gauge theory and bcs theory of superconductivity

7/29/2019 Gauge Theory and BCS Theory of Superconductivity

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IL NU OV O CIMEN TO VOL. 14 D, N. 5 Maggio 1992

Gauge Technique and BCS Theory of Superconductivity(*).R. ACHARYA(1) and P . NARAYANA SWAMY(2)(1 ) Physics Department, Arizona State University - Tempe, AZ 85287(2) Physics Department, Southern Illinois University - EdwardsviUe, IL 62026(ricevuto il 27 Set tem br e 1991; approvato il 16 M arzo 1992)

Summary. m The gauge technique in unbroke n (exact) relativistic quantumelectrodynamics is applied to the nonrelativistic BCS theory exhibitingspontaneously brok en U(1) gauge invariance. In addition to th e BC S-type solution,we f ind an in teres t ing new solution for weak coupling exhibiting a high T c.PACS 74.20.Fg- BCS theory; appl icat ions .PA CS 11.15.Tk - O ther nonp erturb ative techniques.PA CS 12.20 - Qu antum electrodynamics.

T h e p i o n e e r i n g c l as s ic t h e o r y o f B a r d e e n , C o o p e r a n d S c h r i e f f e r [1 ] ( B C S ) w a s t h ef i r s t m ic ro s co p i c t h e o ry w h ich p ro v id ed a s a t i s f ac to ry d es c r ip t i o n o f s u p e rco n d u c t i v i t yi n m e t a l s , w h i c h a l s o l e a d t o s e v e r a l r e m a r k a b l e p r e d i c t i o n s . T h e t w o f o r e m o s tr e s u l t s a r e : 1 ) t h e c r i ti c a l t e m p e r a t u r e Tr is o f t h e o r d e r o f 0D e x p [- - 1 /~ e ~ ] , w h e r e 0Di s t h e D e b y e t e m p e r a t u r e f o r t h e s o l i d a n d ~eff ~--"N ( 0 ) V a d i m e n s i o n l e s s p a r a m e t e rc h a r a c t e r iz i n g t h e s t r e n g t h o f a t t r a c t io n b e t w e e n t h e e l e c t ro n s in a s u p e r c o n d u c t o rn e a r t h e F e r m i s u r f ac e . T h e v a lu e o f Tc is t h u s d e t e r m i n e d b y o n l y t w o p a r a m e t e r s .2 ) T h e B C S f o r m u l a r e l a t i n g T r a n d t h e e n e r g y g a p A(O)/kB T c = = / ] ,, w h e re k s i s t h eB o l t z m a n n c o n s t a n t a n d l n~ , i s th e E u l e r c o n s t a n t . A n e l e g a n t r e f o r m u l a t i o n o f t h eB C S t h e o r y w h i c h p a ra l le l s q u a n t u m e l e c t ro d y n a m i c s (Q E D ) , i n c o r p o r a ti n g t h e r o leo f W a r d i d e n t i t y , is d u e t o N a m b u [ 2 ] . I n N a m b u ' s f o r m u l a t i o n i t b e c o m e st r a n s p a r e n t t h a t t h e B C S t h e o r y r e li e s on t h e l a d d e r a p p r o x i m a t i o n t o t h eD y s o n - S c h w i n g e r ( D S ) e q u a t io n f o r t h e e l e c t r o n p r o p a g a t o r . T h i s a s p e c t i s c l e a r l yp r e s e n t e d b y S c h r ie f f e r[ 3 ] . T h e g e n e r a l f e a t u r e s o f s u p e r c o n d u c t i v i ty a r e i n d e e dm o d e l - i n d e p e n d e n t c o n s e q u e n c e s o f t h e s p o n t a n e o u s b r e a k d o w n o f e le c t r o m a g n e t i cg a u g e i n v a r i a n c e , a s h a s b e e n d e m o n s t r a t e d b y W e i n b e r g [ 4 ] .T h e B C S t h e o r y w a s f u r t h e r d e v e l o p e d a n d r e f in e d b y A n d e r s o n , B o g ol iu b ov ,G o r k o v a n d E l i a s h b e r g [5 ]. T h e w o r k o f E l i a s h b e r g i n p a r t i c u l a r d e a l s w i t h a m o r er e a l is t i c i n t e r a c t i o n b e t w e e n t w o e l e c t r o n s w h i c h i n c l u d es a fu l ly r e t a r d e d a t t r a c t i o n(*) The authors of this paper have agreed to not receive the proofs for correction.

