Self-consistent theory of electron correlation in the Hubbard model

16. S. N. Sokolov, Preprint IFVIs GTF 74-135 [in Russian[, Serpukhov (1974). 17. S. N. Sokolov, Preprint IFVI~ OTF 75-94 [in Russian], Serpukhov (1975). 18. S. N. Sokolov, Preprint IFVI~ OTF 76-50 [in Russian[, Serpukhov (1976); Proe. Fourth Intern. Conf. on Nonloeal Field Theories, April 1976 [in Russian[, Alushta, p. 77-86. 19. S. N. Sokolov, Dokl. Akad. Nauk SSSR, Ser. Mat.-Fiz., 233, 575 (1977). 20. L. L. Foldy, Phys. Rev., 122, 275 (1961). 21. S. N. Sokolov, Teor. Mat. Fiz., 23, 355 (1975). 22. R. G. Newton, Scattering Theory of Waves and Particles, New York (1966). 23. R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That, Benjamin, New York (1964). 24. F. Coester, Meson Theory of Nuclear Forces. Quanta, University of Chicago Press (1970, p. 174-175; F. Coester, S. C. Pieper, and F. J. D. Serduke, Phys. Rev. C, 11, 1 (1975); F. Coester and A. Ostbee, Phys. Rev. C, 11, 1836 (1975). 25. S. N. Sokolov, Teor. Mat. Fiz., 32, 354 (1977).

SELF-CONSISTENT CORRELATION A.L.

THEORY IN

THE

OF

ELECTRON

HUBBARD

MODEL

Kuzemskii

The Dyson equation for the two-time t h e r m a l G r e e n ' s functions is used for a s e l f - c o n s i s t e n t calculation of the s i n g l e - p a r t i c l e G r e e n ' s functions in the Hubbard model. The method makes it possible to obtain a generalized interpolation solution of the Hubbard model valid for a r b i t r a r y relationship between the effective band width and the Coulomb repulsion p a r a m e t e r . Two v a r i a n t s of the t h e o r y make it possible to obtain two exact r e p r e s e n t a t i o n s for the mass o p e r a t o r , which are used to obtain approximate solutions in the atomic and band limits. 1. The method of two-time thermal G r e e n ' s functions [1, 2] is convenient and effective for investigating s y s t e m s of many interacting p a r t i c l e s . Recently, a helpful reformulation of this model has been given; it makes it possible to operate with the exact m a s s o p e r a t o r and p e r f o r m the decouplings in the final stage. This approach is based on the introduction of irreducible G r e e n ' s functions [2-6], which makes it possible, without r e c o u r s e to a truncation of the h i e r a r c h y of equations for the G r e e n ' s functions, to write down the Dyson equation and obtain an exact analytic r e p r e s e n t a t i o n for the mass o p e r a t o r . Approximate solutions a r e constructed as definite approximations for the mass o p e r a t o r . The method of irreducible G r e e n ' s functions has been used in a large number of investigations into the s e l f - c o n s i s t e n t t h e o r y of phonons, the Heisenberg model, the spin-phonon interaction, f e r r o e l e e t r i c i t y , and more (see [2-6] and the l i t e r a t u r e quoted there). In the p r e s e n t paper, we c o n s i d e r the application of this approach to the Hubbard model, which is one of the ones most widely used to describe magnetic p r o perties and the transition f r o m the metallic to the nonmetallic state in transition metals and their c h a l c o genides [7, 8]. The Hamiltonian proposed by Hubbard [7], H=~

t~ja.~+a j o + .U- - ~ l n~oni_~ i~a

depends on two p a r a m e t e r s :

'

the effective band width w =

(1)

2Z.~

ia

N -1

It,~r ~

and the e n e r g y U of the i n t r a - a t o m i c

ij

Coulomb repulsion of the e l e c t r o n s . As their ratio changes, so does the band s t r u c t u r e of the s y s t e m . Thus, to describe t r a n s i t i o n s in the system, it is n e c e s s a r y to obtain an interpolation solution for the Hubbard Hamiltonian valid in a wide range of values of the p a r a m e t e r z = w/U f r o m the atomic limit (z ~ 0) to the band limit (z >> 1). The method of irreducible G r e e n ' s functions makes it possible to c o n s t r u c t such solutions

Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Voi.36, No.2, pp.208-223, August, 1978. Original article submitted June 21, 1977.

692

0040-5779/78/3602-0692,~07.50 9 1979 Plenum Publishing C o r p o r a t i o n

systematically. 2.

