On the degeneracy of the superconductive state

- ~

NuclearPhysics 25 (1961) 622--528; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publi~er

ON THE DEGENERACY

OF T H E S U P E R C O N D U C T I V E

STATE

P. M I T T E L S T A E D T

Max-Planck-lnstitut /i~r Physik und Astrophysih, Miinchen Received 16 J a n u a r y 1961 T h e g r o u n d s t a t e of the BCS t h e o r y is investigated in the limit of infinite volume. Using a t h e o r e m of Bogoljubov it is s h o w n t h a t the energy d e n s i t y of the g r o u n d s t a t e can be obtained exactly using either s t a t e s degenerate with respect to the gauge t r a n s f o r m a t i o n or a s t a t e w i t h a fixed n u m b e r of particles.

Abstract:

1. G e n e r a l F o r m a l i s m

To describe a Fermion m a n y - b o d y system we use the second quantization formalism. The creation and annihilation operators of a Fermi particle with m o m e n t u m k and spin up ( ~' ), and with m o m e n t u m - - k and spin down, ( ~, ) respectively, are designated b y

bt=aLk , b,,=a_k,. The c o m m u t a t i o n relations are [a., a.,]+ = [a~, ab] + = 0,

[a~, a.,] = ~ , , ,

[bk, b.,]+ = [b.', b.',]+ = 0,

Ebb, b,,] = ~ , .

The operator of the n u m b e r of particles is N

=

X

=

k

The bare v a c u u m state

I*

l> is defined b y a

l> =

=

0.

We are discussing here the BCS Hamiltonian 1.2) HBcs

:

•k E (k) (a~ ax + bk*bk) + ~

1

~• I' ( k ,

k')a~,b*waxb k,

where f~ is the volume, I(k, k') the interaction and E ( k ) the kinetic energy of a particle with m o m e n t u m k. For the discussion of this Hamiltonian it has been found to be useful not to introduce the particle-conserving subsidiary condition X (a,~a,,+b~b,,) = k

522

No,

D E G E N E R A C Y OF T H E S U P E R C O N D U C T I V E S T A T E

523

where N O is the t o t a l n u m b e r of particles, b u t to replace HBcs b y H = HBCS--MV, where A is a L a g r a n g i a n multiplier d e t e r m i n e d b y the condition t h a t the expect a t i o n value 2V of N is 2V = N o. Using ~(k) :

E ( k ) - - 2 , we get

1

X Z ( k , k')o~,b~',~b~.

The g r o u n d s t a t e of this H a m i l t o n i a n a n d its e n e r g y are d e s i g n a t e d b y ]0) a n d E, respectively, so t h a t Hi0> = El0>. ~,

W e now i n t r o d u c e new creation a n d annihilation operators ~tk+, /~k+ and /~k b y the canonical t r a n s f o r m a t i o n ~. s), ~tk

=

ukax--vkb~,

ax = uko~k + V~,fl*k,

~,, = v k o 4 , + , ~ , b , , ,

b,, =

,~,~,,--v,,a4,,

where u k a n d vk are real n u m b e r s which satisfy the condition uaEa+viz = 1. The g e n e r a t o r of this t r a n s f o r m a t i o n is

T(u, v) = e 'S,

S = - i y 0~(~,b~'--b,~), k

where cos Ok : u k ,

sin Ok : v

k.

T h e new particles are usually called quasi-particles. T h e o p e r a t o r of the n u m ber of quasi-particles is k

k

The v a c u u m s t a t e of the quasi-particles, the q u a s i v a c u u m [q). [q) ---- e,Sl ) ---- I I ( u a , + v k ~ / ~ ) l

),

k

where axlq) ----/~klq) ---- O. If we use also for the q u a s i v a c u u m s t a t e the condition ( q l N I q ) = 2 ~. v~,2 = N o , k

