VOLUME 60, NUMBER 16 PHYSICAL REVIEW LETTERS 18 APRIL 1988

Conserved Quantities of the One-Dimensional Hubbard Model

Eugenio Olmedilla ' and Miki Wadati
Institute of Physics, College of Arts and Sciences, University of Tokyo, Meguro Ku-, Tokyo l53, Japan

(Received 16 November 1987)

We explicitly provide, in a simple way, the form of the conserved quantities of the one-dimensional
Hubbard model. The method developed is based in a newly found factorization property of the local
transition matrices, which is satisfied by several spin and fermion chain models. In this way, we have

shown a systematic method for deriving the conserved quantities of integrable quantum systems.

PACS numbers: 05.30.—d, 05.50.+q, 75.10.Lp

For several decades the Hubbard model has attracted
the attention of physicists working in both solid-state
physics and mathematical physics. The eigenfunctions
of the model, found using the Bethe Ansatz, were studied

by Lieb and Wu. ' However, the clarification of the alge-
braic structure underlying the model was initiated only
recently. 2 3

The tool for this algebraization is the quantum inverse
scattering method, 4 a method that places the theory of
completely integrable quantum systems and of solvable
statistical mechanical models in a unified framework. To
a quantum integrable system we can associate the local
transition matrix X„(X),which is part of a Lax pair of
operators, such that

R(X,lt) fX„(X) L„(P)]

matrix, for a chain with length N, is defined by

7tv(z) =/tv(X)Xiv-((X) X((X)

n 1

(2)

The relation (1) is also satisfied by Tiv instead of X„.
By use of an appropriate definition of trace in (1) with
7'tv, it is easy to see that there exists a family of com-
muting matrices, denoted as transfer matrices,

[Tiv(X), Tiv(p)l =O, (3)

where Ttv is a trace of 7'tv. The X expansion of Ttv(k)
gives a set of conserved operators [CI] which are involu-
tive:

=[X.(p)eX. (X)]R(X,p); (1) [c;,c,] =o. (4)

X,p are spectral parameters, and %(X,p) is a c-number
matrix. The Yang-Baxter relation (1) is a sufficient con-
dition for the system to be integrable. The monodromy

This is commonly understood as the condition for the
complete integrability of a model.

The Hamiltonian of the one-dimensional Hubbard
model is

0= —g(at+(, a, —a +(,,a,,)+—g(2n t
—1)(2n

~

—1),
m, s 4 m

where s = t and J. The corresponding local transition matrix is

(s)

(6)

with

X(8) =cosh[ —,
' h(8)]1+sinh[ —,

' h(8)]cr'a',

x ((8) =
-ia(e)n (-y(e)a (a (

am

10m t

—iy(8)n t+a( )8ata

X ((8)=
ia(e)n t+y(e)a (at)

1am J

am)

iy(8)n (
—a(8)a (a

Here, 1 is the 4x4 identity matrix and as we are dealing with fermion operators, the direct product in (6) is a
Grassmann product defined by

[A@g] ( 1 ) [P(a)+P(p&t1P(r)g ~ (8)

1988 The American Physical Society 1595



VOLUME 60, NUMBER 16 PHYSICAL REVIEW LETTERS 18 ApRIL 1988

with P(1) =0 and P(2) =1. The coupling constant in

the model (5), U, and the parametrization in (7) are re-
lated by

4sinh[2h(8)] =Usin(28).

At 8 =0 the matrix operators X, are

lnm f lQm f

X 1(0)=
Qm f Qm f Qm f,

We also have

a(8) =y(8+ sr/2) =sin (8+ sr/2). (io)

lnm j Qmiz I(0) — . ,
lQmi QmiQmi,

and their jth-order derivatives at 8=0 are

—
yjQm, s Qm, s «jnm, sr'i', =sgn(s)

6jQm, s Qm, s l yjnm, s,
(i2)

where aj and yj are the jth-order derivatives of a(8) and y(8) at 8=0, respectively, and sgn(s) =+1 for s = f and —
1

for s= i.
In terms of the local transition matrix 2 (8), the monodromy matrix is

I'

7(8)= P S (8),
M= —p

and the transfer matrix is defined by

T(8) =strH (8)] =trl(4T'IiiIcr')7 (8)1,

where str denotes the supertrace of a matrix.
We define the operators

(i4)

H, (8)= g H. ..(8),
m —p

where p and r are the extremes of a chain of length N =r+p+1, with periodic boundary conditions, and

(is)

Hm+ I,m, s(8) I nm+isnm, s , am+ Isam+I, ,samsams+, y(8, )(am+ Iams, sam, + Isam, s),

+a(8) (nm+ I,snm, s+ am+ I,sam+ I,samsam, s ),.

