Systems with separable many-particle interactions. II

Physica 85A (1976) 425-456 t~ North-Holland Publishing Co.

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. II L.W.J. den OUDEN, H.W. CAPEL and J.H.H. PERK Instituut-Lorentz voor Theoretische Natuurkunde, Rijksuniversiteit te Leiden, Leiden, The Netherlands

Received 24 June 1976

The treatment of a previous paper on systems with many-particle interactions is generalized to hamiltonians containing an analytic function of a number of short-range interaction operators V, which are normalized. An exact expression for the free energy per particle in the thermodynamic limit can be obtained from a trial hamiltonian which is linear in the operators V.

1. Introduction In a p r e v i o u s p a p e r ~) w e h a v e c o n s i d e r e d a c l a s s o f s y s t e m s c o n t a i n i n g m a n y - p a r t i c l e i n t e r a c t i o n s of t h e s e p a r a b l e t y p e . A s e p a r a b l e m - b o d y int e r a c t i o n is d e f i n e d b y t h e p r o p e r t y t h a t the i n t e r a c t i o n V(k~ . . . . . kin) b e t w e e n p a r t i c l e s k~ . . . . . km c a n b e w r i t t e n as a p r o d u c t V ( k ~ ) . . . V(km), w h e r e V ( k ) is an o p e r a t o r a c t i n g o n t h e H i l b e r t - s p a c e o f p a r t i c l e k. S e p a r a b l e i n t e r a c t i o n s i n c l u d e in p a r t i c u l a r s o - c a l l e d e q u i v a l e n t - n e i g h b o u r i n t e r a c t i o n s , f o r w h i c h t h e i n t e r a c t i o n s b e t w e e n p a r t i c l e s d o n o t d e p e n d on t h e d i s t a n c e s b e t w e e n t h e particles. For many models with equivalent-neighbour interactions, the free energy p e r p a r t i c l e has b e e n e v a l u a t e d a n d t h e r e s u l t is u s u a l l y of t h e m o l e c u l a r - f i e l d t y p e . E x a m p l e s are, f o r i n s t a n c e , t h e H u s i m i - T e m p e r l e y m o d e l f o r a l a t t i c e gas2), t h e Ising m o d e l w i t h e q u i v a l e n t - n e i g h b o u r i n t e r a c t i o n s 3) a n d o t h e r spin models4), t h e r e d u c e d h a m i l t o n i a n in t h e B C S - t h e o r y of s u p e r c o n d u c t i v i t y ''6) a n d t h e D i c k e m a s e r model7'8). T w o - b o d y s e p a r a b l e i n t e r a c t i o n s h a v e b e e n t r e a t e d f r o m a m o r e g e n e r a l p o i n t o f v i e w in refs. 9, 10. M a n y - b o d y i n t e r a c tions o f this t y p e h a v e also b e e n i n v e s t i g a t e d ~ - ' 3 ) . A m o r e e x t e n s i v e disc u s s i o n o f t h e l i t e r a t u r e h a s b e e n g i v e n in ref. I. B e f o r e w e d e s c r i b e t h e g e n e r a l i z a t i o n s w h i c h will b e d e a l t w i t h in t h e p r e s e n t t r e a t m e n t , w e first g i v e a b r i e f r e v i e w o f t h e m a i n r e s u l t s of o u r p r e v i o u s p a p e r . In ref. 1 w e r e s t r i c t e d o u r s e l v e s to t h e c l a s s of s y s t e m s d e s c r i b e d b y the h a m i l t o n i a n ?EN = N { T N + P (VN)},

(l.1) 425

426

L.W.J. DEN OUDEN, H.W. CAPEL AND J.H.H. PERK

where TN and V~ are normalized* sums of bounded hermitean one-particle operators, i.e.

TN = N ' ~, T(k),

VN = N -1~_, V ( k )

k--I

(1.2)

k--I

and P ( V N ) is a polynomial in the operator VN. In ref. 1 the free energy per particle in the t h e r m o d y n a m i c limit, defined by f [ ~ l -= ~ m fN[~N] --= tim - (/3N) ' l n T r e ~N,

(1.3)

has been e x p r e s s e d in terms of a trial hamiltonian ~tr.N(~), which can be obtained b y linearizing ~N with respect to VN, i.e. f[~g] = rain f[~tr(~)],

(1.4)

where

~,r,N(~) = N { T N + e (~) + e '(~)( VN -- ~)}-

(1.5)

The minimum in the right-hand side of (1.4) is taken o v e r the set of values C d~ satisfying the molecular-field equation s~ = t i m (VN):,~,rN~,,

(1.6)

where ~B)A =--Tr B

e

OA(Tr e OA) ~

(1.7)

is the canonical average of B with respect to A. As a first step in the derivation of eqs. (1.4)-(1.6) we expressed the free energy per particle in terms of a hamiltonian ~ I . N ( ~ ) = ~N + N A ( V s - ~ f , using a fundamental theorem due to Bogoliubov Jr. 9'14) for a hamiltonian with " f e r r o m a g n e t i c " quadratic operators. Applying the Bogoliubov-Peierls inequality, (see e.g. ref. 15), and also a l e m m a for the average value of P ( V N ) we have shown that the free energy per particle in the t h e r m o d y n a m i c limit is given by f[~(] = m in m a x f [ ~ ( ~ l n ) ] -

(1.8)

H e r e ~2,~(~]'0) can be obtained by linearizing the operator ~,,N(~) with respect to VN, cf. eqs. (3.4) and (3.5) of ref. 1. Eq. (1.8) has been proved by deriving a lower and an upper bound for f [ ~ ] which turn out to be equal in the t h e r m o d y n a m i c limit; the lower bound is obvious for sufficiently large A > 0 ; in t h e derivation of the upper bound use has been made of a factorization property for the autocorrelation function of VN. The final result (1.4)-(1.6) has been obtained f r o m (1.8) using again the Bogoliubov-Peierls inequality. * H e r e a n d f r o m n o w on a n o r m a l i z e d o p e r a t o r is an o p e r a t o r a c t i n g on t h e H i l b e r t - s p a c e of an N p a r t i c l e s y s t e m , d i v i d e d b y N.

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. II

427

In the present paper the treatment of ref. 1 will be generalized in two ways. First the polynomial P ( V N ) is replaced by an arbitrary analytic function of operators V~ ). . . . . V~ ). Moreover, these operators V~,~ need not be normalized sums of one-particle operators, but can be of a much more general type, including also short-range operators. In connection with this we can mention the Ising model with two-spin and (long-range) four-spin interactions treated by Oitmaa and Barber16). With regard to this more general type of operators we mention the three specific properties for the operators which have been used in the proof of ref. 1. a) The c o m m u t a t o r between two normalized operators VN and V~ (or TN) tends to zero in the t h e r m o d y n a m i c limit, i.e. [I[VN, V~,]I[~0 ,

ifN~oo.

(1.9)

b) The free energy per particle of a trial hamiltonian which is linear in the operators VN, i.e.

'~r,,,, = N ~ ,

hiV~ )

(1.10)

i

converges in the limit N ~ oo uniformly on a bounded region in the space of the variables hl. c) The autocorrelation function of the normalized operator V~ with respect to the trial hamiltonian tends to zero in the t h e r m o d y n a m i c limit, i.e. lim ((VN - (V,,,)~e.,~))~e 2 .... = 0.

(1.11)

N ~

Eq. (1.11) is essential for the derivation of an upper bound for the free energy per particle. In the case of normalized sums of one-particle operators (1.2) the proof of eqs. (1.9)-(1.11) is trivial, assuming that the dependence of V ( k ) on k is not of a pathological nature. Eq. (1.11) is a direct consequence of the factorization

(V(k)V(k'))~,,r.N = (V(k))g,,.,~(V(k'))x, ....

for k S k'.

(1.12)

Also, in the general case under consideration here, it turns out that the free energy per particle can be obtained by minimizing the free energy of a linear trial hamiltonian over a set ~ analogous to the one in eq. (1.6). Note in connection with the definition of At that the average values of the operators V ") may show discontinuities because of the presence of short-range interactions in the trial hamiltonian. In section 2 we shall give the formulation of the theorem, see in particular subsection 2.4. The proof will be given in section 3 and proceeds along the lines of reasoning of ref. 1. Section 4 contains some remarks and a discussion on possible applications.

428

L.W.J. D E N O U D E N .

H.W. C A P E L

AND J.H.H. PERK

2. F o r m u l a t i o n of the theorem

Before we formulate the main theorem of this treatment, we discuss the extensions to more general operators VN and functions P.

2.1. Generalization of the operators In the introduction we mentioned the three properties a), b) and c) of the operators VN, which have been used in the proof of ref. 1. These properties suggest a close relationship between the generalization of the operators VN and the process of taking the thermodynamic limit. In order to discuss this process, we consider a sequence of (u-dimensional) lattice systems with N particles* located on a subset S2N of an infinite lattice. We shall say that the sequence of systems S~ tends to infinity in the sense of Van Hove, cf. refs. 17, 18, if for each ~N there exists a collection of disjoint equivalent cubes

CM(K), with M ( N ) ditions

K = 1,2,3 . . . . .

(2.1)

sites, L ( N ) cubes being contained in D~, satisfying the con-

1) lira M ( N ) = 2

(2.2)

N ~

2) lim L ( N ) = 0% 3) lim L ( N ) M ( N ) _ N~ N

(2.3) 1.

(2.4)

Furthermore, to be specific, we assume that for N ' > N each cube CM, can be constructed from cubes CM corresponding to N, i.e. 4) CM,(K') = ~f~"~"CM(K),

(2.5)

K

M ' =- M ( N ' ) ,

M=- M ( N ) ,

N ' > N.

The symbol Z ~K'~is used to express that the cubes do not overlap and that the M ' / M values of K which contribute to CM,(K') are determined by K ' . The process of taking the thermodynamic limit has been illustrated in fig. 1. From now on each operator which has a subscript N denotes an operator acting on the direct-product Hilbert space of particles belonging to one of the systems g2N given above. In order to describe the generalization of the operators we d e c o m p o s e an arbitrary hermitean operator NVN into an operator N V ° and an operator NRN. The operator N V ° contains only the interaction between particles lying inside the same cube CM(K) of the set (2.1) * If f2N is not well d e f i n e d for e a c h v a l u e of N, one c a n c o n s i d e r a s e q u e n c e ~2N(.~ w i t h

N(n)~oo, if n ~ , .

