SOLITONS IN LOW TEMPERATURE PHYSICS

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SOLITONS IN LOW TEMPERATURE PHYSICS

K. Maki

To cite this version:

K. Maki. SOLITONS IN LOW TEMPERATURE PHYSICS. Journal de Physique Colloques, 1978,

39 (C6), pp.C6-1450-C6-1455. �10.1051/jphyscol:19786586�. �jpa-00218078�

JOURNAL DE PHYSIQUE

Colloque C6, supplPPnent au no 8, Tome 39, aozit 1978, page C6-1450

SOLITONS IN LOW T E M P E R A T U R E PHYSICS

K. Maki

Department of Physics University o f Southern CaZifornia,Los AngeZes, California 90007, U.S.A.

RLsum6.- Les s o l i t o n s s o n t d e s s o l u t i o n s s p d c i a l e s d e s d q u a t i o n s n o n l i n d a i r e s . 11s s o n t d e s o b j e t s l o c a l i s 6 s q u i s e comportent comme d e s c o r p u s c u l e s c l a s s i q u e s . Les s o l i t o n s p a ~ a i s s e n t j o u e r un r 6 l e de p l u s en p l u s important dans l a physique des b a s s e s temp6ratures.

Abstract.- S o l i t o n s a r e s p e c i a l s o l u t i o n s of n o n l i n e a r e q u a t i o n s . They a r e compact o b j e c t s and behave l i k e c l a s s i c a l p a r t i c l e s . S o l i t o n s appear t o p l a y a more and more important r o l e i n low temperature physics.

INTRODUCTION.- The phenomenon of s o l i t o n s o r s o l i t a - r y waves f i r s t d i s c o v e r e d by J . S c o t t R u s s e l l / I / i n t h e summer of 1834, when he was watching a huge bulge of water moving smoothly w i t h a uniform velo- c i t y along a water channel. He r e p o r t e d t h i s remar- k a b l e o b s e r v a t i o n t e n y e a r s l a t e r and he coined t h e term " s o l i t a r y wave". The i n t e r v e n i n g y e a r s were a p p a r e n t l y s p e n t confirming e x p e r i m e n t a l l y t h e e x i s - tence of t h i s phenomenon.

Probably t h e n e x t important e v e n t on t h i s s u b j e c t i s t h e d i s c o v e r y of Korteweg de V r i e s / 2 / e q u a t i o n i n 1895. This i s one of t h e s i m p l e s t non- l i n e a r e q u a t i o n s and d e s c r i b e s behavior of w a t e r waves i n a long shallow channel ; t h e e q u a t i o n h a s a c l a s s of s o l u t i o n s which d e s c r i b e a bulge o f w a t e r moving w i t h uniform v e l o c i t y . A f t e r t h i s e v e n t t h e

s u b j e c t appeared t o be completely f o r g o t t e n by t h e p h y s i c s community u n t i l modern times. I n t h e l a t e s i x t i e s , t h e i n t r i g u i n g s t a b i l i t y of t h e s e s o l u t i o n s were r e v e a l e d through computer experiments. Indeed, Zabusky and Kruskal 131 c h r i s t e n e d i t " s o l i t o n " t o i n d i c a t e t h i s remarkable s t a b i l i t y . Then Gardner e t a l . 141, d i s c o v e r e d t h a t t h e i n i t i a l v a l u e problem of Korteweg de V r i e s (KdV) e q u a t i o n i s solved com- p l e t e l y i n terms of a few s t e p s of l i n e a r o p e r a t i o n

( t h e i n v e r s e s c a t t e r i n g method). I n r a p i d succes- s i o n i t was shown t h a t a s i m i l a r method a p p l i e s t o o t h e r n o n l i n e a r e q u a t i o n s ; t h e c u b i c S c h r s d i n g e r and sine-Gordon (SG) e q u a t i o n s . It i s now known t h a t a l a r g e c l a s s of n o n l i n e a r e q u a t i o n s a r e amen- dable t o t h e i n v e r s e s c a t t e r i n g method 151. Almost a t t h e same time i t was r e a l i z e d t h a t many problems i n p h y s i c s , chemistry, b i o p h y s i c s and e n g i n e e r i n g reduce t o a few s e t s of n o n l i n e a r e q u a t i o n s , which

l e a d t o an e x p l o s i o n of l i t e r a t u r e on s o l i t o n s . I n t h e l i m i t e d time i t i s impossible t o cover a l l as- p e c t s of r e c e n t developments, even i f I c o n f i n e my- s e l f only t o s o l i t o n s i n low temperature p h y s i c s . T h e r e f o r e , I s h a l l t r y t o p r e s e n t h e r e only t h e g i s t of some of t h e r e c e n t developments

161.