~

P. MITTELSTAEDT

the s t a n d a r d d e v i a t i o n of real particles in this s t a t e is


p(2 )3

dkv*(k)

a n d p = No/~ is the d e n s i t y of particles. To t r a n s f o r m the H a m i l t o n i a n H we use the n o r m a l p r o d u c t of a bilinear o p e r a t o r R with respect to the s t a t e [q>,

R = : R : + . T h e n we write H = Ho+H1, where H1= ~

1

~,'~I(k' k') : a~,b~*, • • a~bk:

and H o is a q u a d r a t i c form in the F e r m i o n operators. T h e r e f o r e the coefficients Uk, Vk can be chosen such t h a t H o is diagonal, i.e. Ho = E0 + X ~ / k ( X ~ a k + f l k t f l ~ ) k

The q u a s i v a c u u m s t a t e is t h e n the g r o u n d s t a t e of H o a n d

Hie> = Eotq>. F o r the following investigations we i n t r o d u c e the gauge t r a n s f o r m a t i o n a k -+ ake-~%

bk -+ bke-~%

a~ ~ a~ e +'%

b~ -> b~*e+'%

with the g e n e r a t o r G(9) = e~N~This gauge t r a n s f o r m a t i o n does not c o m m u t e with the canonical t r a n s f o r m a tion T(u, v) discussed above. The p r o d u c t of these transformations,

T(u, v; qJ) = G(v)T(u, v)G-'(V ), is again a canonical t r a n s f r o m a t i o n T(u, v), where the real v(k) is replaced b y the complex variable v ( k ) e 2~*, a n d the subsidiary condition is replaced b y Uk2--~ -

lVktz - : I.

T h e p a r a m e t e r ~0 is not d e t e r m i n e d b y the diagonalization of H o and is therefore a free p a r a m e t e r in H o and in the s t a t e Jq). T h e total H a m i l t o n i a n is i n v a r i a n t u n d e r this gauge t r a n s f o r m a t i o n [H, G] = 0,

DEGENERACY

OF THE

SUPERCONDUCTIVE

STATE

525

but it is not necessary that the ground state IO) be an eigenstate of G. The decomposition of H into H o + H I is not gauge invariant, for [H o , G ] : # O ,

[H 1 , G ] : # O ,

and the quasivacuum state is not an eigenstate of G. This cart be seen from the explicit formula I%) = G (~0)]q) = 1 I (uk + vk e z'~ a~ b~,)l ). k

Using the canonical transformation T we obtain Iq,) = G(9~)T(u, v)l ). 2. T h e L i m i t o f i n f i n i t e V o l u m e

The limit of infinite volume is of particular interest, for the following theorem has been proved recently b y Bogoljubov 4): lim {(qlHlq)--(OlHlO)} = l,

(1)

1~-¢ OO

where l is a finite positive number independent of ~2. On the other hand, it is easily seen that (qlHxlq) = --dL (qlHl~lq) = +cL where d ~ and c~ are numbers independent of X2. Therefore it follows that the difference HIq) -- EIq) -= (Eo-- E + e l ) I q ) , has a finite norm in the limit of infinite volume: lira IHiq)-- Elq)12 = 12+c 2 - 2ld~.

(2)

0-.*00

If we consider the finite quantities H/O and E/12 instead of H and E, we see that lim H

E

12

This means that the $chrSdinger equation for the ground state energy E is satisfied asymptotically even b y the quasivacuum state Iq). Furthermore it follows, since the gauge transformation operator G (~0) is unitary, that the same relation holds for any state I%) = G(9)lq), i.e. lim H E o.~,-~1%)--~1%)

2=

O.

(4)

526

P.