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. II

429

o,

Fig. 1. Two systems with N and N' (N' > N) particles located on the subsets/~ and/~s, of the infinite lattice D. l'~s contains cubes CM(K), (M = 4, K = 1. . . . . L(N) = 5, N = 37), and /~N' contains cubes CM,(K'), (M' = 16, K' = 1. . . . . L(N') =7, N' = 147). c o r r e s p o n d i n g to ON a n d t h e o p e r a t o r RN c o n t a i n s t h e r e m a i n i n g i n t e r a c t i o n s . (The i n t e r a c t i o n t e r m s in RN i n v o l v e p a r t i c l e s b e l o n g i n g to d i f f e r e n t c u b e s a n d a l s o p a r t i c l e s w h i c h b e l o n g to t h e p a r t o f ON o u t s i d e t h e c u b e s . ) T h e n o r m a l i z e d o p e r a t o r V° can be e x p r e s s e d b y L(N)

NV~N= ~_, M ( N ) V ~ ( K ) ,

(2.6)

K=I

where V~K) is an o p e r a t o r a c t i n g on t h e H i l b e r t s p a c e o f t h e p a r t i c l e s b e l o n g i n g to c u b e C , ~ ( K ) . T h e d e c o m p o s i t i o n o f t h e n o r m a l i z e d o p e r a t o r VN c a n b e w r i t t e n as VN = V ° + RN.

(2.7)

W e n o w r e q u i r e t h a t t h e r e s i d u a l o p e r a t o r RN t e n d s to z e r o in t h e t h e r m o d y n a m i c limit, i.e. lim IIRN[I = 0.

(2.8)

A l s o , to be specific, w e a s s u m e t h a t f o r a s u b d i v i s i o n o f a large c u b e into s m a l l e r o n e s , as g i v e n b y eq. (2.5), t h e i n t e r a c t i o n b e t w e e n d i f f e r e n t s u b c u b e s , i.e. M'RM,t~=-- M ' V ~ , -

z,

(K')

c

MV~(K),

(2.9)

K

is negligible in t h e t h e r m o d y n a m i c limit, i.e. lim s u p IIRM,,,,,,I,,,,,,,,II = o. N ~

(2.10)

N'>N

F u r t h e r m o r e it is a s s u m e d that, f o r sufficiently large v a l u e s o f N, t h e r e is a

430

L.W.J. D E N O U D E N , H.W. C A P E L A N D J.H.H. P E R K

translational invariance, i.e. TV~K)T

' - V~KT)

= 0,

(2.11)

f o r a translation T t r a n s f o r m i n g the c u b e K into one of the o t h e r c u b e s K r * . Eqs. (2.8), (2.10) and (2.11) will turn out to be sufficient for our p u r p o s e . F r o m n o w on o p e r a t o r s satisfying these conditions will be r e f e r r e d to as s h o r t - r a n g e operators. N o t e that this c o n c e p t o f short-range o p e r a t o r is of a rather general nature, cf. the d i s c u s s i o n later on in this section. T h e s e conditions ensure that properties similar to eqs. (1.9)-(1.11) are satisfied, cf. r e m a r k 4 at the end of this section. In particular, f o r s h o r t - r a n g e hamiltonians N V N the free e n e r g y per particle has a well-defined t h e r m o d y n a m i c limit in the sense of Van H o v e , cf. conditions (2.2)-(2.5), i.e. f=

(2.12)

lirn fN = l~rn /~,,N,,

where fN ~- - ( / 3 N ) ' l n T r N e ~NVN

(2.13)

f ~ ------- (/3M) ' In TrM e -~Mv~'

(2.14)

and

are the free energies per particle of the s y s t e m Ou and the cube CM(K) resp. U s i n g eqs. (2.8), (2.10), (2.11) and the B o g o l i u b o v - P e i e r l s inequality'5), we have

[B,,-B,I ~< [IeM,r~[l~0,

(M, M' ~oo)

(2.15)

and

If.

-

f;.,N,f

<~ IIR~II + (1 - L M N

')[f~Mt~ 0,

( S ~ oo),

(2.16)

f r o m w h i c h eq. (2.12) follows. 2.2. D i s c u s s i o n It should be m e n t i o n e d that eqs. (2.8) and (2.10) are satisfied for " ( n o t too) l o n g - r a n g e " interactions, cf. eq. (2B.24) of ref. 19 or eq. (2.2.8) of ref. 18. To see this, note that the action of an arbitrary o p e r a t o r V on the s y s t e m S2N can be written as **'? NVN = ~

E mk

V((.ok),

(2.17)

kEgIN ~kCglN

* Equation (2.1 l) can be relaxed a s s u m i n g that the left-hand side tends to zero for N - ~ . ** Equation (2.17) is equivalent to the definition by Ruelle and Griffiths. For comparison, note that NVN can be written as ~ c a N @(w), where ~(to) --= N(to) V(to), N(to) being the n u m b e r of particles in O).

+In eq. (2.17) we restrict ourselves to open b o u n d a r y conditions. The effect of other b o u n d a r y conditions can be included in the operator RN in eq. (2.7), without affecting the line of reasoning.

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. II

431

w h e r e k labels the sites in I2N and w h e r e the s u m m a t i o n o v e r wk involves the different subsets of gin containing particles w h i c h interact with particle k. In eq. (2.17) an m - b o d y interaction V(w) b e t w e e n m particles kl . . . . . km o c c u r s m times, n a m e l y as one of the terms V(tok,), f o r i = 1 . . . . . m resp. (A set o) containing e.g. two n e i g h b o u r i n g particles 1 and 2 c o r r e s p o n d s to a nearest n e i g h b o u r interaction b e t w e e n 1 and 2 and the c o r r e s p o n d i n g V(to) o c c u r s twice, n a m e l y as a term V(o)k) for k = 1, 2 resp.). As in refs. 18 and 19, we a s s u m e that for sufficiently large values of N the interaction is invariant u n d e r translations f r o m one cube to a n o t h e r and also that the sum of the interactions V(tok) o v e r an infinite lattice is finite f o r fixed k, i.e. A)

TV(~ok)T-'= V(wk~),

(2.18)

w h e r e T is the o p e r a t o r a s s o c i a t e d with the translation f r o m one cube C~,(K) into a n o t h e r cube CM(Kr) and w h e r e ~okr is the subset o b t a i n e d f r o m wk after application of the translation T.

(2.19)

B) ~ Ilv(,ok)ll-< v, < ~, ~k

w h e r e the s u m m a t i o n in the left-hand side is o v e r all finite subsets wk, containing particle k, of the infinite lattice. In view of (2.7), the o p e r a t o r RN does not contain the interaction terms V(to) f o r w h i c h w is included in one single cube. F r o m this we have the inequality

]]RN [1~<

N - LM LM N v, + ~ vM.

(2.20)

H e r e the first term is an u p p e r b o u n d f o r the interactions involving particles w h i c h do not belong to one of the cubes, and MvM is an u p p e r b o u n d f o r the interaction of the particles belonging to a cube CM with particles outside that cube, i.e..

MvM = ~

Y~ [IV(o~k)ll-

k

~k

kEG M

~kcZCI*4

(2.21)

In o r d e r to estimate (2.21) we c o n s i d e r the restricted s u m of interactions with a particle k such that the " d i a m e t e r " D(oJk) of the subset cok is larger than s o m e fixed distance d. F r o m (2.19) it follows that

Y~ IIV(,o~)ll~ld-~0,

ford~oo.

(2.22)

~k D(~k)~d

U s i n g (2.22) we can give s e p a r a t e estimates for the contributions to MVM f r o m particles k E CM lying at a distance larger than d f r o m one of the sides of the cube and f o r the c o n t r i b u t i o n s f r o m the o t h e r particles in C~. W e then have, f o r e a c h value of d, vM <~ Vld + 2t,d M

l]vv

I,

(2.23)

432

L.W.J. DEN OUDEN, H.W. CAPEL AND J.H.H. PERK

In eq. (2.23) it has been used that the n u m b e r of sites in CM at a distance less than d f r o m one o f the sides of the c u b e is b o u n d e d by 2vd M ~-'~/v, w h e r e u is the dimensionality of the lattice. Since eq. (2.23) is valid for all d, we can take as an u p p e r b o u n d for v,~ the infimum of the right-hand side with r e s p e c t to d, w h i c h is a f u n c t i o n of the variable M. As a result we have v~ --. 0,

for M ~ oc.

(2.24)

H e n c e , in view of (2.20), RN tends to zero if we take the t h e r m o d y n a m i c limit in the sense of V a n H o v e , so that eq. (2.8) and also eq. (2.10) are satisfied. This s h o w s that o p e r a t o r s satisfying eqs. (2.18) and (2.19) are included in our treatment. As an e x a m p l e one m a y c o n s i d e r a spin model for a u-dimensional lattice with anisotropic H e i s e n b e r g interactions, described by the hamiltonian (2.25)

N V N = ~_, Sk • Jkk " Sk,. k.k'

The d e c o m p o s i t i o n (2.7) is then d e t e r m i n e d by I,(N)

NV'~ = ~

(2.26a)

~ S~ • dkk' " Sk,~Sk(K)6k,(K),

K--1

k,k'

NRN=~Sk.dkk.'Sk,

1--

,

(2.26b)

where 6~(K)=

1, 0,

if k E C M ( K ) , otherwise.

