COOPER PAIRING IN SPIN-POLARIZED FERMI SYSTEMS

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COOPER PAIRING IN SPIN-POLARIZED FERMI SYSTEMS

A. Leggett

To cite this version:

A. Leggett. COOPER PAIRING IN SPIN-POLARIZED FERMI SYSTEMS. Journal de Physique

Colloques, 1980, 41 (C7), pp.C7-19-C7-26. �10.1051/jphyscol:1980704�. �jpa-00220141�

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JOURNAL DE PHYSIQUE CoZZoque C7, suppZe'ment au n o 7 , Tome 42, juiZlet 1980, page C 7 - 1 9

COOPER PAIRING IN SPIN-POLARIZED FERMI SYSTEMS

A . J . Leggett

SchooZ o f Mathematical, and Physical Sciences, University of Sussex, Brighton, Susssex, BNZ 9QH, Grande Bretagne

Resume.- On discute la formation de paires de Cooper dans les syst&mes de Fermi Z spin polaris6, tels que 3 ~ e + et D + , en insistant particuligrement sur les questions suivant.es

:

(1) Quellessontles conditions de formation des paires de Cooper, etsquelles temperatures cette formation est-elle probable pour les systPmes envisag6s

?

(2)

Quelle est la relation entre les paires de Cooper et des mol6culesdiatomiques?

(3) Quels sont les ph6nomPnes qualitativement nouveaux que l'on attend dans un systgme

2

paires de Cooper, et qu'apparaltra-t-il probablement de nouveau si la formation de paires a lieu dans un systsme

3

spin polaris6

?

Abstract.- I discuss the phenomenon of Cooper pairing in strongly spin-polarized Fermi systems, such as 3 ~ e + and D + , with particular attention to the questions

(1) what are the conditions for Cooper pairing to occur, and at what temperatures is this likely to happen for the systems of practical interest

? (2)

what is the relationship between Cooper pairs and diatomic molecules

? (3)

what are the quali- tatively new phenomena we expect in a Cooper-paired system, and what new physics is likely to emerge if the phenomena occurs in spin-polarized systems

?

In this talk I shall discuss, informally and resulting system is called a "Fermi superfluid".

without detailed derivation, the quest ions

:

What The conditions for its occurrence are: a fairly high is Cooper pairing and under what conditions do we degree of degeneracy, a (weakly) attractive inter- expect it to occur in spin-polarized (and some 0th- action between the fermions, and the absence of too er) quantum systems? What are the similarities and uuch incoherent scattering. It is not necessary differences between Cooper pairs and diatonic mle- that the paired fennions be identical, or evcn that cules? What are the consequences of Cooper pair- they have the same mass, but they mst have at ing and what can we use it for? It should be least approximately the same

?c:nni

mnent~un.

_An

qhasized at the start that the experimental rele- important consequence of this is that pa~.rine;

of

vance of the phenomenon to spin-polarized system, fennions with opposite spin is sur~pressed

by

even a particularly the hydrogen isotopes, is extremely fairly weak spin polarization, since this will inc- sensitive to the

m i m u m density at which they can

rease the up-spin Fermi surface at the expense of

be

stabilized, which at the time of writing is an the down-spin one.

unknown quantity. Taking for the moment a naive view of the

Cooper pairing is, in the crudest tenns, a Cooper pairs

as

simply giant diatomic mlecules, phenomenon which occurs in degenerate Fenni system one would expect them to be described by some "mle- and which involves the fomtion by t m fermions of cu1ar"wave function of the type

a sort of giant diatomic molecules ("Cooper pairs") *(L~-x~, p12p2) which automatically under@ Base andensation. It

where p indicates the spin of a femYon and xl-x2

generally leads

the

complex

Of

phenomena which the relative separation of the two fermions. (Xere go under the generic ~~ of aperfluidit~, and the

md in the subsequent discussion we assume for

s-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980704

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JOURNAL DE PHYSIQUE

p l i c i t y t h a t t h e c e n t r e of mass of t h e p a i r s is a t r e s t ) . I n accordance with t h e F e m i s t a t i s t i c s , t h e wave function should be a n t i s y m t r i c under exchange of p a r t i c l e s 1 and 2. Moreover, f o r a l - mst a l l t h e systems which a r e of i n t e r e s t i n t h e present context t h e t o t a l s p i n of a p a i r is l i k e l y t o be conserved t o a very good approximation. Thus, i n t h e case of two fermions of s p i n

4,

one has t h e p o s s i b i l i t i e s :

a ) S = O , % = e v e n b ) S = l , % = o d d

where % is t h e r e l a t i v e o r b i t a l angular nxrmentum.

