GAUGE GROUP AND PHASES OF SUPERFLUID 3He

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GAUGE GROUP AND PHASES OF SUPERFLUID

3He

V. Golo, M. Monastyrsky

To cite this version:

JOURNAL DE PHYSIQUE Colloque C6, supplément au n" 8, Tome 39, août 1978, page C6-50

GAUGE GROUP AND PHASES OF SUPERFLUID 3He

V.L. Golo and M.I. Mcmastyrsky

Institute of Theoretical and Experimental Physics, Moscow, V.S.S.R.

Résumé.- On propose une classification des phases A et B qui ne fait appel ni à la contrainte d'uni-tarité, ni à des restrictions sur l'aimantation nucléaire et la susceptibilité.

Abstract.- The p-wave phases of superfluid 3He are classified without the unitary constraint or the

constraints on the net nuclear magnetization and the magnetic susceptibility.

In paper/1/ for different phases of superfluid

3

He in the state of p-wave pairing we studied the topology of spaces of the order parameter (i.e. va-cuum manifolds) by means of the gauge group

SO(3) x SO(3)2x U(l) of the transformations

A ->- e^RjAR"1 (1)

Here A is a complex 3x3-matrix of the order parame-ter, R is a rotation matrix in the spin part and R„ is rotation matrix in the orbit part. In the pre-sent paper we suggest an algebraic method which en-ables to get explicit formulae for the order para-meter and to amend some statements of paper/1/.

Under the constraints that the net nuclear ma-gnetization vanished in the absence of an applied magnetic field and the magnetic susceptibility was unchanged from its normal state value N.D. Mermin and G. Stare found six distinct p-wave phases by minimizing the free energy functional/A/,

F = atr(AA+) +3i|trAAt|2 + g2|trAA+|2 +

BsftrCAA'XAA11)^] + B^tr((AA+)2) +

^ s t r C ^ A A ) ^ ) * ) (2) They proved that two of these phases were unstable

against variations of the order parameter, when the constraints were relaxed.

Another condition which enables to find the p-wave phases is the unitary restriction/4-5-6/,

e. ., AK.A .n n = 0 (3)

ljk pi qj p q

where A ., A . are the entries of the order

parame-• p x * q j _ V

ter matrix and n is a real unit vector, n = 1.

P P There exist four p-wave phases with the unitary

res-triction, but only two of them are stable/5-6/. The-se stable phaThe-ses are generally accepted as the A-and B-phases of 3He.

We do not use the unitary assumption or

cons-traints on the net nuclear magnetization and the magnetic susceptibility. We study the order parame-' ter by means of the gauge group and its subgroups. Indeed, in our opinion this problem mainly concerns the symmetries of the system. We do not study the problem of the minimization ; the stability of the phases is to be discussed elsewhere.

Following paper/1/ we assume that all the va-lues A of the order parameter for a p-wave phase are generated by transformations (1) with A = A . If there is no superdegeneracy ordering to the singula-rities of the space of the order parameter, this condition means that the phase is fixed by the value of A . Then the problem of classification of the phases is reduced to the proper choice of A ; which can be described by the subgroup H = H(A ) of the gauge transformations leaving A invariant. Thus we may say that the S0(3) x S0(3), x U(l) - gauge symmetry is broken down to the subgroup H = 8(A ) .

It is easy to indicate the spaces of the or-der parameter where the gauge symmetry is complete-ly broken, e.g. if A is a Jordan matrix of the rank equal to 3 then the subgroup H is trivial . Another example to the effect is a diagonal matrix with un-equal diagonal entries ; it is not hard to prove that for it the subgroup H = H(A ) is trivial. We do not know under what contraints on the coefficients

To prove the statement we note that for any non-unit rotation matrix R we have RA0^A0, A0R^A0. This

means that in S0(3)j 2 w e have no subgroups leaving

Ao invariant. If there were such subgroups in S0(3)j x S0(3>2, we should have RA0R~' = A0 for some R. Since the Jordanian matrix A generates an irre-ducible matrix algebra A and the matrix R must be-long to the commutator 'arlgebra of A_. Schur's lemma requires that R be a unit matrix, Tc.f. H. Weyl "The Classical Groups" for the specific information). The similar arguments are applied for the Jordanian matrix of rank 2, when the subgroup H is also tri-vial.