(2.27)

T h e f a c t o r b e t w e e n b r a c k e t s in the right-hand side of (2.26b) e x p r e s s e s that R~ contains only contributions f r o m particles k and k' which do not belong to the same cube. N o w R~ in (2.26b) tends to zero in the t h e r m o d y n a m i c limit u n d e r rather general conditions. If the interactions dkk, in eq. (2.25) are of finite range D, i.e. dkk. = 0,

if IRk -- R~,[ > D,

(2.28a)

w h e r e Rk and Rk. are the lattice sites c o r r e s p o n d i n g to particles k and k', then v,a = 0 in (2.22) f o r d > D, and RN ~ 0 b e c a u s e of (2.19), (2.20) and (2.23). Also for interactions with a K a c - t y p e d e p e n d e n c e on the distanceZ"), i.e. jkk=jy,,e

~l~k'~,,

y>0,

(2.28b)

R~ tends to zero. 2.3. G e n e r a l i z a t i o n o f the p o l y n o m i a l In ref. 1 we dealt with the simple case of a polynomial P ( V N ) of one normalized sum of one-particle operators. In o r d e r to discuss the generalization to an " a n a l y t i c " f u n c t i o n of more o p e r a t o r s we c o n s i d e r a finite n u m b e r of

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. II n o r m a l i z e d h e r m i t e a n s h o r t - r a n g e o p e r a t o r s V ~ ~, i = 1, 2 . . . . . a s s u m e to be u n i f o r m l y b o u n d e d , i.e.

IIv~,'ll ~ M, < ~ ,

i = 1, 2 . . . . .

n,

433

n, w h i c h we m a y

(2.29)

i n d e p e n d e n t of N * ' * * . A n a r b i t r a r y a n a l y t i c f u n c t i o n P of t h e s e o p e r a t o r s is d e t e r m i n e d b y its series e x p a n s i o n , i.e.

P(v.)

-=

2

m--I il,...

2

.(i, ..... im)v ,' ..

,im--1

V~ .

(2.30)

I n eq. (2.30) u s e has b e e n m a d e of a v e c t o r n o t a t i o n V~ ~ ( V ~ ' . . . . .

V~').

(2o31)

F u r t h e r m o r e , it is a s s u m e d that

p(i ......

i,) = p(i~ . . . . .

ira)*,

(2.32)

so that the o p e r a t o r P ( V ~ ) is h e r m i t e a n . F o r the coefficients p ( i , . . . . . i~) it is sufficient to i m p o s e the " a n a l y t i c i t y " condition: ~ ( M ÷) = po < oo,

(2.33)

where

•~7~(71) =- ~, rrl--1

~ i I .....

IP(i, . . . . .

i~)l'0,, • • • r~,~,

(2.34)

i~--I

for s o m e v a l u e s of M ~ > M,, i = 1 . . . . .

n.

2.4. T h e o r e m I n s e c t i o n 2.1 a n d 2.3 the d e f i n i t i o n s of a n a l y t i c f u n c t i o n s of s h o r t - r a n g e o p e r a t o r s h a v e b e e n given. W i t h t h e s e d e f i n i t i o n s the f o l l o w i n g t h e o r e m c a n be formulated.

T h e o r e m : L e t the h a m i l t o n i a n of the s y s t e m ON be g i v e n b y an " a n a l y t i c " f u n c t i o n of n o r m a l i z e d " s h o r t - r a n g e " o p e r a t o r s , i.e. ,~gN = N P (VN).

(2.35)

H e r e the o p e r a t o r P ( V N ) is defined b y (2.30) a n d the coefficients p ( i , . . . . . ira) satisfy eqs. (2.32) a n d (2.33). F u r t h e r m o r e the v e c t o r VN, cf. (2.31), d e n o t e s a finite set of n o r m a l i z e d h e r m i t e a n o p e r a t o r s V~ ~, i = 1. . . . . n, w h i c h are u n i f o r m l y b o u n d e d , cf. (2.29) a n d w h i c h s a t i s f y the s h o r t - r a n g e c o n d i t i o n s (2.8), (2.10) a n d (2.11). T h e n the free e n e r g y per particle in the t h e r m o d y n a m i c limit exists in the * Note that for the operator given by (2.17) IIVNII~v,, independent of N, c[. (2.19). ** Non-hermitean operators JN, as have been used in refs. 9, 14, can also be treated using the decomposition J = {~(J*+ J)} + i{~i(J'- J)}.

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L.W.J. DEN OUDEN, H.W. CAPEL AND J.H.H. PERK

s e n s e o f V a n H o v e a n d is g i v e n b y

f - ~ tim fN[Y(N] = m i n lim fN[Y(tr.N(~)].

(2.36)

T h e " t r i a l " h a m i l t o n i a n ~tr,N(~) in eq. (2.36) is d e f i n e d by*

Y(~.~(~) = N {P (~) + P ' ( ¢ ) " (VN - ~)}.

(2.37)

H e r e P ( ~ ) c a n be o b t a i n e d f r o m (2.30) r e p l a c i n g the o p e r a t o r s V ~ ~. . . . . V ~ ~b y the real p a r a m e t e r s sc, . . . . . ¢,. F u r t h e r m o r e , u s e has b e e n m a d e o f a v e c t o r n o t a t i o n , in w h i c h the d o t i n d i c a t e s the s c a l a r p r o d u c t o f t w o n - d i m e n s i o n a l v e c t o r s , e.g. p'(~)

• (v~

-

~:)

9P(~) ~-(V~'-

-=

~,).

(2.38)

i=I

T h e set M in eq. (2.36) is d e f i n e d b y the v a l u e s £ s a t i s f y i n g t h e r e l a t i o n lira l i m e • (VN)~,~.~(~) ~ "~vN <~ e v~O N~

• ~ <~

lim l i m e vJ, O N ~

•

(V~)~e~,.,~(~) ,,.. N,¢~,

(2.39)

f o r a n y unit v e c t o r e.

R e m a r k 1: T h e t h e r m o d y n a m i c limits in (2,36) a n d (2.39) a r e well d e f i n e d in the s e n s e o f V a n H o v e . In o r d e r to s e e this, w e i n t r o d u c e

6 ( h ) =-lirnfN[h • NVN],

(2.40)

w h e r e , a c c o r d i n g to (2.12), t h e r i g h t - h a n d side c o n v e r g e s to a c o n t i n u o u s c o n c a v e f u n c t i o n o f h ( u n i f o r m l y on a b o u n d e d r e g i o n in h - s p a c e ) . F i r s t w e r e w r i t e (2.36) as f = min {P (~:)- ~" P'({~) + ~b(P'({~))},

(2.41)

which implies that the thermodynamic limits in eq. (2.36) exist**. Furthermore, eq. (2.40) ensures that ~b(P'(~) - re), for a fixed direction e, is a concave function of ~,, so that the "left" and "right" derivatives with respect to v are monotonic decreasing functions of v. Equation (2.39) can now be reformulated in terms of t h e s e d e r i v a t i v e s , i.e. lim - d ~'0

(lb'

q~(p, (~) _ re) ~< e - ~: ~< lim - ~ v].O

~0(P' (~) - re),

(2.42)

OP

so t h a t the t h e r m o d y n a m i c limits in eq. (2.39) a r e well defined.

R e m a r k 2: If, f o r e a c h d i r e c t i o n e, t h e " l e f t " a n d " r i g h t " d e r i v a t i v e s o f t h e f r e e e n e r g y q J ( P ' ( ~ : ) - re) a r e e q u a l , eqs. (2.39) a n d (2.42) c a n be r e p l a c e d b y

* Note that ~t',r.N(g) can be obtained linearizing ~N with respect to VN. ** In section 3.6 it will be shown that g/~ 0.

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. II

435

the "molecular-field equation" (2.43) Equation (2.43) is a natural generalization of the molecular-field equation (3.33) of ref. 1, where we considered the special case of normalized sums of one-particle operators. In the present treatment, h o w e v e r , the operators VN can contain arbitrary short-range interactions, so that the trial hamiltonian (2.37) can give rise to first-order transitions at a fixed value of ~. In that case eq. (2.39) should be used rather than the molecular-field equation (2.43).

Remark 3: In the formulation of the t h e o r e m we have omitted the operator TN, cf. (2.35), in contrast with the t r e a t m e n t of ref. 1. This is no restriction, since in the case of the hamiltonian ~N = N{TN + P(VN)}, one can always define a new function Q of the n + 1 operators V~ ~. . . . . TN, which is linear in the operator TN, i.e.

TN + P(VN)=- Q(VN, TN).

(2.44) V~n~, (2.45)

The trial hamiltonian corresponding to NN is given by

~g~r.N(~, ~n+,) = N{TN + P (~) + P ' ( ~ ) • (VN - ~)},

(2.46)

cf. (2.37). N o t e that the right-hand side of (2.46) does not depend on the p a r a m e t e r ~,+1 associated with the o p e r a t o r TN.

Remark 4: In the introduction we have mentioned the three specific properties a), b) and c) of the operators VN, which have been used in the treatment of ref. 1. Similar properties are valid for the short-range operators of section 2.1 and will be used in the p r o o f in section 3. The c o m m u t a t i o n p r o p e r t y (1.9) is a direct c o n s e q u e n c e of eqs. (2.8), (2.10), (2.11) and will be discussed in more detail in appendix A. The t h e r m o d y n a m i c limit, cf. (I.10), has been investigated in this section, c[. eqs. (2.12)-(2.15). F u r t h e r m o r e the short-range conditions ensure the validity of a factorization p r o p e r t y of the autocorrelation function similar to (1.11), which is n e c e s s a r y in the derivation of an u p p e r bound for the free energy per particle, c[. section 3.4.

3. Proof of the t h e o r e m

In this section the theorem, f o r m u l a t e d in subsection 2.4, will be proved in six subsequent steps similar to those in ref. 1. H e r e , h o w e v e r , the t h e o r e m deals with the general case of analytic functions of normalized short-range operators, as defined in subsections 2.1 and 2.3.

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L.W.J. DEN OUDEN, H.W. CAPEL AND J.H.H. PERK

3.1. A p p l i c a t i o n

o f the f u n d a m e n t a l

theorem

As a first step we apply a f u n d a m e n t a l t h e o r e m due to B o g o l i u b o v Jr. 9 14). Theorem:

C o n s i d e r an N - p a r t i c l e s y s t e m described by the hamiltonian (3.1)

Y(N = N ( T N - A V e ) ,

w h e r e A > 0 and the o p e r a t o r s T~, V~ -= (V~' . . . . . Vk7)) f o r m a set of (n + 1) n o r m a l i z e d b o u n d e d h e r m i t e a n o p e r a t o r s with c o m m u t a t o r s that tend to zero in the t h e r m o d y n a m i c limit, i.e.