I t t u r n s out t h a t f o r % = 0 (s-wave p a i r i n g ) t h e p r o p e r t i e s of t h e Cooper p a i r s , and hence of t h e whole system, a r e i s o t r o p i c , whereas f o r %

+

^{0 (with}

one exception which is not relevant i n t h e present context) t h e p r o p e r t i e s a r e a n i s o t r o p i c and one speaks of an "anisotropic superf luid"

.

One f i n d s t h a t i n d i l u t e unpolarized systems t h e energetics wi1.l always favour R = 0 p a i r i n g ( j u s t as t h e groundstate of a diatomic molecule i n a 1-state always has angular rnxnentum zero). On t h e o t h e r hand, f o r a spin-polarized system, even a d i l u t e one, R = 0 p a i r i n g (which is associated, as above, with t o t a l s p i n zero) is suppressed and gen- e r a l l y speaking t h e p a i r s w i l l form i n an R = 1 s t a t e , thereby giving rise t o a n i s o t r o p i c s u p e r f l u i - d i t y .

In Table 1 , I review some a c t u a l and possible (laboratory) Fenni superfluids. The last column i n d i c a t e s t h e c r i t i c a l temperature at which one n i @ t expect t h e onset of Cooper pairing. Under t h e heading of "actual" s u p e r f l u i d s one might perhaps a l s o include t h e Bose-condensed e x c i t a t i o n s reported by A.Mysyrowicz at t h i s Conference, since a f t e r all an exciton is nothing but a bound s t a t e of tm fennions ( e l e c t r o n p l u s hole). / I / .

ltYo aspects of t h e numbers i n Table 1 deserve c o m n t . F i r s t , it should be emphasized t h a t t h e c a l c u l a t i o n of t h e c r i t i c a l temperature of a Femi

- - - - ~

Table 1

Actual and possible F e m i s u p e r f l u i d s

System S*

-

R Tc( OK)

( a ) A c t u a l

e l e c t r o n s i n 0 0 s 20

superconductors

( b ) Possible

3He i n 'He 0 0

-

IO-~-I.O-~ ?

3 ~ e + i n ' ~ e 0 0 ?

3 ~ e + 1 1(3?) l 0 - ~ - 1 0 - ~ ?

D + ~ 2 1

r

D+3 1 0

*

In a l l case5 except t h e f i r s t , S is t h e nut- l e a r spin. In t h e case of deuterium t h e r e is a l s o

-

an e l e c t r o n i c s p i n contribution, which i n t h e "spin- polarized" s t a t e is by d e f i n i t i o n always 1.

s u p e r f l u i d is i n general an extremely t r i c k y busi- ness, s i n c e it depends exponentially on p a r m t e r s such as t h e e f f e c t i v e i n t e r a c t i o n a t t h e Fenni surface which themselves a r e o f t e n not well known.

Hence one should t r e a t t h e numbers quoted f o r 3He+, and t o a lesser extent f o r 3 ~ e i n ' ~ e ( a t t h e maxi- m concentration,

-

1%) with considerable caution.

However, an exception t o t h e general r u l e is t h e case of a very d i l u t e gas where t h e two-particle s- wave s c a t t e r i n g length as is known; i n t h i s case t h e critical temperature f o r s-wave p a i r i n g should be given t o a very p d approximation, by t h e form- u l a

T c 1 . 6 T F e x p - . n & k F l a s 1 ) ( a s < O ) ( 1 ) where TF is t h e Fenni temperature and kF t h e Fermi nWnentum.

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For a d i l u t e system with p-wave p a i r i n g , t h e f a c t o r kFla

I

i n t h e exponent is replaced by a fac-

S

t o r of o r d e r q l b l , where b is a q u a n t i t y with t h e dimensions of volume which is t h e p-wave analogue of t h e s c a t t e r i n g length. Because of t h e much sharper dependence of t h e exponent on density i n t h e p-wave case, p-wave p a i r i n g is l i k e l y t o o m i n d i l u t e s y s t e m , i f a t a l l , only a t p r e s e n t l y m a t - t a i n a b l e temperatures.

The second comnent concerns spin-polarized deuterium. This is c l e a r l y a special case i n t h e context of t h e above discussion, i n t h a t t h e deuterium atom, though a fermion, has nuclear s p i n 1.

This i n v a l i d a t e s t h e considerations given above f o r fennions o f s p i n

4.

