d y n a m i c s o f m a g n e t i c v o r t i c e s

8/3/2019 d y n a m i c s o f m a g n e t i c v o r t i c e s

http://slidepdf.com/reader/full/d-y-n-a-m-i-c-s-o-f-m-a-g-n-e-t-i-c-v-o-r-t-i-c-e-s 1/38

Nuclear Physics B360 (1991) 425-462

North-Holland

D Y N A M I C S O F M A G N E T I C V O R T I C E S *

N. PAPANICOLAOU and T.N. TOMARAS

De partm ent o f Physics , Unicersi ty of Crete, an d Research C enter o f Crete,

71409 Iraklion, G reece

Received 13 December 1990

The canonical conservation laws of linear and angular momentum in the ferromagnetic

contin uum have been known to be plagued by certain ambiguities which are resolved in this

paper by constructing conservation laws as suitable moments of a topological density. The

resulting canonical structure is then shown to be analogous to that encountered in the familiar

Hall effect and explains the unusual features of the dynamics of magnetic vortices without

resorting to a detai led solution of the underlying nonline ar equations. Thus, in the absence of

external magnetic fields, a magnetic vortex is shown to be spontaneously pinne d around a fixed

guiding center. The guiding center would drift in a direction perpendicular to an applied

magnetic field gradient, provided that dissipation can be neglected, with a Hall velocity that is

calcu lated explicitly in terms of the initial configuration of the vortex. In the presence of

dissipation, the vortex undergoes skew deflection at an angle ~ ~ 90° with respect to the applied

field gradient. The angle 6 is related to the winding number of the vortex according to the

well-known golden rule of b ubble dynamics.

I . I n t r o d u c t i o n

Topolog ica l so l i tons a r i s e in a va r ie ty o f phys ica l p rob lems and have been the

s u b j e c t o f m u c h s t u d y o v e r t h e l a s t t h i r t y y e a r s o r s o . A m o n g t h e b e s t k n o w n

e x a m p l e s a r e d o m a i n w a l l s a n d m a g n e t i c b u b b l e s i n a f e r r o m a g n e t i c c o n t i n u u m ,

vor t i ces in HeI I o r in a supe rconduc to r , topo log ica l de fec t s in l iqu id c rys ta l s , a s

we l l a s skyrmions and monopoles which a re pa r t i c le - l ike so lu t ions in gene r ic

m o d e l s o f h i g h - e n e r g y p h y si cs .P e r h a p s t h e m o s t u n e x p e c t e d f e a t u r e o f t o p o l o g i c a l s o l i t o n s i s t h e i r u n u s u a l

d y n a m i c a l r e s p o n s e t o e x t e r n a l p r o b e s . T h u s m a g n e t i c b u b b l e s a r e n o t o r i o u s f o r

the i r skew d e f lec t ion un de r the in f luence o f a ma gne t ic f ie ld g rad ien t [1 , 2 ], whi le

v o r t ic e s i n a s u p e r c o n d u c t o r b e h a v e i n a s i m i la r fa s h i o n [ 3 ]. F u r t h e r m o r e r e c e n t

theore t i ca l s tud ie s in re la t iv i s t i c f i e ld theor ie s [4 -6 ] have demons t ra ted tha t

s k y r m i o n s a n d m o n o p o l e s w o u l d e x h i b it s c a t te r i n g p a t t e r n s t h a t a r e h i g h l y u n u s u a l

f rom the po in t o f v iew of fami l i a r s ca t t e r ing p roces ses o f o rd ina ry pa r t i c le s , bu t

s t ro n g l y r e m i n i s c e n t o f t h e s k e w d e f l e c ti o n o f m a g n e t i c b u b b l e s.

* Dedicated to the m e m o r y of our colleagu~:s Basilis Xanthopoulos and Stcphanos Pnevmatikos.

