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A note on Bose condensation
VoLume 37A, number 3
PHYSICS LETTERS
A NOTE
ON B O S E
22 November 1971
CONDENSATION
D. SHERRINGTON
Physics Department, Imperial College, London SW7, UK Received 11 October 1971
A recent paper of Kohn and Sherrington on Bose condensation in Fermi systems and its relation to superfLuidity is specialized to systems whose constituents may be considered essentially as true bosons. The notion of g e n e r a l i z e d Bose condensation in s y s t e m s of i n t e r a c t i n g f e r m i o n s was r e c e n t l y f o r m a l i z e d by Kohn and S h e r r i n g t o n [1]. It was indicated that two c l a s s e s of condensate a r e p o s s i b l e with quite d i f f e r e n t types of long r a n g e o r d e r . Only one of t h ese c l a s s e s g i v e s r i s e to superfluidity. Although the bas i c c o n s t i t u e n t s of all m a c r o s c o p i c physical s y s t e m s a r e f e r m i o n s it is often convenient to cons i d e r s y s t e m s in w h i c h the f e r m i o n s bind strongly t o g e t h e r into b o s o n - l i k e c o m p o s i t e s (of type I in the notation of ref. [1]), such as h e l iu m , as being s y s t e m s of i n t e r a c t i n g bosons. In this note we indicate how the concepts of [1] m a y be c o n s i d e r e d d i r e c t l y in the i n t e r a c t i n g boson model. J u s t as in the f e r m i o n c a s e we can c o n s i d e r two c l a s s e s of condensate defined by the conditions that the boson density m a t r i c e s in m o m e n t u m r e p r e s e n t a t i o n have t e r m s of the following c h a r a c t e r s
Type l:
A ( p ~ , p ~ , . . . pnl p(n)t p l ' p 2 ' " " " pn ) = :~N6(Pl+P2 + ' ' ' + p n - k )
G (rest ofp variables),
Type I1:
A(pl, p~,..,
)×
8(pl+p2+'''+pn-p'I-p½"''-pn
(1)
p~lp(n) IPl, P 2 , ' " "Pn ) = ),'N5 (pi+p~+... +P'n/2 -Pn/2+l - P:n/2+2"'" -Pn- k) ×
6 ( P l + P 2 + " " +Pn-P'I-P~."" -P~)
G' ( r e s t of p v a r i a b l e s ) ,
(2)
w h e r e N i s the n u m b e r of p a r t i c l e s in the s y s t e m , ~,).' a r e n u m b e r s of o r d e r unity ( c o m p a r e d with N " l ) and G , G ' a r e s m o o t h functions of the r e m a i n i n g independent m o m e n t u m v a r i a b l e s not w r i t t e n e x p l i c i t l y in the d e l t a functions and have unit t r a c e [1]. In both eqs. (1) and (2) it is the f i r s t delta function which i n d i c a t e s the type I or type I I c h a r a c t e r ; the second d e l t a function m e r e l y e x p r e s s e s o v e r a l l m o m e n t u m c o n s e r v a t i o n . The f a c t o r N i n d ic a t e s the m a c r o s c o p i c nature of the condensation. The d e g r e e of the condensation is given by the o r d e r n of the lowest density m a t r i x which has the above s t r u c t u r e : n may be even or odd for a type I condensate but must be even for type II. k is the wave v e c t o r of the condensate. J u s t as in the f e r m i o n c a s e , type I condensation l ead s to off diagonal long r an g e o r d e r in c o o r d i n a t e spa c e whilst type I I condensation leads only to diagonal (or n o r m a l ) long r an g e o r d e r *. F u r t h e r m o r e , the a n a l y s i s of ref. [1] (section 6) c o n c e r n i n g the p o s s i b i l i t y of superfluidity does not depend on whether the constituent p a r t i c l e s a r e f e r m i o n s or bosons and thus the c o n c l u s i o n s may be taken o v e r d i r e c t l y : a s y s t e m e x h i b i t s s u p e r f l u i d c h a r a c t e r (in the s e n s e defined in ref. [1]) if and only if one or m o r e of its de nsit y m a