a,Bl,..,B5

t h e s e phases can b e minima of f r e e ener- gy (2) and how t h e d i p o l e i n t e r a c t i o n energy chan- ges t h e form of t h e o r d e r parameter ; i t p r e s e n t s a r a t h e r complicated a n a l y t i c a l problem. The magne- t i c f i e l d c o n t r i b u t i o n t o t h e f r e e energy i s more t r a c t a b l e . I f t h e coherence l e n g t h

5,

t h e magnetic l e n g t h

5

and t h e d i p o l e i n t e r a c t i o n l e n g t h

5

sa- H D t i s f y t h e c o n s t r a i n t 5<<5 <<5 we may c a n c e l o u t H D

the d i p o l e i n t e r a c t i o n terms f o r t h e systems having t h e s c a l e R, CH<<R<<SD. Then i t i s p a r t i c u l a r l y in- t e r e s t i n g t o know what degeneracies of t h e o r d e r pa-

-+

rameter s t i l l remain a f t e r t h e magnetic f i e l d

H

is taken i n t o account. The answer t o t h i s q u e s t i o n shows t h a t t h e phases w i t h t h e t r i v i a l subgroup H(A ) a r e h i g h l y a n i s o t r o p i c . Let us w r i t e down t h e magnetic energy c o n t r i b u t i o n i n t h e form/5/,

a t+ -+ FH = g $ * . ~ .H H = gH A H.AH P l 91 P 9 (4) s i n c e A = R ~ A ~ we have R ~ ~ ~ ~ ,

+

We s h a l l minimize F w i t h r e s p e c t t o H' and t h e n H

we s h a l l make some conclusions concerning t h e o r d e r parameter. To s i m p l i f y t h e c a l c u l a t i o n s we suppose t h a t A i s a d i a g o n a l m a t r i x , (AoIij =

XiGij

X12X2fX3 The c a s e of A. b e i n g a Jordan m a t r i x of rank 2 o r 3 i s s i m i l a r . Now we may w r i t e down FH

Since we a r e i n t e r e s t e d i n t h e symmetry of t h e o r d e r parameter we may minimize (6) w i t h r e s p e c t t o t h e

+

d i r e c t i o n of H' o r j u s t t h e same under t h e c o n s t r a i n t H' = c o n s t . The answer i s s t r a i g h t f o r w a r d , t h e r e a r e

+

t h r e e d i r e c t i o n s of t h e f i e l d H' which minimize (6) and which correspond t o t h e axes of A

.

Now we re- t u r n t o e q u a t i o n (4) and want t o f i n d what cons- t r a i n t s a r e imposed on t h e o r d e r parameter b by t h e o r i g i n a l magnetic f i e l d H. Equation (5) t e l l s , t h a t we may p u t t h e answer f o r t h e minimization problem f o r (6) i n t h e form A = R A R-I where R;' i s a r o t a -

+

2 0 - I +

t i o n m a t r i x such t h a t H' = R l l H minimizes (6) and R2 i s any. The r o t a t i o n m a t r i x R i s d e f i n e d by t h e

-f 1

v a l u e of H' up t o a S O ( 3 ) m a t r i x m u l t i p l e

RH,

which

+

l e a v e s H' i n v a r i a n t . T h e r e f o r e we may conclude t h a t i n t h e magnetic f i e l d t h e degeneracy of t h e phase is n o t d e f i n e d i n t h e unique way l i k e i n t h e c a s e of t h e A- and B-phase, b u t t h a t i n t h e s p i n p a r t t h e space of t h e o r d e r parameter is t h r e e l i n k e d c i r c l e s

1 1 .I 1

(S 9 s 9s

llinked

= L(S ) . The whole space JH of t h e

' 1

o r d e r parameter i s a product of L(S ) and t h e group

x SO(3) x U(1).

