(p) -e-b(*)74 ~1, PAA?PI p3(*, P> eb(*)a( CL)-e-‘r(*)y( P) PA *, PFL) = a’( *) - r*( Pet> ’ -e’r(*)y( PI +e-‘a(*)a( PI Ps(A*F) P*(A, PI = a2(A) - Y2(P) ’ P6(A,P.) e-*[eb(*)y(*) -e-b( P)Y( ~11 Pz(** P> = a*(*) -Y*(P) ’ where h = h(A) + h( p), 1 = h(A) - h( p). There are P,( A, cL)P4( A* l.L) + P9( ATP)PlO( A, p) = &(JL
k)Ps(
k
PI
-
P6( A,
PI2
= P2(A,
(A-1) (~4.2) (A.3)
(A.4) (A-5) (A4 (A.7)
r&itiOnS
P2(
*,
among the pi functions,
PY~
(‘4.8)
I-&
(A.91
(A.lO) We found that there is a set of relations “dual” to (A.l)-(A.7). Introducing a transformation @, which keeps A unchanged and substitutes I_L+ in for k, we get the following trai%fOimatiOn rules, PI(
JL ct)Ps(
A3 CL) + P3(
Ps -
PI --bP2
P4
A? PCL)P4( A* F.)
‘P3 -
P4 ----b--f
P6
PI
P6
P2
Ps -+--,
P3
P6 --+--.
P2
P2
P6
P2
P6
=
2P2(A,
P9 -+_’
d2*
PI0
P2
P6’
PI0 -+-P2
P9
P3 -+--
PS
P6 ’
P2
P6 ’
Explicitly, we obtain the relations P5( A,
P)
-
P4( *,
F) =
P6cA7 P3( A+ II)
-
-e"cY(A)y(~)+e-hy(A)cu(~),
Pd
A, LL)
(A.12)
=ehr(h)~(~)-e-ha(h)y(EL). P6(*, _
PIOh
P) CL) =
P6( A,
(A.11)
II)
cc)
--eha(
A)a(
II>
-
e-h?/(h)y(
P),
(‘4.13)
224
S. Murakumi,
F. Giihmunn
/Physics
- Pd*, CL)= -e”r( *)y( P) -f+WM L%t*,PI
Letters A 227 11997) 216-226
CL),
(A.14)
I%(**PL) -e*cYt*)yt~)--e-*yt*)cwt~) a2(A) - cxyp) ’ Pd*? Pu>=
(A.15)
Pd** PI = -eSt+t 1.4-e-WM d(A) - a’( /_L) Pet*?FCL)
(A.16)
1-4 ’
- P,(*, 1-4= e-‘[eW*M*) +e-+4 PM 1-41 a”( A) - a’( p)
Pd*~ l-4
which are shown by direct calculation.
Appendix
B. Asymptotic
expansion
The relations
E+
l.“( A) =R(
MM(
(A$)-(A.lO)
of the elements
We shall explain below details of expansion satisfies the recursion relation
(A.17)
’ are invariant
of the monodromy
of the monodromy
under @.
matrix
matrix y(A)
in terms of u(A)-‘.
ym,,(A)
A)’
where .P”2(A)=V(A)- “I- ‘_?q h)V( A)“’ (
e-ku,
ci COtq%“l + ‘1,” 1
(i cot A)““’
imp(A)
c,, 1 -e sin A
(i cot A)“‘“’ - ‘IS”,
CL,t (i
I
-imp(A)
cot A)n”‘se
4 t C”,1 -c:,,(icot
sin A cos A e-hw A)-‘I’“‘-e CDS A
-ic,,(icot
c “Z
A)
T
““.,
e-h(A)
1
____g=P(A)
-ic mT ‘m 1 -cot2”k sin~A
sin A
i cm 1 sin
(i cot A)
imk(A)
,w
I
ic,r
A cos A
- )I,,,f
eh(h)
+ II..‘
e-h(N _ i(i cot A)- “‘“7 $, 1 =ee-
in,k(h) (i cot A)- “m’ -e sin A
(icot Inlk(A)
A)-“‘“‘c,,,~I_~
(B .2)
I”,k(l)
Sl” A
(i COt A) - ‘1,”t
- “P”I /
Here we introduced e’kfA) =
_e2Mh)
new functions cot
*,
eia(Af =
_e-2MN
which we adopted from the recent analytic follows from (3.5) and (B.3) that sin k(A) = - fu( A) + +ilJ,
Cot
Bethe ansatz for the Hubbard
sin p(A)
(B.3)
*
= - +u( A) - +iU.