t r i c e s is s i n g u l a r in a m o m e n t u m sum v a r i a b l e ( p l + P 2 +. .. +pn+p~+p~+... +p~). In cons e q u e n c e it follows f r o m the d i s c u s s i o n of [1] that for bosons too only type I condensation leads to superfluidity. Note that the condensation (of type I) need not be e i t h e r h o m o g e n e o u s or single p a r t i c l e (n = 1) to get superfluidity. As e x a m p l e s of the above situations we may quote (i) liquid helium 4 which below Tc contains a type I condensate (believed to be of d e g r e e 1 and wave v e c t o r z e r o ) giving r i s e to s u p e r f l u i d i t y , (ii) solid h e l i u m 4 which by i t s v e r y c r y s t a l l i n e c h a r a c t e r has type I I c o n d e n s a t e s (of d e g r e e 2 and k equal to all * We might emphasize that the concepts of ODLRO or DLRO are weLLdefined in terms of the density distribution of physical particLes. It is meaningless to taLk about types of LRO in the densities which are not subject to experimental measurement, such as was suggested by Parmenter and Henson [4]. 223
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PHYSICS
LETTERS
22 November 1971
the reciprocal lattice vectors) and which may possibly simultaneously have a type I condensate leading to s u p e r f l u i d i t y e v e n in t h e s o l i d s t a t e [2,3]. T h e a u t h o r w o u l d l i k e to t h a n k P r o f . R. S t r o f f o l i n i f o r t h e h o s p i t a l i t y of the I s t i t u t o di F i s i c a T e o r i c a d e l l ' U n i v e r s i t ~ di N a p o l i , d u r i n g a v i s i t to w h i c h t h i s p a p e r w a s w r i t t e n .
References [1] [2] [3] [4]
224
W.Kohn, D. Sherrington, Revs. Mod. Phys. 42 (1970) 1 G. V. Chester, Phys. Rev. A2 (1970) 256. A.J. Leggett, Phys. Rev. L e t t e r s 25 (1970) 1543. R. H. P a r m e n t e r and W. R. Henson, Phys. Rev. B2 (1970) 140.
PHYSICS LETTERS
A NOTE
ON B O S E
22 November 1971
CONDENSATION
D. SHERRINGTON
Physics Department, Imperial College, London SW7, UK Received 11 October 1971
A recent paper of Kohn and Sherrington on Bose condensation in Fermi systems and its relation to superfLuidity is specialized to systems whose constituents may be considered essentially as true bosons. The notion of g e n e r a l i z e d Bose condensation in s y s t e m s of i n t e r a c t i n g f e r m i o n s was r e c e n t l y f o r m a l i z e d by Kohn and S h e r r i n g t o n [1]. It was indicated that two c l a s s e s of condensate a r e p o s s i b l e with quite d i f f e r e n t types of long r a n g e o r d e r . Only one of t h ese c l a s s e s g i v e s r i s e to superfluidity. Although the bas i c c o n s t i t u e n t s of all m a c r o s c o p i c physical s y s t e m s a r e f e r m i o n s it is often convenient to cons i d e r s y s t e m s in w h i c h the f e r m i o n s bind strongly t o g e t h e r into b o s o n - l i k e c o m p o s i t e s (of type I in the notation of ref. [1]), such as h e l iu m , as being s y s t e m s of i n t e r a c t i n g bosons. In this note we indicate how the concepts of [1] m a y be c o n s i d e r e d d i r e c t l y in the i n t e r a c t i n g boson model. J u s t as in the f e r m i o n c a s e we can c o n s i d e r two c l a s s e s of condensate defined by the conditions that the boson density m a t r i c e s in m o m e n t u m r e p r e s e n t a t i o n have t e r m s of the following c h a r a c t e r s
Type l:
A ( p ~ , p ~ , . . . pnl p(n)t p l ' p 2 ' " " " pn ) = :~N6(Pl+P2 + ' ' ' + p n - k )
G (rest ofp variables),
Type I1:
A(pl, p~,..,
)×
8(pl+p2+'''+pn-p'I-p½"''-pn
(1)
