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Broken symmetries and hydrodynamics of superfluid 3P2-neutron star matter
Physica 107B (1981) 53-54 North-Holland Publishing Company
BE 6
BROKEN SYMMETRIES AND HYDRODYNAMICS OF SUPERFLUID3p2-NEUTRON STAR MATTER Helmut
Fachbereich
Physik,
Brand and H a r a l d
Universit~t
Essen,
4300
Pleiner
Essen,
W-Germany
3 We i n v e s t i g a t e the s p o n t a n e o u s l y b r o k e n s y m m e t r i e s of P2 n e u t r o n star m a t t e r and we p r e s e n t the n o n l i n e a r h y d r o d y n a m i c e q u a t i o n s of this superfluid. A p a r t from broken g a u g e symmetry, a fact w h i c h is common to all superfluids, we find that the total r o t a t i o n a l s y m m e t r y of spin and orbit space is s p o n t a n e o u s l y broken, a u n i q u e feature of P2 n e u t r o n star matter. As ha~ ~ e c o m e clear d u r i n g the last ~ e a r s ~ - - there exists very p r o b a b l y a P9 s u p e r f l u i d p h a s e of n e u t r o n s in the in£eri o r of n e u t r o n stars and it is the p u r p o s e of the p r e s e n t c o n t r i b u t i o n to c l a r i f y the nature of the s p o n t a n e o u s l y b r o k e n c o n t i n u o u s symmetries of that phase and to d e r i v e the c o r r e s p o n d i n g n o n l i n e a r h y d r o d y n a m i c equations. Our c o n s i d e r a t i o n s are r e l e v a n t to n e u t r o n stars b e c a u s e i) the h y d r o d y n a m i c e q u a t i o n s give a m a c r o s c o p i c d e s c r i p t i o n (including m a g n e t i c fields) of ~P2 n e u t r o n star m a t t e r and 2) we find a c o u p l i n g t e r m b e t w e e n the d e n s i t y of linear m o m e n t u m and the v o r t i c i t y of the m a g n e t i z a t i o n density w h i c h will p r o b a b l y have an imp o r t a n t i n f l u e n c e on the rotational d yn a m i c s of a n e u t r o n star. The o r ~ e { p a r a m e t e r of 3p~ n e u t r o n star m a t t e r - ' ~ is a complex, t~aceless, s y m m e t r i c 3x3-matrix, d e n o t e d by A ~. Under r o t a t i o n s of the orbital coordinates A . t r a n s f o r m s like a vector w i t h r e s p e c t U ~ o the index i and u n d e r rotations in spin space A transforms llke a v e c t o r w l t h respect to the index ,
,
1
,
The o p e r a t o r for the o r d e r p a r a m e t e r is d e f i n e d in ref 4 (for a c o r r e s p o n d i n g ~ e f i n i t i o n in the s u p e r f l u i d phases of He we r e f e r to ref 5). For the n o r m a l i zed, u n i t a r y e q u i l i b r i u m o r d e r p a r a m e t e r we have A
A°~I = N e i ~ [ u
M
A
ui + r~ ~ i - ( l + r ) w
A
wi]
w here N is a n o r m a l i z a t i o n c o n s t a n t and w here the p a r a m e t e r r is chosen in such a way that t h e 3 G i n z b u r g L a n d a u free e n e r g y is m i n i m i z e d . 6, %% and ~ from a t r i a d of unit vectors in spin or real space, r e s p e c t i v e l y . It seems i m p o r t a n t to notice that A . is
03784363/81/0000-0000/$0250
© North-HollandPublishingCompany
symmetric with respect of real and spin space A°
to an i n t e r c h a n g e indices
= A°