Expansion of H +I,(8),

( ) gJ
Hm+ I,m, s(8) = Z Hm+ I,m, s .

I
ij-o '' Jt

gives
- (0)0m+i, m, s =1,

and for j~ 1

~ (j) — r' +
ssm+I, m, s yj(am+I, sam, s am+ Isams)+, aj(n, m+ 1,snm, s+ am+ I,sam+Isamsam, s). , ,

We further denote

k =(2n I
—1)(2n I

—1).

With these notations, the Hubbard Hamiltonian (5) is

Zm, sHm+I, m, s+ 4 UZmhm.

(2o)

(21)

With these definitions it is easy to prove the following relations between the local transition matrices X (8) and the

H, (8):

+1,(8)X,(0) =X +1,(0)X,(0)H +1,(8), X +1,(0)X~,(8) =H +I,, (8)X +I,,(0)X,(0); (22)

we also have for j~0
a'X J, (0) =(—l)jX j,(0)(2n, —1), X j,(0)a'=( —1)j(2n, —1)X,, (0), (23)

1596



VOLUME 60, NUMBER 16 PHYSICAL REVIEW LETTERS 18 APRIL 1988

and

(2n +(,s —I )Hm'+)m, s,=( —I )'H~'+), m;(2n +); —I)(j) j (j)

For a chain with periodic boundary conditions, definition (14) after taking account of (6) and (13) gives us

N r —
1 r —j+1

T(e) = g cosh Jh (8) sinh Jh (8) g g g T I' (8)T
~
(8).

j=p k, = —
p

—1+j k2= —
p —2+j k, = —

p

Here k=lk(, k2, . . . , kjl, k( & k2. » k), and

T,"(8)=S«~&,,, (e) ~'&k, s(e, ) ' ' ' o +k, ,s(e) ' ' ' o +k s(e) ' ' ' +—p, s(e)f,

and in particular

(24)

(2S)

(26)

T, (e)=str Q X„,(e) .
n= —p

It is convenient to introduce the following quantity:

H, "=H, , (,,+ +—( —1) Hk, k, —),+(—1) Hk, k, (,+ — +(—1) Hk k —( z+ +H p, , —

We observe that

0 (2I)k 0 (2I)
s s

After making use of (11), (23), and (24), we obtain

N —
1 J

Tp(e) =T, (0) 1+ g C,",,8" Q (2nl,.—1).
n 1 m k

The symbol = means equality up to order N 1 in 8, an—d

(27)

(28)

(29)

(3o)

0 (1)k

0 (3)k

H (4)k

Cf,, =

H(2))r/2(yH(1)toH (1)k

0 (2)k 0 (2)k
oH (1)kg 0 (1)ko

2! ' '
2I

0 (3)k 0 (3)k
+ ~ o H (1)k+0 (1)ko

3t 3!

pH (1)koH (1)koH (1)k
s s S

H (2)k H (2)k H (2)k H (2)k
+ s o s + s oH, 'koH k+0 'ko s oH 'k

21 21 2! ' '
2I

"(2)k
+H(l)koH(1)ko +0 ' "oH(1)koH(1)"oH(1)ks s

21
s s s $ (31)

where o denotes the products of Hp "/j! in such a way that if two or more suffixes of the H's are connected [like

(j+1,j) and (j,j—I)] their ordering cannot be broken. For instance, we have

) oH(j );,(j ) H(j ) H(j ) H(j ) ~(j,) 0(j,) H(j, ) H(j, )06,5,s 84,3,s 02, 1,s 81,6,s ~ 2, 1,s01,6,s06, 5,sH4, 3,s +4,3,S02, 1,SH1,6,SH6, 5,S

We call such a product the connected product and o is called the circle product. In general, C„", is the sum of the cir-
cle products of H, j "/j! among themselves whose exponent sum gj is n.