model by Yue and Deguchi
[36]. It
03.4)
S. Murukumi. F. GAhmann/Physics
In the limit I m I --) cc, the above matrix L?,,(A) converges iteratively we obtain y(A) as = . fd2”,+
F(A)
,( h)P”,(
Lefrers A 227 f1997) 216-226
225
in the weak sense to the identity matrix. Solving (B. 1)
h).L&_ ,( A) . ‘. (B-5)
k
k>l
To expand _T?,,(A) in terms of u( Al-‘, we consider the limit Im( A) -+ x and, to begin with, expand function in terms of e2’A. For e- 2h(A) there are two possible choices of branch, e -2h(A)=
-fU
sin 2A +
To achieve convergence this choice we get e2h(h)
_e2ih
C’ ,-i/HA)
=
+
O(e6”),
!_{,*,A _ ze4iA u
(B.6)
of the matrix elements
4i
=
each
Y:ls (cu, /3 = 2, 3) we have to take the lower sign in (B.6). For
eikfh) = +!,,ZiA
+qe6iA)),
&
+ ze4iA + qe6’A))_
= qIe*iA +qe6iA)).
(B.7)
Now the leading terms in the sums in (B.5) are of order e2ih, e4ih, . . . Thus, from the first two sums in (B.51, we get the expansion of the matrix A(A) up to order e4ih. Then the last equation in (B.7) yields the required expansion in (u(A))- ’ up to second order.
References [I] F.H.L. EBler, V.E. Korepin and K. Schoutens, Phys. Rev. Lett. 67 (1991) 3848; Nucl. Phys. B 372 (1992) 599; 384 (1992) 43 1. [2] O.J. Heilmamt and E.H. Lieb, Ann. NY Acad. Sci. 172 (1971) 583. 131 C.N. Yang, Phys. Rev. Lett. 63 (1989) 2144; C.N. Yang and SC. Zhang, Mod. Phys. Lett. B 4 (1990) 759. [4] E.H. Lieb and F.Y. Wu, Phys. Rev. Lett. 20 (1968) 1445. [5] V.E. Korepin and F.H.L. EBler, eds., Exactly solvable models of strongly correlated electrons (World Scientific, Singapore, 1994). [6] C.N. Yang, Phys. Rev. Lett. 19 (1967) 1312. [7] M. Gaudin, Phys. Lett. A 24 (1967) 55. [8] B.S. Shastry, Phys. Rev. Lett. 56 (1986) 1529; 56 (1986) 2453. [9] B.S. Shastry. J. Stat. Phys. 50 (1988) 57. [IO] M. Wadati, E. Ohnedilla and Y. Atutsu, J. Phys. Sot. Japan 56 (1987) 1340. [l I] E. Olmedilla, M. Wadati and Y. Ahutsu, .I. Phys. Sot. Japan 56 (1987) 2298. [I21 E. Olmedilla and M. Wadati, Phys. Rev. Lett. 60 (1988) 1595. [ 131 M. Shiroishi and M. Wadati, J. Phys. Sot. Japan 64 (1995) 57. [I41 P.B. Ramos and M.J. Martins, preprint hep-th/9605141 (1996). [I51 D.B. Uglov and V.E. Korepin. Phys. Lett. A 190 (1994) 238. [16] V.G. Drinfeld, in Proc. of ICM (Berkeley, California, 1986) p. 269. [ 171 V. Chari and A.N. Pressley, L’enseignement mathematique 36 (1990) 267; J. Reine. Angew. Math. 417 (1991) 87. [IS] A. LeCtair and F.A. Smimov, Int. J. Mod. Phys. A 7 (1992) 2997. [ 191 F.H.L. EBler and V.E. Korepin, Phys. Rev. Lett. 72 (1994) 908; Nucl. Phys. B 426 ( 1994) 505. [20] M. Jimbo, K. Miki, T. Miwa and A. Nakayashiki, Phys. Lett. A 168 (1992) 256. [21] F. Gebhard and A.E. Ruckenstein. Phys. Rev. Lett. 68 (1992) 244; P.A. Bares and F. Gebhard, Europhys. Lett. 29 (I 995) 573. [22] F. Giihmann and V. Inozemtsev, Phys. Lett. A 214 (1996) 161. [23] S. Murakami and M. Wadati, J. Phys. Sot. Japan 65 (1996) 1227. [34] S. Murakami and M. Wadati, Connection between Yangian symmetry and the quantum inverse scattering method, preprint ( 1996).
