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Journal o f Statistical Physics, Vo l. 65, Nos. 1 /2, 1991

U p p e r B o u n d o n t h e C o n d e n s a t e i n t h e

H a r d - C o r e B o s e L a t t i c e G a s

B f i l i n t T 6 t h I'2

Received April 14, 1991," inal May 6, 1991

Using methods developed by G. Roepstorf, we prove an upper bound for the

amount of condensate in a hard-core Bose lattice gas.

KEY WOR DS : Bose-Einste in condensation; quantum correlation inequalities.

1 , I N T R O D U C T I O N

R o e p s t o r f f (4) p r o v e d t h e f o l l o w i n g i n e q u a l i t y :

< A ' A > ~ > < [ A , A * ] > { e x p ~ < [ c * ' i < [ ~ ," C ] ] > < [ ~ ,* ] > C , ] > i - l } ' (1.~

w h e r e < . - . ) s t a n d s f o r t h e r m a l e x p e c t at io n :

T r ( e e l l A )

< A > T r ( e - ~ H ) ( l .2 )

a n d A a n d C a r e a n y t w o b o u n d e d o p e r a t o r s . I n a s e c o n d p a p e r , (5) t h e

s a m e a u t h o r u s e d t h i s i n e q u a l i t y t o p r o v e a n u p p e r b o u n d o n t h e a m o u n t

o f c o n d e n s a t e f o r a B o s e g a s i n R d, d / > 3 , w i t h a r b i t r a r y p a i r i n t e r a c t io n .

E x p l o i t i n g t h e s a m e m e t h o d s , w e g i v e a n u p p e r b o u n d f o r t h e a m o u n t

o f c o n d e n s a t e f o r a hard-core Bose l a t t i c e ga s . A s , due t o t he ha rd -core

c o n d i t i o n , t h e c o m m u n i c a t i o n r e l a ti o n s o f t h e e m e r g i n g c re a t i o n a n d

a n n i h i l a t i o n o p e r a t o r s a r e d i f fe r en t f r o m t h e s t a n d a r d b o s o n i c o n e s , th i s

1Department of Mathematics, Heriot-Vfatt University, Edinburgh, Riccarton, EH144AS,

Scotland.

2 On leave from the Mathematical Institute of the Hungarian Academy of Sciences, Budapest,

H-1053 Hungary.

373

00224715/91/10o0-0373506.50/0 9 1991 Plenum PublishingCorporation





C o n d e n s a t e i n H a r d - C o r e B o s e L a t t i c e G a s 3 7 5

w h e r e x (p , f l ) is th e u n i q u e s o l u t i o n o f th e e q u a t i o n

a n d

c o t h / ~ ( 1 / 2 - p ) r ai n{ p , 1 - p } D ( p )_ ~ , ~ ] d X

d p

(2.6)

d

D (p ) = ~ (1 - c o s P i ) (2.7 )i = l

R e m a r k s . ( 1 ) A s i n o n e a n d t w o d i m e n s i o n s , E q . ( 2 . 6 ) h a s n o

p o s i t iv e s o l u t i o n ; t h e t h e o r e m o f M e r m i n a n d W a g n e r is re c o v e r e d : in le ss

t h a n t h r e e d i m e n s i o n s t h e r e i s n o B o s e - E i n s t e i n c o n d e n s a t i o n i n t h e s e

m o d e l s .

( 2 ) F o r t h e s a k e o f c o m p a r i s o n : th e u p p e r b o u n d p r o v e d in re f. 5 ,

app l i ed t o t he l a t t i c e ga s c a se , w ou ld re su l t i n a s i i l a r p ropos i t i on , w i th

x (p , f i ) b e i n g t h e u n i q u e s o l u t i o n o f t h e e q u a t i o n

f 1 - 1exp t ip D (p ) I dp (2 .8 )

X = P - - ( -~ ) ~) d . E _ ~ , ~ ] a X

R o e p s t o r f f 's u p p e r b o u n d is c o m p l e t e ly i n d e p e n d e n t o f th e i n t e ra c t io n . I n

o u r d e r i v a t i o n th e h a r d c o r e is ta k e n i n t o a c c o u n t , h e n c e th e i m p r o v e m e n t .

( 3) A p a r t f r o m b e i n g s o m e w h a t s t ro n g e r , o u r u p p e r b o u n d h a s t h e

a d v a n t a g e o f s h o w i n g t h e s a m e s y m m e t r y as t h e c o n d e n s a t e i ts el f:

P c ( P , fl ) = pc(1 - p, f l ) an d x ( p , fl) = x(1 - p, fi) (2.9)

Proo f . W e a p p l y R o e p s t o r f f ' s in e q u a l i t y w i t h t h e f o l l o w i n g c h o ic e s o ft h e o p e r a t o r s :

H ~ = H o - - ~ ~ ' , [ a + ( x ) + a ( x ) ]x E A

( 2 . 1 o )

w i t h p e r i o d i c b o u n d a r y c o n d i t i o n s o n t h e d i s c r e t e t o r u s A . N o t e t h a t a

s y m m e t r y - b r e a k i n g te r m i s a d d e d t o t h e H a m i l t o n i a n (2 .3 ). ( - - - ) ~ , ~ w i ll

d e n o t e t h e r m a l e x p e c t a t i o n w i t h r e s p e c t t o t h i s H a m i l t o n i a n . W e h a v e

A = c i ( p ) = }-" ei"Xa(x),X G A

A * = c i + ( - p ) = ~ e-ipXa+(x)x ~ A

(2.11)

822/65/1-2-25



3 7 6 T b t h

c = , ~ ( p) = Y ' ~ ' ~ ~ n ( x ) ,

x~A (2.12)