p~lp(n) IPl, P 2 , ' " "Pn ) = ),'N5 (pi+p~+... +P'n/2 -Pn/2+l - P:n/2+2"'" -Pn- k) ×
6 ( P l + P 2 + " " +Pn-P'I-P~."" -P~)
G' ( r e s t of p v a r i a b l e s ) ,
(2)
w h e r e N i s the n u m b e r of p a r t i c l e s in the s y s t e m , ~,).' a r e n u m b e r s of o r d e r unity ( c o m p a r e d with N " l ) and G , G ' a r e s m o o t h functions of the r e m a i n i n g independent m o m e n t u m v a r i a b l e s not w r i t t e n e x p l i c i t l y in the d e l t a functions and have unit t r a c e [1]. In both eqs. (1) and (2) it is the f i r s t delta function which i n d i c a t e s the type I or type I I c h a r a c t e r ; the second d e l t a function m e r e l y e x p r e s s e s o v e r a l l m o m e n t u m c o n s e r v a t i o n . The f a c t o r N i n d ic a t e s the m a c r o s c o p i c nature of the condensation. The d e g r e e of the condensation is given by the o r d e r n of the lowest density m a t r i x which has the above s t r u c t u r e : n may be even or odd for a type I condensate but must be even for type II. k is the wave v e c t o r of the condensate. J u s t as in the f e r m i o n c a s e , type I condensation l ead s to off diagonal long r an g e o r d e r in c o o r d i n a t e spa c e whilst type I I condensation leads only to diagonal (or n o r m a l ) long r an g e o r d e r *. F u r t h e r m o r e , the a n a l y s i s of ref. [1] (section 6) c o n c e r n i n g the p o s s i b i l i t y of superfluidity does not depend on whether the constituent p a r t i c l e s a r e f e r m i o n s or bosons and thus the c o n c l u s i o n s may be taken o v e r d i r e c t l y : a s y s t e m e x h i b i t s s u p e r f l u i d c h a r a c t e r (in the s e n s e defined in ref. [1]) if and only if one or m o r e of its de nsit y m a t r i c e s is s i n g u l a r in a m o m e n t u m sum v a r i a b l e ( p l + P 2 +. .. +pn+p~+p~+... +p~). In cons e q u e n c e it follows f r o m the d i s c u s s i o n of [1] that for bosons too only type I condensation leads to superfluidity. Note that the condensation (of type I) need not be e i t h e r h o m o g e n e o u s or single p a r t i c l e (n = 1) to get superfluidity. As e x a m p l e s of the above situations we may quote (i) liquid helium 4 which below Tc contains a type I condensate (believed to be of d e g r e e 1 and wave v e c t o r z e r o ) giving r i s e to s u p e r f l u i d i t y , (ii) solid h e l i u m 4 which by i t s v e r y c r y s t a l l i n e c h a r a c t e r has type I I c o n d e n s a t e s (of d e g r e e 2 and k equal to all * We might emphasize that the concepts of ODLRO or DLRO are weLLdefined in terms of the density distribution of physical particLes. It is meaningless to taLk about types of LRO in the densities which are not subject to experimental measurement, such as was suggested by Parmenter and Henson [4]. 223
Volume 37A, n u m b e r 3
PHYSICS
LETTERS
22 November 1971
the reciprocal lattice vectors) and which may possibly simultaneously have a type I condensate leading to s u p e r f l u i d i t y e v e n in t h e s o l i d s t a t e [2,3]. T h e a u t h o r w o u l d l i k e to t h a n k P r o f . R. S t r o f f o l i n i f o r t h e h o s p i t a l i t y of the I s t i t u t o di F i s i c a T e o r i c a d e l l ' U n i v e r s i t ~ di N a p o l i , d u r i n g a v i s i t to w h i c h t h i s p a p e r w a s w r i t t e n .
References [1] [2] [3] [4]
224
W.Kohn, D. Sherrington, Revs. Mod. Phys. 42 (1970) 1 G. V. Chester, Phys. Rev. A2 (1970) 256. A.J. Leggett, Phys. Rev. L e t t e r s 25 (1970) 1543. R. H. P a r m e n t e r and W. R. Henson, Phys. Rev. B2 (1970) 140.