226
S. Murakami,
F. Giihmunn/Physics
Letters A 227 (19971216-226
12.51 L.D. Faddeev
1261 [27] [281 (291 [30] [311 [32] [33]
[34] [35] [36]
and L.A. Takhtajan, Hamiltonian methods in the theory of solitons (Springer, Berlin, 1987). L.D. Faddeev, Sov. Sci. Rev. Math. Phys. C 1 (1980) 107. H.B. Thacker, Rev. Mod. Phys. 53 (1981) 253. F. Giihmann and S. Murakami, Algebraic and analytic properties of the one-dimensional Hubbard model, preprint (1996). F.D.M. Haldane, in: Proc. 16th Taniguchi Symp. on Condensed matter physics (Springer, Berlin, 1994). A.G. lzegin and V.E. Korepin, Sov. Phys. Dokl. 26 (1981) 653. V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions (Cambridge Press, Cambridge, 1993) p. 160. H. Grosse, Lett. Math. Phys. 18 (1989) 151. A. Polychronakos, Phys. Rev. Lett. 69 (1992) 703; J.A. Minahan and A. Polychronakos, Phys. Len. B 302 (1993) 265; D. Bernard, M. Gaudin, F.D.M. Haldane and V. Pasquier, J. Phys. A 26 (1993) 5219. D.B. Uglov, Phys. Lett. A 199 (1995) 353. K. Hikami and S. Murakami, Phys. Lett. A 221 (19%) 109. R. Yue and T. Deguchi, preprint cond-mat/9605039 (1996).
Univ
PHYSICS ELSEVIER
LETTERS
A
Physics Letters A 227 ( 1997) 2 16-226
Yangian symmetry and quantum inverse scattering method for the one-dimensional Hubbard model Shuichi Murakami a,1,Frank Gijhmann b*2 a Department o/Applied Physics. Fuculty ofEngineering. University ofTokyo. Hongo 7-3-l. Bunkyo-ku, Tokyo 113, Japun b Department ofPhysics, Faculty of Science, University ofTokyo, Hongo 7-3-l. Bunkyo-ku, Tokyo 113. Japan ’ Received 3 October
1996; accepted for publication 13 December Communicated by A.R. Bishop
1996
Abstract We develop the quantum inverse scattering method for the one-dimensional Hubbard model on the infinite interval at zero density. The R-matrix and monodromy matrix are obtained as limits from their known counterparts on the finite interval. The R-matrix greatly simplifies in the considered limit. The new R-matrix contains a submatrix which turns into the rational R-matrix of the XXX-chain by an appropriate reparametrization. The corresponding submatrix of the monodromy matrix thus provides a representation of the Y(su(2)) Yangian. From its quantum determinant we obtain an infinite series of mutually commuting Yangian invariant operators which includes the Hamiltonian. Keywords:
Hubbard
model; Quantum
inverse scattering;
Yangian
1. Introduction Among the exactly solvable Id quantum systems the Hubbard model has probably the most interesting applications in solid state physics. Its elementary excitations, their dispersion relations and S-matrix at half filling [I] have been calculated exactly by use of the coordinate Bethe ansatz in conjunction with the SO(4) symmetry of the model [2,3]. The story of this development is long and originates in the seminal paper [4] of Lieb and Wu. A recent overview is offered tiy the reprint volume [S]. From the point of view of the coordinate Bethe ansatz the one-dimensional Hubbard model appears similar to the fermionic nonlinear Schrijdinger model. In fact, Lieb and Wu in their article used this analogy to obtain the Bethe ansatz equations from Yang’s earlier result [6,7]. Algebraically, however, the model seems to be more complicated. Nearly 20 years passed before the basic tools of quantum inverse scattering method (QISM), R-matrix and L-matrix were derived by Shastry [S;9] and by Olmedilla et al. [ 10-121, and it was shown only
’ E-mail: [email protected]. 2 E-mail: [email protected]. 3 Address from October 1996: Physikalisches 0375-9601/97/$17.00 Copyright PII SO375-9601(96)00953-X
lnstitut der UniversitZt Bayreuth,
TPI, 95440 Bayreuth,
0 1997 Elsevier Science B.V. All rights reserved
Germany.
S. Murukumi, F. Giihmutm/Physics
Letters A 227 (1997) 216-226
217
recently that the R-matrix satisfies the Yang-Baxter equation (YBE) 1131. The R-matrix and monodromy matrix of the Hubbard model have unusual features. The monodromy matrix is 4 X 4 rather than 3 X 3, as one might have guessed naively from the fact that there are two levels of Bethe ansatz equations or from the analogy with the fermionic nonlinear Schrijdinger model. It seems to be impossible to find a parametrization of the R-matrix, such that it becomes a function of the difference of the spectral parameters. For these reasons an algebraic Bethe ansatz is difficult and was performed only recently by Ramos and Martins [ 141. There is another algebraic structure related to the Hubbard model on the infinite line. As was discovered by Uglov and Korepin [15] the Hubbard Hamiltonian commutes with two independent and mutually commuting representations of the Yangian Y(su(2)). Yangians are the quantum groups connected to rational solutions of the YBE [ 16-181. The Yangian invariance of the Hubbard Hamiltonian became likely after the observation that the S-matrix of elementary excitations at half filling is essentially a direct sum of two rational solutions of the YBE, each corresponding to the XXX spin chain [19]. There is some hope that the Yangian symmetry might be used to obtain excitation spectrum and n-point correlators of the Hubbard model in a way similar to the calculation of these quantities for the XXZ-chain by usage of its II,@) symmetry [20]. Such a kind of approach might also be applicable to an extended class of non-nearest-neighbour Hubbard models [21], which have recently been shown to be Yangian symmetric, too [22], and for which a QISM approach is unlikely to exist. The main concern of this Letter is to show how QISM and Yangian symmetry of the Hubbard model are connected. We benefit from the experience of one of the authors with the fermionic nonlinear Schrodinger model [23,24]. It turns out that the situation in case of the Hubbard model is to a large extent analogous. The Yangian symmetry is shown, when the model is considered on the infinite interval. Below the R-matrix and monodromy matrix are obtained as limits from their known counterparts on the finite interval. The R-matrix greatly simplifies in the considered limit. The new R-matrix contains a submatrix which turns into the rational R-matrix of the XXX-chain by an appropriate reparametrization. The corresponding submatrix of the monodromy matrix thus provides a representation of the Y(su(2)) Yangian. This representation is identified as the Yangian representation constructed earlier by Uglov and Korepin using ad hoc methods. From the quantum determinant of the considered submatrix of the monodromy matrix we obtain an infinite series of mutually commuting Yangian invariant operators which includes the Hamiltonian.