Assuming t h a t t h e e l e c t r o n i c s p i n s are c ~ n p l e t e l y polarized, we can consider two main cases: ( a ) D+l, i n which only t h e lowest nuc- l e a r Zeeman s t a t e is appreciably populated. I n t h i s case t h e (nuclear) s p i n of t h e Cooper p a i r is 2 ( t h e t o t a l s p i n is 3!) and, bearing i n mind t h e Fenni s t a t i s t i c s , we see t h a t t h e o r b i t a l angular mcmentum must be odd. For a d i l u t e system t h e e n e r g e t i c a l l y favoured p a i r i n g s t a t e is a p s t a t e , but it is l i k e l y t h a t t h i s w i l l cccur only a t m a t - t a i n a b l y low temperatures f o r t h e reason given above.

( b ) Df3, i n which a l l t h r e e nuclear Zeeman states a r e (nearly) equally populated. Depending on t h e density and f i e l d t h i s may be t h e equilibrium s t a t e o r possibly a long-lived metastable s t a t e . I n t h i s case t h e favoured p a i r i n g is with R = 0 but nuclear s p i n 1

-

a unique case. Since t h e system is l i k e - l y t o be very d i l u t e , we can use t h e f o m l a (1) and s u b s t i t u t e t h e experimental value of t h e

316

s c a t t e r i n g length f o r deuterium,

-

3.7

a.

^This

gives a p p m x i m t e l y

Tc

-

^lOOn

213

exp

-

1/(Gn

'/3

) ( 2 ) where t h e number density n is measured i n 1-3. ( I t

is necessary, here, t o remember t h a t t h e r e l a t i o n

between kF and n is modified from t h e f a m i l i a r one because of t h e t r i p l e spin degeneracy). Thus, f o r example, f o r n = 10'' t h e c r i t i c a l tenqxra- ture is mobsemably low ( & 1 0 - * 0 ~ ) but f o r n = 1021 ,-3 ~t . would be already of t h e order of 1'~.

Let me now t u r n t o t h e second t o p i c of t h i s t a l k : i n what ways are Cooper p a i r s l i k e and un- l i k e diatomic m l e c u l e s which have s u f f e r e d Bose condensation? There are a number of obvious quali- t a t i v e differences: i n a l l known F e m i s u p e r f l u i d s , t h e p a i r "radius" is very m c h l a r g e r than t h e mean spacing between p a r t i c l e s , whereas t h e naive concept of a diatomic molecule would s e e m t o imply t h e opposite assumption; t h e standard BCS theory /2/ of Cooper p a i r i n g invokes heavily t h e degeneracy of t h e Fermi s e a , whereas f o r diatomic mlecules t h i s plays no r o l e ; and, i n t h e a n i s o t r o p i c case, t h e e x c i t a t i o n spectrum o f t h e paired system is gene- r a l l y a n i s o t r o p i c 131, whereas f o r a diatomic m l e - cule it is i s o t r o p i c whatever t h e angular mxnentum state. Nevertheless I believe t h a t t h e r e m y be some sense i n which it is l e g i t i m a t e t o view diatomic molecules and Cooper p a i r s as t h e tm ends of a continuous spectrum of possible behaviour of a Fermi system with a t t r a c t i v e i n t e r a c t i o n s . To i n v e s t i g a t e t h i s p o i n t , let u s consider t h e follow- i n g m d e l system ( f o r a mre d e t a i l e d account of t h i s model and t h e c a l c u l a t i o n s based on it, see r e f . 141.) W e imagine a system of N fermions of s p i n contained in u n i t volume, and with ( f o r t h e mment) no spiri p o l a r i z a t i o n and no e x t e r n a l mag- n e t i c f i e l d . The p o t e n t i a l between t h e fermio~ls has a core which is f a i r l y strongly repulsive, p l u s a weakly a t t r a c t i v e t a i l which e f f e c t i v e l y c u t s o f f a t some c h a r a c t e r i s t i c radius ro (which might i n p r a c t i c e be a few 1 ) ; t h e o v e r a l l p o t e n t i a l is repulsive ( i . e.

/v(

r)@ >. 0 ) . However, t h e d e t a i l s of t h e p o t e n t i a l aresuch t h a t it is e i t h e r j u s t enough, o r not q u i t e enough, t o bind two p a r t i c l e s

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C7-22 JOURNAL DE P H Y S I Q U E

i n f r e e space i n t o a diatomic molecule. I n e i t h e r case t h e s-wave s c a t t e r i n g length as is very much larger than t h e c h a r a c t e r i s t i c range ro of t h e p o t e n t i a l . (as > 0 f o r a bound s t a t e , < 0 i f t h e s t a t e is not q u i t e bound). I f t h e two-particle state is bound, its energy is given approxinately

2 2

by E =

-

fi /mas. Wenow assume t h a t N is such t h a t t h e m a n spacing between p a r t i c l e s , say R, is l a r g e compared t o ro; however, we W e no assumptions about t h e r a t i o R/as and moreover imagine t h a t by varying t h e d e t a i l s of t h e p o t e n t i a l we can vary t h i s quantity continuously from p o s i t i v e t o negative values. The v i r t u e of t h i s model is t h a t , when s u i t a b l y scaled, mst of t h e p r o p e r t i e s of t h e system should be i n s e n s i t i v e t o t h e d e t a i l s of t h e p o t e n t i a l and functions only of t h e s i n g l e dimen- s i o n l e s s v a r i a b l e R/as.