C * = h ( - p ) = ~ ' e - i p x n ( x )x E A

The commuta to r s appear ing in (1 .1 ) wi l l be

[A , A * ] = [~ (p ), f i + ( - p ) ] = [ A I - 2 ~ n(x) (2.13)x E A

[ A , C * ] = [ fi( p) , h ( - p ) ] = ~ a(x) (2.14)x ~ A

[ c , [ H , C * ] ] = [ ~ ( p ) , [ H ~ , ~ ( - p ) ] ]

= Y ~ I ~ ] ( 1 - c o s p . 3 ) a+ ( x) a (x + 3 )x~A t.6~A,161=l

+ g [ a § (x ) + a ( x ) ] (2 .15)

A n d t h e co r r e sp o n d i n g ex p ec t a t i o n s a r e

< [ A , A * ] ) = < [ d ( p ) , a + ( - - p ) ] ) A , ~ = I A [ [ 1 - - 2 < n ( O ) ) A , A (2.16)

< [ A , C * ] > = < [ a (p ) , h ( - p ) ] >A,~= I11 <a(0)>A ,~ (2.17)

< [ c , [ H , C * ] ] > = < [~ (p ), [H ~ , ~ ( - p ) ] ] > ~,o

= I A I { ~

( 1 - - c o s p '3 ) < a + ( O ) a ( 6 ) ) A ,,+ ~ < a (O ) ) a , ,}OEA , 16l = 1

( 2 . 1 8 )

We u se t h e i n eq u a l i t y

{ a + ( 0 ) a(6 ))A,~ <~m i n { { a + ( 0 ) a ( O ) ) A , ~ , {a (O)a+(0) )a , . }

= m in{{n (0) )A .~ , 1 - - (n (0 ) )A.~} (2 .19)

to ge t

< [ C , [ H , C * ] ] ) ~ < IA I ( D ( p ) m i n {< n ( 0 ) ) A , ~ , 1 - - < n ( 0 ) > A , e } +e<a(O)>A,e)

(2.20)



C o n d e n s a t e in H a r d - C o r e B o s e L a t t ic e G a s 3 7 7

Pluging a l l these in (1 .1) , we ge t

IAI - ' ( a + ( - p) gt(p) )A,~

> ~(1 -- 2 ( n ( 0 ) ) a . ~ )

{ ( f l ( D ( p ) m i n { ( n ( O ) ) A , * , l- - ( n ( O ) ) A ,~ } ~ } ~\ + e ( a ( O ) ) a , ~ ) ( 1 - - 2 ( n ( O ) )a , ~ ) ]

x ex p ( a ( 0 ) ) 2 ~ 1 (2.21)

A n d n o w t a k i n g t h e l i m i t s A .* Z d, e N 0 a n d i n t e g r a t i n g o v e r t h e c u b e

I - r e , r e i d ( i n t h i s o r d e r ! ) l e a d s u s t o t h e i n e q u a l i t y

1 { e x p f l D ( p ) m i n { p , l _ p } ( l _ 2 p ) } - 1P - f i c > ~ ( 1 - 2 p ) ~ f E . . . 3d , 2 - 1 d p

W h e r e

(2.22)

t / = l i m l i m ( a ( O ) ) A , , ( 2 . 2 3 )t z N O A ~ Z d

1f i c = l i m l i m d T , 2 ~ , ( a + ( x ) a ( y ) ) A , ~ (2.24)

e ~ , O A / * Z I Z I ] x , y ~ A

[ -A l t e rna t ive ly , one can ge t (2 . 22) by summing ove r p v 0 i n (2 . 21) and then

t a k i n g t h e l i m i t A 7 z d . ] U s i n g t h e i n e q u a l i t i e s

Pc ~< I/2 ~< tSc (2 .2 5)

( the f i r st one o f w h ich i s p rov ed in t he l a s t s ec t i on o f r ef . 5 ; t he se cond one

is t r iv ia l ) we a rr ive a t

P c < ~ ( p , f l ) (2.26)

a s g iven in (2 . 6 ) . A s men t ioned in re f . 3 , t he bound

p c < ~ p ( 1 p ) (2.27)

e a s il y f o l lo w s f r o m t h e m e t h o d o f r ef . 2 , a p p l i e d t o t h e h a r d - c o r e B o s e

l a t t i c e g a s c a s e . W e d o n o t r e p e a t h e r e t h a t a r g u m e n t .

A C K N O W L E D G M E N T

T h i s w o r k w a s s u p p o r t e d b y t h e B r i t i s h S c i e n c e a n d E n g i n e e r i n g

R e s e a r c h C o u n c i l .



3 7 8 T b t h

R E F E R E N C E S

1 . O . Pen rose and L . Onsage r , Bose -Eins te in condensa t ion and l iqu id he l ium, P h y s . R e v .

104:576--584 (1956).

2 . O . Penrose , T w o inequal i t ies for c lass ica l and qu ant um systems of par t ic les with hard core ,

P h y s . L e t t . 11:224-226 (1964).

3 . O. P enrose , B ose-E inste in co nden sat ion in an exact ly so luble sys tem of in terac t ing

particles, J . S t a t . P h y s . (1991).

4 . G. Roepstorff, Co rre la t ion inequal i t ies in qu ant um s ta t is t ica l mec hanics and the ir appl ica t ion

i n t h e K o n d o p r o b l e m , C o m m u n . M a t h . P h y s . 46:253-262 (1976).

5 . G. Roepstorff , Bo unds for a Bose condensate in d imen sions v >~ 3 , J . S t a t . Ph ys . 18:191 206

(1978).

Communicated by D. Szdsz