[[T~I[ ~ MT,

IIV~'II~M,,

II[T~, V~']ll = e , ( N ) ~ 0 , Ill

v~,', v~,']ll

= ,,,(N)--,

if N ~ ~,

0,

ifN~,

(3.2) i,j=

1. . . . .

n.

T h e n the free e n e r g y per particle in the t h e r m o d y n a m i c limit is given by f ~ tim f~ [~,~ ] = m in l i ~ fN [~,.~ (~)],

(3.3)

w h e r e the trial hamiltonian 9(,,N(~) is defined by Y(,,N(g) = N ( T N - 2 A ~ .

VN + Ag2).

(3.4)

In eqs. (3.1), (3.3) and (3.4) use has been m a d e of the same v e c t o r notation as in s u b s e c t i o n 2.4. H e r e it is u n d e r s t o o d that the right-hand side of (3.3) c o n v e r g e s u n i f o r m l y for values of ~ satisfying I~c,[<~M,,

i = 1. . . . .

n.

(3.5)

N o t e that we can restrict o u r s e l v e s to (3.5), since f r o m the B o g o l i u b o v Peierls inequality, see e.g. ref. 15, it follows that fN [~N ] ~< fN [ ~(,.N (~)1 - A ((VN -- ~)'->~,~,,,.

(3.6)

E q u a t i o n (3.6) implies that at the absolute m i n i m u m of (3.3) we m u s t have = tim (YN),,N,*,,

(3.7)

i.e. the limit in the right-hand side of eq. (3.7) exists and is equal to ~.

T h e f u n d a m e n t a l t h e o r e m has b e e n f o r m u l a t e d and p r o v e d by B o g o l i u b o v Jr. u n d e r the a s s u m p t i o n that the c o m m u t a t o r s in eqs. (3.2) tend to zero as N ' in the t h e r m o d y n a m i c limit. In ref. 1 we have given a simplified p r o o f for n = 1 u n d e r the w e a k e r a s s u m p t i o n [I[T~, VNlll = ~ ( N ) ~ 0 , if N ~ o ~ , cf. (3.2). T h e e x t e n s i o n to a finite n u m b e r of o p e r a t o r s VN is trivial. In order to see this, the hamiltonian (3.1) can be written as ~,~ = N(T~7 1)__

AVON,2),

(3.8)

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. I1

437

where T~" ' ~

(3.9)

TN -- A ( V ~ 2 + • • • + V~7 ,~2)

and so on. Equation (3.3) follows applying the fundamental theorem for one operator V~~, (i = n . . . . . 1), n times successively. We now turn to the hamiltonian (2.35). As in ref. 1, we first subtract a " f e r r o m a g n e t i c " quadratic operator - A V~,, i.e. Y(N = N ( T ~ - A V ~ ) ,

a >0,

(3.10)

where T~-~ P(VN)+

(3.11)

AV~.

F r o m the conditions in section 2 it follows that N operators T k and V~ ~. . . . . V~7~ satisfy conditions (3.2). For details we refer to appendix A. The application of the fundamental theorem leads to

f = min lim fN [ Y(,,N (g)],

(3.12)

where Y(,.N(~) = N ( P ( V N ) +

AV~-

2Ag.

VN + ag2).

(3.13)

The right-hand side of eq. (3.12) converges uniformly for all £ satisfying eq. (3.5), as will be proved in subsection 3.5. The hamiltonian Y(,.N(g) will be c o m p a r e d with a trial hamiltonian Y(2.N(~[II) which can be obtained linearizing Ygl.N(g) with respect to the operators V~. The explicit expression for Y(2,N(~ITI) is given by Y(2.N(~1~1) = N [P (~l) + A (~1 - g)2 + {p,(n) + 2A (~1 - g)}" (VN - ~q)]. (3.14) For normalized short-range (hermitean) operators V~ ~. . . . . V~7~, each of which satisfying eqs. (2.8), (2.10), (2.11) and (2.29), it has been proved that the free energy per particle corresponding to a hamiitonian 3~2,N(~1~) has a well-defined t h e r m o d y n a m i c limit, i.e. f [ ~ ( g i n ) ] -- tim fN [ ~ . ~ (gin)I,

(3.15)

cf. eq. (2.12). The right-hand side of (3.15) converges uniformly for all p a r a m e t e r s ~ and ~q belonging to the box ~ defined by

xE~CC,[x~[~M,

i = 1. . . . . n.

(3.16)

In the derivation of u p p e r and lower bounds for fN [Y(,.N(~)]- f~ [Y(~.~(gllq)] use will be made of a lemma, which is a generalization of the l e m m a in section 3(ii) of ref. 1. The fundamental theorem (3.3) which applies to general operators TN and VN, satisfying eqs. (3.2), can also be generalized to include c o n c a v e functions of a n u m b e r of operators VN2'). This can be shown using the l e m m a Remark:

438

L.W.J. DEN OUDEN, H.W. CAPEL AND J.H.H. PERK

in s u b s e c t i o n 3.2. F o r d e t a i l s w e r e f e r to a p p e n d i x B. In the p r e s e n t t r e a t m e n t w e can r e s t r i c t o u r s e l v e s to the c o n c a v e f u n c t i o n - A V ~ in eq. (3.1), w h e r e A s h o u l d be so large t h a t P ( V ) + A V ~ is a c o n v e x f u n c t i o n . F r o m this it c a n be s h o w n that the f r e e e n e r g y p e r p a r t i c l e c a n be e x p r e s s e d in t e r m s o f the m i n m a x of the f r e e e n e r g y of the trial h a m i l t o n i a n (3.14), w h i c h can be o b t a i n e d f r o m (3.13) l i n e a r i z i n g P ( V ) + A V 2 with r e s p e c t to a p a r a m e t e r ~q. 3.2. L e m m a L e t V ~'~. . . . . V '~' be a set o f b o u n d e d h e r m i t e a n o p e r a t o r s . L e t P ( V ) be an a n a l y t i c f u n c t i o n of t h e s e o p e r a t o r s , as d e f i n e d in s u b s e c t i o n 2.3. L e t p be a d e n s i t y o p e r a t o r a c t i n g on the s a m e H i l b e r t s p a c e as the o p e r a t o r s V. T h e n , f o r all p a r a m e t e r s ~q w i t h ]~,] ~< M,, w h e r e M satisfies M , / > ]]V("]], w e h a v e the inequality FTrp{P(V)-

P(o)-P'(71).

O\~P P2 m ~' m a x e • 0~c~/ ,,

P~

(V

rl)}[<~p2Trp(V-

17)2+p3,

(3.17)

(3.18)

• e,

Onc)~qOrl ., , I MH[ V, V] H,

(3.19)

w h e r e e is a real unit v e c t o r a n d .~ is defined as in eq. (2.34). F o r the d e t a i l s o f the p r o o f w e r e f e r to a p p e n d i x B. 3.3. D e r i v a t i o n o f a l o w e r b o u n d i n o r d e r to d e r i v e a l o w e r b o u n d f o r fN [~,.N (~)] w e c o n s i d e r t h e d i f f e r e n c e b e t w e e n t h e h a m i l t o n i a n s Y(,.~(~) a n d ~92.N(~t'If~) , i.e. ~ , , ~ (~) - ~ 2 " ( g l n ) = N { P ( V N )

- P(n) -P'(~I)"

(VN -- ~ ) + ,4(VN

7/)2}.

(3.20) F r o m the B o g o l i u b o v - P e i e r l s i n e q u a l i t y w e h a v e

>/fN [Y(~.N(~1~/)] + (A - p_~)((V~ - T/)2)~, ~(~,- p , ~ .

(3.21)

In the d e r i v a t i o n of (3.21) use h a s b e e n m a d e of t h e l e m m a (3.17) w i t h V--- VN; p2 a n d p3.N are g i v e n b y (3.18), (3.19) resp. M o r e o v e r , p3.~0,

ifN~.

(3.22)

b e c a u s e of eqs. (2.29) a n d (3.2). F o r sufficiently large v a l u e s of A, i.e. A > p,,

(3.23)

eq. (3.21) r e d u c e s to fN [~,.N (~)] ~ f~ [ ~-'.~ (~:]~q)] -- P3.~

f o r all ~: a n d ~q b e l o n g i n g to the b o x ~ .

(3.24)

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. II

439

3.4. Derivation of an upper bound In section 3(iv) of ref. 1 the u p p e r bound for fN [YG.N(~)] was derived using the B o g o l i u b o v - P e i e r l s inequality, the l e m m a and the factorization p r o p e r t y (1.12) with 9(,r.N = Y(,-.N(~I~?). This p r o p e r t y could be p r o v e d easily using that the normalized operator VN and Yf,_.N(~Irl) are linear combinations of oneparticle operators V(k). In the general case under consideration here the operators VN are normalized short-range operators and the validity of an equation like (I.!1) is by no means obvious. Yet, a factorization property seems to be a basic ingredient for the derivation of an upper bound. For that reason the averages will be taken with respect to the " b a r e " trial hamiltonian ~.N(~lrl), which is a sum of operators acting on the Hilbert spaces associated with the cubes CM(K) originating f r o m the subdivision of the s y s t e m g/N, cf. (2.1)-(2.5). Since the operators V~2 occurring in the trial hamiltonian ~2.N (~lrl) are normalized short-range operators, use can be made of the decomposition properties (2.6)-(2.8) for each of the V~~, i = 1. . . . . n. The " b a r e " trial hamiltonian ~,_.~(~llq) can be obtained f r o m (3.14) replacing the V~2 by the corresponding operators V~", i.e. ~ . N (~]r/) = ~_-.N(~]~q)+ NR~..N(~]~I),

(3.25)

where

~t~2.~(~l'o)= N [ P ( r t ) + A ( n -

~)2 +{P'(rt)+ 2 A ( r t - ~)}.(V°~-rt)]

(3.26)

and R2.~(~la~) = {P'(n) + 2 A ( n - ~?)}" RN.

(3.27)

Using the B o g o l i u b o v - P e i e r l s inequality it follows that fN [Y(,N] ~< fN [Yt~2N]+ N '(Y(,N -- Yta~'-N)~ ~
-- ~2)N)~N

+ N '(~_,~- Y(2N)~-~.