We can now write down an ansatz /I/ f o r t h e wave function of t h e N-body system which reduces t o t h e description of a set of noninteracting, Bose- condensed diatomic molecules i n t h e l i m i t &/as + + m

and t o t h a t of a Cooper-paired system i n t h e opposite l i m i t R/a ⁺

-

. I t is t h e following:

S

$ ( r l u l , r20 2 . . . . r N o N ) =

A'F('101r202) Y (r3u3r44a4)

- - . . .

( 3 ) 'f (rN-loN-lrNON)

where A is an a n t i s y m z t r i z a t i o n operator. Whether o r not t h e wave function (3) is a reasonable approximation t o t h e t r u e ground-state wave function of t h e system, it is of saw i n t e r e s t t o study how t h e t r a n s i t i o n between t h e two l i m i t s takes place.

Using o u r knowledge o f t h e correct form i n t h e s e l i m i t s , we a s s m t h a t t h e "moleculart1 wave function

is of t h e form

ir:(r1x201u2)= ~ i . ( + + - + + ) y , ( l r ~ - r ~ l ) 1 ( 4 ) i n an obvious notation, i . e . it corresponds t o a spin s i n g l e t , R = 0 state, with t h e c e n t r e o f mass a t rest.

To do any useful c a l c u l a t i o n s with t h e wave function (3) it is necessary t o use t h e standard BCT, t r i c k of relaxing p a r t i c l e n-r conservation and minimizing, instead of H, t h e q u a n t i t y H

-

iJN where IJ, is t h e chemical p o t e n t i a l . (For t h e subsequent s t e p s , see e.g. r e f . /5/). I f we then i n t r o - duce t h e Fourier transform, c j , of t h e function and define t h e complex q u a n t i t i e s

%,

vk such t h a t

then it t u r n s out t h a t t h e function ( 3 ) is just t h e N-particle projection of t h e particle-nonconserving BCS-type function

where t h e function

6k

is a s t a t e vector i n t h e "occ- upation" space associated with t h e p a i r of plane- wave s t a t e s (kL

,-&+).

This space is four-dimen- s i o n a l and is spanned by t h e b a s i s v e c t o r s 10,0>, 11,1>, 10,1> and 11,0>, where f o r example 11,0>

l a b e l s t h e s t a t e i n which tile plane-wave s t a t e ( k + ) is occupied and t h e s t a t e (-kJ) is empty. The l i n e a r combination

$

(eqn. ( 6 ) ) is t h e groundstate within t h i s space; t h e e x c i t e d s t a t e s are t h e two

"broken-pair" states 11,0> and

1

0,1> and t h e "exci- ted-pair" state v* k 10,0>

-

^L$ 11,1>.

A many-body wave function of t h e form (3) (with Qgiven by t h e s i n g l e t , s-wave form ( 4 ) ) is completely parametrized by t h e set of q u a n t i t i e s

It is t h e Fourier transform of t h i s q u a n t i t y , F ( r ) , ( r a t h e r than

(g))

which p l a y s t h e role o f a

wave

function f o r t h e r e l a t i v e motion of t h e Cooper p a i r s . Indeed t h e expectation value of any two-particle operator of t h e general form A =

4 1

A(ri

-

r .)

i j -J

(e.g. t h e p o t e n t i a l energy) is given, a p a r t from Hartree-Fock-type t e r n which are of no g r e a t inte-

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rest i n t h e present context, by t h e expression

which m y be compared with t h e corresponding expres- s i o n f o r an i s o l a t e d diatomic m l e c u l e

(F(r)++(r)).

When F(x) has its e q u i l i b r i m value, t h e ener- g i e s of t h e e x c i t e d states of t h e p a i r

(&+ -&+)

are

,

given by Ek (broken p a i r ) and 2Ek ( e x c i t e d p a i r ) where Ek and t h e associated q u a n t i t y Ak are impli- c i t l y defined by t h e equations

F~ =

ak/mk

(8a)

2 2

i

Ek = ( ( E ~ - u ) ⁺

lak!

