14. 15.

V. V. Nesterenko and A. M. Chervyakov, "Singular Lagrangians. Classical dynamics and quantization," Preprint R2-86-323 [in Russian], JINR, Dubna (1986). S. S. Schweber, An Introduction to Relativistic Quantum Field Theory, Evanston, Illo

16.

A. Barut and R. Raczka, Theory of Group Representations and Applications, Warsaw

17.

(1977). F. Bopp, Z. Naturforsch. T e l l A, ~, 197;

(1961).

18. 19.

20. 21. 22.

~, 537 (1948).

J. Weysenhoff, Acta Phys. Pc1., 11, 49 (1953). B. Sredniawa, "Relativistic equations of motion of "spin particles"," in: Cosmology and Gravitation, Spin, Torsion and Supergravity (eds. P. G. Bergmann and V. de Sabbata), Plenum Press, New York (1980), pp. 423-434. M. Pavsic, Phys~ Lett. B, 221, 264 (1989). M. Rivas, J. Math. Phys., 30, 318 (1989). A. J. Hanson and T. Regge, Ann. Phys., 87, 498 (1974).

REMARK ON THE MEAN FIELD LIMIT FOR MULTICOMPONENT GIBBS SYSTEMS WITH NEUTRALITY CONDITION V. V. Gorunovich and V. I. Skripnik The mean field limit is obtained for classical and quantum (Boltzmann statistics) neutral Gibbs systems of charged particles interacting through a two-body integrable or nonintegrable potential. The mean field method, based on the formalism of the introduction of a small parameter e in the interaction potential and corresponding to renormalization of the correlation functions, is usually employed to justify the derivation of kinetic equations [I]. For Gibbs systems of uncharged particles, it was used in [2] to construct solutions of the Kirkwood--Salsburgequations in the form of asymptotic expansions with respect to ~. In the case of Coulomb systems, this method is associated with the Debye-Hfickel limit [3-5]. Essentially, this limit amounts to allowance for the fluctuations of first order in ~ in the mean field method (analog of the central limit theorem). The aim of this note is to emphasize the fact that in multicomponent systems the neutrality condition plays an important part in the mean field method in the case of both integrable and nonintegrable potentials. The condition is necessary when one is considering systems with Coulomb interaction potential~ However, Kennedy [3] and Fontaine [4] used a special case of this condition -- a symmetry condition. Therefore, our note is essentially an extension of [2] and [3,4], since it covers the cases of both integrable and nonintegrable potentials. The obtained result is a weighty indication that there are no phase transitions in the considered systems in the mean field limit. i.

Main Result

Suppose that in a volume A c R v (v = 2, 3) there are particles of M species with charges el, ..., eM, activities zl/s, .... ZM/S (where s > 0), respectively, and masses ml . . . . , mM" The particles interact through a two-body potential

V,j(x--x'; e) =e~e~e. V(x--x') ; i, ]6 {1 . . . . , M}, w h e r e V ( x -- x ' ) i s a p o s i t i v e - d e f i n i t e integrable, as in [2], or nonintegrable, We d e n o t e

by ~ c and

p~A , ~ ( ( e )

im ;

( 1.1 )

function ( a t t h e s a m e t i m e , V ( x -- x ' ) as in [3-5], function); V(O) < ~ .

(x)~),

where

(x)m=(x~, . .

xm)

and (e)('~=(el,

may b e a n e~),

ifi{i,...~M}, respectively, the classical grand partition function and the correlation functions of the considered system. We also define the pressure in the system: Institute of Mathematics, Ukrainian SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 86, No. 2, pp. 257-261, February, 1991. Original article submitted April 24, 1990.

178

0040-5779/91/8602-0178512.50 9 1991 Plenum Publishing Corporation

~,.= T ~ -

(I .2)

m -~,..

Here, ~ = I/kBT , T is the temperature, k B is Boltzmann's constant, and IAI is the volume

of t h e r e g i o n A. THEOREM i.