and that the only i m a g i n a r y c o n t r i b u t i o n to the e q u i l i b r i u m s t r u c t u r e is the phase factor e ±~. The former p r o p e r t y has d r a s t i c c o n s e q u e n c e s on the h y d r o d y n a m i c v a r i a b l e s and on the h y d r o d y n a m i c equations: Every v e c t o r in real space can be t r a n s f o r m e d into a v e c t o r in spin space by m u l t i p l i c a t i o n with one of the factors 6 6. ~ ~. or %% %% . Of c o u r s e . p ~ s ~ i b l e for the the same ~ r ~ c'e d d r e1 is t r a n s f o r m a t i o n of vectors (or tensors) in spin space into c o r r e s p o n d i n g q u a n t i ties in real s p a c e . 3 T h i s is a feature w h i c h is u n i q u e to P9 n e u t r o n star m a t t e r and w h i c h has ~ot been found for any h y d r o d y n a m i c s y s t e m studied so far and it simply means that spin and orbit indices can be i n t e r c h a n g e d at will! As c a n d i d a t e s for v a r i a b l e s c h a r a c t e r izing b r o k e n symmetries we have to consider the p h a s e d e v i a t i o n 6~ and the dev i a t i o n s from the unit vectors of the t r i a d (6, ~, ~) in spin or orbit space or e q u i v a l e n t l y the r o t a t i o n angles 60. about these axes in th~ c o r r e s p o n d i n g space. As it turns out the c o r r e s p o n ding o ~ e r a t o r s Rave the same s t r u c t u r e as in - H e - A o r ~ - H e - B w i t h o u t e x t e r n a l m a g n e t i c field- and only the c o n t e n t s of the s t r u c t u r e m a t r i x is different. For the c o m m u t a t i o n r e l a t i o n s for the total p a r t i c l e n u m b e r N and for the total angular m o m e n t u m J.=L. + S (where L is the 1 1 1 a n g u l a r m o m e n t u m o p e r a ' o r and S i the oper a t o r for the spin) we find <[6~,
N]> = - 2i
<[68i,N]>
= O
<[6~,Ji]>
= O
<[~Sf~]>
= i 6ik
53
54
It should be s t r e s s e d that c o m m u t a t o r s w i t h L. or S do hot,give rise to a canonica~ r e l a t i o n in 5p~ neutron star matter.' Thus we have i ~ e n t i f i e d the hyd r o d y n a m i c v a r i a b l e s c h a r a c t e r i s i n g spontaneously broken continuous symmetries of -P~ n e u t r o n star matter: One variable, the phase d e v i a t i o n 6~ or the s u p e r f l u i d v e l o c i t y v~ c h a r a c t e r i s e s the b r o k e n gauge i n v a r z a n c e and the three v a r i a b l e s 6 8 are c h a r a c t e r i s i n g the s p o n t a n e o u s l y l . . . . b r o k e n total r o t a t l o n a l lnvarzance, z.e. total a n g u l a r m o m e n t u m J =L + S. serves 1 1 1 as the g e n e r a t o r of a b r o k e n symmetry! The l a t t e r p r o p e r t y is unique to -P? n e u t r o n star m a t t e r and has not beeH found for any h y d r o d y n a m i c s y s t e m s t u d i e d p r e v i o u s l y . It has to be cont r a s t e d e.g. to the s p o n t a n e o u s l y b r o k e n r e l a t i v e spin orbit ~ y m m e t r y of the Bphase of s u p e r f l u i d -He. In this phase one has <[S i,
~0j]>
= - <[L i,
<[Ji'
6@j]>
= O
~0j]>
or
w h e r e a s in all o t h e r h y d r o d y n a m i c systems w ~ t h b r o k e n r o t a t i o n a l s y m m e t r i e s (e.g. He-Al, nematics, a n t i f e r r o m a g n e t s ) these s y m m e £ r i e s are s p o n t a n e o u s l y broken in real and spin space separately• To set up the h y d r o d y n a m i c e q u a t i o n s the p r o c e d u r e is as follows. We start with the Gibbs r e l a t i o n w h i c h defines the t h e r m o d y n a m i c c o n j u g a t e q u a n t i t i e s and then we w r i t e down the c o n s e r v a t i o n and q u a s i - c o n s e r v a t