) (8b)

A l l t h e above statements are q u i t e generally t r u e once we assume t h e ansatz ( 3 ) f o r t h e many-body wave function, i r r e s p e c t i v e of whether o r not we a r e i n t h e usual "Cooper-pairw l i m i t . However, it should b e strongly enphasized t h a t once we are outside t h i s l i m i t we can no longer

assume

t h a t t h e chemi- c a l p o t e n t i a l p is simply t h e f r e e Fenni energy ^{E ~ ;} it must, i n f a c t , be determined s e l f - c o n s i s t e n t l y from t h e equation

1

nk = N, using t h e f a c t t h a t t h e

k

number of p a r t i c l e s nk i n t h e plane-wave state

&

(with e i t h e r s p i n ) is given ( i n t h c groundstate (6)) by

Now f o r t h e equation determining ??(r);we f i r s t note f o r o r i e n t a t i o n t'hat t h e f a m i l i a r Zchrijdinger equation f o r a diatomic fimlecule can be w r i t t e n a f t e r lburier transformation i n t h e f o m

( 2 - Eo) ~ ~$Ik +

1

V(k-k') qk, = 0 (10) k'

I t t u r n s out t h a t t h e q u a n t i t y Fk obeys t h e equa- t i o n ( t h e f a m i l i a r BCS gap equation l i g h t l y disgui- sed)

2EkFk +

1

V(k

-

k l ) F k , = 0

k' ( l l a )

This is s nonlinear equation because, by ( 3 ) and ( l l a ) , Ak depends on Fk i t s e l f by t h e r e l a t i o n

A, = -

1

^V(k

-

k ' )Fk, (12)

k'

Now, it is f a i r l y obvious by inspection of e q n s . ( l l ) and (9) and t h e u s e of some simple renor- m l i z a t i o n tricks /4/ t h a t both t h e chemical poten- t i a l p and t h e q u a n t i t y Ak a r e a t most of o r d e r of lnsgnitude of t h e f r e e Fermi energy

6Y/h

o r t h e q u a n t i t y h2/2ma2. I f we now consider values of k of t h e order of

ri

1

^,

then by our i n i t i a l hypothesis (kF ro

-

^r0/t^<<1 , ro/as << 1 ) we f i n d t h a t Ek is very nuch l a r g e r than e i t h e r of t h e s e two energies and is hence l a r g e compared t o p and

1 ak 1 ,

and a l s o t o Eo

-

^h2/2ma:. Under these conditions t h e EG3 gap equation ( l l a ) simply reduces t o t h e schr8dinger equation ( l o ) , s o w e f i n d t h e h g o r t a n t r e s u l t t h a t t h e short-range behaviour of t h e p a i r wave function

1

t i o n o f an i s o l a t e d diatomic molecule. This resclt is probably q u a l i t a t i v e l y v a l i d f o r cases more gen- e r a l than t h e simple &el considered here ( c f . r e f . / 3 / ) .

Let u s now consider t h e s o l u t i o n of equations ( l l a ) and (9) i n t h e t m l i m i t i n g cases. QLute generally it t u r n s o u t t h a t f o r k << ro t h e q u a n t i t y

41

tends t o a constant, A. I n t h e case R/as ⁺+ (two-particle state bound, very d i l u t e system) we f i n d t h a t A ⁺0 , p +

-

h 2 / 2 m a ~ (h a l f t h e binding energy of t h e molecule) and t h e l?€S equation reduces t o t h e SchriSdinger equation f o r &l

5.

Thus i n t h i s l i m i t o u r wave function simply describes a Bose condensation o f noninteracting d i a t h c molecules, a s indeed we should expect a p r i o r i . I n t h i s l i m i t t h e p a i r radius (which is j u s t as) is by hypothesis much less than t h e i n t e r p a r t i c l e spacing.

I n t h e opposite l i m i t , &/as ⁺-- m (dibte s y s t a n with very weak a t t r a c t i o n ) we obtain t h e standard

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C 7 - 2 4 JOURNAL DE PHYSIQUE

BCS r e s u l t s : p tends t o t h e f r e e Fenni energy cF

-

n2$/2m, while A b e m s exponentially small:

A = const. cF exp

- IT/@

kFlasl) (13) We f i n d t h a t i n t h i s l i m i t t h e p a i r "radius" is of o r d e r +ivF/A (vF = Fenni v e l o c i t y ) and hence is very much g r e a t e r than t h e i n t e r p a r t i c l e spacing.