CANONICAL NONLINEAR EQUATIONS.- Let us f i r s t w r i t e down some of t h e n o n l i n e a r e q u a t i o n s which appear most f r e q u e n t l y i n l i t e r a t u r e / 7 / .

a ) Korteweg de V r i e s (KdV) e q u a t i o n

@ t ⁺ a@@x ⁺ v@xxx = 0 b ) Cubic Schrodinger e q u a t i o n

i+t + QXX + v1@I2

0

= 0 C ) @'

-

f i e l d t h e o r y

d) Sine-Gordon (SG) e q u a t i o n

@ t t

-

@xx + m2 s i n @ = 0

As a l r e a d y mentioned, KdV e q u a t i o n d e s c r i b e s t h e long water waves i n a shallow channel. The c u b i c Schrsdinger e q u a t i o n d e s c r i b e s t h e s p a t i a l conforma- t i o n of t h e e l e c t r i c f i e l d i n n o n l i n e a r o p t i c s and p o s s i b l y s u p e r f l u i d ' ~ e i n l i n e a r c a p i l l a r y . The

41' -

f i e l d theory d e s c r i b e s t h e l a t t i c e d i s p l a c e m e n t o f t h e system w i t h s t r u c t u r a l phase t r a n s i t i o n / 8 , 9 / . F i n a l l y , t h e SG e q u a t i o n i s u b i q u i t o u s i n low tempe- r a t u r e physics / l o / . Some examples a r e given i n S e c t i o n 111. A l l t h e above e q u a t i o n s p o s s e s s i n ad- d i t i o n t o l i n e a r s o l u t i o n s around t h e e q u i l i b r i u m c o n f i g u r a t i o n , s o l i t o n s o l u t i o n s which a r e summari- zed i n Table 1 . Furthermore t h e s e e q u a t i o n s e x c e p t t h e c a s e c a r e amendable t o t h e i n v e r s e s c a t t e r i n g method /5/.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19786586

Equation Dispersion of

Radiations Soliton Solution

Cubic

Schrzdinger w = k2

SG w2 = k2 + rn2 O = 4 tan-' {exp

[

my ( x - ut)

1

1

TABLE I

The s o l i t o n s o l u t i o n s a r e c h a r a c t e r i z e d by : 1) They a r e l o c a l i z e d s o l u t i o n s ;

2) I n t h e absence of e x t e r n a l ~ e r t u r b a t i o n , th e y move w i t h c o n s t a n t v e l o c i t y . For a small p e r t u r - b a t i o n t h e y respond l i k e c l a s s i c a l p a r t i c l e s ; 3) They a r e q u i t e s t a b l e a g a i n s t p e r t u r b a t i o n . Often

t h e i r s t a b i l i t y i s guaranteed by t h e t o p o l o g i c a l c o n s e r v a t i o n ; t h e y c a r r y t o p o l o g i c a l charges of which sum has t o be conserved throughout physi- c a l p r o c e s s e s .

Because of t h e s e p r o p e r t i e s , s o l i t o n s appear a s a new c l a s s of elementary e x c i t a t i o n s , which resemble remarkably elementary p a r t i c l e s c a r r y i n g conserved quantum numbers. I n f a c t t h e r e i s t h e f a s c i n a t i n g p o s s i b i l i t y t h a t a l l known p a r t i c l e s a r e s o l i t o n s of some underlying f i e l d s and the conserved quantum numbers l i k e e l e c t r i c charge, baryon charge, e t c . , a r e n o t h i n g b u t t o p o l o g i c a l charges.