i o n laws for the h y d r o d y namic v a r i a b l e s thus d e f i n i n g the reversible and i r r e v e r s i b l e currents. As conserved q u a n t i t i e s we have the d e n s i t y p, the e n e r g y d e n s i t y e (or e n t r o p y d e n s i t y 0), the d e n s i t y of linear m o m e n t u m g and the d e n s i t y of total a n g u l a r momentum. The four v a r i a b l e s c h a r a c t e r i s i n g spont a n e o u s l y b r o k e n s y m m e t r i e s have a l r e a d y been d i s c u s s e d above• To close the system of the h y d r o d y n a m i c e q u a t i o n s we proceed in three steps. First we e s t a b l i s h the free e n e r g y from w h i c h the t h e r m o d y namic c o n j u g a t e q u a n t i t i e s can be obt a i n e d by d i f f e r e n t i a t i o n . In the s e c o n d step we e x p a n d the i r r e v e r s i b l e currents into the t h e r m o d y n a m i c conjugates. In the last step we derive the r e v e r s i b l e currents. To achieve this, it is necessary, however, to take into a c c o u n t the a n h o l o n o m i t y r e l a t i o n for the hydrod y n a m i c v a r i a b l e s ~0~ (cf ref 4 for the details) • Quite a n a l o±g o u s r e l a t i o n s have been gi v e n p r e v i o u s l y for biaxial nematics by the authors. F u r t h e r m o r e we dem o n s t r a t e a novel p r o c e d u r e to g u a r a n t e e c o n s e r v a t i o n of total a n g u l a r m o m e n t u m locally. This p r o c e d u r e involves the t h e r m o d y n a m i c c o n j u g a t e force of the total a n g u l a r m o m e n t u m density, a q u a n t i t y
which
is local
in space!
A f t e r the d e r i v a t i o n of the compl~te nonlinear h y d r o d y n a m i c e q u a t i o n s of -P? neutron star m a t t e r it is p o s s i b l e to ~heck that e.g. the n o n l i n e a r h y d r o d y n a m i c e q u a t i o n s for biaxial nematics are contained in the p r e s e n t ones as a special case. A very i n t e r e s t i n g static c o u p l i n g w h i c h will p r o b a b l y influence the r o t a t i o n a l dynamics of a n e u t r o n star is found in the e q u a t i o n for the d e n s i t y of linear momentum gi =
"'" + ~ij
ejkl
?k hl
where h I is the t h e r m o d y n a m i c c o n j u g a t e to the m a g n e t i z a t i o n density• The three p h e n o m e n o l o g i c a l c o e f f i c i e n t s w h i c h are i n v o l v e d in ~.. lead to a c o u p l i n g between the v o r t ~ i t y of the m a g n e t i z a t i o n and the d e n s i t y of linear momentum. As h y d r o d y n a m i c e x c i t a t i o n s of the l i n e a r i zed e q u a t i o n s we find three pairs of prop a g a t i n g s p i n - o r b i t - w a v e s w h i c h are c o u p l e d in an i n t r i c a t e m a n n e r to firstand second sound. These c o m p l i c a t e d c o u p l i n g s can be t r a c e d back to the bia x i a l i t y in real and spin space of the p r e s e n t system. To sum up we have d i s c u s s e d the b r o k e n s y m m e t r i e s and the h y d r o d y n a m i c equations of P2 n e u t r o n star m a t t e r and we have found several p r o p e r t i e s w h i c h are unique toa-P? superfluid. The most i m p o r t a n t one a m o n g - t h e s e seems to be the c o n c l u s i o n that total angular m o m e n t u m can serve as the g e n e r a t o r of a s p o n t a n e o u s l y b r o k e n symmetry!