If the neutrality condition

Zz~e~=O.

is satisfied for arbitrary z i ~ 0

and ~ > 0, the following limits hold: M

9 (e.Pi,~)=~c ~EZG hm

(1.3)

lim(e m .p~A,8 ( ( e ) , " ; ( x ) ~ ) ) = z , , X . . . X z ~ , .

(1.4)

e-->o Q

A, 8

Now suppose ~A,~ and p~ ((e)~; (x)~, (x)~) d e n o t e t h e grand p a r t i t i o n f u n c t i o n and t h e r e d u c e d d e n s i t y m a t r i x ( d i a g o n a l e l e m e n t s ) o f t h e quantum s y s t e m w i t h Boltzmann s t a t i s t i c s in which t h e p a r t i c l e s w i t h c h a r g e s ez, . . . , e M and a c t i v i t i e s i ~ / E , . . . , ~M/r i n t e r a c t , as in t h e c o r r e s p o n d i n g c l a s s i c a l system c o n s i d e r e d above, t h r o u g h t h e two-body p o t e n t i a l (1.1). We d e n o t e by PQ,~ t h e p r e s s u r e in t h e quantum s y s t e m , which i s d e f i n e d in t h e same ~q way as PC, a, b u t w i t h t h e o n / y d i f f e r e n c e t h a t ~l~:c in ( 1 . 2 ) must be r e p l a c e d by ~,e. Suppose a l s o ~ = ~ , i6{1 . . . . . M}. THEOREM 2.

If the neutrality

condition

Lz~e~=O i s s a t i s f i e d

for arbitrary

zi > 0

and $ > 0 t h e f o l l o w i n g l i m i t s h o l d : M

lim(8"P:~)= ~-'. Z z i ~ d x ~ dPA'~ ((o);

(I.5)

9 (x)~, ( x ) , ~ ) ) = I I ~,~Y d e ~a,~,~ (o,~), lim(e".p:" ((e)J';

(1.6)

g~O

h=l

where dpA,,2(~o) is the conditional Wiener measure associated with the kernel of the operator exp(--XA A) (A A is the Laplacian with Dirichlet boundary condition on 8A; ~=~2/2~, is Planck's constant) and defined on trajectories ~(t) such that ~(t=0)=x, o(t=%)=x; o(t)EA;f~= X /)tv, where I~ is the single-point compactification of R~.

t(io, ~,]

.

Proof of the Theorems

We give here the proofs of the theorems formulated in Sec. i. of Theorem I.

We first give the proof

For the considered multispecies classical systems we have (see, for example, [6]) _

.

=c . . . .

.

ZI!4

~.~ NJ...-N--~..-~ 2v d(x)Nexp[--~UN((e)JN;(x)N)];

. ~

NM=O

AN

(2.1)

N=N~+. . .+N,~; d(x)~=dx, . .dx~;

Z~b: ~A,

8

m

oo

Ni=O

NM~O

zN,

N M

ZM

i d (x')N exp [-- tSUN((eh'G (e)jN; (x),,. (X')N)].

(2.2)

A

Here, UN((e)N; (x) N) is the potential energy of the N particles with charges e z ..... eM, ..., eM at t]he points xi,...,xi,...,x~-,++~.~,_,+~,...,x~.,respectively, interacting

el,

9 ..,

179

through the potential (1.2). In what follows, we shall use the already standard approach to the study of systems of charged particles based on use of the sine-Gordon transformations and functional integration with respect to a Gaussian measure whose covariance is the interaction potential of the particles (see [4-6] and the references there). Since V(x -- x') is a positive-definite function, there exists a probability measure du@ and a Gaussian random variable @(x) for V x ~ A such that ~d~O(x)@(x')=V(x-x~). Then, using the sine-Gordon transformation N

~d~|

(2.3)

N~+...+N~_~
where ]k=i, i~{1 . . . . . M}, i f (Jk = 1 for 1 < k < Nz), we can r e a d i l y obtain the following expressions for the partition function (2.1) and the correlation

functions (2.2):

~.,~-=~ - ~ d ~ ~xp[C.,~(~)] ; m

9~

((e)~; (x)~)=

(2.4)

m

dBr exp[iu

z~ (e).e-~(~f,~) -*

(2.5)

k:l

h~i M

s

: ~ dx ZzJ(e) exp[iefl/-~q)(x)];s A

z~(g):z~ exp[-~- e~ZeV(O)].