W e can, of course, solve e q n s . ( l l a ) and (9) f o r q u i t e general values of R/as and so s t ~ i d y t h e trans- i t i o n between t h e

two

l i m i t s . Is t h e r e any point a t which a q u a l i t a t i v e change occurs? One might a t f i r s t s i g h t expect t h a t such a point might occur at Rias = 0 , which is t h e point a t which t h e two-particle s t a t e i n f r e e space becomes bound; but i n f a c t nothing s p e c i a l happens a t t h i s point. Indeed, t h e formal s o l u t i o n s t o e q n s . ( l l a ) and (9) are q u i t e continuous throughout t h e whole range. However, t h e r e $ i n f a c t a p i n t a t which at least t h e physical s i g n i f i c a n c e of some of t h e r e s u l t s changes, namely t h e point at which t h e chemical p o t e n t i a l

u

passes through zero. To see t h i s we go back t o eqn.(8b) and note t h a t Ek is t h e (minimum) energy of e x c i t a t i o n of t h e p a i r state

(If+, - ^&+).

^{I t}

follows t h a t t h e minimum e x c i t a t i o n energy of t h e system a s a whole ("energy gaptt) is t h e minimum value of Ek as

If

v a r i e s . Now, f o r ^p> 0 t h i s minimum value (which always occurs i n t h e region k << r i l , where Ak = A) is j u s t / A / i t s e l f

-

hence t h e conventional name "energy gap1' f o r t h e q u a n t i t y A , i n BCS theory. On t h e o t h e r hand, f o r

u

< 0 t h e energy gap is

not 1

^A

I

but r a t h e r t h e q u a n t i t y

(

1 u 1

⁺

1

^A

1

2)t. W e would t h e r e f o r e expect scme of t h e high-order themdynamic d e r i v a t i o n s t o have s i n g u l a r i t i e s a t t h e p i n t ^p= 0 , and it is no doubt q u i t e possible t h a t t h e ansatz ( 3 ) breaks d m

corn

p l e t e l y i n t h e neighbowhood of t h i s point.

It should be added t h a t t h e whole s i t u a t i o n hems a g r e a t d e a l mre complicated a t f i n i t e temperatures. I n t h e BCS l i m i t t h e temperature a t

which p a i r s a r e formed is i d e n t i c a l t o t h e tempera- t u r e a t which they undergo Bose condensation. I n t h e opposite l i m i t of diatomic m l e c u l e s , however, it is obvious t h a t t h e m l e c u l e s d i s s o c i a t e only around a temperature very much higher than t h a t a t which they Bose-condense. (Dissociation does not

correspond t o a phase t r a n s i t i o n i n t h e usual sense).

It is p o s s i b l e t o generalize t h e d e l t o t h e case of p-wave p a i r i n g ; f o r example, i f we consi- d e r N f e m i o n s all with s p i n +& i n u n i t v o l m , then it is clear t h a t t h e o r b i t a l wave function of t h e p a i r must be odd, so t h a t t h e e n e r g e t i c a l l y favoured p a i r i n g s t a t e is a p s t a t e , and by a s u i t a b l e choice of p o t e n t i a l it is possible t o arrange t h a t t h e system be c l o s e t o t h e onset of t h e two-particle bound s t a t e . The problems of renormalization of t h e p o t e n t i a l , e t c . , are r a t h e r mre complicated than i n t h e s-wave case, but t h e general p a t t e r n of t h e r e s u l t s is s i m i l a r ; i n p a r t i c u l a r e q n s . ( l l ) and (9) still apply. The "gap" A k is now no lon- g e r constant i n t h e region k ro << 1 , but is of t h e general form A. k.c, where

- - c

^isa r e a l o r complex u n i t vector. I t imnediately follows t h a t t h e exci- t a t i o n energy Ek is a n i s o t r o p i c and f o r y > 0 has nodes a t t h e p o i n t s where

k.c

= 0. For y < 0 , on t h e o t h e r hand, t h e energy gap is f i n i t e f o r a l l d i r e c t i o n s ; f o r mall departures f m t h e diatomic- m l e c u l e l i m i t t h e main e f f e c t of t h e many-body i n t e r a c t i o n s is t o give t h e e x c i t a t i o n s on aniso- t r o p i c e f f e c t i v e mass. The quantity F ( r ) is always a n i s o t r o p i c (with approximately pwave symnetry) and reduces i n t h e diatomic-mlecule l i m i t t o t h e Sch*

dinger wave function of a molecule i n a pstate, a s we expect.