I ) 5 5 0 - 3 2 1 3 / 9 1 / $ ( 1 3 . 5 ( I ,c> I ~)9 1 - E l s c v i u r S c i e n c e P u b l i s h e r s B . V . ( N o r l h - H o l h m d )



426 N. Papanicolaou, T.N . Toma ras / Magn etic cortices

I t i s c l e a r t h a t a f u n d a m e n t a l s i m p l i c i t y u n d e r l i e s t h e d y n a m i c s o f t o p o l o g i c a l

s o li to n s , th e f u ll e x t e n t o f w h i c h h a s n o t y e t b e e n d e t e r m i n e d . T h e p u r p o s e o f th i s

a r t ic l e i s t o p r e s e n t s o m e n e w r e s u l t s i n t h a t d i r e c t i o n u s i n g m a g n e t i c v o r t ic e s a n d

b u b b l e s a s e x a m p l e s . T o b e s u r e , t h e e a r l i e r w o r k o n m a g n e t i c b u b b l e s i s a l r e a d y

o n r e a s o n a b l y s o l i d t h e o r e t i c a l g r o u n d s . Y e t w e b e l i e v e t h a t t h e e a r l i e r t h e o r e t i c a la r g u m e n t s h a v e n o t b e e n s u f f i c ie n t ly a p p r e c i a t e d b y t h e w i d e r p h y s i c s c o m m u n i t y ,

p r o b a b l y b e c a u s e t h e y h a d n o t s u c c e e d e d i n r e v e a l i n g t h e f u l l s i m p l i c i t y o f t h e

u n d e r l y i n g d y n a m i c s .

O u r s t a r t i n g p o i n t i s t h e s e m i - p h e n o m e n o l o g i c a l o b s e r v a t i o n t h a t t h e m o t i o n o f

b u b b l e s a n d v o r t i c e s i n e x t e r n a l f i e l d g r a d i e n t s c l o s e l y r e s e m b l e s t h e f a m i l i a r H a l l

m o t i o n o f e l e c t r o n s i n e x t e r n a l m a g n e t i c a n d e l e c t r i c f i e l d s [ 7 ] . I n o u r e f f o r t t o

m a k e t h i s a n a l o g y p r e c i s e , w e w e r e l e d t o i n t r o d u c e s o m e r a d i c a l c h a n g e s i n t h e

c a n o n i ca l d e f in i ti o n o f th e c o n s e r v a t io n l aw s o f m o m e n t u m a n d a n g u l a r m o m e n -

t u m w h i c h h a ve b e e n k n o w n t o be p l a g u e d b y c e r t ai n a m b i g u it ie s . U n a m b i g u o u s

c o n s e r v a t i o n la w s a r e c o n s t r u c t e d h e r e a s s u i ta b l e m o m e n t s o f a t o p o l o g i c a l

c u r r e n t d e n s i ty . T h e r e s u l t in g c a n o n i c a l s t r u c t u r e is t h e n s h o w n t o b e c o m p l e t e l y

a n a l o g o u s t o t h a t e n c o u n t e r e d i n t h e H a l l e f f e c t a n d e x p l a i n s t h e g r o s s f e a t u r e s o f

v o r t e x d y n a m i c s w i t h o u t g r e a t e f f o r t.

T h e m a t h e m a t i c a l b a s i s f o r t h e s t u d y o f t h e t e r r o m a g n e t i c c o n t i n u u m i s p r o -

v i d e d b y t h e L a n d a u - L i f s h i t z e q u a t i o n w h i c h i s b r i e f l y d e s c r i b e d i n s e c t . 2 a s a n

i n t r o d u c t i o n t o o u r m a i n t a sk , th e c o n s t r u c t i o n o f u n a m b i g u o u s c o n s e r v a t i o n l a w s

g i v e n i n s e c t . 3 . S u b s e q u e n t s e c t i o n s a r e t h e n d e v o t e d t o a d e t a i l e d d i s c u s s i o n o f

t h e i m p l i c a t i o n s f o r d y n a m i c s . H e n c e t h e p o s s i b l e t y p e s o f s t e a d y m o t i o n i n t h e