2. Quantum
inverse scattering
The Hamiltonian
method on the infinite interval
of the one-dimensional
Hubbard
model is
(2.1) where c-
and cl’ (r are annihilation and creation operators of electrons of spin CT at site j of a Id lattice, and is the particle number operator. Since we want to study finite excitations over the zero density vacuum IO> of the infinite interval, we normalized the Hamiltonian such that fi IO> = 0. Starting point for the QISM for the Hubbard model is the exchange relation [ 111 njc = c;L :jg
(2.2) where a,, denotes the Grassmann
direct product (2.3)
218
S. Murakami, F. Giihmonn/Physics
Letters A 227 (1997) 216-226
with grading P(1) = P(4) = 0, P(2) = P(3) = 1. We adopt the expressions for the matrices 9 and 2 in terms of two parametrizing functions a(A), -y(A) from Ref. [I 1I. For later convenience, however, we shift the arguments of &A) and -y(A) by i T, such that we simply have (Y(A)= cos A, y(A) = sin A. The L-matrix is I
-eh(AYmtfml
-if,,
-%AA)=
+
‘mT _
,-h(A,,+
f ml mt
=-
sin 2A
c
mf
icmtcmLeh(*)
eeh(*)cmT
CnT
ckl
I
icmTg,,
(2.4)
7 eeh(*)gm
ml
t f,
.L
ct ml
= ~(h)(l - n,,)
sinh 2/z(A)
icm,fml
ewh(*)fmT g,,
CL1
ieh( h)cf
\
where f,,(A)
-fmtcml
1
g mt
‘m1
L ig,,
gm1
+ i&A)nmV, g,,(A)
ckl
--ET,,
&Iieh(*)
= o(A)(l - nmo) - iy(A)n,,,
and h(A) is defined as
U (2.5)
4’
Due to space limitations we do not reproduce the R-matrix here. It is 16 X 16 and contains 36 nonvanishing entries, only ten of which are different modulo signs. The ten different entries are denoted by pi, i = 1,. . . , 10, in Ref. [ 1 I]. They are rational functions of a(A), y(A) and e ‘(*) . We provide a list and some basic formulae which have been used in our calculations in Appendix A. The (m - n)-site Hubbard model (m > n) is characterized by the monodromy matrix ym,(A> =%-,(A)%,(A)
. . .-%(A).
(2.6)
It has been shown in Refs. [8,11] that the logarithmic derivative of the graded trace of Y,,,(A)
at A = 0 reproduces the Hamiltonian (2.1) under periodic boundary conditions. Like -E”,(A) the monodromy matrix Ym,( A) satisfies (2.2). In contrast to the classical case [25] a formulation of QISM on the infinite interval [26,27] has so far only been possible for zero density of elementary particles. This is due to the complicated structure of the finite density (finite band filling) vacua formed by an infinite number of interacting particles. For the Hubbard model, there are four simple vacua, the empty band, the completely filled band and the half filled band with all spins up or all spins down. In the following we will consider the empty band 10) (c,, 10) = 0) as reference state. Zero particle density means considering only states with a finite number of particles in the empty band. Expectation values of the L-matrix with respect to this space have a finite limit for ]m ( + x, which formally can be obtained by setting normal ordered products of operators equal to zero [26,27],
_Y’(A) + V(A) = diag( - y( A)*eh(*), CI( A)y( A)emh’*‘, a( A)y( A)eChCA), -CI( A)2eh(“‘). V(A) can be used to split off the asymptotics Fm,( A) = V(A) -?7-,,(
(2.7)
of Ym,,(A>. We expect the matrix
A)V( A)”
(2.8)
to have a finite limit for m --* x, n + --tc. This limit, y(
A) = m !iF+Ey_J
A).