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d u e t o p h o n o n s an d t h e s c reen ed C o u l o m b rep u l s i o n , i n co n t r a s t t o t h e s i m p l ec o n s t a n t p a i r in g i n t e r a c t io n i n t h e B C S t h e o r y [5 , 6 ]. N e v e r t h e l e s s t h es t ro n g -co u p l i n g t h eo ry [5] i n it s cu s t o m ary fo rm i g n o re s v e r t e x co r r ec t i o n s [7]. Th i sn eg l ec t is u s u a l l y j u s t i f i ed b y ap p ea l i n g t o a t h e o re m d u e t o M i g d a l [8] , a cco rd i n g t ow h i c h s u c h c o r r e c t i o n s a r e o f o r d e r ( m * / M ) 1 /2 , w h e r e m * is th e e f f e c ti v e e l e c t r o nm a s s a n d M i s th e m a s s o f t h e i on . I t h a s b e e n e m p h a s i z e d b y M a h a n [ 7 ], h o w e v e r ,t h a t M i g d a l ' s t h e o r e m m a y b e u n r e l i a b l e s i n c e s u p e r c o n d u c t i v i t y i t s e l f ( i . e . , t h eC o o p e r i n s t ab i l i t y ) i s c au s ed b y a v e r t ex co r r ec t i o n .I t i s t h e p u r p o s e o f t h i s n o t e t o i n c o r p o r a t e t h e v e r t e x c o r r e c t i o n s d u e t o p h o n o ni n t e r a c ti o n s b y m a k i n g u s e o f t h e g e n e r a l i z e d W a r d - T a k a h a s h i ( W T ) i d e n t i t y [3 ] f o rt h e s u p e r c o n d u c t o r , i . e .(1 ) q ~ F ~ ( p + q , p ) = ~ s G - l (p ) - G - l ( p + q )~ 3 ,w h e r e v i( i = 1 , 2 , 3 ) a r e t h e P a u l i m a t r i c e s a n d s u m m a t i o n o v e r r e p e a t e d i n di c e s