absen ce o f ex t e r na l f i e l d s a r e a na l yze d i n s ec t. 4. Th e e f f e c t o f an ex t e r na l f i e l dg r ad i en t i s s t ud i ed i n s ect . 5 , and t he e f f ec t o f d i s s i pa t i on i n sec t . 6. I n t he

c o n c l u d i n g s ec t. 7 w e c o m p l e t e s o m e o f t h e t h e o r e t i c a l a r g u m e n t s a n d c o n t e m p l a t e

g e n e r a l i z a t i o n s t o o t h e r f i e l d t h e o r i e s . F i n a l l y , f o r c o m p a r i s o n p u r p o s e s , w e

s u m m a r i z e i n a p p e n d i x A t h e c a n o n i c a l s t r u c t u r e a s s o c i a t e d w i t h t h e u s u a l H a l l

m o t i o n .

2 . T h e L a n d a u - L i f s h i t z e q u a t i o n

T h e c o n t i n u u m a p p r o x i m a t i o n i s a p p r o p r i a t e f o r th e s t u d y o f m a c r o s c o p i c

d o m a i n s i n w h i c h s u b s t a n t i a l s p a t i a l v a r i a t i o n s o c c u r o n l y o v e r a l a r g e n u m b e r o f

l a tt ic e s p a ci n g s. H e n c e a f e r r o m a g n e t i c m e d i u m is c h a r a c t e r i z e d b y t h e m a g n e t i z a -

t i on M = M ( x , t ) w h i ch i s som e f unc t i on o f pos i t i on x a nd t i m e t and o beys t he

L a n d a u - L i f s h i t z e q u a t i o n [ 8 ]

c~M= M x F , M 2 = 1 , ( 2 . 1 )



N. Papanicolaou, T.N. Tomaras / Magnetic cortices 427

w h i c h d e s c r i b e s p r e c e s s i o n o f t h e m a g n e t i z a t i o n a r o u n d t h e e ff e c ti v e m a g n e t i c

f ie ld

F = A M - A M + H . ( 2 . 2 )

H e r e t h e f i rs t t e r m is t h e e x c h a n g e f i e ld , A b e i n g t h e u s u a l L a p l a c e o p e r a t o r . T h e

s e c o n d t e r m i s a n a b b r e v i a t i o n f o r t h e v e c t o r

A M = a ( M l , M z , 0 ) , a > 0 , ( 2 . 3 )

a n d a c c o u n t s f o r u n i ax i a l m a g n e t i c a n i s o tr o p y . T h i s c h o i c e o f t h e a n i s o t r o p y f i e ld

c o r r e s p o n d s t o t h e m o s t i n t e r e s t in g c a s e o f a m a g n e t w i t h a n e a s y a x is , b u t o t h e r

ch o ices can b e t r ea t ed a lo n g s imi l a r l i n es . F in a l ly t h e t h i rd t e rm in eq . (2 .2 ) i s t h e

m a g n e t o s t a t i c f i e l d p r o d u c e d b y t h e m a g n e t i z a t i o n a n d s a t i s f i e s t h e M a x w e l l

e q u a t i o n s

V × H = 0 , V . ( H + 4 r r M ) = 0 , ( 2 . 4 )

w h e r e t i m e d e r i v a t i v e s a r e n e g l e c t e d b e c a u s e t y p i c a l v e l o c i t i e s o c c u r r i n g i n m a g -

n e t i c s t ru c tu res o f p rac t ica l i n t e res t a r e sm al l e r t h an th e v e lo c i ty o f li g h t b y sev era l

o r d e r s o f m a g n i t u d e .

N o t e t h a t t h e c o n s t a n t m a g n i t u d e o f th e m a g n e t i za t io n v e c t o r M c a n b e se t

eq u a l t o u n i ty , a s was a l r ead y d o n e in eq . (2 .1 ), b y a su i t ab le r esca l in g o f t h e t ime

v a r i a b l e a n d a c o r r e s p o n d i n g c h o i c e o f u n it s f o r t h e m a g n e t i c f ie l d . Si m i la r ly a n

ex c h an g e co n s t an t w as sca l ed o u t o f t h e f i rs t t e rm in eq . (2 .2) b y a r esca l in g o f t h es p a t i a l c o o r d i n a t e s .