Among t h e above n o n l i n e a r e q u a t i o n s t h e s i n e - Gordon e q u a t i o n i s remarkable f o r i t s a d d i t i v e topo- l o g i c a l charge. This i s c l o s e l y r e l a t e d t o t h e f a c t t h a t t h e vacuum of t h e sine-Gordon f i e l d i s i n f i n i - t e l y degenerate ; t h e vacuum i s given by

$ = 0, f 2n,

+

471,

...

A s o l i t o n given by

u = 4 tan-1 (ef my ( X

-

u ~ ) ) , w i t h y = ( 1

-

U2)-1/2/2;

c a r r i e s t h e t o p o l o g i c a l charge (1)

Q = -

,,

1

^i,

^dx

^{mX =I}

₂₇₁

^{'(-1 - rn} (--;I

⁼

^{* i}

(2) The sum of t h e t o p o l o g i c a l charges i n t h e presence of many s o l i t o n s and a n t i s o l i t o n s i s t h e n given

C 1

Qtotal = i

Q, _A

⁼

-

_2* ( 4 (rn)

-

^{$ (-my) (3)}

which should c l e a r l y be conserved throughout a l l p h y s i c a l p r o c e s s . This e s s e n t i a l l y g u a r a n t e e s t h e s t a b i l i t y of t h e s o l i t o n .

Furthermore, t h e s o l i t o n - s o l i t o n s c a t t e r i n g and t h e s o l i t o n - a n t i s o l i t o n s c a t t e r i n g a r e e x a c t l y given by 1 1 1 1

4

s i n h (myx) t a n - = 4 u cosh (myut) and

tan

2

= ,-I s i n h (myut)

4 cosh (myx) (5)

r e s p e c t i v e l y where Y = (1

-

u 2 ) - ^'12.

Moreover, t h e bound s t a t e s of s o l i t o n and a n t i s o l i - ton a r e given by

where

Aw, sinh(wr t ) t a n

2

=

-

4

wr

cash (Amox)

These s o l u t i o n s a r e o f t e n c a l l e d b r e a t h e r s , s i n c e they d e s c r i b e l o c a l i z e d o s c i l l a t i o n of

Gt.

More g e n e r a l l y N-soliton, M-antisoliton s o l u t i o n s can be c o n s t r u c t e d e x a c t l y / l o / .

The quantum v e r s i o n of t h e sine-Gordon equa- t i o n i n t h e (1. l ) space-time dimension h a s a number of i n t r i q u i n g p r o p e r t i e s . For example Coleman 1121 e s t a b l i s h e d t h a t t h e SG theory i s e q u i v a l e n t t o t h e massive T h i r r i n g model. Recently t h e e x a c t S - m a t r i c e s d e s c r i b i n g s o l i t o n - s o l i t o n and s o l i t o n - a n t i s o l i t o n s c a t t e r i n g have been o b t a i n e d by Zamolodchikov 1131.

SINE-GORDON SOLITONS.- The sine-Gordon s o l i t o n s appear most commonly i n condensed m a t t e r physics.

We s h a l l s t a r t w i t h t h e s i m p l e s t example / l o / . 1) Mechanical models.

Perhaps t h e s i m p l e s t model i s a s e r i e s of pendula connected by e l a s t i c s t r i n g , which i s most r e a d i l y made by a t t a c h i n g p i n s on a r u b b e r band. The a n g l e

$. t h e i ' t h pendulum makes from t h e v e r t i c a l d i r e c - t i o n obeys t h e e q u a t i o n of motion

where I and m a r e t h e moment of i n e r t i a and t h e mass of t h e pendulum and K i s t h e e l a s t i c c o n s t a n t . I n t h e continuum l i m i t , eq. ( 7 ) reduces t o

$ t t = C: $xx

-

W: s i n $ (8)

where ^,

c,

= ( K / I ) " ~ a ,

w,

= (gm1.0 ^~2

and a i s t h e s p a c i n g between two pendula. A s o l i t o n i n t h e p r e s e n t model i s a 360" t w i s t of t h e rubber band.