- - 0 , 1 , 2 , 3 i s i m p l ied . W e u s e t h e n o t a t i o n c o n v en t i o n i n wh i ch t h e m e t r i c i s d e f i n edb y g,y = [1 , - 1 , - 1 , - 1 ] a n d h = c = 1 . T h e i n v e r s e o f t h e e l e c t r o n p r o p a g a t o r i s g i v e nb y t h e f o r m(2 ) G - 1 ( p ) = Z ( p ) P o I - s p ~ 3 - A p ~1i n t h e t h e o r y o f s u p e r c o n d u c t i v i t y i n N a m b u ' s n o t a t i o n [9 ] , w h e r e A p i s t h e g a pf u n c ti o n s ig n i fy i n g b r o k e n s y m m e t r y a n d ~p i s t h e n o n r e l a ti v i s ti c e n e r g y m e a s u r e dr e l a t i v e t o t h e F e r m i e n e r g y , ~ p = p 2 / 2 m * - ~ . T h e r e n o r m a l i z a t i o n f a c t o r Z ( p )s a t i s fi e s co u p l ed i n t eg ra l eq u a t i o n s [3 ] an d h as t h e g en e ra l fo rm(3 ) Z ( p ) = 1 + O ( ~ e~ ) .In t h i s wo rk , we s h a l l u t i l i z e t h e l e ad i n g o rd e r ap p ro x i m a t i o n fo r t h er e n o r m a l i z a t i o n f a c t o r y o n l y , Z ~ 1 (pa i r ing ap prox im at ion [3] ), w h ich wi l l ena b le u st o d eco u p l e t h e eq u a t i o n s a t i s f ied b y t h e g ap fu n c t i o n Ap . In t h e p a i r i n ga p p r o x i m a t io n , G - I ( p ) is g i v e n b y a s u m o f g r a p h s r e p r e s e n t e d b y t h e e q u a t i o n(4) G -1 (p ) = Po I - ~p ~3 - i ( 2 7 : ) - 4 f d 4 k v3 G ( p + k ) ~8 V ( k ) .T h i s is t h e l a d d e r a p p r o x i m a t i o n to t h e D S e q u a t i o n in f ie l d t h e o r y . W e o b s e r v e t h a tt h e f r e e v e r t e x i s g i v e n b y [3 ]( 5 ) r ~ ( p ' , p ) = ~ 3 ( ~ = 0 ) .W e c a n w r i t e d o w n t h e f u ll D S e q u a t i o n a n a lo g o u s t o Q E D , w h i l e s ti ll r e t a i n i n g t h ep a i r i n g f o rm o f t h e p o t en t i a l , b y r ep l ac i n g o n e o f t h e f ac t o r s o f v3 i n eq . (4 ) w i t h t h ed r e s s e d v e r t e x f u n c t i o n I ' o ( p , k ) . I t i s w o r t h n o t i n g t h a t t h e b a r e v e r t e x ~3 i sr e n o r m a l i z e d b y t h e r e s i d u a l i n t e r a c t i o n [ l O ] a n d y i e l d s t h e d r e s s e d v e r t e x / ' 0 .F o l l o w i n g S c h r ie f f e r 's n o t a ti o n [3 ] , t h e D S e q u a t io n , w h i c h t a k e s u s b e y o n d t h el a d d e r a p p r o x i m a t i o n , t h u s r e a d s( 6 ) G -1 (P) = P01 - ~p v3 - i(27:)-4 f d tk ~3G ( p + k )F o (p , k ) V ( k ) .T h e s a l ie n t f e a t u r e s o f t h e g a u g e m e t h o d i n v e n t e d b y S a l a m [ l l , 12 ] c o n s i st o fex p re s s i n g t h e l o n g i t u d in a l p a r t o f t h e v e r t e x fu n c t i o n wh i ch i s a s o l u ti o n o f eq . (1) i n


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GAUGE T E CHNI QUE AND BCS THEORY OF SUPERCONDUCTIVITY 489t e r m s o f t h e e l e c t r o n p r o p a g a t o r , t h e r e b y l e a di n g to a c losed s y s t e m o f e q u a t io n s f o rt h e e l e c t r o n p r o p a g a t o r . T h e g a u g e t h e o r y , b a s e d o n t h is t e c h n i q u e , i s m a n i f e s t l yn o n p e r t u r b a t i v e a n d t h i s e n s u r e s i t s v a l i d i t y a s a s t r o n g - c o u p l i n g t h e o r y . T h e g a u g ea p p r o x i m a t i o n c o n s i s ts o f d is c a r d i n g t h e u n k n o w n t r a n s v e r s e p i e c e, F w ( n o td e t e r m i n e d b y th e W T i d e n t i ty ) , w h i c h s a t is f i e s(7 ) q ~ ' F T ( p + q , p ) = - - O , F = / ' L + F Ta n d r e t a i n i n g t h e l o n g it u d in a l p a r t F L . W e s h a l l o m i t t h e s u p e r s c r i p t L i n w h a tf o ll o w s f o r si m p li c it y . W e c a n w r i t e d o w n t h e s o l u t io n f o r / ' , i n a s t r a i g h t f o r w a r dm a n n e r , t h u s