REFERENCES [1] Hoffberg, M., Glassgold, A.E., Richardson, R.W. and Ruderman, M., Phys.Rev. Lett. 2_44775 (70) [2] Sauls, J.A. and Serene, J.W., Phys. Rev. D I 7 1 5 2 4 (78) [3] Muzikar, P., Sauls, J.A. and Serene, J.W., Phys.Rev. D21, 1494 (80) [4] Brand, H. and Pleiner, H. to be published [5] Brand, H., D~rfle, M. and Graham, R., Ann. Phys. (N.Y.) 1 1 9 4 3 4 (79) [6] Leggett, A.J., Rev.Mod. Phys. 4_/733 (75) [7] Graham, R. and Pleiner, H., J.Phys. C__99279 (76) [8] Liu, M. and Cross, M.C., Phys.Rev. Lett. 4__ii250 (78)
BE 6
BROKEN SYMMETRIES AND HYDRODYNAMICS OF SUPERFLUID3p2-NEUTRON STAR MATTER Helmut
Fachbereich
Physik,
Brand and H a r a l d
Universit~t
Essen,
4300
Pleiner
Essen,
W-Germany
3 We i n v e s t i g a t e the s p o n t a n e o u s l y b r o k e n s y m m e t r i e s of P2 n e u t r o n star m a t t e r and we p r e s e n t the n o n l i n e a r h y d r o d y n a m i c e q u a t i o n s of this superfluid. A p a r t from broken g a u g e symmetry, a fact w h i c h is common to all superfluids, we find that the total r o t a t i o n a l s y m m e t r y of spin and orbit space is s p o n t a n e o u s l y broken, a u n i q u e feature of P2 n e u t r o n star matter. As ha~ ~ e c o m e clear d u r i n g the last ~ e a r s ~ - - there exists very p r o b a b l y a P9 s u p e r f l u i d p h a s e of n e u t r o n s in the in£eri o r of n e u t r o n stars and it is the p u r p o s e of the p r e s e n t c o n t r i b u t i o n to c l a r i f y the nature of the s p o n t a n e o u s l y b r o k e n c o n t i n u o u s symmetries of that phase and to d e r i v e the c o r r e s p o n d i n g n o n l i n e a r h y d r o d y n a m i c equations. Our c o n s i d e r a t i o n s are r e l e v a n t to n e u t r o n stars b e c a u s e i) the h y d r o d y n a m i c e q u a t i o n s give a m a c r o s c o p i c d e s c r i p t i o n (including m a g n e t i c fields) of ~P2 n e u t r o n star m a t t e r and 2) we find a c o u p l i n g t e r m b e t w e e n the d e n s i t y of linear m o m e n t u m and the v o r t i c i t y of the m a g n e t i z a t i o n density w h i c h will p r o b a b l y have an imp o r t a n t i n f l u e n c e on the rotational d yn a m i c s of a n e u t r o n star. The o r ~ e { p a r a m e t e r of 3p~ n e u t r o n star m a t t e r - ' ~ is a complex, t~aceless, s y m m e t r i c 3x3-matrix, d e n o t e d by A ~. Under r o t a t i o n s of the orbital coordinates A . t r a n s f o r m s like a vector w i t h r e s p e c t U ~ o the index i and u n d e r rotations in spin space A transforms llke a v e c t o r w l t h respect to the index ,
,
1
,
The o p e r a t o r for the o r d e r p a r a m e t e r is d e f i n e d in ref 4 (for a c o r r e s p o n d i n g ~ e f i n i t i o n in the s u p e r f l u i d phases of He we r e f e r to ref 5). For the n o r m a l i zed, u n i t a r y e q u i l i b r i u m o r d e r p a r a m e t e r we have A
A°~I = N e i ~ [ u
M
A
ui + r~ ~ i - ( l + r ) w
A
wi]
w here N is a n o r m a l i z a t i o n c o n s t a n t and w here the p a r a m e t e r r is chosen in such a way that t h e 3 G i n z b u r g L a n d a u free e n e r g y is m i n i m i z e d . 6, %% and ~ from a t r i a d of unit vectors in spin or real space, r e s p e c t i v e l y . It seems i m p o r t a n t to notice that A . is
03784363/81/0000-0000/$0250
© North-HollandPublishingCompany
symmetric with respect of real and spin space A°
to an i n t e r c h a n g e indices
= A°