JOURNAL DE PHYSIQUE

2) Propagation of d i s l o c a t i o n i n c r y s t a l s . I n 1939 degrees of freedom, t h e amplitude and t h e phase. I n Frenkel and Kontrova 1141 s t u d i e d a one-dimensional p a r t i c u l a r i n a weakly pinned condensate a t low tem- model shown i n f i g u r e 1 t o d e s c r i b e motion of a s l i p p e r a t u r e s , i t s dynamics i s d e s c r i b e d by t h e

i n t h e c r y s t a l . Lagrangian d e n s i t y

At@)

^{= N, {;}

^{@ : -} i

^C:

^{4: -}

^{v($)} (10)}

where $ i s t h e phase of t h e CDW condensate, No(=

-1,

av 1

F

C o and wo a r e t h e d e n s i t y of s t a t e s , t h e pha- son v e l o c i t y and t h e pinning frequency. The s i m p l e s t

__t pinning p o t e n t i a l may be g i v e n by

F i g . 1 : A propagating s l i p i n one-dimensional chain It c o n s i s t s of two c h a i n s of atoms ; t h e lower chain i s assumed t o be f i x e d a t t h e r e g u l a r l a t t i c e posi- t i o n , w h i l e the upper c h a i n of atoms can move. The lower c h a i n c r e a t e s a p e r i o d i c p o t e n t i a l of t h e form A (1

-

cos ( 2 a $ k / a ) ) f o r t h e k-th atom i n t h e upper c h a i n w i t h $k displacement from t h e e q u i l i - brium l a t t i c e p o s i t i o n . The e q u a t i o n of motion f o r

@k i s then given by

2" s i n (2n

4 1 8 )

+ K

[mkil

^{- 2}

m.($k)tt =

- y-

where K i s t h e e l a s t i c c o n s t a n t between two atoms i n t h e upper chain. Again i n t h e cantinuurn l i m i t eq. (9) reduces t o eq. (8), w i t h C o = (Kim) "2 a , t h e v e l o c i t y of t h e sound i n t h e c r y s t a l and w ₀ = (2mt1A)l!~

Here t h e s o l i t o n corresponds t o t h e s l i p ( o r t h e d i s l o c a t i o n ) and has t h e minimum energy

The p r e s e n t model d e s c r i b e s a l o s s l e s s motion of d i s l o c a t i o n ; t h e s l i p moves uniformly w i t h o u t s l o - wing down. The p r e s e n t model i s extended t o c r y s t a l models w i t h two and t h r e e s p a t i a l dimensions.

3 ) $I

-

s o l i t o n s i n quasi-one dimensional charge d e n s i t y wave (CDW) condensatk.

The low temperature p r o p e r t i e s of quasi-one dimen- s i o n a l conductors l i k e K2Pt (CN),,Br 3H20 (KCP),

0 . 3

t e t r a t h i o f ulvalene- t e tracyanoquindime thane (TTF- TCNQ) and NbSe3 a r e of p a r t i c u l a r i n t e r e s t . They undergo a F r i j h l i c h t r a n s i t i o n a t low temperatures.

The a s s o c i a t e d CDW have been seen i n t h e X-ray s c a t - t e r i n g experiment. As p o i n t e d o u t by Lee, Rice and Anderson 1151, t h e ~ r i j l i c h CDW condensate h a s two

V ($) =

p

1 (1

-

cos (N$)) (1 I)

w i t h N i n t e g e r .

The Lagrangian (10) t h e n g i v e s t o a sine-Gordon e q u a t i o n . I n p a r t i c u l a r , R i c e , Bishop, Krumhansland T r u l l i n g e r 1161 p o i n t e d o u t t h a t t h e $ - s o l i t o n p l a y s an important r o l e i n t h e e l e c t r i c c o n d u c t i v i t y a t low temperatures. Equation (10) t e l l s us t h a t t h e

@ s o l i t o n h a s mass M = 2 (2/N12 N m / C

$ 0 0 0 (12)

Furthermore, i t c a r r i e s e l e c t r i c charge

Q = (2e)/N (13)

which f o l l o w s from t h e e x p r e s s i o n of t h e e l e c t r i c charge 1161

q ( x , t ) = @x (x, t ) (14)

where q ( x , t ) i s t h e charge d e n s i t y due t o t h e @ f i e l d . Therefore, a t low t e m p e r a t u r e s , where no t h e r - m a l l y a c t i v a t e d q u a s i - p a r t i c l e s a r e around, t h e e l e c t r i c conduction i s dominantly due t o t h e $-Soli-

t o n s

O$ = k e 2 m o t / ( E t B T )

''1

(ns i n )

'

^{exp (- E}

₄

^{/k T)} _B ^{( 15)}

where R is t h e e l e c t r o n i c mean f r e e p a t h ,

c 2

and ns i s t h e s u p e r f l u i d d e n s i t y .