(8 ) F , (p ' , p) = [(~' - ~) 1 + (P0 - Pg ) r8 ] ( p ' + p ) ,( p , 2 _ p 2 ) (P ' - P)~ i re (A + 4 ' )(p ' - - p)~w h e r e ~ ' = ~ p ,. T h i s i n v a r ia n t f o r m , i n a d d i ti o n t o b e i n g u n i q u e , p o s s e s s e s a ll t h ed e s i r a b l e p r o p e r t i e s e n j o y e d b y t h e g e n e r a l d e c o m p o s i t io n f o r m i n s c a la r Q E D [1 1].T h e s e c o n d t e r m o n t h e r . h . s , i s s i n g u l a r a t p ' - p = 0 a n d r e p r e s e n t s t h e G o l d s t o n ep o l e s i g n i f y i n g s p o n t a n e o u s l y b r o k e n U ( 1 ) s y m m e t r y , w h o s e p r e s e n c e i s e s s e n t i a l f o rt h e e x i s t e n c e o f a n o n v a n i s h in g g a p f u n c t i o n A . W e n e e d o n l y F 0 , t h e ~ = 0 c o m p o n e n to f t h e v e r t e x f u n c ti o n , w h i c h is g i v e n b y(9 ) k oFo (0, k) = ~-~ [(Sk + ~) 1 -- ko r8 + it2 (A + zl ')] .I n s e r t i n g t h i s i n eq . ( 6) a f t e r s e t t i n g p = 0 ( w i t h n o l o s s o f g en e r a l i t y ) , w eo b t a i n(10) G -1 (0) = ~r8 - i (2=) -4 j dt k rs G(k)Fo (0, k ) V ( k) .Using eq . (2 ) , wi th Z ( p ) = 1 , an d a s s u mi n g a co n s t an t p a i r i n g p o t en t i a l V (k ) = V a n d ac o n s t a n t g a p f u n c t i o n Ap = A, w e t h u s d e r i v e

(11) AZ l- -- - i ( 2 r 0 - 4 V f d a k r 3 G ( k ) ~ - 2 2 { ( s k + t z) l - k o r 8 + 2 i A t 2 } .A f t e r s o m e a l g e b r a , t h i s l e a d s t o t h e s u p e r c o n d u c t i v i t y g a p e q u a t i o n

f ko(12) 1 = 3i(2=) -4 V ~ dakJ (ko - k 2 )(ko - E ~ ) 'w h e r e E k = V ~ + A 2. T h is r e s u l t r e p r e s e n t s t h e m a j o r c o n s e q u e n c e o f t h e g a u g et h e o r y a t z e r o te m p e r a t u r e a n d m a y b e c o n t r a s te d w i t h t h e c o r r e s p o n d i n g e q u a t io n i nt h e B C S c a s e :

f 1(12a) 1 = i(270-4V | dakJ ( k o '34 - I I N u o v o C i m e n t oD


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490 R. ACHARYA and P. NARAYAN A SWAMY

w h i ch , a f t e r i n t eg r a t i o n o v e r k o l ead s t o [ 10 ]

(12b) l = V----~- d3 k ~2(2r:)3T o p ro c e e d w i t h a n e x t e n s io n t o f in it e te m p e r a t u r e s , w e m a y e m p l o y t h ef i n i t e - t em p er a t u r e f ie l d t h e o r y d ev e l o p ed b y D o l an an d J ac k i w [1 3] i n t h e r ea l - t im ef o r m a l i s m w h e r e i n t h e b o s o n a n d f e r m i o n p r o p a g a t o r s a r e r e s p e c t i v e l y g i v e n b y t h eex t en s i o n s :