and that the only i m a g i n a r y c o n t r i b u t i o n to the e q u i l i b r i u m s t r u c t u r e is the phase factor e ±~. The former p r o p e r t y has d r a s t i c c o n s e q u e n c e s on the h y d r o d y n a m i c v a r i a b l e s and on the h y d r o d y n a m i c equations: Every v e c t o r in real space can be t r a n s f o r m e d into a v e c t o r in spin space by m u l t i p l i c a t i o n with one of the factors 6 6. ~ ~. or %% %% . Of c o u r s e . p ~ s ~ i b l e for the the same ~ r ~ c'e d d r e1 is t r a n s f o r m a t i o n of vectors (or tensors) in spin space into c o r r e s p o n d i n g q u a n t i ties in real s p a c e . 3 T h i s is a feature w h i c h is u n i q u e to P9 n e u t r o n star m a t t e r and w h i c h has ~ot been found for any h y d r o d y n a m i c s y s t e m studied so far and it simply means that spin and orbit indices can be i n t e r c h a n g e d at will! As c a n d i d a t e s for v a r i a b l e s c h a r a c t e r izing b r o k e n symmetries we have to consider the p h a s e d e v i a t i o n 6~ and the dev i a t i o n s from the unit vectors of the t r i a d (6, ~, ~) in spin or orbit space or e q u i v a l e n t l y the r o t a t i o n angles 60. about these axes in th~ c o r r e s p o n d i n g space. As it turns out the c o r r e s p o n ding o ~ e r a t o r s Rave the same s t r u c t u r e as in - H e - A o r ~ - H e - B w i t h o u t e x t e r n a l m a g n e t i c field- and only the c o n t e n t s of the s t r u c t u r e m a t r i x is different. For the c o m m u t a t i o n r e l a t i o n s for the total p a r t i c l e n u m b e r N and for the total angular m o m e n t u m J.=L. + S (where L is the 1 1 1 a n g u l a r m o m e n t u m o p e r a ' o r and S i the oper a t o r for the spin) we find <[6~,
N]> = - 2i
<[68i,N]>
= O
<[6~,Ji]>
= O
<[~Sf~]>
= i 6ik
53
54
It should be s t r e s s e d that c o m m u t a t o r s w i t h L. or S do hot,give rise to a canonica~ r e l a t i o n in 5p~ neutron star matter.' Thus we have i ~ e n t i f i e d the hyd r o d y n a m i c v a r i a b l e s c h a r a c t e r i s i n g spontaneously broken continuous symmetries of -P~ n e u t r o n star matter: One variable, the phase d e v i a t i o n 6~ or the s u p e r f l u i d v e l o c i t y v~ c h a r a c t e r i s e s the b r o k e n gauge i n v a r z a n c e and the three v a r i a b l e s 6 8 are c h a r a c t e r i s i n g the s p o n t a n e o u s l y l . . . . b r o k e n total r o t a t l o n a l lnvarzance, z.e. total a n g u l a r m o m e n t u m J =L + S. serves 1 1 1 as the g e n e r a t o r of a b r o k e n symmetry! The l a t t e r p r o p e r t y is unique to -P? n e u t r o n star m a t t e r and has not beeH found for any h y d r o d y n a m i c s y s t e m s t u d i e d p r e v i o u s l y . It has to be cont r a s t e d e.g. to the s p o n t a n e o u s l y b r o k e n r e l a t i v e spin orbit ~ y m m e t r y of the Bphase of s u p e r f l u i d -He. In this phase one has <[S i,
~0j]>
= - <[L i,
<[Ji'
6@j]>
= O
~0j]>
or