(2.9)
will be the monodromy matrix on the infinite asymptotics W( A, p) of _T$(A) @,Zm( p) for operators. Then W(A, p)-“(S,,(A) BsYmn( just the tensor product V(A) @,vV( p). Due to W( A,
~112.21
W(k
PL)~~.~~=W(A~
W(
=
+L
I-&,~,
=
-iy(Ah(
P)uAx=~~(A)@(PCL)~
A, ~u),~,~,= -eh(h)*h(CI).
interval. To derive an exchange relation for &A), consider the large 1m ) again by omitting all normal ordered products of Fermi p))W(A, /.L)” is expected to have a finite limit. W(A, /_L)is not normal ordering there appear additional off-diagonal elements, CL),
WA, W(A,
~u)14.23= ~cL)23,41
-W(k =
-W(k
P)w~= PL)s~A,
-WA)a(pu), =
-ia(A)y(p),
S. Murakami,
It follows from the exchange 9(
relation
F. Giihmann/Physics
for the monodromy
um( P, A)-%A, = [%,(
relation
~u)u,(A.
(2.10)
and the definition
#m,(A)
P) @.,%(A)]Cr,(
matrix that
A, CL)*
A, P)W( A, P) = W( P> A)S(
Using once more the exchange
219
Letters A 227 (1997) 216-226
(2.8) of ym,,(A) we obtain
@G%,(P)]
~3 A)-‘s(A,
P)U,(A,
(2.11)
CL),
U~(A.~)=W(A.CL)-~[V(A)~~,V(~)~]. Postponing the discussion by use of the definitions u,(A,
of convergence
CL) = m!“,,Lim(A. _
we arrive at the exchange $+‘(A,
p)[y(A)
for a while we formally
5@‘)(
P),
A, CL) = U,(
relation for the monodromy S,$=(
(2.12)
c()] = [y(p)
take the limits
I-L, A)-‘9’(
matrix y(h)
e$-(A)]R”‘-‘(A,
The matrices (/+(A, pu), their inverses and the R-matrices &-(“(A, pi’s by utilizing the formulae provided in Appendix A. The non-zero
&(A. ~)ap.up= 1,
u&L
m, - n + x in (2.11). Then
A, P)&(
on the infinite
interval, (2.14)
CL).
CL) can be calculated as functions of the matrix elements of CT,(A, p) are
P)IW = Ps
-
P4
’
-ii3 u,(
A,
lu)m
=
U&L
PL),w,
(2.13)
A. /J).
=
-
PI0
ik
’
u,(
A,
CL),~.B
=
-
u,
( A,
CL)WU
=
___
P3_PI
*
where pi = pi( A, p). The corresponding matrices &” * ) are given in (2.15) (Scheme I>. The reader is urged to compare (2.15) with the R-matrix on the finite interval [ 111. Instead of the 36 nonvanishing elements of the original R-matrix we have only 18 nonvanishing elements here, which brings about simpler commutation relations between elements of the monodromy matrix. From our experience with the finite interval case we know that the monodromy matrix is of the following block form, /&(A)
y(A) =
C,,(A)
C,*(A)
012(A)
\
B,,(A) A,,(A) An(A) ButA) &I( A> fbd A) A**(A) &2(A) ’
,&,(A)
C,,(A)
C&A)
Q,(A)
I
(2.16)
where on the finite interval A(A) corresponds to the su(2) Lie algebra of rotations, D(A) corresponds to the q-pairing su(2) Lie algebra and the blocks B(A) and C(A) are connected to each other by particle-hole and gauge transformations [28]. Using the explicit form of the matrices &( * ) the exchange relation (2.16) implies
PA A, P) P,O(A, CL) z
P?( A,
CL) P~(A,
PL) -
PZ( A,
[A,&% A,,( P)] =A+( P>A~G(A)-Av~t+%d
E.L). (2.17)
~1
i.e. the commutation relations between the matrix elements of A(A) are decoupled from the rest of the algebra. This fact will be crucial for the derivation of the Y(su(2)) Yangian representation below.
220
S. Murakumi,
@“)(A,
p)=si+‘(h.
0
0
0
0
0
0
Giihmann/Physics
Letters
A 227
(1997)
216-226
CL)
0
0
31
F.