(3.28)

H e r e use has been made of the notations ~,N ~ ~,.~(~),

Y(,.N=- Y(,-.N(~In),

Yt~-?N--- ~(~.N(~lrl)"

(3.29)

Introducing rN =-- 2 max OP(rt) + 2 A ( ~ -

~,)[IIR~'II,

(3.30)

we have /N [Y(,~] ~
(3.31)

F r o m (3.20) and the l e m m a (3.17) it follows that fN [Yt',N'] ~< fN [Y(2N] + (A + p~)(( VN -- ~q)2)~ + 2r~ + p3.N,

(3.32)

440

L.W.J. DEN OUDEN, H.W. CAPEL AND J.H.H. PERK

w h e r e the a v e r a g e in the right-hand side is taken o v e r the " b a r e " trial h a m i l t o n i a n ~.~(~l~q) rather than the h a m i l t o n i a n ~.N(~I~q). Using the d e c o m p o s i t i o n (2.7) for the o p e r a t o r s V~,', i = 1 . . . . . n, the a v e r a g e in the right-hand side of (3.32) can be e x p r e s s e d as ( ( v ~ - ,~)~)~N = ( ( v ' ~ , ) ~ - n ) ~ + ( ( ~ - ( v ' - ) ~ ) ~ ) ~

(3.33)

+ ( R N " VN + VN " R N -- 2~q • R N -- R ~ ) ~ N .

Since the o p e r a t o r s WN and also the h a m i l t o n i a n ) i f ~ . ~ ] ~ ) are linear c o m b i n a tions of o p e r a t o r s V~M(K), cf. (2.6), acting on the cubes CM(K), we have the factorization property (reaM(K) Y~M(K'))~N = ( W M ( K ) ) ~ (V~M(K')) ~N

(3.34)

f o r different cubes K ¢ K ' . F r o m (3.33) and (3.34) it can be s h o w n that (( VN -- rl)2)~N <~ ((V°N)~N -- rl)2 + M 2 L ( N ) ' + 4 M " IIRNI[.

(3.35)

In the derivation of (3.35) use has been m a d e of the inequality M ( N ) L ( N ) <~ N, c[. s u b s e c t i o n 2.1, and the n o r m e s t i m a t e (2.29) for the o p e r a t o r s VN [and also for the V ~ K ) ] . Substituting (3.35) into (3.32), we have fN [~,.N (~)] ~< fN [ ~2.N (~:l r/)] + (A + P2)((V~)~.,,~t,,) - ~q)2 + qN,

(3.36)

where q~=--(a + p 2 ) ( M 2 L ( N )

' +4M'[[RNII) + 2 r ~ + p 3 . N ~ O ,

ifN~,

(3.37)

i n d e p e n d e n t of ~: and lq, cf. (2.3), (2.8), (3.22) and (3.30). C o n s i d e r n o w the v e c t o r f i e l d ~b(rl) -= ( I ~ ) ~ ~1,) for a gives 1 ..... point

(3.38)

fixed N and a fixed ~ satisfying 1~:~1~
~b(TIN(~))= ~qN(~).

(3.39)

(Eq. (3.39) can also be seen in an elementary way looking for the maximum of jN[ 2.u(~llq)], which is a differentiable function of ~1 on the box @, taking into account that the matrix W'(B)+ 2A1 is positive definite. If fN[~.N(~llq)] assumes its maximum on the boundary of the box, eq. (3.39) follows noting that the derivative of f N along the direction ~b0q)-lq, pointing inwards, is nonnegative.) Inserting (3.39) in (3.36), we have the upper bound f~ [Y(,.N (~)] ~< f~ [9~2.N(~:l~q~ (~))] + q~ u n i f o r m l y for ~: in the box @.

(3.40)

S Y S T E M S W I T H S E P A R A B L E M A N Y - P A R T I C L E I N T E R A C T I O N S . II

441

3.5. Minimax formulation for the free energy From (3.24) and (3.40) we have for all ~ in the box g~ - - p 3 . N ~ fN [ ~ , , N ( ~ ) ] I

max fN[Ygz,N(~ln)] ~ qN,

(3.41)

where p3.N and qN are independent of ~ and ~q and tend to zero for N ~ ~, cf. (3.22) and (3.37). Using eq. (3.15) it follows that the free energy per particle corresponding to Y(,.~(g) has a well-defined t h e r m o d y n a m i c limit, i.e. f[Y(,(~)] ~ tim fN [Yg,.N(~)] = max f[Y(2(~l~l)].

(3.42)

Moreover, this limit converges uniformly for all ~ satisfying (3.5). Hence, f r o m (3.12), the free energy per particle corresponding to the original hamiltonian (2.35) has a well-defined t h e r m o d y n a m i c limit given by f = l~m fN [~N ] = min m a x f[~2(~laq)l.

(3.43)

In eq. (3.43) the free energy has been expressed in terms of the free energy of a trial hamiltonian Y(2.u(gbq) which is linear in the operators V. This hamiltonian ~.~(gln) contains a p a r a m e t e r A which must be sufficiently positive, cf. (3.23). The final result, however, cannot depend on A. This will be shown in an explicit way in the following subsection.

3.6. Molecular-field equation In order to derive the theorem, as formulated in eqs. (2.36), (2.37) and (2.39), we investigate the values ~m E ~ at which f[~2(gllq)] as a function of lq, for fixed g ~ ~ , assumes its absolute m a x i m u m , i.e. f[Y(2(~l~qm)] = max f[~2(~Jlq)].

(3.44)

F r o m (3.21), taking the t h e r m o d y n a m i c limit and using (3.23), (3.42) and (3.44), it follows lim (( VN - n,,)2)~,.~,~ = 0,

(3.45)

and eq. (3.44) defines a unique function rim - ~,,(~) = luim=(VN)~e,.~,,

(3.46)

which implies that the limit in the right-hand side exists for all values ~ ~ g~. F u r t h e r m o r e aqm(~) is a continuous function of ~, since f[~2(~llq)] is a continuous function of ~ and ~q. The unicity of ~lm in eq. (3.46) is a c o n s e q u e n c e of the (strict) convexity of the operator function P ( V ) + A V z, cf. (3.23). T h e r e f o r e phase transitions will occur as a c o n s e q u e n c e of nonanalytic

442

L.W.J. DEN OUDEN. H.W. CAPEL AND J.H.H. PERK

b e h a v i o u r w i t h r e s p e c t to the ~:'s, w h i c h a r i s e f r o m t h e l i n e a r i z a t i o n of t h e c o n c a v e f u n c t i o n - A V 2. In o r d e r to a r r i v e at the g e n e r a l i z e d m o l e c u l a r - f i e l d e q u a t i o n (2.39), w e first d e r i v e a c o n d i t i o n s i m i l a r to eq. (2.39) f o r t h e p a r a m e t e r s rl,,(~). F o r t h a t p u r p o s e w e c o m p a r e t h e f r e e e n e r g i e s f[~-,(~51~m)] a n d f[YG(~[~I)] f o r fixed ~5 ~ ~ a n d a n a r b i t r a r y v a l u e lq ~ ~ . U s i n g (3.14) a n d (3.15), f[Y(2(~l~)] c a n b e e x p r e s s e d as

f[ff~'2(~[~)] = ~b(h(~l)) - TI • h(71) + P 0 q ) + A ( ~ - s¢)2,

(3.47)

where

h (7) --- P ' ( ~ ) + 2,1 (~q - ~:).

(3.48)

and where ~b(h ) = l i r a - (N/~) ' I n T r e ~"'NVN

(3.49)

cf. (2.40). F r o m (3.44) a n d (3.47) we h a v e 0/> f [ ~ : ( ~ l n ) ] - f[Y(2(~l~q,,)] = ~ ( h ( r l ) ) - + ( h ( ~ ° , ) ) -~q,, " { h ( n ) - h ( , , , , ) } + ~ ( n - v , , , , ) ' ( P " ( ~ , ) + 2 A 1 ) . ( n - n , , ) ,

(3.50)

rt, b e i n g a p o i n t b e t w e e n rt a n d rl .... F o r all rl ~ ~ we c a n d e f i n e a u n i t v e c t o r e b y t h e r e l a t i o n h('q)-h,,,

=

re,

h,, ~h(Tq,,,),

u>O.

(3.51)

w h e r e ~ is the l e n g t h of t h e v e c t o r in t h e l e f t - h a n d side. N o t e t h a t f r o m eq. (3.51) 7/ c a n be s o l v e d as a u n i q u e f u n c t i o n of ~,e, s i n c e the d e r i v a t i v e o f h 0 q ) . i.e. t h e m a t r i x P " 0 q ) + 2A 1, is p o s i t i v e d e f i n i t e . S i n c e t h e last t e r m in (3.50) is p o s i t i v e , we h a v e e • lq,,, ~< - v ]{qt(h,, - ue) - ~(h,,,)},

u > 0.

(3.52a)

A s i m i l a r i n e q u a l i t y , f o r n e g a t i v e u, is o b t a i n e d u s i n g t h e r e p l a c e m e n t e --+ - e. u--,-u, -~'

i.e. l{~b(hm-ue)-~b(h,.)}<~e.~q.,,

v
(3.52b

N o w O(h,,, ue) as a f u n c t i o n of u f o r fixed h,, a n d a fixed u n i t v e c t o r e is a c o n t i n u o u s a n d c o n c a v e f u n c t i o n of o n e v a r i a b l e . T h i s i m p l i e s t h a t f o r e a c h v a l u e u the left a n d right d e r i v a t i v e s exist. M o r e o v e r . t h e s e d e r i v a t i v e s are m o n o t o n i c d e c r e a s i n g . F r o m eqs. (3.52). t a k i n g the l i m i t s u 1' 0 a n d u $ 0, we a r r i v e at the i n e q u a l i t y lira

,,~,,

d=_ th(h,, _ u e ) <~ e . lq,, <~ l i m

du

,,~,,

d

duu O ( h , , , - ue).

w h i c h m u s t b e satisfied f o r e v e r y u n i t v e c t o r e.