E$ = M$

This a c t i v a t e d form of t h e c o n d u c t i v i t y has been observed 1171 i n both KPC and TTF-TCNQ. Unfortunate- l y , however, t h i s i s t h e o n l y evidence s o f a r favo- r i n g t h e $ s o l i t o n h y p o t h e s i s . I n o r d e r t o p u t t h i s h y p o t h e s i s on f i r m e r groudd more works b o t h i n theo- r y and experiment a r e c l e a r l y d e s i r a b l e .

4 ) Magnetic f l u x propagation along a Josephson junc- t i o n . F i r s t l e t u s c o n s i d e r two superconductors se- p a r a t e d by a t h i n i n s u l a t i n g b a r r i e r ( s a y oxide f i l m of t h i c k n e s s 10

-

20 AO). As shown by Josephson 1181 i n 1962 t h e p a i r of e l e c t r o n s can t u n n e l from one superconductor t o a n o t h e r l e a d i n g t o t h e weak cou- p l i n g beatween two superconducting wave f u n c t i o n s Y and

Y

a c r o s s t h e j u n c t i o n . I n p a r t i c u l a r when

@ (s

@, - 4,)

0 , where and @2 a r e phases of Y1 and Y,, t h e r e flows a s u p e r c u r r e n t J p e r u n i t a r e a a c r o s s t h e j u n c t i o n

J = J s i n @ (1 6)

where J i s a c o n s t a n t depending on t h e temperature and t h e c h a r a c t e r i s t i c s of t h e j u n c t i o n . Further- more, i f a v o l t a g e V i s a p p l i e d a c r o s s t h e j u n c t i o n ,

t h e phase d i f f e r e n c e

4

i n c r e s e s a s

These a r e t h e c e l e b r a t e d dc and a c Josephson e f f e c t r e s p e c t i v e l y . Now l e t us c o n s i d e r a l a r g e a r e a Josephson j u n c t i o n , which extends i n t h e x-y p l a n e . The l a r g e a r e a Josephson j u n c t i o n i s c h a r a c t e r i z e d i n terms of a t r a n s v e r s e ( s c a l a r ) v o l t a g e V and a t r a n s v e r s e c u r r e n t -f J ( t h e s u r f a c e c u r r e n t along t h e j u n c t i o n ) , which obey / l o /

-f -f

W = L J t ( 1 8 4

-f -f

V

.

^{J =}

^-

^{C V t}

^-

^{J~ s i n @} ( l a b )

+

-f

where J and V a r e t h e two dimensional v e c t o r s , t h e c a p a c i t a n c e C and t h e r e a c t a n c e L of t h e j u n c t i o n a r e

and ^K i s t h e d i e l e c t r i c c o n s t a n t , d i s t h e gap b e t - ween two superconductors, and

X

i s t h e London pe- n e t r a t i o n depth. E l i m i n a t i n g V and J faom eq. ( l a ) , -f

we have

LC @ t t

-

@xx

-

$yy =

-

2e LJ, s i n @ (20) t h e two dimensional SG e q u a t i o n . The s t a t e of t h e j u n c t i o n i s completely determined by

@.

Of t e c h n i c a l importance i s t h e l i n e a r j u n c t i o n , which c o n s i s t s of two t h i n t a p e s of superconductors s e p a r a t e d by a t h i n oxide l a y e r . I n t h i s c a s e eq. (20) reduces t o t h e (one-dimensional) SG e q u a t i o n . The s o l i t o n h e r e d e s c r i b e s moving magnetic f l u x l i n e s a l o n g t h e junc- t i o n . The f l u x l i n e c a r r i e s a s i n g l e f l u x quantum

m

2e and

O o 24 2 x 10-' gauss cm2

These p r o p a g a t i n g f l u x quanta may be used a s a s i g n a l u n i t i n a f u t u r e high speed computer.