1 1 2ir:~(k~' k~ )(13) k" k, + i~ k" k, + i~ ex p [fl lko I - 1]an d(14) 1 ~ 1 + 2ir:~(k~ - E ~)

ko - E~ + i~ k~ - E~ + i~ ex p [fllk01 + 1] "W e o b s e r v e t h a t t h e f i n i t e - te m p e r a t u r e b o s o n p r o p a g a t o r , e q . (1 3), i s c o n s i s t e n t w i t heq s . (1 ) an d ( 8). I n s e r t i n g t h e s e i n eq . ( 1 2), an d a f t e r an i n t eg r a t i o n o v e r t h e v a r i ab l ek 0 , w e o b t a i n

3___~V(1 5) 1 - | d 3 k -2(2=) 3 J

E~ - k 2 Ek tgh ~ E k - -w h e r e ~ = l / k s T an d k 2 = 2 m * ~. Eq u a t i o n (1 5) co n t a i n s o u r f u n d a me n t a l r e s u l t o ft h e g a u g e t h e o r y o f s u p e r c o n d u c t i v i t y f o r a n y t e m p e r a t u r e . W e s h al l h e n c e f o r w a r do m i t t h e s u b s c r i p t k i n E an d ~ f o r co n v en i en ce . W e r e co v e r eq . ( 12 ) f o r T = 0 K .E q u a t i o n ( 1 5 ) s h o u l d b e c o n t r a s t e d w i t h t h e B C S t h e o r y r e s u l t ,

( 1 6 ) 1 = f d 3 k E t g h ( l z E2(2r:)aW e s h a ll n o w p r o c e e d t o a n a ly z e t h e c o n s e q u e n c e s o f t h e g a u g e t h e o r y r e s u l t ,eq . (1 5), f o r s u p e r co n d u c t i v i t y an d d e s c r i b e t h e p o i n t s o f d e p a r t u r e f r o m t h ec o n c lu s io n s o f t h e B C S t h e o r y .

I n t r o d u c i n g t h e d e n s i t y o f s t a t e s N ( 0 ) o n t h e F e r m i s u r f a c e a n d a c u t - o ffr e p r e s e n t e d b y t h e D e b y e f r e q u e n c y ~ [1 4], w e c a n c a s t e q . (1 5) in t h e f o r m

(17) 1= VN(O) d~ ~ 2 + A 2 _ 2 m .- -~o

W e m a y n o w s p e c ia li ze t o t h e c r i ti ca l t e m p e r a t u r e T = Tc , ~ = fie w h e n t h e g a p A =


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GAUGE TECHNIQUE AND BCS THEORY OF SUPERCONDUCTIVITY 491

= z l(T r v a n i s h e s . I n t h i s c a s e w e o b t a i nto

1 = 1 f d~ [ ~ t g h ( 1 / ~ c ~ ) _ 2 V ~ - ~ c t g h ( l ~ e 2 V ~ - ~ ) ]( 18 ) 3 V N ( 0 ) 2 r _ 2 m * ~- - to

W e c a n n o w d e t e r m i n e T r f r o m e q . (1 8) b y r e s o r t i n g t o t h e a p p r o x i m a t i o n [1 5]I x , x ~ l ,( 1 9 a ) t g h x - - 1 , x > 1 ,I 1 / x , x < ~ l ,( 19 b ) c t g h x = [ 1 , x > 1 .

S t r a i g h t f o r w a r d i n t e g r a t i o n g i v e s , i n t h i s a p p r o x i m a t i o n ,1 - l + ~ m ~ c l n + In -(20) 3vN (0) 1;m* o g

- a r c t g ~ + a r c t g 1 l l n l + m * / ~m * f l r 2 1 - m * ~1 In 1 - m * 2 f l~ l 1 ~ / - s ~ I