(3.53)

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. II

443

Equation (3.53) is a necessary condition for ~q,,. Moreover, eq. (3.53) determines the maximizing ~q~(~:) in (3.44) completely, since for each ~ ~ the solution of (3.53) is unique. This can be shown as follows. Assume that there is another value ~q satisfying (3.53). Then we can introduce a unit vector ~, so that h(~l) - h(aq~) = -/x~,

# > 0.

(3.54)

From (3.53), (3.54) and the concavity of tO, we should have

~- ~q >~lim,,~,, ~d t o ( h " - ~ - v ~ ) ~ > l i m - v ~ o d~

(3.55)

to(h'-v~)>~'Tlm"

On the other hand, f r o m (3.54) and (3.48), we have

~.(n

- n~)=

; d ~ ' ~ • ~ d, n ( ~ ' ) o

=-

I

djx'~.{W'('~(~'))+2A1}

o

'.~-~x(A

'

+p2)

'<0, (3.56)

cf. (3.18). Equation (3.56) is incompatible with (3.55), so that for each ~: E the function 71,,(~), as defined by (3.44), is also determined by (3.53). We finally consider the value ~ ~ ~ at which f [ ~ , ( ~ ) ] , as defined by (3.42), assumes its absolute minimum. In view of (3.7), we have

= tim (V~)~, ~,~,

(3.57)

so that, cf. (3.46), = nm(~).

(3.58)

From (3.43) it follows that the free energy per particle corresponding to the original hamiltonian (2.35) is given by f = rain f[N,_(~[~)],

(3.59)

t5~,u

where ~ff is the set of all values ~ C ~ satisfying eq. (3.58) with lqm(~) uniquely determined by (3.53) [which is equivalent to (3.44)]. For all ~ ~ off we have the relation l, i~o m - ddu t o ( P ' ( ~ ) - r e ) <~ e • [~ <~ limv~o- 4a u to(P' (~) - ue),

(3.60)

which follows f r o m (3.53) and (3.58) using that h,, = P'(~:). Note that the relation (3.60) gives an equivalent definition for the set d~, since, for all satisfying (3.60), eq. (3.53) has the unique solution ~qm(~)= ~ (and therefore

444

L.W.J. DEN OUDEN, H.W. CAPEL AND J.H.H. PERK

f[YG(~lrl)l<~f[~2((glg)]

for all ~q E ~ , cf. (3.44)). Equation (3.60) is also equivalent to the definition of the set A/ given previously in eq. (2.39), of. (2.42). Using (3.59) and (3.14) the free energy per particle in the thermodynamic limit, corresponding to the hamiltonian (2.35), is given by f = min lim f,~ [ N { P (~) + P' ( ~ ) . ( V,~ -- so)}].

(3.61)

Hence, the theorem (2.36) has been proved.

4. D i s c u s s i o n

In this section we shall give a brief summary of the main theorem of this treatment, a number of applications and a comparison with the fundamental theorem due to Bogoliubov Jr. Furthermore, we shall add a remark on the Dicke-maser model and some comments in connection with the theorem (2.36) and on (time-independent) correlation functions.

4.1. Summary In the present paper we considered a sequence of systems J2N with N particles located on a subset of an infinite lattice. The systems are described by hamiltonians of the type ~N = NP(VN), where P(VN) is an analytic function, of. (2.30), (2.32)-(2.34), of a finite number of normalized short-range operators VN, ~'~ . . . . V~"' satisfying the requirements (2.8), (2.10), (2.11) and (2.29). The free energy per particle is defined by f~ -= - ( N / 3 ) ' i n Tr exp ( - / 3 ~ ) , where the trace is taken over the direct-product Hilbert space of N one-particle Hilbert spaces; the thermodynamic limit is taken in the sense of Van Hove, cf. eqs. (2.2)-(2.5). It has been proved that in the thermodynamic limit the free energy per particle f can be expressed in terms of the trial hamiltonian ~tT.~(~)= N{P(~:)+ P ' ( { ) . ( V N - ~:)}, cf. (2.37), which can be obtained linearizing the operator P(V~) with respect to VN. The free energy per particle f is given by f = min Hm fN [~,,.N(~)], cf. (2.36), where the minimum should be taken over the set of values s¢: ~ ~4/ satisfying the "generalized" molecular-field equation (2.39). In eq. (2.39) it has been taken into account that the trial hamiltonian can contain arbitrary short-range interactions, so that in the thermodynamic limit it can give rise to phase transitions for fixed parameter values ~:. In particular, if, for a certain value of ~, Yg,r.N (~) gives rise to a first-order transition, at least one of the first derivatives of the free energy has a discontinuity and the average values (VN) are not well defined in the thermodynamic limit. In such cases symmetry breaking terms should be added to the hamiltonian in order to define the thermodynamic quantities. In eq. (2.39) the definition of the set has been given unambiguously adding terms like ue • NVN to the hamiltonian,

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. 11

445

where I~1--,0 and e is an arbitrary unit vector. In simple cases, when the averages (VN) are well defined in the t h e r m o d y n a m i c limit, eq. (2.39) reduces to the molecular-field equation (2.43). 4.2. Applications a) As a first application we mention a general treatment of the problem of competing interactions. Let us consider a spin system on a lattice, described by the hamiltonian ~N = Y(SR,N+ Y(~R,~,

(4.1)

where YfsR.~ is a short-range hamiltonian of the type Y(s,,~ = •

Sk" J ~ " S k . - / x B . ~] Sk,

k,k'

(4.2)

k

and Y(LR.Nis a long-range hamiltonian of the equivalent-neighbour type ~,~,~ = - N ' ~ Sk • J" Sk..

(4.3)

k,k'

In eq. (4.2), the exchange interaction d~k m a y be assumed to be of finite range, cf. (2.28a), whereas in (4.3) the tensor J is independent of k and k'. Also, more terms can be added to eq. (4.2) such as biquadratic exchange terms like --Zk.k' (SD2Kkk,(SZk,) 2 and crystalline field terms - D Ek (S~) 2, cf. e.g. ref. 23. In the case of a long-range interaction like (4.3) the trial hamiltonian is the hamiltonian of a system with short-range interactions in an effective magnetic field. The problem of competing interactions can then be solved to the extent that the free energy of the trial hamiltonian can be evaluated. Well-known examples of exactly solvable models are: 1) The one-dimensional Ising model with a nearest-neighbour coupling Jkk' and an equivalent-neighbour coupling J between the z - c o m p o n e n t s of the spins in the presence of a magnetic field in the z-direction. This model, which gives rise to interesting critical behaviour, has been treated in detail by Nagle and Bonner24). 2) The one-dimensional X Y model with a nearest-neighbour coupling between the x- and y - c o m p o n e n t s of the spins and an equivalent-neighbour coupling between the z-components, B being in the z-direction. This model has been investigated by Suzuki ~) and Gibberd2~). Such models can also be useful in the discussion of properties of three-dimensional systems consisting of weakly coupled one-dimensional chains27'~8). 3) Another example can be obtained f r o m the two-dimensional super-exchange antiferromagnet introduced by Fisher 29) adding an equivalent-neighbour interaction of the Ising type. The trial hamiltonian is the hamiltonian of the Fisher antiferromagnet in an external magnetic field. This model has been investigated by Hall and Stell3"). In the three examples mentioned a b o v e the free energy could be evaluated exactly. The present treatment can lead to more general results, since it is not

446

L.W.J. DEN OUDEN, H.W. CAPEL AND J.H.H. PERK

necessary to know the detailed properties of the trial hamiltonian. In fact, if one wants to investigate the influence of a (small) long-range interaction like (4.3) on the critical behaviour, use can be made of approximate expressions for the free energy of the trial hamiltonian based on series expansions 3') or scaling hypotheses~2). b) As a second application one can consider hamiltonians containing manybody interactions, i.e.

5~N = Y(S,,N + Y(MB,N,

(4.4)

where Y(SR,N is a short-range hamiltonian such as e.g. eq. (4.2) and Y(M,,N contains m a n y - b o d y interactions (i.e. three-body, four-body, and so on). If one assumes that YgM..N has the form N P ( V ~ ) , where the V~L~ e.g. are normalized sums of one-particle operators or normalized short-range twobody interaction operators, then the free energy per particle in the thermodynamic limit can be expressed in terms of a trial hamiltonian which is a normal short-range two-body hamiltonian. In the evaluation of the free energy use can be made of approximate expressions as mentioned above. Recently, there has been much interest in many-particle interactions. Fourparticle interactions have been shown to be important in compressible spin systems, in which there is a coupling between the spins and the lattice, cf. e.g. ref. 33. Also we can mention the exactly-solvable B a x t e r - W u model with three-spin interactions3'). Moreover, m a n y - b o d y interactions are important in the framework of the renormalization theory, see e.g. ref. 35. An example of (4.4) is the model treated in ref. 16 by Oitmaa and Barber in which YG,.N is a nearest-neighbour Ising hamiltonian (B = 0) and ~gMB.Nis a ferromagnetic four-spin interaction proportional to - N '(YG~,~)L The free energy per particle can be expressed in terms of a trial hamiltonian containing only Ising nearest-neighbour interactions. Using the exact solution for the two-dimensional Ising model and an approximate formula for the internal energy of the three-dimensional Ising model, Oitmaa and Barber showed that even in the presence of an arbitrary small ferromagnetic four-spin interaction the (secondorder) phase transition of the Ising model is changed into a first-order transition. Similar considerations can be given for more general spin models using approximate results for the free energy of the trial hamiltonian. 4.3. Relation with the f u n d a m e n t a l theorem In the proof of the main theorem of this paper use has been made of a fundamental theorem due to Bogoliubov Jr. 9~'4) and the assumption that the operators V are normalized short-range operators, cf. (2.8), (2.10), (2.1 i) and (2.29). Note that the theorem of Bogoliubov Jr. applies to a larger class of operators T and V, which may contain in particular separable interactions, cf. eqs. (3.2). This is an essential feature in the derivation of (2.36). Theorem (2.36), however, cannot be extended to such general operators. The short-

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. 1I

447

range properties have been used in an essential way in the derivation of the u p p e r b o u n d for the free energy in subsection 3.4 and also in connection with the t h e r m o d y n a m i c limit of the free energy of the trial hamiltonian, cf. (3.15). In order to see that our theorem does not hold w i t h the same conditions on

the operators, we consider the hamiltonian ?~,N = N ( T N + w W ~ ) ,

(w > 0),

(4.5)

where TN and WN are general operators satisfying eqs. (3.2). F r o m the B o g o l i u b o v - P e i e r l s inequality it follows that f =- firn fN [Y(N] >/g,

(4.6)

where g ~ max lira [,,,[TN + 2 ~ w W ~ - w~2],

(4.7)

assuming that the t h e r m o d y n a m i c limits in (4.6) and (4.7) exist. If T~ and WN are normalized short-range operators, we can apply eq. (2.36) with P (V'g', V~ ~) = V~g~+ w ( V ~ ) ~, V~g~= TN, V~ ' = WN. As a result we would have f = g. In general, however, [ can be larger than g, even if the t h e r m o d y n a m i c limits in (4.6) and (4.7) exist, i.e. f > g,

(4.8)

which implies that theorem (2.36) cannot be extended to more general operators satisfying (3.2). As an example of eq. (4.5) we consider the hamiltonian ~N=N(-vV~+wW~),

(v,w>O;T~=---vV~),

(4.9)

where VN and W~ are normalized short-range operators with IIV~II<~M~, IIw~ II~< M~. F r o m (4.7) and the f u n d a m e n t a l t h e o r e m it follows that g = max rain ~b(sc, ~),

(4.10)

cl,(~, "0) =- I~m f N [ N ( - 2~vVN + v~ 2 + 2 ~ w W N - wr/2)].