Now we t u r n t o t h e l e s s d e g e n e r a t e phases w i t h t h e subgroup H(Ao) b e i n g n o n - t r i v i a l . The pre- vious a n a l y s i s i n d i c a t e s t h a t we must exclude a l l A having Jordan blocks of rank more t h a n 1. Our main i d e a is t o s t u d y t h e p o s s i b l e forms of A _by t h e i r rank and e i g e n v a l u e s . To g e t around t h e d i f - f i c u l t i e s generated by t h e complex eigenvalues we apply t h e f o l l o w i n g t r i c k . We change t h e m a t r i x A i n t o a m a t r i x D by means of t h e transformation

D =

x-'A

x

( 7 )

w i t h

a non-singular m a t r i x X. Let us c o n s i d e r t h e group G1(3I1 x Gl(3) ₂ of p a i r s of complex non-sin- g u l a r 3 x 3 m a t r i c e s and i t s a c t i o n on complex 3 x 3 m a t r i c e s

I n t h e group G1(3l1 x Gl(3) we c o n s i d e r t h e sub- 2

group HC which l e a v e s t h e m a t r i x D i n v a r i a n t , i .e.

D = SID

s;'.

The o r i g i n a l subgroup H = H(A ) of S 0 ( 3 I 1 x S0(3)2 x U(1) i s transformed by ( 7 ) i n t o a subgroup of H s i n c e we have

C

-1 -1

s , i l (mx-')xs;'

= x ~ 1 ( x s l x ~ 1 ) ~ o ( x s 2

x

) x It i s important t h a t H and t h e r e f o r e i t s conjugate i n H _C a r e compact groups. We may take t h e m a t r i x

x

such t h a t t h e m a t r i x D should be t h e most simple.

A s we have demonstrated, i f A h a s a Jordan block of rank 2 o r a l l i t s eigenvalues a r e d i s - t i n c t , t h e n t h e subgroup H = H(A ) i s t r i v i a l . The- r e f o r e we may c o n s i d e r now t h e o p p o s i t e c a s e , when ( i ) a t l e a s t two of t h e eigenvalues a r e equal o r ( i i ) t h e m a t r i x A i s of rank 1 . Under t h e s e cons- t r a i n t s we s h a l l w r i t e down t h e o r d e r parameter.

(1) A l l . e i g e n v a l u e s a r e e q u a l ; (A

0 13

H(Ao) = SO(3) ; t h e o r d e r parameter space i s SO(3) x U(1). It is t h e B-phase.

(2) Two e i g e n v a l u e s of A a r e e q u a l ,

X1=X2#o

We may t a k e t h e m a t r i x D i n t h e form

D =

[!;;I

The subgroup HC i s p a i r s of non-singular m a t r i c e s (S 19s 2) : ( s l ) i j = ( s 2 ) i j ; ( s l ) i 3 = (S1)3i =

Gi3

The cdmpact subgroups a r e t h e images of t h e diagonal subgroup i n t h e group S0(3)] x S0(3)2 x U(1). The space of t h e o r d e r parameter i s

The case = 0 i s o f t e n s i n g l e d o u t ( c . f . / 5 / ) ; i t 1

i s c a l l e d t h e 2D-phase.

(3) Two e i g e n v a l u e s o f A a r e e q u a l t o z e r o . The arguments s i m i l a r t o t h e p r e v i o u s no show t h a t t h e subgroup H(A ) = S0(2)1

x

S0(2)2 and t h e s p a c e of t h e o r d e r p a r a m e t e r is

SO(3l1 x S0(3)2 x U(1)

/

S0(2)1 x S0(2)2 T h i s i s t h e p o l a r p h a s e / 5 / .

( 4 ) The m a t r i x A is of r a n k I and is n o t sym- m e t r i c , t h e n we may choose D i n one of t h e two forms

The m a t r i c e s D a r e i n v a r i a n t under t h e t r a n s f o r - 132

mations

1 The c o r r e s p o n d i n g compact subgroups a r e isomorph t o SO(2). Hence we o b t a i n t h e subgroups SO(2) x

111,

S0(2)2 x (11 of t h e gauge group SO(3) x S0(3)2 x U(1). F o r t h e A-phase we need t h e subgroup SO(2) x

1 ( 1 ) ; t h e o r d e r parameter A i s of t h e form

where , A 2 a r e c o o r d i n a t e s of t h e complex v e c t o r

h,

t h e r e a l and imaginary p a r t o f

h

b e i n g u n i t vec- t o t s

h',xss.