2 r e * t i c 1 7 m ' 2 ~ c + g In V ~ - ~T h i s c o m p l i c a t e d t r a n s c e n d e n t a l e q u a t i o n m a y b e s o l v e d n u m e r i c a l l y i n o r d e r t od e t e r m i n e T c b u t w e s h a ll in s t e a d u s e t h e f o ll o w i n g a p p r o a c h . I t i s e v i d e n t t h a t T c i sd e t e r m i n e d b y ~ , V N ( O ) a n d o n e a d d i t io n a l p a r a m e t e r , n a m e l y th e e f f e c t iv e m a s s m * .W e c a n t h u s o b t a i n d i f f e re n t s o l u ti o n s d e p e n d i n g o n th e s iz e o f m * . I t s u f f i c e s t oe x a m i n e t h r e e d i s t in c t c a se s i n t e r m s o f t h e d i m e n s io n l e ss p a r a m e t e r m * f l c : a)m * # e >> 1 ; b ) m * f l r ~ 1 ; c ) r e * f ie


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492 R. ACHARYA and P. NARAYAN A SWAMY

m* flc


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G A U G E T E CH N I Q U E A N D BCS THEORY OF SUPERCONDUCTIVITY 493

o r

(32) z l = 2 ~ e x p [ 3 V N (0 )I ]

w h i c h is t h e p r e d i c t i o n fo r th e g a p f u n c t io n , o r m o r e p r e c i s e ly t h e z e r o t e m p e r a t u r ev a l u e , A (0 ). W e c a n n o w c o m b i n e th i s w i t h e q . ( 2 5) a n d o b t a i nA(O) 4- - ~ 1 . 4 7 2 ,(33) kB Te e

a r e s u l t i n d e p e n d e n t o f t h e v a l u e o f VN ( O ) . T h i s m a y b e c o n t r a s t e d w i t h t h e B C St h e o r y r e s u l t[ zl(0) ] ~ --~ ~ 1 .7 6 4 .(34) k ~ Jsc s Y

I t i s a m u s i n g t o n o t e t h a t e q . (3 3) i s a l so in t e r m s o f a tr a n s c e n d e n t a l n u m b e r .T h i s w o r k c le a r ly d e m o n s t r a t e s t h a t t h e g a u g e t e c h n i q u e , b y g o i n g b e y o n d t h el a d d e r a p p r o x i m a t i o n , c a n y i e l d a h i g h v a l u e o f Tr even f o r w e a k c o u p l i n g , Z e . ~< 1 / 3 .I n c o n c lu s io n , w e m a k e t h e f o ll o w i n g i n t e r e s t i n g o b s e r v a t io n . T h e g a u g e t h e o r y is an o n p e r t u r b a t i v e m e t h o d a n d i s t h u s v a l i d fo r al l v a l u e s o f t h e e f f e c t iv e c o u p l i n g , ~ e, == VN ( O ) . O n e i s t h e r e f o r e f r e e t o c o n s i d e r t h e l i m i t Z e,--~ 0% i n c o n t r a s t t o t h ew e a k - c o u p l i n g l a d d e r a p p r o x i m a t i o n [1 9]. I n t h i s l im i t th e n , w e o b t a i n t h e s a t u r a t e dv a l u e o f e q . ( 2 5 ), t h u s

1( 35 ) Max (k B Tc) = ~ eo~ ,w h i c h i m p l i e s t h e l i m i t i n g v a l u e (0D ~ 2 3 0 K ) :( 36 ) M a x ( T e ) = 3 1 2 K ,w h i c h a ll o w s t h e p o s s i b il it y o f r o o m t e m p e r a t u r e s u p e r c o n d u c t i v i ty . T h e e x t e n s i o n o ft h e p r e s e n t w o r k t o i n c lu d e a m o r e r e a li s t ic r e t a r d e d p a i r i n g p o t e n t i a l w il l l e ad t o t h eg e n e r a l iz e d E l i a s h b e r g e q u a t i o n s [20 ] w h i c h r e q u i r e e x t e n s i v e n u m e r i c a lc o m p u t a t i o n a n d w i ll b e a d d r e s s e d e l s e w h e r e .

W e t h a n k P r o f s . A . S a l a m a n d J . S t r a t h d e e f o r t h e i r v a l u a b l e c o m m e n t s .