So f a r i n d e r i v i n g eq. (20) t h e v o l t a g e l o s s a c r o s s t h e j u n c t i o n i s completely n e g l e c t e d . I f t h e l o s s

mechanism i s i n c l u d e d , t h e moving f l u x w i l l decele- r a t e o r sometimes t h e f l u x may be a n n i h i l a t e d by c o l l i d i n g w i t h t h e a n t i f l u x ( i . e . , t h e magnetic f l u x w i t h o p p o s i t e s i g n ) .

5) S o l i t o n s i n s u p e r f l u i d 3 ~ e . S u p e r f l u i d 3 ~ e con- s i s t s f o two d i s t i n c t phases ( o r t h r e e i n t h e pre- sence of magnetic f i e l d s ) ; 3 ~ e - ~ and 3 ~ e - ~ . The

o r d e r parameters d e s c r i b i n g t h e condensate of 3 ~ e - ~ and 3 ~ e - ~ a r e s i m i l a r t o t h a t of a superconductor except t h a t t h e p a i r i n g t a k e s p l a c e i n t h e s p i n t r i - p l e t and t h e P-wave s t a t e . T h i s i m p l i e s l a r g e de- g r e e s of freedom a s s o c i a t e d w i t h t h e i n t e r n a l confi- g u r a t i o n of t h e condensate (non-Abellian o r d e r para- m e t e r ) . Here I s h a l l d e s c r i b e t h e s i m p l e s t s o l i t o n

t h e $-soliton i n 3 ~ e - ~ . The t e x t u r e of s u p e r f l u i d 3 ~ e - ~ may be c h a r a c t e r i z e d by two u n i t v e c t o r s

2

and

k ,

where

2

d e s c r i b e s t h e d i r e c t i o n of t h e s p i n component of t h e condensate and

?,

denotes t h e symme- t r y a x i s of tihe q u a s i - p a r t i c l e energy gap. For sim- p l i c i t y we s h a l l c o n s i d e r t h e case of a s t a t i c f i e l d a p p l i e d i n t h e z d i r e d t i o n , which f o r c e s t h e

2

vec- t o r t o l i e i n t h e x-y plane. Furthermore, we assume t h a t l i e s i n t h e y d i r e c t i o n . (The d i p o l e energy f a v o r s t h a t

^R

l i e s i n t h e same d i r e c t i o n a s

2 ) .

I n t h i s circumstance 3 ~ e - ~ i s c h a r a c t e r i z e d by two or- d e r parameters A+ and A+ t h e up s p i n p a i r s and t h e down s p i n p a i r s . S i n c e t h e amplitudes of A+ and A+

a r e almost c o n s t a n t , dynamics of A+ and AS a r e des- c r i b e d by t h e i r phases @+ and @+. I n p a r t i c u l a r @ =

@

-

^$+ d e s c r i b e s t h e magnetic p r o p e r t i e s of 3 ~ e - ~ ; 4-

t h e l o c a l m a g n e t i z a t i o n and t h e m a g n e t i z a t i o n cur- r e n t a r e given by

++ s p i n

where

xN

and Ps a r e t h e s t a t i c s p i n s u s c e p t i b i - l i t y and t h e s p i n s u p e r f l u i d d e n s i t y t e n s o r of 3 ~ e - A. We n o t e t h a t @ a l s o ' s p e c i f i e s t h e d i r e c t i o n of

.. 2

from t h e y a x i s ; $ = 28 where I3 i s t h e a n g l e d makes from t h e y a x i s .