F i n a l l y , let me d i s u c s s b r i e f l y some of t h e mre s t r i k i n g manifestations and consequences of Cooper p a i r i n g i n a F e d system. First t h e r e are phenomena associated with t h e c e n t r e e f - m i s s motion

(8)

of t h e p a i r s ; most of t h e s e occur f o r any s p i n and angular m n t u r n of t h e p a i r s , The most spectacu- lar phenomenon is t h a t of s u p e r f l u i d i t y ( p e r s i s t e n t c u r r e n t s , f r i c t i o n l e s s flow through "superleaks", and t h e associated phenmnon of anomalous rotation- al i n e r t i a ) ; i n addition such s y s t e m a r e expected t o show anomalously low entropy, convective heat t r a n s f e r and t h e phenomenon of second sound. I n a d d i t i o n , i f t h e s p i n of t h e p a i r s is nonzero, one would expect metastable "spin supercurrents" and i f t h e o r b i t a l angular m~lentum is nonzero, " o r b i t a l supercurrents" associated with s i t u a t i o n s i n which t h e o r i e n t a t i o n of t h e anisotropic wave function v a r i e s i n space, and possibly a f i n i t e o r b i t a l angu- l a r mcmr~turn i n equilibrium. /6/.

A second c l a s s o f s t r i k i n g e f f e c t s occurs only i n Cooper-paired s y s t e m where t h e p a i r s have S

3

0 and/or !L $ 0 , and is associated with t h e i n t e r n a l s t r u c t u r e of t h e p a i r wave function. Because t h e p a i r function picks out a p a r t i c u l a r o r i e n t a t i o n o r set of o r i e n t a t i o r ~ s , one g e t s a v a r i e t y of phenom ena s i m i l a r t o those observed i n l i q u i d c r y s t a l s , e . g . t h e occurrence of various types of topological s i n g u l a r i t i e s . In a d d i t i o n , even when t h e orienta- t i o n is s p a t i a l l y uniform, many of t h e p r o p e r t i e s of t h e system w i l l be anisotropic. Another proper- t y p e c u l i a r t o t h e a n i s o t r o p i c case is t h e existence of various types of c o l l e c t i v e e x c i t a t i o n corresponding t o deformation of t h e i n t e r n a l structure of t h e p a i r wave function; some of t h e s e e x c i t a t i o n s m y play a very important role i n nuclear magnetic resonance o r i n t h e absorption of ultrasound.

But perhaps t h e mst f a s c i n a t i n g prospect open- e d up by t h e existence of new types of system with Cooper p a i r s formed i n an a n i s o t r o p i c state is t h e p o s s i b i l i t y of amplification of ultra-weak e f f e c t s , which is a direct consequence of t h e f a c t t h a t Cooper p a i r s are by t h e i r very n a t u r e automatically Bose-condensed. Let me f i n i s h by i l l u s t r a t i n g t h i s

phenomenon b r i e f l y with t h r e e examples from o u r only e x i s t i n g anisotropic s u p e r f l u i d , l i q u i d 3 ~ e below 3 mK. (Of these t h r e e examples, t h e f i r s t is w e l l established experimentally, t h e second may have been observed acd t h e t h i r d is as yet s p e c u l a t i v e ) .

(1) The nuclear dipole-dipole i n t e r a c t i o n . For a gas of ordinary diatomic molecules i n a rela- t i v e p s t a t e t h i s would tend t o o r i e n t the nuclear s p i n s perpendicular t o t h e d i r e c t i o n of o r b i t a l angular mmsntum (two m g n e t s have lower energy when they l i e i n t h e plane of r e l a t i v e motion).

However, t h e associated energy advantage is a t most of order I O - ~ O K , which is t i n y compared t o t h e thermal energy a t T

-

¹mK. IIencs i n an ordinary gas t h e nuclear dipole energy is a very s n a l l perl;u- rbation indeed. However, i n s u p e r f l u i d 3He w e a r e dealing not with ordinary diatomic molecules but with Cooper p a i r s , and t h e l a t t e r , being Pose-condensed, must a l l have t h e same r e l a t i v e motion as w e l l as t h e same centre-of-mass motion. Hence t h e eriergy advantage gained by t h e "right" configuration is not

-

I O - ~ O K , but

-

I O - ~ O K x N , t h e t o t a l number of p a i r s i n t h e system. This is very l a r g e corn- pared t o KT, so t h e p a i r s do indeed o r i e n t t h e i r s p i n s perpendicular t o t h e i r o r b i t a l angular m n -

tum.

(2) Electronic ferromagnetism. I 1 a homo- p o l a r diatomic m l e c u l e r o t a t e s , it generates a mall magnetic moment which is proportional t o t h e extent t o which t h e average p o s i t i o n of t h e e l e c t m i i s on one of t h e at- f a i l s t o coincide with t h a t of t h e nucleus. (An i n t r i n s i c a l l y chemical e f f e c t ) . For a rare-gas dimer, t h i s magnetic m n t is extremely a l l , and i n f a c t t h e energy of o r i e n t a t i o n i n any a t t a i n a b l e magnetic f i e l d is t i n y compared t o t h e thermal energy KT. Consequently an ordinary gas of r o t a t i n g m l e c u l e s would have t h e individual m l e c u l e s oriented at random i n even t h e strongest f i e l d s

.