(4.11)

l~l~ Mw 16J~Mv

On the other hand, we find in a similar way as in section 3 that f = min lira J : u [ N ( - 2 ~ v V u + v~2+ wW~)] =

min max d~(¢, ~).

I~l~Mv [~1~ Mv¢

(4.12)

The operators VN and WN can be chosen such that eqs. (4.12) and (4.10) lead to the inequality (4.8), which implies that eq. (2.36) is not valid for the hamiltonian (4.5) with the operator TN = V~ ~= - v V ~ . Explicit examples have been given in refs. 36, 1. A very simple example can

448

L.W.J. DEN OUDEN, H.W. CAPEL AND J.H.H. PERK

be o b t a i n e d c h o o s i n g ~rk,

VN=WN=SN=--N

V>W>0,

(4.13)

k=l

w h e r e ~rk = +- 1 refers to the spin o f particle k. In this case we have ~b(sc, 7/) = vsc 2 - wrl 2 - fl-' In 2 c o s h 2 f l ( v ~ - w r l ) .

(4.14)

F o r sufficiently tow t e m p e r a t u r e s 4~(~, ~) a s s u m e s its m i n m a x for the value ~0 = rio ~ 0, cf. (3.58) i.e. f ~ f ( f l ) =- ( v - w)~Zo - fl ' l n 2 c o s h 2 f l ( v - W)~o,

(4.15)

w h e r e ~o ~ 0 is the solution of the e q u a t i o n ~o = tanh 2 f l ( v - w)s%. This can be s h o w n in various w a y s , e.g. by applying the f u n d a m e n t a l t h e o r e m directly to the hamiltonian YfN = - N ( v - w ) S ~ . U s i n g simple inequalities similar to eqs. (145) and (146) of ref. 36, it follows that 4~(~, rl) a s s u m e s its m a x m i n for the values rl~ = 0, ~ ~ 0, i.e. g =- g ( f l ) = v~ 2, - fl-~ in 2 c o s h 2 f l v ~ ,

(4.16)

w h e r e ~:, ~ 0 is the solution of the e q u a t i o n ~:~ = tanh 2 f l v ~ . The m a x m i n p r o c e d u r e leads to a value g w h i c h is equal to the free e n e r g y per particle in the case that w = 0. This value is smaller than the c o r r e c t value (4.15), as follows also f r o m the explicit relation g ( f l ) = v "- w f

4.4. D i c k e - m a s e r

~-

(4.17)

w fl "

model

In eq. (2.35) we c o n s i d e r e d a hamiltonian containing b o u n d e d hermitean n o r m a l i z e d short-range o p e r a t o r s V• acting on the N - p a r t i c l e d i r e c t - p r o d u c t Hilbert space hN. A n i m p o r t a n t model with u n b o u n d e d o p e r a t o r s is the D i c k e - m a s e r model in which the N - p a r t i c l e s y s t e m interacts with a (finite) n u m b e r s of h a r m o n i c oscillators representing the m o d e s of a quantized radiation fieldT"8). In this s u b s e c t i o n we c o n s i d e r the hamiltonian ~o,a~a~ + N ~

=

~' + A~aJN (A *~ a,~JN

(4.18)

) + NP(VN).

H e r e the o p e r a t o r s a~ and a, are b o s o n creation and annihilation operators, (o)~ > 0), acting on the b o s o n Hilbert space hB. The o p e r a t o r s WIN~.I)

~

1 (~)* + J ~ ) , ~(JN

1" (~)~" - J ~ N W N( ~ . 2 ) =--~I(JN ) x] ,

(a = 1,

" " • ,

s),

(4.19)

and V%~, (i = 1. . . . . n), are b o u n d e d hermitean normalized short-range operators acting o n the N - p a r t i c l e Hilbert space hN. T h e total hamiltonian 9(~ is defined on the Hilbert space hB (~)h~. F r o m the t r e a t m e n t in ref. 8 it follows that the free e n e r g y per particle f

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. II

449

corresponding to the hamiltonian (4.18) can be expressed as f = f ~ Nlim ~

(fiN) ' In Tr

(4:20)

e -oe~,

where*

~u = N{P(VN)--

,~]=~]A" + 12w (WL""2 ,

W~'z)~)}"

(4.21)

In the right-hand side of eq. (4.20), the hamiltonian ~ and the traceoperation refer to the N-particle Hilbert space hN rather than the full Hilbert space h~ (~) hN. On the basis of the theorem (2.36) the free energy per particle f = f can be expressed in terms of a trial hamiltonian which is linear in the operators V and W. 4.5.

Comments

So far we did not take into account the presence of an operator TN in the hamiltonian (2.35), since the operator function P(VN) may contain terms linear in the operators V, cf. remark 3 in subsection 2.4. It may be worthwhile, however, to consider hamiltonians of the type N{TN +P(VN)}, since the operator TN is not necessarily bounded. This can be of interest in connection with possible generalizations of (2.36). To the hamiltonian (2.35) one can always add a term NTN =-N(Tk+ T~), where T/~ is a bounded short-range operator and where the unbounded part T~ is a sum of one-particle operators, so that the trial hamiltonian ~2,~(£1~q) converges uniformly for all ~ and ~ ~. The boundedness of TN is not necessary for the fundamental theorem, see refs. 9, 14. Moreover, since the unbounded part NT~ is a sum of one-particle operators, it is included in the " b a r e " trial hamiltonian ~.~(£1~q) without contributing to the quantity r~ in (3.30) and the upper bound (3.36), (3.37). (Also it is not always necessary to require that the operators VN are bounded; one may imagine situations in which the N-particle Hilbert space h~ contains a subspace hk, which is a direct-product of one-particle subspaces, so that the contributions arising from the c o m p l e m e n t of hL to the free energy per particle are negligible in the thermodynamic limit.) Furthermore one may also take into account an infinite number of operators V, provided that eq. (2.33) is satisfied. Until now we have restricted ourselves to quantum systems in which the partition function of an N-particle system is given by Tr exp ( - f l ~ N ) , where Tr involves the trace over a direct-product Hilbert space. The theorem (2.36) also applies to continuous classical systems in which the trace-operation is replaced by an integration like e.g. f I I ~ II~ dXk~, where X labels the internal variables Xk~ (such as e.g. spin components) corresponding to particle k. * The hamiltonian Jt°Ngiven in ref. 8 differs from eq. (4.21) by commutator terms which can be neglected in the thermodynamic limit as a consequence of eq. (3.2) and the Bogoliubov-Peierls inequality.

450

L.W.J. DEN OUDEN, H.W. CAPEL AND J.H.H. PERK

Equation (2.36) remains valid, if all integrations are restricted to finite regions, the norm of an interaction potential V between particles 1,2 . . . . . m being replaced by the maximum of the corresponding function V({X,,~} . . . . . {X,,,,}). If the integrations extend over an infinite volume, the hamiltonian ~ should contain e.g. a function Ek T({Xk~}) ensuring the convergence of the free energy per particle in the t h e r m o d y n a m i c limit. As an example we can mention the n-component classical spin model introduced by Emery~7), which is of interest in connection with critical exponent renormalization~"). This model has been solved exactly in the limit n~ using the Laplace method. The solution can be obtained from (2.36) noting that the thermodynamic limit N - ~ ~ in (2.36) corresponds to the limit n~ in ref. 37. In classical spin systems use can be made of the Laplace method and long-range interactions can often be treated from a more general point of view using a Kac-type of potential, as in eq. (2.28b), taking the limit 7 $ 0 after the thermodynamic limit, of. ref. 39. (Kac-like potentials have also been used in the case of Ising systems"'"') and for more general quantum spin models42)). Finally we mention two models of classical oscillators with anharmonicity of infinite range4~), for which the exact solution for the free energy can also be obtained from (2.36).

4.6. Correlation functions Theorem (2.36) has some direct applications for time-independent correlation functions of normalized short-range interactions. For this purpose we consider a hamiltonian

Y(N = N { P (¥'N) + AQ( VN )},

(4.22)

where P ( V ~ ) and Q(V~) are analytic functions of a number of normalized short-range operators and A is a real parameter. The free energy per particle corresponding to YfN in the thermodynamic limit is given by, cf. (2.36),

f ( h ) ~ lim f N [ N { P ( V N ) + AQ(VN)}] = min lim f,~ [ N { P (~) + P ' (~)- ( VN -- ~) + AQ(~:) + AQ' (~r). (VN -- ~)}1. (4.23) In the absence of first-order transitions, ~:--=~ ( A ) ~ M satisfies the molecularfield equation (2.43). Taking the derivative with respect to A, it follows that lim ( Q ( V ~ ) ) ~ . , . ~ , -

df(A ) ~ - ~ ,, = Q(~(O))

Relations like (4.24) have also been established in an algebraic approach ~''~).

