ON THE SPIN DYMAMICS OF SUPERFLUID 3He- A1 AND A

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ON THE SPIN DYMAMICS OF SUPERFLUID

3He-A1 AND A

H. Pleiner

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-19

ON THE SPIN DYNAMICS OF SUPERFLUID 3H

e

- A, AND A

H. Pleiner

Vnivevsit&t Essen GSS, FB Physik, MS Essen, W.-Germany

Résumé.- Pour la phase A et en particulier pour la phase Ai de He superfluide nous comparons la théorie de deux liquides de spin de Leggett et Takagi avec les théories hydrodynamiques de Plei-ner et Graham. Nous indiquons quelques différences physiquement importantes et nous montrons comment les équations de Leggett et Takagi peuvent être modifiées pour éviter l'usage des rela-tions de commutation douteuses.

Abstract.- We compare fort the A- and especially for the Ai- phase of superfluid 3He the two-fluid spin theory of Leggett and Takagi with hydrodynamic theories of Pleiner and Graham. We point out some physically relevant differences and show how to amend Leggett and Takagi's equa-tions in order to avoid the use of questionable commutation relations.

Recently, Legett and Takagi /l/(hereafter called LT),/2/ have given a phenomenological descrip-tion of the spin dynamics in the superfluid phases of 3He. In addition to the hydrodynamic spin variables they include as a macroscopic variable S (the spin of the pairs), with t =jZ {N(k)3(-k) + S(k)N(-k)}. It turns out, that S is (apart from a constant factor) identical to the variable W = ig {T(k)xT+(k)} ^ u s e d in the linearized hydrodynamic (macroscopic) theories of the Aj-phase and the A-phase in high magnetic fields by Pleiner and Graham /3/ and Pleiner / 4 / . After having chosen a certain set of macroscopic va-riables (which can only be done on intuitive grounds outside the true hydrodynamic region), only symmetry arguments and thermodynamics are used in /3/ and / 4 / , which, therefore, provide ibr a framework, within which the phenomenological ideas and approximations of LT can be discussed. In the A-phase, however, the theories are not directly comparable, since LT's theo-ry is valid for low magnetic fields and high frequen-cies, while in /4/ high fields and low frequencies are considered. The theories are compatible in the hydrodynamic regime, which is obtained from- LT by adiabatic elimination of the variable S and from / 4 /

P

by the limit H -K3. In the A-phase the regions of va-lidity of both theories coincide, since the necessary external magnetic field breaks rotational invariance in spin space already externally and the spin order

parameter is just w (or S ) , while a variable cor-responding to 8 (or d in the notation of LT) of the A-phase does not exist in the Aj-phase (Cf. 121). We will discuss, therefore, the physically relevant differences contained in LT (Eqs. (4.15) and (4.16),

the damping term -T*1 S -X S included) and in /3/ (Eqs.(6.1) and (7.3) for k=0) :

i) There is a difference in the transverse NMR shift which in LT is given by A(co2)=ax*1Y2 and in /3/ by

aY^X-^f)"

1

.

The difference arises from the energy, which splits the A-transition (e= c2 w.H) and which is neglected

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by LT . However, the numerical difference seems to be small.

ii)The most problematic point of the derivation of the equations of motion by LT is the use of the com-mutation relation

( V 'Pi) =i^

ijk

S

pk

(Eq.(3.44)of M/)

instead of the correct one

(

S

pi'

S

pj]

= iMe

ijk

Q

'k

w i t h

3'=* ^ft(

k

)N(k)f(-k) +

N(-k)N(-k)3(k)}

(Eq.(3.46) of /l/)in order to get a closed set of equations. However, it is possible to use the cor-rect commutation relation and to obtain a closed set of equations by means of a formalism due to Forster /5/. The only difference will be, that the prefactor of the S x S term in the equation of S will take a

P p different value than that of LT. This prefactor can

be evaluated as an element of the frequency matrix/5^

For a normalized w one finds by inspection w= Tjj-y S (mo=XH equilibrium magnetization den-sity, V=volume,Xcf. /I/, y= gyromagnetic ratio). (2)

Their choice of the static susceptibilities im-plies for the susceptibility T of /3/

(Cf. Equations (4. 6 ) , (4.12) and (4.14))

T = Xj m02A (l-A)"1- X//X21(X//-X|)H2 = C1H2 ; this T has nothing to do with the decay time T of LT.

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For t h e a p p r o p r i a t e p r e f a c t o r i n t h e l i n e a r i z e d t h e o r y , a s ( c f . Eq. (5.1) of / 3 / ) y o u f i n d (3)

i n s t e a d of

which one would g e t u s i n g t h e i n c o r r e c t commutation r e l a t i o n . T h e r e seems t o be no s y s t e m a t i c approxima- t i o n , i n which (2) c o u l d b e o b t a i n e d from ( 1 ) . S i n c e i n t h e A1-phase t h e d i p o l e - t o r q u e

%

i s p r o p o r t i o n a l _-+ t o G w (cf.Ch.7 o f 1311, t h e p r e f a c t o r o f t h e RD-term Y % i n t h e e q u a t i o n f o r S i s a l s o governed by a s a n d , P t h u s , changed by t h e u s e o f ( 1 ) . I n t h e A1-phase LT's e q u a t i o n (4.16) now t a k e s t h e form w i t h 61=f; m:

X

(1-XY1 ( a 3 - ym-i ) . I n e q u a t i o n (3) we have n e g l e c t e d c n ( c f . t h e d i s c u s - s i o n under i ) ; f o r t h e most g e n e r a l c a s e c f . E q u a t i o n (5.4) of 1 3 1 ) Using E q u a t i o n ( 2 ) , 6 l would b e e q u a l 1 t o

w

(1+

7 Z,X;~X~)

and LT's r e s u l t s would b e r e o b t a i - L ned. One o f t h e p h y s i c a l l y r e l e v a n t consequences o f u s i n g (1) i s r e l a t e d t o t h e second t r a n s v e r s e NMR mode ( w ~ , ~ o f E q u a t i o n (7.5) of 131) a t t h e f r e q u e n - c y 61 ( a p a r t from d i p o l e - d i p o l e c o r r e c t i o n s ) . The q u a n t i t y 61 a l s o e n t e r s t h e l i n e w i d t h o f t h e Larmor mode ( E q u a t i o n (7.8) o f / 3 / ) , which i s , t h u s , a f f e c - t e d by u s i n g ( 1 ) i n s t e a d of ( 2 ) , t o o . i i i ) We b r i e f l y mention, t h a t i n t h e Al-phase f l u c - t u a t i o n s of t h e s p i n o r d e r p a r a m e t e r 6 ( o r

3

) a l o n g -+ P t h e m a g n e t i c f i e l d H, i . e . 6w ( o r 6S ) a r e i n t h e PZ l i n e a r i z e d t h e o r y e q u i v a l e n t t o f l u c t u a t i o n s of t h e a b s o l u t e v a l u e o f t h e o r d e r p a r a m e t e r , i . e . 6151

',

a s c a n b e s e e n by t h e d e f i n i t i o n o f

O ( 4 ) . I t

i s v e r y l i k e l y , t h a t t h e r e l a x a t i o n t i m e o f S i s d i f f e r e n t PZ from t h a t of S s i n c e t h e l a t t e r d e s c r i b e f l u c - PX,Y' t u a t i o n s o f t h e d i r e c t i o n o f t h e s p i n o r d e r parame- t e r . Thus. LT's r e l a x a t i o n time T seems t o be a t e n -

1 1

- -

1

- -

s o r (-) = - (6.7H.H.) +

-

H.H. i n t h e Al-phase.

i j T 1J 1 J T Z 1 J

Although a d i r e c t comparison between t h e two t h e o r i e s i s n o t p o s s i b l e f o r t h e A-phase ( s e e a b o v e ) ,

( 3 ) ~ h e r e i s , however, t h e p o s s i b i l i t y , t h a t

a3

a c q u i - r e s a n a d d i t i o n a l c o n t r i b u t i o n from t h e memory m a t r i x ( c f .

151).

some remarks c a n b e made. The above d i s c u s s i o n under p o i n t i i ) c o n c e r n i n g t h e u s e o f t h e c o r r e c t commuta-

-+

t i o n r e l a t i o n f o r S i s e q u a l l y v a l i d f o r t h e A- P +

+

4 p h a s e and t h e p r e f a c t o r of t h e S xS term i n t h e S P P e q u a t i o n i s g i v e n a s i n ( 3 ) . S i n c e i n t h e A-phase

+

t h e d i p o l e - t o r q u e RD i s p r o p o r t i o n a l t o

sox

n ( o r

+

-+ d O x d i n LT' s n o t a t i o n ) , t h e commutation r e l a t i o n

+

[Spi,dj] g o v e r n s t h e p r e f a c t o r o f t h e R _D-term i n t h e ?p-equation. LT o n l y u s e t h a t r e l a t i o n f o r i

#

j ( E q u a t i o n (3.45) o f / l / ) a n d n e g l e c t i t f o r i = j ( t h e q u a n t i t y c a l l e d P by LT). However, t h i s i s no a p p r o x i m a t i o n , b u t e x a c t , s i n c e t h e f r e q u e n c y ma- t r i x element <

(

S . , d . > f o r i = j v a n i s h e s e x a c t l y PI

I1

by symmetry ( i n c o n t r a s t t o <

i

Spi, cijkd;dk

1

>).Thus, i n t h e A-phase LT's e q u a t i o n (4.16) r e a d s ( 6 l g i v e n Acknowledgment : I am i n d e b t e d t o Prof.R.Graham f o r may h e l p f u l d i s c u s s i o n s . R e f e r e n c e s / I / L e g g e t t , A.J.,Takagi,S.,Ann.Phys.= (1977) 79. / 2 / L e g g e t t , A.J.,Takagi,S.,Phys.Rev.Lett.z (1975) 1424. / 3 / P l e i n e r , H.,Graham, R . , J . P h y s . g (1976) 4109 1 4 1 P l e i n e r , H., J.Phys.

Clq

(1977) 2337 /5/ F o r s t e r , D., "Hydrodynamic F l u c t u a t i o n s , Broken Symmetry, and C o r r e l a t i o n F u n c t i o n s " (Benjamin) 1975.

-+