F o l l o w i n g [9, 10], we i n t r o d u c e the s i n g l e - p a r t i c l e G r e e n ' s functions

Gko(t)=((ako(t);a~))=--iO(t)~ w h e r e ~ = ( k T ) - ' , 1ko((0) is the s p e c t r a l intensity. G r e e n ' s function h a s the f o r m

e-'~'G~o((o) ~

w-e-

~---|

; (e '+t)]ko(O) ),

The equation f o r t h e F o u r i e r t r a n s f o r m

(o)-e~)Gk.(o))= i + U_--:-_Z 2i

Gk.(o) of the

((aa+.,oa.+,-oaq-.[a~.>>.. + +

(2)

Pq

B y definition, we introduce an irreducible Green's function that does not contain renornmlizations of the a v e r a g e field,

ir )| = >.--Sp,o Gk..

(3)

The i r r e d u c i b l e G r e e n ' s function in (3) is defined in such a w a y that it c a n n o t be r e d u c e d to G r e e n ' s functions of l o w e r o r d e r with r e s p e c t to the n u m b e r of f e r m i o n o p e r a t o r s by an a r b i t r a r y p a i r i n g of o p e r a t o r s c o r r e s p o n d i n g to one instant of t i m e . Substituting (3) in (2), we obtain HF

IIF U

{r

~

G,,o (co) =Gk. (o)) +G~o -~- )

+

)..

(4)

Pq

H e r e , we have i n t r o d u c e d the notation l

HF

U

a ~ ~ (0~)= (~-~'o~ )-,~o =~,~+~

N~--J q

.

To obtain the Dyson equation, we must express the Green's function '~on the right-hand side of (4) in terms of the total Green's function Gab(co). For this, we differentiate this function with respect to the second time: -i(d/dt)~r= -----O.

Ir i .

(24)

k

Here, t ~ U2 ~[~.h,q-(~+y,~)]' v'1 N~q-N~q ~., / Uo.(k)=U( 1 + - z2N ~ ~ j Pq

Ua

A=V,~ hpq

'

,

,

N~,,-N~,

[~,,~- (~.+un) ]'

[n(~h~q)-n(,~+Un)].

In general, one can a s s u m e that (24) is a generalization of S t o n e r ' s c r i t e r i o n for the o c c u r r e n c e of f e r r o m a g n e t i s m in the Hubbard model. The relation (24) takes a f o r m c l o s e s t to this c r i t e r i o n when one can a s s u m e Uo,(k)~Uctf(O). Then f r o m (24) we obtain 695

Uo,tDo(e,)-A (el) >1.

(25)

It can be seen f r o m this result that the o c c u r r e n c e of m a g n e t i s m in the s y s t e m is determined by the electron density of states. The o c c u r r e n c e of the r e n o r m a l i z e d quantity Uef f and the p a r a m e t e r A is natural when a more a c c u r a t e allowance is made for the c o r r e l a t i o n than in the H a r t r e e - F o c k approximation, which leads to S t o n e r ' s c r i t e r i o n . It is well known that in the H a r t r e e - F o c k approximation the tendency of the s y s t e m to f o r m a magnetically o r d e r e d state is s t r o n g l y o v e r e s t i m a t e d . As we have a l r e a d y said above, in the band limit the r e p r e s e n t a t i o n (17) for the m a s s o p e r a t o r can be used for iterative calculation of the higher (in the interaction p a r a m e t e r U ) o r d e r s of the m a s s o p e r a t o r , for which it is n e c e s s a r y to substitute (23) in (17). Thus, the proposed s e l f - c o n s i s t e n t p r o c e d u r e makes it possible to c o n s t r u c t a perturbation theory f o r the m a s s o p e r a t o r . Because the resulting e x p r e s s i o n s a r e c u m b e r s o m e , we do not give them here but r e f e r to [11], in which the use of a two-pole r e p r e s e n t a t i o n for the spectral density (two delta functions) as initial approximation is also considered. 4. In the region of resonance of the single-particle and collective excitations the binary approximation breaks down. In particular, to describe a magnetically ordered state it is particularly important to take into account the contribution of the spin-flip processes to the mass operator. The method developed here permits this in a simple and perspicuous manner. For this, we perform in (15) the following differenttime decouplings. ~ + I r , c--(aj~+a~o(t)>+ + (t) a~o(t) >>

-

~"

.

1-

B'+~'"] •

tB_.,[

}

(36)

zl

~. a r e determined f r o m the condition -~ 0.

Calculating the c o r r e s p o n d i n g c o m m u t a t o r s in (37) and equating the t e r m s proportional to prectively, we obtain (i r l )

(37)

5ir and 5fl, r e s -

{A.}~=a(+) (n_J~)-~, {B.}~={