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. II

451

Acknowledgment This investigation is part of the research p r o g r a m m e of the Stichting voor F u n d a m e n t e e l O n d e r z o e k der Materie (F.O.M.), which is financially supported by the Nederlandse Organisatie v o o r Zuiver Wetenschappelijk Ond e r z o e k (Z.W.O.).

Appendix A N, In this appendix we show that the normalized short-range operators V"' i = 1. . . . . n, as defined in section 2.1, have c o m m u t a t o r s which tend to zero in the t h e r m o d y n a m i c limit, i.e.

]][V~', V~qlJ ~- E , j ( N ) ~ 0 ,

if N - ~ ,

(A.I)

cf. (3.2). For this purpose we use the decomposition (2.6), (2.7) and (2.8) of the operators VN, i.e. VN=V°~+RN,

[IRB'II=-r,(N)~O,

(N~oo),

L(N)

V°N = N ' ~

M(N)V~I().

(A.2)

K-1

Then L(N)

I ] [ ~ , V°~]II~< L ( N ) -2 ~

II[V~(K), V~(K)III.

(A.3)

K--I

Using the norm estimate (2.29), i.e.

IIW211~< M,, IIV "(K)II <- M :

~ M, + ri(N),

(A.4)

it follows that (A.I) is satisfied, since e,~( N ) ~< 2{L ( N ) 2MIM~ + M i r ~ ( N ) + r~(N)M} + r , ( N ) r j ( N ) } - , 0, (N ~ ~).

(A.5)

In the derivation of (3.12) use has been made of

[l[Tk, V~2][I----E , ( N ) ~ 0 ,

with T k = P (VN) + AV~.

(A.6)

Equation (A.6) is obvious f r o m (2.30), (2.33) and (2.34), which lead to II[P(V~), v~lll<~p, • II[v~,

v~]ll,

(A.7)

where p, ~

°21

,

(A.8)

O=Bvl

and ~,(N) ~< 2 (P,,J + AM~)e,j(N)--->O, ]=1

(N--->oo).

(A.9)

452

L.W.J. DEN OUDEN, H.W. CAPEL AND J.H.H. PERK

R e m a r k : It m a y be n o t e d t h a t the c o m m u t a t o r s ( A . I ) t e n d to z e r o as N ' u n d e r r a t h e r g e n e r a l c o n d i t i o n s . In o r d e r to be specific, use will be m a d e of the n o t a t i o n (2.17), i.e. VN = N ' ~

Y~

k

V(wk),

(A.10)

~k

kEt~N

~cglN

o)k d e n o t i n g a s u b s e t c o n t a i n i n g p a r t i c l e k. If eq. (2.19) is r e p l a c e d b y the stronger condition N(,ok)[I V"'(~o~)ll ~< v, < 2,

(i = 1 . . . . .

n),

(A.11)

~k

w h e r e N(oJD is the n u m b e r of p a r t i c l e s in t h e s u b s e t ~ok, t h e n

[I[VN, VN]II <~ 2 N 'vv.

(A.12)

In o r d e r to see this, eq. (A.10) is r e w r i t t e n as

VN = N ' ~

N(w)V(w),

(A.13)

o) C l ~ N

noting t h a t e a c h V(o)) o c c u r s N(oJ) t i m e s in the s u m m a t i o n o v e r k in eq. (A.10). F r o m (A.13) w e h a v e

Iltv~, v~]ll =

N ~ Y~ ~C~

=N

2

=1

N(o))N(o)')ll[V(o~), V(~o')]ll

Z N ~'Cg~ N

E ~kC~

N

N(oJk)N(~o'k) Z N ( w k A w'k) [][ V(~ok), V(~o ~)]ll, ~'kCIIN

(A.14)

s i n c e the c o m m u t a t o r s o f V(oJ) a n d V(oJ') o n l y c o n t r i b u t e if w a n d w' h a v e at l e a s t o n e p a r t i c l e k in c o m m o n . F r o m (A.14) eq. (A.12) is o b v i o u s .

Appendix B In this a p p e n d i x the p r o o f o f the l e m m a (3.17)-(3.19) will be given. W e s t a r t with an a n a l y t i c f u n c t i o n P (V), as d e f i n e d in s u b s e c t i o n 2.3, a n d an a r b i t r a r y d e n s i t y o p e r a t o r p. T h e n w e a p p l y T a y l o r ' s t h e o r e m with L a g r a n g e ' s f o r m f o r the r e m a i n d e r , i.e. &(t) = &(0) + t&' (0) + ½t2&"(~-),

(B. 1)

f o r s o m e v a l u e r with 0 < r < t, to t h e f u n c t i o n &(t) -= Yr p P ( V , ) ,

(B.2)

where

V,=-- ~q + t ( V -

rl).

(B.3)

C h o o s i n g t = 1, w e h a v e

X=-Trp{P(Y)-P('q)-P'Uq)'(V-~q)}=~Tr f o r s o m e ~- with 0 < r < 1.

p

P(Vt)

, :~},

(B.4)

SYSTEMS WITH SEPARABLE MANY-PARTICLE INTERACTIONS. II

453

According to (2.30) we write P (V,) = ~ e.l~l

~ i I .....

p(i, . . . .

" ""(','

, lm)vt

• • .

~1('~),

(B.5)

--t

im~l

so that in view of eq. (B.3)

~~5 P ( V,) =

p(i, . . . . . i.,) m~2

i I .....

im=l

V~',' a=l

x ( V ('°'- n,o)

b=a+l

V?') (V ('°- n,~) \l~a+l

V~'') • |

(B.6)

I

We now shift the factor (V ('o)- r/~°) to the left and the factor (V " . ) - ~ , ) to the right. This leads to d2 ] ½~-~P(V,)

=(V-~).P(2).(V-n)+P

(3),

(B.7)

t=T

where (2) __

P,~

-

p(i, m=2

i I .....

. . . . . ira) ~,

im=l

a=l

6,.i.6j.,~ b=a+l

V(.~')

(B.8)

/=1 I~o,b

The contribution arising f r o m p o ) can be estimated as follows:

[TrpP(3)l~llP'~'ll~ ~ m=3

~

Ip(il .....

im)[

il.....im=l

{~2~=o+, ~=,Zil[v"~, V"oqllllv('o-n,~ll (n,=, IIv'>'ll) + Y~ ~ IIv"°)- ~,o1111[7%,, v"~'] it( IIv~,' ,)} x

$~a,b,c

a=l

<~ ~ m~3

b=a+l

~ il .....

c=b+l

1=1 I ~a,b,c

IP(i, . . . . . im)l im=l

<~O~01qO~ ° ~ ( n ) ,,:M iMll[V, VIH=-p3,

(B.9)

cL (2.29), (2.33), (2.34). Note that the last line of eq. (B.9) is identical to (3.19). We now proceed to derive an inequality for the quantity X defined by (B.4). From (B.7) and (B.9) we have

iX I ~< iTr p ( V _ ~q). p(2,. ( V - n)l +p3 ~<2 1TrP ( V " ) i,i= 1

- n~)P~}'( V ° -

"/J)[+ P3.

(B.10)

454

L.W.J. DEN OUDEN, H.W. CAPEL AND J.H.H. PERK

The first term in the right-hand side of (B.10) can be estimated using the Schwarz inequality* ITr A B C [ <~(Tr A*A)~{Tr C*(B*B)C}~ <~(Tr A*A)~[[Bll(Tr C*C)~,

(B. 11)

A = p ~ ( V ('~-~,),

(B.12)

with B=PCj2',

C = ( V °~--0j)p ~.

We then have IXl ~< 2

[{Tr p( V ( ' ) - ~/,)2}~'llp/j2'l[{Trp( V " -

n,)~}~1+ p3,

(B. 13)

i,i= 1

where , O~(rl)

=-

P,7',

(B.14)

cf. (2.29), (2.33), (2.34) and (B.8). F r o m (B.13) and (B.14) we conclude

p 2 { ~ Tr p ( V ( ; ' - -0;y/

P~,

(B.15)

where p2 is the largest eigenvalue of the matrix p ~ . Hence, the lemma (3.17)-(3.19) has been proved. As an application of the lemma, we note that the theorem by Bogoliubov Jr., @ section 3.1, can be generalized to an N-particle system described by the hamiltonian Y(N = N { T N + P (V~)},

(B. 16)

where P ( V ) is an analytic and c o n c a v e operator function. H e r e T~, V~ are normalized hermitean operators satisfying (3.2). Then the free energy per particle, corresponding to (B.16), in the t h e r m o d y n a m i c limit is given by f -= tim f~ [ggN] = min lim f~ [Y(,r.~ (~)],

(B.17)

where the trial hamiltonian is defined by ~(,.~(~) = N { T N + P (~) + P ' ( ~ ) . (V~ - ~)},

(B. 18)

provided that the t h e r m o d y n a m i c limit in the right-hand side of (B.17) exists. This can be proved, following the line of reasoning in section 2 of ref. 1, keeping in mind that VN, ~ and M y are vectors in this case. The upper bound is a direct c o n s e q u e n c e of the Bogoliubov-Peierls inequality and the concavity of P ( V ) , i.e. fN [Y(N] - fN [ ~ . . ~ ) 1 ~< (P ( VN ) -- P (~) - P'({f) • (VN -- ~)),,~,~.~(,, ~< 0.

(B. 19)

*Equation (B. I1) can also be considered as a special case of the H61der inequality for operators, cf. e.g. ref. 45.

SYSTEMS WITH SEPARABLE

MANY-PARTICLE

I N T E R A C T I O N S . II

455

The lower bound can be found in a similar way as in ref. 1, noting that, after applying the lemma (3.17)-(3.19), eq. (2.15) of ref. 1 should be replaced by aN(v)~rnin

fN[~tr,,(~)-- Nv

. V N ] - - fN[Y(N -- N v

<- p.~((VN - (VN)~N-,,,v VN)2)~, N v . V , + p3.N,

" VN]

(B.20)

where v is an n-dimensional vector. In the absolute minimum in eq. (B.17), eq. (3.7) holds (provided that P(V)+eV 2 is concave for some e > 0 ) .

References I) 2) 3) 4) 5) 6) 7) 8)

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