Short Notes K149

phys. stat. sol. (b) 6CJ,K149(1975)

Subject classification: 13 and 18

Institut fur Theoretische Physik der Universitat Kiel

A New Upper Bound for the Free Energy

of the Hubbard Model Based on the Cluster Approach

BY

W. -H. STEEB and E. MARSCH

In this letter it is shown that the cluster approach of Caron and m a t t (1) and

Monecke (2) can be modified in such a way that it leads to an upper bound for the

free energy of the Hubbard model. In this approach the Hubbard Hamiltonian in its

simplest form

+ + g C n n H =CJij CidCjd id i-d ijd id

I is replaced by H = Hi , where i

Here J.. is assumed real and nonzero only if sites i and j are nearest neighbours.

Note that J > 0 . The physical meaning of this Hamiltonian has ,been widely discussed

by Caron and Pratt (1). They have calculated the ground state,

9

This result ist not comparable with the exsrct ground state E

sufficiently low we find that E

ciently large we have E CP>EH . Furthermore the limiting case of U = 0 leads to

E C d N = -35 for the simple cubic lattice (z = 6 ) . An exact calculation shows

E (U = O)/N = (1/N) c I&(k)\ = -25 . Now the aim of this paper is to give an upper

bound for the free energy within the cluster approach. Moreover we want to compare

the result with other upper bounds.

If the ratio U/J is H' On the other hand, if the ratio U/J is suffi- < E C P H '

H k

Upper bounds for the free energy of the Hubbard model have been considered by

Kaplan and Bari (3), Dichtel e t al. (4), Elk (5), Steeb and Marsch (6) and Grensing

K15C physica status solidi (b) 69

and Koppe (7). A lower bound ,is given by Steeb (8). The approximation based on the

inequality

52 f Tr{ (. - p N e ) Wt} f ( l / p ) T r (Wt In Wt, , (4)

where W is the so-called grand-canonical trial-density matrix. According to equa- t tion (2) we choose W as follows:

Wt=exp - P C H i ( a , a * ) T r e x p ( . . . ) . (5) I/ t

2

L 3 + i5 and both Q and M play the role of real variational parameters We put rx= (compare equation (2)). After a straightforward calculation we obtain for the Helm-

holtz free energy in the half-filled case (p = U/2)

1

Further we get a self-consistent equation which is the same as Monecke (2) has

This yields a trivial solution Iccl= 0 and a non-trivial one, i.e. In1 > 0 . Obviously,

at

and equation (6); the ground state in this approach is

= 0 we obtain the atomic limit. At T = 0, we can explicitly solve equation (7)

This results does not show a discontinuity in the slope at the critical ratio U/2J2.

In Fig. 1 we have plotted the various ground state solutions versus the para-

meter U for the simple cubic lattice. Here also the ground state E

by Dichtel e t al. (4) is considered. We find that E

For the sake of completeness we have also plotted the simple Hartree-Fock energy,

namely E

which was found

is always the lowest upper bound. A

A

/N = -(I/N) Z jc(k)l+ u/4 . In contrast to Caron and ma t t (I), our k H F

Short Notes K151

Fig. I . The various ground state

ling constant U/J; (1) E modi-

H F fied cluster approach, (2) E

simple Hartree-Fock, (3) E anti- A ferromagnetic solution, (4) E

ground state of Caron and Pratt

U/J - energies as functions of the coup- 0 5 75

CL

C P

modified cluster solution shows a continuous slope at the transition point.

Fig. 2 shows the free energy as a function of temperature for the ratioU/J=4.

while above

both a re equal. The critical temperature can be determined with the aid ofequa- CL' F ~ ~ ' Clearly, below a critical temperature T

T

tion (7). F and F in this case a re lower than F for all temperatures. For

comparison we have plotted F and F (compare reference (8)).

we always have F C

C

A HF CL

KIN LB Furthermore it should be noted that the entropy of our solution is equal to

In2 at the absolute zero of temperature. It seems that this behaviour i s afea- -K B ture of the approximation. Remember that the entropy of the atomic limit, i. e.

J = 0 , has the same disadvantage that it contradicts the Nernst theorem.

Monecke used that cluster approach to discuss the metal-insulator transition

based on a formula which he derived for the frequency-dependent conductivity (9).

This formula makes sense only if one inserts the exact occupation values (n

of the true solution of the Hubbard model into the effective mass tensor > Kd

His proof that the conductivity looks like that of free particles d PV

depends essentially on the nature of the exact eigenfunctions of the Hubbard-Hamiltonian.

Therefore it seems to us a very crude approximation to take<% ,,>from the cluster approach

In a more consistent way one should calculate d (w) by use of the cluster -trial- PV

K152 physica status solidi (b) 69

Fig. 2. Free energies as a function of temperature (U/J = 4); (1) FAL

atomic limit J = 0 , (2) FCL modified

cluster solution, (3) F antiferromag-

netic approach, (4) F Hartree-Fock

(5) FLB lower bound, (6) F

ic term U = 0

A

HF kinet-

KIN

- -Hamiltonian to obtain from linear

response theory the conduction prop-

erties of this model. As noted above,

the antiferromagnetic solution has the lowest ground state for all values of the coup-

ling ratio U/J. Therefore, we also obtain for the s . ~ . lattice an insulator in the

region U 5 2 ZJ = 12 J (reference (lo)), whereas Monecke (2) predicted a metallic

behaviour. For U > 125 results an insulator.

4 h, T?--

References

(1) L.G. CARONand G. W. PRATT, Rev. mod. Phys. 40, 802 (1968). G. W. PRATT and L. G. CARON, J. appl. Phys. 39, 485 (1968).

(2) J. MONECKE, phys. stat. sol. (b) g, K81 (1972); 51, K21 (1973); 65, 231(1974). (3) T. A. 'KAPLAN and R. A. BARI, J. appl. Phys. g, 875 (1970). (4) K. DICHTEL, R. J. JELITTO, and H. KOPPE, 2. Phys. 246, 2% (197J);

- 251, 373 (1972)-

(5) K. ELK, Ann. Phys. 30, 272 (1973) (6) W:H. STEEB and E. MARSCH, phys. stat. sol. (b) E, 403 (1974). (7) D. GRENSING and H. KOPPE, to published in Z . Naturf.

(8) WFH. STEEB, to be published in J. Phys. C.

(9) J. MONECKE, phys. stat1 sol. (b) 5l-, 369 (1972).

(10) E. MARSCH aridW.-H. STEEB, Z. Naturf. a, 1655 (1974). (Received March 26, 1975)