The l o c a l s p i n c o n s e r v a t i o n then r e q u i r e s

s i n @ (23)

Where H i s t h e t o t a l Hamiltonian of t h e system, ED i s t h e n u c l e a r d i p o l a r i n t e r a c t i o n energy and QA i s t h e l o n g i t u d i n a l resonance frequency of 3 ~ e - ~ . The r h s of e q r (23) i m p l i e s t h a t t h e l o c a l s p i n conser- v a t i o n i n 3 ~ e - ~ i s broken due t o t h e n u c l e a r d i p o l e

C6-

1454 JOURNAL DE PHYSIQUE

i n t e r a c t i o n , which t r a n s f e r t h e up s p i n p a i r s i n t o the down s p i n p a i r s and v i c e v e r s a . This i s v e r y si- m i l a r t o t h e Josephson e f f e c t , although t h e d i p o l e energy i s bulk a g e n t . S u b s t i t u t i n g eq. (22) i n t o eq.

( 2 3 ) , we have

@ t t

- cq

(@xx ⁺ @ z z )

- crI

+yy =

- "

^{s i n @ (24)}

where CL and CII a r e t h e s p i n wave v e l o c i t i e s perpen- d i c u l a r and p a r a l l e l t o

x.

Equation (24) i s t h e t h r e e dimensional SG e q u a t i o n . I n t h e p r e s e n t c i r - cumstance s o l i t o n s a r e moving domain w a l l s i n t h e t h r e e dimensional space. These s o l i t o n s may be crea- t e d m a g n e t i c a l l y 1191. So f a r we have assumed t h a t

^R

i s f i x e d and uniform i n space. I n a c t u a l s i t u a t i o n s t h e

^d

s o l i t o n i s n o t s t a b l e , s i n c e the energy of t h e domain w a l l i s reduced by allowing t h a t

a

^{is a l s o}

space dependent. The lowest energy c o n f i g u r a t i o n thus o b t a i n e d i s t h e t w i s t composite s o l i t o n . There a r e o t h e r composite s o l i t o n s depending on t h e geome- t r i c c o n s t r a i n t 1201 ( s e e f i g u r e 2 ) .

t h a t SG s o l i t o n s play an important r o l e i n a v a r i e t y of phenomena i n low temperature p h y s i c s . Obviously t h e most u r g e n t q u e s t i o n i s t h a t i f t h e n o t i o n of s o l i t o n can be extended t o t h e h i g h e r dimensions. To e x p l o r e t h i s p o s s i b i l i t y , I would l i k e t o s t a r t w i t h an e l e g a n t n e g a t i v e theorem by Derrick 1231, i n t h e case of a r e a l s c a l a r f i e l d t h e o r y , t h e r e i s no con- f i n e d s o l i t o n i n more t h a n one space dimension. The s c a l a r f i e l d t h e o r y i n d-dimensional space can b e d e s c r i b e d by a Hamiltonian;

w i t h

I n o r d e r t o have a s o l i t o n l i k e s o l u t i o n t h e vacuum @

i

I v ( + ~ ) = 01 have t o be degenerate. Suppose we f i n d a confined s o l u t i o n @ ( x ) i n t h e d-dimensional space w i t h energy E. Then l e t us c o n s i d e r a new so- l u t i o n

Oh

= @(Ax). The energy of $A is o b t a i n e d a s

+

^{+ V}

⁽⁴

^(Ax)) I

= h2-d (KE)@

+

(PE)@

i

(26)

*---

where

a *

_{V ($ ( X I )}

a-

- --

(27) Since b o t h (KE) and (PE) a r e nonnegative, eq. (26)

+) ^{@ @}

a)

i s minimized o n l y f o r h + when d

2

2 , implying

Fig. 2 : Two t y p i c a l composite s o l i t o n s i n super- f l u i d 3He-~, a ) t w i s t s o l i t o n ; and b ) s p l a y s o l i t o n . S o l i d arrows i n i i c a t e t h e d i r e c t i o n of

II

and broken arrows t h a t of d.

These composite s o l i t o n s have d i s t i n c t s a t e l l i t e f r e q u e n a i e s i n t h e n u c l e a r magnetic resonance and a r e r e a d i l y i d e n t i f i e d 1211. I n 3 ~ e - ~ on t h e o t h e r hand, t h e condensate i s c h a r a c t e r i z e d by a r o t a t i o n matrix%

(g,

0 ) , where n -+ i s t h e a x i s of r o t a t i o n and 8 i s t h e r o t a t i o n a n g l e around n. Then

+

8 i s u s u a l l y f i x e d by t h e d i p o l e i n t e r a c t i o n energy a t 0 = =

I

+

cos-I (-

$.