In particular:

—H( )/2!+H, oH, C3 =H, /3. +H /2!oH~ +H oH /21+H~ oK, oH, . (32)

Finally, we arrive at

N

T(e) =T(0) g cosh Jh (8) sinh~h (8)

N —
1

=T(0) 1+ g (.„8" .
n=l

X Z
kl = p 1+jk2= —p —2+j

r —1+1

kj= —
p

N —
1 N —

1 J
I+ 2 C" 8- I+ Z C"ie' »»,

m=1 q=l t=1

(33)

It is straightforward to write down the explicit forms of the conserved quantities C„. The first three conserved quan-

1597



VOLUME 60, NUMBER 16 PHYSICAL REVIEW LETTERS 18 APRIL 1988

tities are

C i =H, C2=CP1+C21+C|1Cii+—g (Cil+Cti)hk+(N+2)U Q2

4k- —p
32'

r r r —
1

P3 C31+C31+C21C/1+CtlC21+ —Q (C21 +Col +C)1Cji )hk+ P X, (C&I '+C)1 ')hkqhkq
4 k--p l6 k, —p+lk2 -p

U2 U2
+N (Ci 1+Cl 1)+ N (34)1 U

3 2'

Formula (33) is also true with T(0) to the right instead
of the left.

For the infinite chain, we have expressions similar to
(25), (30), and (33) for the monodromy matrix in place
of the transfer matrix. Here we only mention that both
trace and supertrace of the monodromy matrix give us

an infinite set of conserved quantities. The properties
that assure these results are

trA =str[(cr'8 cr'8 cr'8 ')Al,

[%(8,8'), cr'8 cs'8 cr'8 cr'] 0,

(35)

(36)

where the matrix %(8,8') has been given in Ref. 3.
In brief, we wish to remark on the simplicity of the

purely algebraical method used to derive the conserved
quantities (conserved operators). This method has
proved to be systematic, and by using the quantum in-

verse scattering method formulated on a lattice, we
have been able to discuss spin models and fermion mod-
els on an equal footing. 5

The key relations in our treatment are the commuta-
tion relations among the local transition matrices and the
H operators [see (22)-(24)1. We have obtained a com-
pact formula for the transfer matrix which acts as the
generator of the conserved quantities. Then, simply by
expanding formula (25) in a power series of the spectral
parameter, we can write down a sequence of conserved
quantities.

Note that besides the N —1 conserved quantities that
can be obtained by use of our method, the transfer ma-
trix evaluated at 8 0, T(0), is also a conserved quanti-

ty.
Our conserved quantity C 2 (fermion model) and

1 Shastry's (boson model) second current (H2 —ij in Ref.
2) essentially coincide, up to a term that goes with U2.
We are unaware of a publication by Shastry which
would allow us to discuss this in more detail.

We wish to thank Dr. Yasuhiro Akutsu and Dr. Ki-
yoshi Sogo for valuable discussions and critical reading
of the manuscript. One of us (E.O.) is very grateful for
the hospitality of the Institute of Physics, University of
Tokyo, and for the financial support by the Japan So-
ciety for the Promotion of Science.

Permanent address: Departamento de Metodos Mate-
maticos de la Fisica, Facultad de Ciencias Fisicas, Universidad
Complutense de Madrid, 28040 Madrid, Spain.

'E. Lieb and F. Y. Wu, Phys. Rev. Lett. 20, 1445 (1986).
2B. S. Shastry, Phys. Rev. Lett. 56, 1529, 2453 (1986).
M. Wadati, E. Olmedilla, and Y. Akutsu, J. Phys. Soc. Jpn.

56, 1340 (1987); E. Olmedilla, M. Wadati, and Y. Akutsu, J.
Phys. Soc. Jpn. 56, 2298 (1987).

4E. Olmedilla and M. Wadati, J. Phys. Soc. Jpn. (to be pub-
lished).

5L. D. Faddeev, Sov. Sci. Rev. Math. Phys. C 1, 107 (1980);
H. B. Thacker, Rev. Mod. Phys. 53, 253 (1981). P. P. Kulish
and E. K. Sklyanin, in Integrable Quantum Field Theories,
edited by J. Hietarinta and C. Montonen, Lecture Notes in

Physics Vol. 151 (Springer-Verlag, Berlin, 1982), p. 61; L. A.
Takhtadzhan and L. D. Faddeev, Russian Math. Surveys 34,
11 (1979); M. Wadati, in Dynamical Problems in Soliton Sys
terns, edited by S. Takeno (Springer-Verlag, Berlin, 1983), p.
68; M. Wadati and Y. Akutsu, "Exactly Solvable Models in

Statistical Mechanics, " edited by M. Laksh mann, Lecture
Notes in Physics (Springer-Verlag, Berlin, to be published).

1598