0 PIP4
0
iPI
0
0
0
0
0
o
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
PI P4
0
0
0
ip,,
0
0
0
0
0
-ipI
0
0
0
0
0
0
0
0
0
P4
0
0
0
0
0
0
0
0
0
0
0
-ipI
0
0
0
0
0
oo-
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
o-o
0 7 0
P3_PI
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
P9PlO
o
o
0
0
0
0
0
0
0
0
P3 -PI 0
0
0
0
ip,
p4 P9
0
0
0
P3P4_PI
0
0
0
P4
P5 - P4
0
0
0
0
0 -PI
0
0
0
P9PlO
o
0
0
0
P3 P4 - P: -0 P3_PI
P3_PI
0
0
0
0
0
0 0
P4 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ip, p4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ip9
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
ip,
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
PI
PI-P,
0
0
0
PO
Scheme 1. Eq. (2.15)
So far we avoided to comment on the analytic structure of the exchange relation (2.14). This is indeed a delicate point. Along with the calculation of the limits U*(A, p) we obtain convergence conditions. The convergence conditions for U+( A, ~1 and U_ (A, ~1 are complementary. This situation is typical for the QISM on the infinite interval. Thus the first equation in (2.15) is only formal for the present time. We conjecture, however, that analytic continuation in A and p respects the exchange relation (2.14) with the possible exception of A = p (modulo periods), where singular terms (like 6(A - ~1) may destroy the first equality in (2.15). We know from our experience with the fermionic nonlinear Schradinger model [23,24] that such singular terms are irrelevant for the derivation of a Yangian representation from the exchange relation (2.14). 3. Yangian symmetry The Y(su(2)) Yangian the following relations,
16-181 algebra is generated
by six generators
Qt (n = 0, 1; a = 1, 2, 3), satisfying
Q,h] =fu”=Q;
(3.1)
[Q& Q;] =fu”QL
(3.2)
[Q;,
[[Qr,
Q;],
[Q;,
Q;]]
+ [[Q;.
Qf],
[Q& Q:]]
= ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Qd, Q;}, (3.3)
where
K
is a nonzero
constant,
cr ’ (a = 1, 2, 3) are he Pauli matrices,
fuhc = igobc
is the antisymmetric
S. Murukami. F. Giihmann/Phy.sics
221
Letrers A 227 (1997) 216-226
tensor of structure constants of su(2), and Aahcdef = fodkfbe’fcfmfk’m. Here and in the following we are using implicit summation over doubly occurring indices. The bracket { ) denotes the symmetrized product (3.4) Now we will show how to obtain a representation matrix (2.16). Introducing the reparametrization
of Y(su(2))
from the submatrix
A(A) of the monodromy
u( A) = - 2i cot 2 A cash 2 h( A),
(3.5)
the prefactor on the 1.h.s. of (2.17) becomes
&At CL)P,O(A, CL) 4 A) - 4 CL) icI ’ p3(A, P)P~(A, P) - idA P)*=
(3.6)
and we can write (2.17) in matrix form as (3.7)
(iu+(~(A)-u(E*-))~)[A(A)~A(~)1=[A(~)~A(A)l(i~+~o(A)-u(~))~).
Here LP is a 4 X 4 permutation matrix. (icI + {u(A) - u( ~)}LP) is the rational R-matrix of the XXX spin chain. (3.7) implies that A,,( A) is a generating function of the Y(su(2)) Yangian. The Yangian generators Q,” and Q; are the first coefficients in the asymptotic expansion [29,24] 1 A,p(A)=l+iUC nzO
u(A)‘+’
i 0=l
Q:k$
+
(3.8)
Qk,,
In order to obtain compact expression for Q,” and Qf we introduced the abbreviations 4 5 ’ = - u2, G2 = a’, (73 = (T 3. There are several possibilities to perform the limit u( A> --) =. However, we found that only one of these yields finite results for Q: and Qf. We have to take Im(A1 -+ 3~ and further have to choose the proper branch of solution in Eq. (2.5) which determines h(A) as a function of A. Some of the details of the calculation are given in Appendix B. The final result is (3.9)
(3.10) In our conventions the constant K in (3.3) is equal to iU. Comparing Korepin [ 151, we find complete equivalence,
(3.9), (3.10) with the result of Uglov and
E,=Qh+iQ,‘,
F,=Qh-iQ6,
H,=pQ;,
(3.11)
E, =Ql
F, =Qi - iQy,
H, = 2Q:.
(3.12)
+iQf,
where the expressions on the 1.h.s. are taken from the paper of Uglov and Korepin. Associated with the exchange relation (3.7) we can consider the quantum determinant
[30,31],
Det, A( A) = A,,( u(A)) A,,( u(A) - ic/) - A,,( u(A)) A2,( LJ(A) - iv), which is in the center of the Yangian [Det,A(A),
A,p( P)]
=O,
and provides a generating [Det, A( A), Det, A( p)]
function = 0.
of mutually
(3.13) commuting
operators, (3.14)
’ The reader should not worry about this notation. II results from our choice of the L-matrix, which we took from Ref. [I I ] to facilitate comparison with earlier work. It is easy to introduce a slight change of the L-matrix, compatible with the exchange relation, such that 6” is replaced by 17” in (3.8).