The n v e c t o r i s c o n t r o l l e d then by t h e superweak i n t e r a c t i o n s , which g i v e r i s e t o t h e n

-+

s o l i t o n s . The n s o l i t o n -+ i s c e r t a i n l y the most l i k e - l y c a n d i d a t e , which allows a simple i n t e r p r e t a t i o n of unusual t r a n s v e r s e N M R observed by Osheroff 1221 i n 3 ~ e - ~ .

SOLITONS I N HIGHER DIMENSION.- So f a r we have seen

t h a t t h e r e i s no confined o b j e c t e x c e p t i n t h e one space dimension. This n e g a t i v e r e s u l t may be more simply o b t a i n e d from t o p o l o g i c a l c o n s i d e r a t i o n . There i s no homotopy mapping between d i s c r e t e p o i n t s ( i . e . , t h e vacua of

4)

t o a sphere i n t h e d dimensional space 1241. Therefore, i n o r d e r t o c o n s t r u c t exten- t e d o b j e c t s i n t h e h i g h e r dimensional space, t h e b a s i c f i e l d h a s t o be n e c e s s a r i l y multicomponent.

For example, s u p e r f l u i d ' ~ e o r a superconductor a r e c h a r a c t e r i z e d by a complex s c a l a r (two component) f i e l d and v o r t e x l i n e s i n t h e above system may be considered a s s o l i t o n s i n t h e two space dimension ( i . e . , t h e l i n e a r s o l i t o n ) , s i n c e t h e i r s t a b i l i t y i s guaranteed by t o p o l o g i c a l c o n s e r v a t i o n . A screw d i s l o c a t i o n i n t h e c r y s t a l i s a n o t h e r two dimensio- n a l s o l i t o n i n t h e v e c t o r c r y s t a l f i e l d theory. Simi- l a r l y , monopoles and i n s t a n t o n s 1251 i n gauge theo- r i e s a r e s o l i t o n s i n t h e t h r e e and t h e f o u r space dimension r e s p e c t i v e l y . I n our t h r e e dimensional s p a c e , t h e one dimensional s o l i t o n , t h e two dimen-

s i o n a l s o l i t o n and t h e t h r e e dimensional s o l i t o n

1111

S e e n e r , A., Donth, H . , and ~ o c h e n d b r f e r , A., a r e p l a n a r s o l i t o n , l i n e a r s o l i t o n and p o i n t l i k e Z phys.

134

(1953) 173 ; P e r r i n g , J.K., and Skyrme, T. H.R., Nuclear Phys.

21

(1962) 550.

o b j e c t s a s shown i n f i g u r e 3 .

/12/ Coleman, S., Phys. Rev. D

11

(1975) 2088.

1131 Zamolodchikov, A.B., Pisma JE TF

2

(1977) 499.

F i g . 3 : The d-dimensional s o l i t o n s embedded i n t h e t h r e e d i m e n s i o n a l s p a c e . a ) p l a n a r s o l i t o n ; b) l i n e a r s o l i t o n ; c ) p o i n t - l i k e s o l i t o n ; and

d)

i n s t a n t o n .

I t i s a p l e a s u r e t o thank a l l of t h o s e who h e l p e d me t o u n d e r s t a n d s o l i t o n r e l a t e d problems.

I n p a r t i c u l a r , I have b e n e f i t e d g r e a t l y f r m d i s c u s - s i o n s w i t h Pradeep Kumar, Y.R. Lin-Liu and S t e v e T r u l l i n g e r . I am a l s o g r a t e f u l t o Hans B o z l e r , C h r i s Gould, Doug Osheroff and J o h n Wheatley, who p r o v i d e d me w i t h c o n t i n u o u s s t i m u l i i n t h i s new v e n t u r e . The p r e s e n t work i s s u p p o r t e d by t h e N a t i o n a l S c i e n c e Foundation under Grant No. DMR76-21032.

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66.

1221 O s h e r o f f , D.D., i n Quantum F l u i d s and S o l i d s , e d i t e d bv. T r i c k e v . S.B., Adams, E.D.,

. .

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