R E F E R E N C E S[1] J. B A R D E E N , L. COOPER and J . SCHRIEFFER:Phys . Rev . , 108 , 1 175 ( 1957) .[2] Y. NAlVIBU:Phys . Rev . , 117, 648 (1960).[3 ] J . SCHRIEFFER:Theoryof Supercond uct ivi ty ( W. B en j am i n I nc . , N ew Y or k , N . Y . , 1964),see especia l ly p . 157, 232 and C hap t . 7 ; S . ENGELSBERGand J . SCHRIEFFER:P h y s . R e v . ,131, 993 (1963).14] S. WEINBERG:Prog. Theor. Phys. Suppl . , 86, 43 (1986), Fes t shcri ft honor ing Y . N am bu onhis 65th birthday.


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494 T. LETARD I, S. BOLLANTI, P. DI LAZZARO, F. FLORA, N. LI SI an d c. E . ZHENG[5] P. W. ANDERSON:Phys. Rev. , 112, 1900 (1958); N. BOGOLIUBOV,V. TOLMACHEVa n d D .SHIRKOV: New Method in the Theory of Superconductivity (Acad . Sci . USSR, Moscow,1958); L. GORKOV:Soy. Phys . JETP, 7, 505 (1958); G. EL IASHBERG :Soy. Phys. J E T P , 11,696 (1960).[6] D. SCALAPINO,J . SCHRIEFFER and J . WILKINS: Phys. Rev. , 148, 263 (1966).[7] G. MA HAN : Many Particle Physics ( P l e nu m P r e s s , N e w Y o r k , N . Y . , 1 9 9 0 ) , 2 n ded i t ion .[8 ] A. MIGDAL: Soy. Phys . JETP, 7, 996 (1958).[9 ] P. AL LEN and B. MITROVIC:Solid State Phys., 37, 1 (1982).[10] P. LITTLEWOODand C. VARMA: Phys. Rev. B, 26, 488 3 (1982).[11] A. SALAM: Phys. Rev. , 13 0 , 12 87 (1963) ; R . DELBOURGO:Nuovo Cimento A, 49, 484(1979).[12] S ee also R. ACHARYA and P. NARAYANA SWAMY: Phys. Rev. D, 26, 2797 (1982).[13] L . DOLAN and R. JACI~W: Phys. Rev . D , 9 , 332 0 (1974).[14] See, e.g., A. FETTER and J . WALECKA: Quantum Theory of M any Particle Sy stems(McGraw-Hill Book Compan y , Ne w Yo rk , N .Y. , 1971) , p . 333.

[15] C. TSU EI, D. N EW NS, C. C ttI and P. PATTNAIK:Phys. Rev. Let t . , 65, 2724 (1990).[16] M. SCADRON:Ann . Phys . , 148 , 257 (1983). Se e also, Y . N AMBU: From Symmetr ies toStrings, a Symposium to honor S. Okubo (W orld Scien t i fic , S ingapore , 1990) , p . 1 .[17] M. TINKHAlgand C. LOBB:Solid State Phys ., 42 , ed i te d by H . EHRENREICHet al. (A cad emicP ress , N e w Y o rk , N .Y . , 1 9 8 9 ) , p . 9 1.[18] D . TILLEY and J. TILLEY:Superfluidity and Superconductivity, 3rd ed . , Chap t . 11 (AdamH i lg e r C o . , N ew Y o rk , N .Y . , 1 9 9 0 ) .[19 ] W e a re o f co u rse aw are o f th e f ac t t h a t i n th i s l im i t , o n e sh o uld r ea l ly co n s id e r th e co u p ledp ro b lem in v o lv in g Z(p) and A(p).[20] M. CRISAN: Theory of Superconductivity (World Scien t i f ic , S ingapore , 1989) , p . 87 .

gauge theory and bcs theory of superconductivity

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