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Letters A 227 (1997) 216-226
The second of these equations is of course a consequence of the first one. Performing again the asymptotic expression in terms of u(A), 1
Det,A(h)
= 1 +iU i n=O u(A)n+l rnT
(3.15)
we obtain I, = 0, I, = ifi, i.e. the Hamiltonian is among the commuting operators. All the conserved operators are Yangian invariant by construction. It will be interesting to investigate their relation to the formerly known conserved quantities [8,9,12,32], which were obtained for the finite periodic model. In closing this section we shall add a comment. The Hubbard Hamiltonian on the infinite interval is invariant under the transformation Cj,? --) ( I)+ J.t ’ u+ -lJ. (3.16) ‘i. 1 + c./,I ’ The Yangian generators Q,” and Qf, however, are transformed into a pair of generators Q,” and Qp of a second, independent representation of Y(su(2)) [ 151. These two representations mutually commute. Therefore they can be combined to a direct sum Y(su(2)) @ Y(su(2)). The reason why we get only one of these representations from our QISM approach is that, in order to perform the passage to the infinite interval, we refer to the zero density vacuum IO). This vacuum has lower symmetry than the Hamiltonian. It is invariant under the su(2) Lie algebra of rotations, but does not respect the q-pairing su(2) symmetry of the Hamiltonian. A fully su(2) @ ~$2) invariant vacuum would be the singlet ground state at half filling [19]. It seems to be yet a formidable task to formulate the QISM with respect to this state.
4. Concluding
remarks
and discussion
We have developed the QISM for the Hubbard model on the infinite interval with respect to the zero density vacuum. The R-matrix (2.15) thus obtained is greatly simplified in comparison with the R-matrix of the finite periodic model. Particularly, it reveals a hidden rational structure, which arises from a certain combination (3.6) of the functions pi. This structure was discovered earlier by Ramos and Martins [14] as part of the exchange relation for the Hubbard model on the finite interval. Note, however, that our reparametrization (3.6), which is essentially unique, differs from that given in Ref. [14]. A comparison is obstructed by the fact that the authors do not show their parameters cyj. Along with the simplified R-matrix we obtained the aymptotic expansion (3.8) of the submatrix A(A) of the monodromy matrix, which naturally provides a representation of Y(su(2)) and generates an infinite series of mutually commuting Yangian invariant operators including the Hamiltonian. There is a number of interesting open problems related to the QISM on the infinite interval. The analytic properties of the R-matrices 2 “( * ) (2.15) and of the monodromy matrix (2.9) deserve further investigations. Only the submatrix A(h) of the monodromy matrix has a limit for Im(h) -+ m. All other matrix elements diverge. It is therefore not clear at the present stage of investigation how to obtain creation operators of elementary excitations that are compatible with the Yangian generators Q,” and Qr. Creation operators are indispensable for the discussion of irreducible Yangian representations [23,24]. Another interesting task will be the construction of Dunk1 operators [33] associated with the Yangian representation discussed in this Letter. Dunk1 operators are building blocks of Yangian generators [33,35,23,24]. They are useful for the investigation of eigenstates. For the one-dimensional Hubbard model, although several attempts [34,35] have been made, no satisfactory Dunk1 operator is known.
Acknowledgement
We are grateful to Professor Miki Wadati for continuous encouragement and comments. We would also like to thank K. Hikami, V.E. Korepin and M. Shiroishi for fruitful discussions and comments. This work has been
S. Murukumi, F. Gijhmann/Physics
223
Letters A 227 (1997) 216-226
supported by the Japan Society for the Promotion of Science and the Ministry of Science, Culture and Education of Japan. Appendix A. Relations between the elements of the R-matrix In this appendix we collect functional relations among the elements of the R-matrix, which have been used in the calculation of the matrix U+(A, CL).We begin with the defining relations of the matrix elements, which are
Pi *9 P> =eb(*)a( tt) +e-‘r(*)y(p)9 PA** I*) p4(**Pu) =e’r(*)+Q +eW*Mh4 P*(*, I*> p9(*, P*) = -e'cu(A)y( p) +e-'r(A)a( CL), P*(A,FL) Plot*, P) =e’r(*)a(p) -e-b(*)74 ~1, PAA?PI p3(*, P> eb(*)a( CL)-e-‘r(*)y( P) PA *, PFL) = a’( *) - r*( Pet> ’ -e’r(*)y( PI +e-‘a(*)a( PI Ps(A*F) P*(A, PI = a2(A) - Y2(P) ’ P6(A,P.) e-*[eb(*)y(*) -e-b( P)Y( ~11 Pz(** P> = a*(*) -Y*(P) ’ where h = h(A) + h( p), 1 = h(A) - h( p). There are P,( A, cL)P4( A* l.L) + P9( ATP)PlO( A, p) = &(JL
k)Ps(
k
PI
-
P6( A,
PI2
= P2(A,
(A-1) (~4.2) (A.3)
(A.4) (A-5) (A4 (A.7)
r&itiOnS
P2(
*,
among the pi functions,
PY~
(‘4.8)
I-&
(A.91
(A.lO) We found that there is a set of relations “dual” to (A.l)-(A.7). Introducing a transformation @, which keeps A unchanged and substitutes I_L+ in for k, we get the following trai%fOimatiOn rules, PI(
JL ct)Ps(
A3 CL) + P3(
Ps -
PI --bP2
P4
A? PCL)P4( A* F.)