O n e m a y f o r m a l l y s o l v e t h e m a g n e t o s t a t i c e q u a t i o n s ( 2 . 4 ) t o w r i t e

H = v f (V'M)(x',t)i x = x ' l d3x ' , ( 2 . 5 )

wh ich may th en b e in se r t ed in eq s . (2 .2 ) an d (2 .1 ) t o o b ta in an eq u a t io n th a t

i n v ol v es o n l y th e m a g n e t i z a t io n . T h e l o n g - r a n g e n a t u r e o f th e m a g n e t o s t a t i c f ie l d

(2 .5 ) is a t ech n ica l h in d ran c e b u t c l ea r ly n o t t h e o n ly d if f icu l ty wi th t h i s h ig h lyn o n l i n e a r e v o l u t i o n e q u a t i o n . N e v e r t h e l e s s , a n a ly t ic a l so l u ti o n s h a v e b e e n f o u n d i n

i m p o r t a n t s p e c ia l c a se s . F o r i n s ta n c e , t h e s t u d y o f o n e - d i m e n s i o n a l s o l u ti o n s

(d o main wa l l s ) i s e s sen t i a l l y co mp le t e [9 ] an d p ro v id es mu ch in tu i t i o n fo r t h e

an a ly s i s o f h ig h er -d im en s io n a l s t ru c tu res su ch as mag n e t i c b u b b les [1 , 2 ].

Th e f i r s t an d mo s t imp o r t an t s t ep i s t o f i n d s t a t i c so lu t io n s , wh ich sa t i s fy eq .

(2 .1 ) wi th t h e t ime d e r iv a t iv e ab sen t . Th e s imp les t s t a t i c so lu t io n i s t h e o n e wh ere

th e mag n e t i za t io n i s u n i fo rm, p o in t in g in t h e p o s i t i v e o r n eg a t iv e t h i rd d i r ec t io n .

On e i s a l so i n t e res t ed in s t a t i c mag n e t i c s t ru c tu res i n wh ich th e mag n e t i za t io n

ap p rec i ab ly d ev ia t es f ro m th e u n i fo rm co n f ig u ra t io n o v er a mo re o r l e s s f i n i t e



4 2 8 N. Papanicolaou, T.N. Toma ras / Magn etic cortices

r e g i o n , s o t h a t t h e i r e n e r g y b e f i n it e . I t i s t h u s i m p o r t a n t t o e x a m i n e a t t h i s p o i n t

t h e e n e r g y f u n c ti o n a l a s s o c ia t e d w i th t h e L a n d a u - L i f s h i t z e q u a t i o n .

T h e e n e r g y fu n c t io n a l w il l b e d e n o t e d b y W a n d c o n s is t s o f t h r e e t e r m s

c o r r e s p o n d i n g t o t h e t h r e e t e r m s i n t h e e f f e c t i v e f i e ld F o f e q . (2 . 2) :

w = + w . + W i n, ( 2 . 6 a )

w h e r e W e i s t h e e x c h a n g e e n e r g y

~ = f we d3x , W e - ½ ( a iM ' O i M ) , ( 2 . 6 b )

Wa i s t h e a n i s o t r o p y e n e r g y

= f W a d 3 x , w~ = ½ a ( M ? + M ~ ) , ( 2 . 6 c )

a n d Wm i s t h e m a g n e t o s t a t i c e n e r g y

1f ,

m - - w m d 3 x , Wm = 8 r r H 2 ( 2 . 6 d )

w h e r e i t i s u n d e r s t o o d t h a t t h e m a g n e t o s t a t i c f i e l d is e x p r e s s e d i n t e r m s o f t h e

m a g n e t i z a t i o n t h r o u g h e q . ( 2 . 5 ) . N o t e t h a t w e h a v e u s e d v e c t o r n o t a t i o n a s m u c h

a s p o s s i b l e , e x c e p t w h e r e a m b i g u i t i e s m a y a r i s e . T h u s , i n e q . ( 2 . 6 b ) , O d e n o t e sd er iv a t iv e w i th r esp e c t to th e sp a t i a l co o rd in a te x i , w i th i = 1 , 2 o r 3 , an d th e u su a l

s u m m a t i o n c o n v e n t i o n f o r r e p e a t e d i n d i c e s i s i n v o k e d .