w h e r e a s in all o t h e r h y d r o d y n a m i c systems w ~ t h b r o k e n r o t a t i o n a l s y m m e t r i e s (e.g. He-Al, nematics, a n t i f e r r o m a g n e t s ) these s y m m e £ r i e s are s p o n t a n e o u s l y broken in real and spin space separately• To set up the h y d r o d y n a m i c e q u a t i o n s the p r o c e d u r e is as follows. We start with the Gibbs r e l a t i o n w h i c h defines the t h e r m o d y n a m i c c o n j u g a t e q u a n t i t i e s and then we w r i t e down the c o n s e r v a t i o n and q u a s i - c o n s e r v a t i o n laws for the h y d r o d y namic v a r i a b l e s thus d e f i n i n g the reversible and i r r e v e r s i b l e currents. As conserved q u a n t i t i e s we have the d e n s i t y p, the e n e r g y d e n s i t y e (or e n t r o p y d e n s i t y 0), the d e n s i t y of linear m o m e n t u m g and the d e n s i t y of total a n g u l a r momentum. The four v a r i a b l e s c h a r a c t e r i s i n g spont a n e o u s l y b r o k e n s y m m e t r i e s have a l r e a d y been d i s c u s s e d above• To close the system of the h y d r o d y n a m i c e q u a t i o n s we proceed in three steps. First we e s t a b l i s h the free e n e r g y from w h i c h the t h e r m o d y namic c o n j u g a t e q u a n t i t i e s can be obt a i n e d by d i f f e r e n t i a t i o n . In the s e c o n d step we e x p a n d the i r r e v e r s i b l e currents into the t h e r m o d y n a m i c conjugates. In the last step we derive the r e v e r s i b l e currents. To achieve this, it is necessary, however, to take into a c c o u n t the a n h o l o n o m i t y r e l a t i o n for the hydrod y n a m i c v a r i a b l e s ~0~ (cf ref 4 for the details) • Quite a n a l o±g o u s r e l a t i o n s have been gi v e n p r e v i o u s l y for biaxial nematics by the authors. F u r t h e r m o r e we dem o n s t r a t e a novel p r o c e d u r e to g u a r a n t e e c o n s e r v a t i o n of total a n g u l a r m o m e n t u m locally. This p r o c e d u r e involves the t h e r m o d y n a m i c c o n j u g a t e force of the total a n g u l a r m o m e n t u m density, a q u a n t i t y
which
is local
in space!
A f t e r the d e r i v a t i o n of the compl~te nonlinear h y d r o d y n a m i c e q u a t i o n s of -P? neutron star m a t t e r it is p o s s i b l e to ~heck that e.g. the n o n l i n e a r h y d r o d y n a m i c e q u a t i o n s for biaxial nematics are contained in the p r e s e n t ones as a special case. A very i n t e r e s t i n g static c o u p l i n g w h i c h will p r o b a b l y influence the r o t a t i o n a l dynamics of a n e u t r o n star is found in the e q u a t i o n for the d e n s i t y of linear momentum gi =
"'" + ~ij
ejkl
?k hl
where h I is the t h e r m o d y n a m i c c o n j u g a t e to the m a g n e t i z a t i o n density• The three p h e n o m e n o l o g i c a l c o e f f i c i e n t s w h i c h are i n v o l v e d in ~.. lead to a c o u p l i n g between the v o r t ~ i t y of the m a g n e t i z a t i o n and the d e n s i t y of linear momentum. As h y d r o d y n a m i c e x c i t a t i o n s of the l i n e a r i zed e q u a t i o n s we find three pairs of prop a g a t i n g s p i n - o r b i t - w a v e s w h i c h are c o u p l e d in an i n t r i c a t e m a n n e r to firstand second sound. These c o m p l i c a t e d c o u p l i n g s can be t r a c e d back to the bia x i a l i t y in real and spin space of the p r e s e n t system. To sum up we have d i s c u s s e d the b r o k e n s y m m e t r i e s and the h y d r o d y n a m i c equations of P2 n e u t r o n star m a t t e r and we have found several p r o p e r t i e s w h i c h are unique toa-P? superfluid. The most i m p o r t a n t one a m o n g - t h e s e seems to be the c o n c l u s i o n that total angular m o m e n t u m can serve as the g e n e r a t o r of a s p o n t a n e o u s l y b r o k e n symmetry!
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