MITTELSTAEDT

Therefore it seems t h a t the ground state of H is a s y m p t o t i c a l l y degenerate with respect to the operator G(9)). The states [q,) are not eigenstates of the particle n u m b e r N b u t satisfy the subsidiary condition (q,[N ]qq,) = N o. Furthermore, we shall show t h a t the i m p o r t a n t relation (3) is also valid for a state having a well-defined n u m b e r of particles a n d which is an element of the closed linear manifold {q~}, formed b y the states ]q~). For this purpose, we decompose the quasivacuum state Iq) =

I I u,,II ( : + e k ) , k

k

where ek = (Vk/U~)a~,b~ into a sum of states with a fixed number of particles. This gives Iq) --- I l u k ( l + ~ + ~ * , ~ k + . . It

~

.)1)

~
= [qo) + [q,) + [q,) + . . . . The states ]qo), Iq2), Iq4) are eigenstates of N with the eigenvalues 0, 2, 4, respectively. On account of the relation ek2 = 0 we can write the states ]q~) in a more compact form,

]q~) = kII uk N.T1(~k)N I )" The n o r m of these states is

(q~[qa~) = I kl u x ~k 1 ( k t (Z. . . ( k

v * ( k l ) v Z ( k , ) . . , v* (k2v) u~.(kl)u2(k,) • • • u2(kN) •

N

If we introduce the projection operator PN' = ~

ean-n')q'dq~,

which projects onto the state with the particle number N', we obtain I

f~n

PNIq) = [qN) -= 2-~do e-~*lq*)d9 and

(q~lqN) = (qP~vq). I n s t e a d of the state [qn), we use here the normalized states

1

iN ) = ~ PNIq) with ( N I N ) = 1 a n d Cu 2 = (qPnq). The state IN) is a normalized eigenstate of N, and is an element of the closed linear manifold {q~}.

D E G E N E R A C Y OF T H E S U P E R C O N D U C T I V E S T A T E

5~7

Using the definition

(H--E)Iq> = 1,~>, we obtain

1

H[N>--EJN> = C--NPN[8> and

I H [ N > _ E I N > I 2 _ <6[PN[~> .

(5)

To prove the relation (3) for the states IN> we must discuss the fight-hand side of (5) in the limit of infinite volume. We shall see, that in this limit relation (3) is valid not only for the special state IN0> but for any state IN> with a particle number N which is not too far from the average value No. 1) N is a finite number. If N o = is the mean value of particles in the state lq>, and the density p = No/~ is a ~-independent number, we obtain the asymptotic formulas for large D,

"" od2~Ve -~°,

<61P~16> ~ fl~N+~e-~,

where ~, fl, y are Y2independent numbers. Therefore the fight-hand side of (5) is asymptotically proportional to Y2~ and relation (3) cannot be proved for the state IN>. 2) The distribution of particles in the state lq> is such that the standard deviation is proportional to V ~ o . Therefore the coefficients C~~ = , for values of N, which are not too far from the average value No, are of the order of magnitude N0-½. If we use states IN) which have this property, we obtain asymptotically

IHIN> -- E[N> 1~--~ const. <61PN[6> ~/Q. The value of <6[PN[6> is bounded for large L2 because

<61PNI6> ~-- <61~>, and because (01d) is bounded for large D according to (2). Therefore we obtain lim H E Drool~ IN>-- ~ IN>,J

=

0.

In the limit of infinite volume the mean value N Obecomesinfinite, and because of its definition the eigenvalueN of the state [N> also becomes infinite; relatto. (6) shows that the only relevant result which can be derived in the limit of infinite volume, i.e. relation (3), may be obtained either with the states ]q~>, with an indeterminate number of particles, or with a state IN>, which is an eigenstate of N and G(~).

528

P. MITTELSTARDT

The author is greately indebted to Professor Garsiorowicz, Professor Lfiders, Professor Heisenberg, Dr. Mitter and Dr. Yamazaki for many stimulating discussions. References 1) J. Ba.rdeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108 (1957) 1175 2) J. G. Valatin, Nuovo Cim. 7 (1958) 843 3) :N. N. Bogoljubov, V. V. Tolmochov and D. V. Schirkov, A new method in the theory of superconductivity (New York, 1960) 4) N. N. Bogoljubov, Report a t the Utrecht Conference on the many-body problem (June 1960), private communication b y K. Yamazaki