We consider first the expression (2.4) for the grand partition function.

simple t r a n s f o r m a t i o n s

(2.6)

j~ 1

(with allowance for the n e u t r a l i t y

After some

c o n d i t i o n ) '~ac,~ can be represented

in the form M

(2.7) M

A

8

j=t M

(2.8) It is readily seen that lim

~ ( z j (8)-zj)

s-~O d : l

limit

ej=0.

(2.9)

~/8

Since [$1~A.~(@)l~
gence theorem (for 3~A,~(O) with respect to d~@), M

limo~o diz~YdA,~(O)= d~|

d~r

[

~

.

The assertion (1.3) follows from the definition (1.2) and Eqs. (2.7), (2.8), and (2.10). For the correlation functions (2.5), we have

p~;~ ((e)s (x)~) =

z~ (e). ~-~ + k~it

(2.ii) ~'=I

180

=

Applying Lebesgue's theorem, as in (2.10), we obtain M

(2.12) It

can be s e e n t h a t

(1.4)

follows

directly

from (2.11)

and ( 2 . 1 2 ) .

With a l l o w a n c e f o r t h e s i n e - g o r d o n t r a n s f o r m a t i o n f o r t h e c a s e o f quantum s y s t e m s of charged particles ( s e e [ 6 ] ) , t h e p r o o f o f Theorem 2, l i k e Theorem 1, i s b a s e d on t h e u s e o f L e b e s g u e ' s t h e o r e m and i s s i m i l a r t o t h e p r o o f g i v e n a b o v e . I n c o n c l u s i o n , we n o t e t h a t o u r p r o o f s o f Theorems 1 and 2 b a s e d on t h e u s e o f Lebesgue's theorem is significantly simpler than the method used to obtain the similar results in [4,5].

LITERATURE CITED i.

2. 3. 4. 5. 6.

H. N. T. J. P. J.

Spohn, Rev. Mod. Phys., 52, 569 (1980). Grewe and W. Klein, J. Math. Phys., 18, 1729 (1977). Kennedy, Commun. Math. Phys., 92, 269 (1983). R. Fontaine, Commun. Math. Phys., 103, 241 (1986). Brydges, Commun. Math. Phys., 73, 197 (1980). Frohlich and Y. M. Park, Commun. Math. Phys., 59, 235 (1978).

TRANSFORMATION OF ELECTRON SPECTRUM IN THE HUBBARD MODEL V. M. Zharkov A new functional representation for strongly correlated systems is used to study the evolution of the electron spectrum in the Hubbard model with increasing U. It is established that the parametric instability of the ground state of a strongly correlated metal with 1/4 filled band can be described by linear transformations of the dynamic fields. It is shown that the operator of the transformations has a nontrivial kernel. It is noted that the considered system can serve as a point of departure for interpreting structural transitions of 2k F and 4k F types in the quasione-dimensional metal MEM(TCNQ) 2. Introduction The Hubbard model has recently become very popular. This is undoubtedly due to the unflagging interest in the standard problems of this model (metal-insulator transition, ferromagnetism, antiferromagnetism), and also to the appearance of problems with superconductivity that are new for the model. Despite the formal simplicity, the existence in the literature of many different approaches that, at the first glance, often do not match each other, suggests that the deepest properties of this model have not yet been uncovered. As an example, we may mention the fairly developed but, apparently, only with difficulty reconcilable functional approach to the formulation given in [i] and the atomic description method of [2] (the recently appeared slave boson approach must evidently be included in the first approach, since, as will be seen below, it is not capable of explaining the characteristic features of the atomic description). The quantization scheme for the Hubbard model proposed in [3] uses coherent states for the dynamic symmetry group~ In Perm State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 86, No. 2, pp. 262-27], February, 1991. Original article submitted May 7, 1990.

0040-5779/91/8602-0181512.50 9 Plenum Publishing Corporation

181