T h i s c o n s t r a i n t i s v e r y i m p o r t a n t s i n c e i t i n f l u e n c e s t h e t o p o l o g i c a l t y p e of t h e s p a c e of t h e o r d e r parameter. Indeed, we s e e t h a t t h e s u b s i - d i a r y c o n d i t i o n on t h e r e a l and imaginary p a r t of

h

r e d u c e s t h e s p a c e of t h e o r d e r parameter 2 t o t h e p r o d u c t S x SO(3).

CONCLUSIONS.- We want t o emphasize t h a t t h e method we used t o c l a s s i f y t h e p-wave phases can b e succes- s f u l l y a p p l i e d t o h i g h e r p a i r i n g s t a t e s . The neces- s i t y t o s t u d y h i g h d i m e n s i o n a l r e p r e s e n t a t i o n s of SO(3) x SO(3) = SO(4) i n t r o d u c e s some cumbersome d e t a i l s which can b e s u c e s s f u l l y overcome.

P-wave phases w i t h c o m p l e t e l y b r o k e n gauge symmetry, i f t h e y do e x i s t , s h o u l d have some i n t e - r e s t i n g p r o p e r t i e s . They s h o u l d b e h i g h l y a n i s o t r o - p i c . I n t h e magnetic f i e l d t h e s p a c e of t h e o r d e r p a r a m e t e r i s t h e p r o d u c t of SO(3) and t h r e e l i n k e d c i r c l e s i n a n o t h e r copy of SO(3). T h i s p r o p e r t y sug-

g e s t s t h e e x i s t e n c e of compl'icated domain s t r u c t u r e s ( c . f . 11-21)

ACKNOWLEDGEMENTS.- W e a r e t h a n k f u l t o M. Vuorio who s u g g e s t e d t h a t t h e p-wave p a i r i n g would impose cons- t r a i n t s on t h e subgroups H(Ao) and G. Volovik and V. Mineev, who p o i n t e d o u t t h a t t h i s c o n s t r a i n t would i n f l u e n c e t h e topology of t h e o r d e r parameter s p a c e . We a r e i n d e b t e d t o D. Alexeevsky and G. Mor- g u l i u s f o r c o n s u l t a t i o n s c o n c e r n i n g t h e r e l e v a n t a l g e b r a i c problems.

References

/ I / Golo,V.L., Monastyrsky,M.I., P r e p r i n t ITEP-173 (1976) ( t o a p p e a r i n Ann. I ' I n s t . H. PoincarB, 28 N 1 (1978) 75)

-

/ 2 / Anderson,P.W., Brinkman,W.F., The Helium L i q u i d s (1975) 315

/ 3 / Mermin,N.D., Stare,G., P-wave models f o r t h e A-phase o f s u p e r f l u i d 3 ~ e , p r e p r i n t , 1974

/ 4 / Mermin,N.D., S t a r e , G . , Phys. Rev. L e t t .

30

(1975) 35

151 Leggett,A.J., Rev. Mod. Phys.

2

(1975) n02 / 6 / Ambegaokar,V., Proc. Canadian Summer S c i . (1974)

These a r e t h e f o l l o w i n g subgroups H(A ) which we d i f f e r e n t i a t e w i t h r e s p e c t t o t h e imbgdding i n t o t h e gauge group.

1. H ( A ~ ) = ( I ) ; t h e t r i v i a l subgroup.

2. H(Ao) = SO(3) ; t h e d i a g o n a l subgroup i n SO(3) x

-

SO(3). The B-phase i s c o n t a i n e d i n t h i s c l a s s . 3 . H(Ao) = SO(2) = ( ( R ; R ) , R b e l o n g s t o ~ 0 ( 2 ) 1 . The

u n s t a b l e 2D-phase i s c o n t a i n e d i n t h i s c l a s s . 4. H ( b ) = SO(2) x SO(2) = {(RI ;R2), S0(2)!,2

S 0 ( 3 ) 1 2) The p o l a r phase is c o n t a i n e d I n t h l s c l a s s .

'