‘P3 -
P4 ----b--f
P6
PI
P6
P2
Ps -+--,
P3
P6 --+--.
P2
P2
P6
P2
P6
=
2P2(A,
P9 -+_’
d2*
PI0
P2
P6’
PI0 -+-P2
P9
P3 -+--
PS
P6 ’
P2
P6 ’
Explicitly, we obtain the relations P5( A,
P)
-
P4( *,
F) =
P6cA7 P3( A+ II)
-
-e"cY(A)y(~)+e-hy(A)cu(~),
Pd
A, LL)
(A.12)
=ehr(h)~(~)-e-ha(h)y(EL). P6(*, _
PIOh
P) CL) =
P6( A,
(A.11)
II)
cc)
--eha(
A)a(
II>
-
e-h?/(h)y(
P),
(‘4.13)
224
S. Murakumi,
F. Giihmunn
/Physics
- Pd*, CL)= -e”r( *)y( P) -f+WM L%t*,PI
Letters A 227 11997) 216-226
CL),
(A.14)
I%(**PL) -e*cYt*)yt~)--e-*yt*)cwt~) a2(A) - cxyp) ’ Pd*? Pu>=
(A.15)
Pd** PI = -eSt+t 1.4-e-WM d(A) - a’( /_L) Pet*?FCL)
(A.16)
1-4 ’
- P,(*, 1-4= e-‘[eW*M*) +e-+4 PM 1-41 a”( A) - a’( p)
Pd*~ l-4
which are shown by direct calculation.
Appendix
B. Asymptotic
expansion
The relations
E+
l.“( A) =R(
MM(
(A$)-(A.lO)
of the elements
We shall explain below details of expansion satisfies the recursion relation
(A.17)
’ are invariant
of the monodromy
of the monodromy
under @.
matrix
matrix y(A)
in terms of u(A)-‘.
ym,,(A)
A)’
where .P”2(A)=V(A)- “I- ‘_?q h)V( A)“’ (
e-ku,
ci COtq%“l + ‘1,” 1
(i cot A)““’
imp(A)
c,, 1 -e sin A
(i cot A)“‘“’ - ‘IS”,
CL,t (i
I
-imp(A)
cot A)n”‘se
4 t C”,1 -c:,,(icot
sin A cos A e-hw A)-‘I’“‘-e CDS A
-ic,,(icot
c “Z
A)
T
““.,
e-h(A)
1
____g=P(A)
-ic mT ‘m 1 -cot2”k sin~A
sin A
i cm 1 sin
(i cot A)
imk(A)
,w
I
ic,r
A cos A
- )I,,,f
eh(h)
+ II..‘
e-h(N _ i(i cot A)- “‘“7 $, 1 =ee-
in,k(h) (i cot A)- “m’ -e sin A
(icot Inlk(A)
A)-“‘“‘c,,,~I_~
(B .2)
I”,k(l)
Sl” A
(i COt A) - ‘1,”t
- “P”I /
Here we introduced e’kfA) =
_e2Mh)
new functions cot
*,
eia(Af =
_e-2MN
which we adopted from the recent analytic follows from (3.5) and (B.3) that sin k(A) = - fu( A) + +ilJ,
Cot
Bethe ansatz for the Hubbard
sin p(A)
(B.3)
*
= - +u( A) - +iU.
model by Yue and Deguchi
[36]. It
03.4)
S. Murukumi. F. GAhmann/Physics
In the limit I m I --) cc, the above matrix L?,,(A) converges iteratively we obtain y(A) as = . fd2”,+
F(A)
,( h)P”,(
Lefrers A 227 f1997) 216-226
225
in the weak sense to the identity matrix. Solving (B. 1)
h).L&_ ,( A) . ‘. (B-5)
k
k>l
To expand _T?,,(A) in terms of u( Al-‘, we consider the limit Im( A) -+ x and, to begin with, expand function in terms of e2’A. For e- 2h(A) there are two possible choices of branch, e -2h(A)=
-fU
sin 2A +
To achieve convergence this choice we get e2h(h)
_e2ih
C’ ,-i/HA)
=
+
O(e6”),
!_{,*,A _ ze4iA u
(B.6)
of the matrix elements
4i
=
each
Y:ls (cu, /3 = 2, 3) we have to take the lower sign in (B.6). For
eikfh) = +!,,ZiA
+qe6iA)),
&
+ ze4iA + qe6’A))_
= qIe*iA +qe6iA)).
(B.7)
Now the leading terms in the sums in (B.5) are of order e2ih, e4ih, . . . Thus, from the first two sums in (B.51, we get the expansion of the matrix A(A) up to order e4ih. Then the last equation in (B.7) yields the required expansion in (u(A))- ’ up to second order.
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Univ