I n 3 ~ e - ~ , however, t h e r o t a t i n g "molecules"

(9)

C 7 - 2 6 JOURNAL DE PHYSIQUE

a r e Cooper p a i r s and t h e r e f ~ ~ r e Bose-condensed, s o they a l l have t h e same a x i s (and sense) of r o t a t i o n , and t h e l i q u i d t h e r e f o r e acquires a magnetic m w n t proportional t o t h e t o t a l n m b e r o f p a i r s , t h a t is it behaves l i k e a f e r r o m g n e t .

( 3 ) P a r i t y v i o l a t i o n . I f one is looking f o r t h e e f f e c t s of t h e p a r i t y v i o l a t i o n characteris- t i c of t h e weak i n t e r a c t i o n , an obvious l i n e is t o search f o r an electric dipole m n t , o n an elementary p a r t i c l e , atom o r m l e c u l e i n a s t a t i o n a r y s t a t e . By t h e Wigner-Eckart theorem such a dipole mment would have t o l i e along t h e t o t a l angular momentum vector J :

fi

= c g , and such a r e l a t i o n s h i p between t h e p o l a r vector

d

and t h e a x i a l vector J would cer- t a i n l y require v i o l a t i o n of p a r i t y conservation (P).

Unfortunately it would a l s o v i o l a t e time-reversal invariance,@) and it is generally believed t h a t t h e strength of t h a t p a r t of t h e weak i n t e r a c t i o n which v i o l a t e s P

and

T js only

-.

of t h a t which vio- l a t e s P alone. However, suppose t h a t an atomic o r molecular system were characterized by

two

independent angular m,ientun v e c t o r s

4

and

2

(say, an o r b i t a l and s p i n angular mmentum). Then we can f o m t h e hypothesis d

-

= cL

-

x S, and t h i s v i o l a t e s

-

P but

not

^T. Now, c a l c u l a t i o n shov~s t h a t any such dipole m n t would have t o be very weak indeed, s o t h a t even i n t h e strongest p o s s i b l e electric f i e l d s its o r i e n t a t i o n energy could not compete with kT.

So a gas of independent atoms o r mlecules having t h i s c h a r a c t e r i s t i c m u l d be completely d i s o r i e n t e d ( t h e vectors I-,

2

,and

&

x

2

would point i n r a n c h d i r e c t i o n s ) and no e f f e c t m u l d be observable.

Once again, however, Bose condensation makes an e s s e n t i a l difference i n 3He-B; t h e Cooper p a i r s t u r n out t o have a f i n i t e expectation value o f t h e vector L x g , and because of t h e Bose condensation t h e

-

d i r e c t i o n of t h i s vector rmst b e t h e same f o r all p a i r s . Consequently one p r e d i c t s a t o t a l e l e c t r i c

d i p o l e m n t due t o parity-violating e f f e c t s which, although c e r t a i n l y very m a l l , is rracroscopic i n t h e sense of being proportional t o t h e t o t a l mass of l i q u i d .

I f t h e spin-polarized systems 3 ~ e + and D+ _do indeed be- s u p e r f l u i d a t a t t a i n a b l e temperatures, one would expect a number of s i m i l a r amplification e f f e c t s . I n p a r t i c u l a r , t h e s e e f f e c t s which depend strongly on t h e "chemistry" of t h e Cooper p a i r s should i n p r i n c i p l e be much stronger i n D+ than i n t h e much mre cherzically i n e r t 3 ~ e . However,

wi-

n s t t h i s must be set t h e l i k e l y reduced density of t h e f o m r system, a s w e l l as t h e f a c t t h a t t h e mere existence of a s t r o n g e l e c t r o n i c p o l a r i z a t i o n may tend t o mark r.mre s u b t l e o r i e n t a t i o n a l e f f e c t s . Clearly, a g r e a t deal depends, here as elsewhere, on t h e

maximum

density a t which spin-polarized system can be s t a b i l i z e d .

This work has benefited from discussions with M. G. McClure, A. A. Abrikosov, P. Nozi&res and with m y of t h e p a r t i c i p a n t s a t t h e S.P.O.Q.S. w n f e r - ence

.

References

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27 (1968) 521.

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108

(1957) 1175.

/3/ Anderson, P.W. and Marel, P . , Phys.Rev. 123 (1961) 1911.

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,

A. J.

,

i n Lecture Notes of t h e 1979 Kaspacz Winter School, Springer-Verlag, t o be published.

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161. h l e d n , N.D., and Muzikar, P . , Phys. Rev. B . 2 1 (1980) 980.