I t i s n o t d i f f i cu lt t o sh o w th a t th e e f fec t iv e f i e ld o f eq . (2 .2 ) i s r e l a t ed to th e

e n e r g y b y

F = - 6 W / 6 M . ( 2 . 7 )

F u r t h e r m o r e s t at ic s o lu t io n s a r e s t a t i o n a r y p o i n ts o f W , p r o v i d e d t h a t t h e c o n -

s t r a i n t M 2 = 1 is t a k e n i n t o a c c o u n t . T h e r e f o r e t h e e x i s t e n c e o f n o n t r i v ia l s t a t i c

s o l u t i o n s i s s u b j e c t t o l i m i t a t i o n s d e r i v i n g f r o m t h e w e l l - k n o w n s c a l i n g a r g u m e n t o f

D er r i ck [1 0 ] .

T h u s i f w e a s s u m e t h a t M ( x ) i s a s t a t i c so lu t io n w i th e n er g y W = We + W., + Wm,

t h e e n e r g y o f t h e c o n f i g u r a t i o n M ( x / A ) , w h ere A i s so m e c o n s ta n t , i s g iv en b y

W ( ~ . ) = A W + ,~ .3 (W + W m ). B y o u r h y p o t h e s i s , A = 1 is a s t a t i o n a r y p o i n t o f W ( A ) ,

i .e. W '(A = 1) = 0 , or

W~ + 3(W~, + W in) = 0 , ( 2 . 8 )

w h ich m ay b e th o u g h t o f as a v i r ia l r e l a t io n th a t mu s t b e sa t i s f i ed b y an y s t a t i c

s o l u ti o n . O n t h e o t h e r h a n d , a ll te r m s o n t h e l e f t - h a n d s i d e o f e q . (2 . 8) a r e



N. Papanicolaou, T.N. Tomaras / Ma gnetic cortices 429

n o n - n e g a t i v e , a s i s e v i d e n t f r o m e q . (2 .6 ) a n d t h e c h o i c e o f t h e a n i s o t r o p y f i e ld

( 2 . 3 ) . The r e f o r e , f o r any s t a t i c so l u t i on w i t h f i n i t e ene r gy , t he v i r i a l r e l a t i on ( 2 . 8 )

c a n b e s a t i s f i e d o n l y i f W e - W - W m = 0 , w h i c h i m p l i es t h a t a n y s u c h s o l u t i o n

c o i n c i d e s w i t h t h e u n i f o r m c o n f i g u r a t i o n e x c e p t p o s s ib l y a t i so l a t e d p o i n t s . O n e

m u s t c o n c l u d e t h a t n o n t r iv i a l s t a t ic s o l u t io n s w i t h f i n it e e n e r g y d o n o t e x i s t in t h et h r e e - d i m e n s i o n a l f e r r o m a g n e t i c c o n t i n u u m w i t h o u t b o u n d a r i e s .

S u c h a d i s a p p o i n t i n g c o n c l u s i o n i s a v e r t e d i n p r a c t i c e b y t h e t ' m it e t h i c k n e s s o f

t h e f i lm t o g e t h e r w i t h t h e e f f e c t o f t h e m a g n e t o s t a t i c f ie l d a n d o f a n a p p l i e d b ia s

f i e l d . A s a r e s u l t , a r i c h v a r i e t y o f i n t e r e s t i n g s t a t i c s o l u t i o n s e m e r g e , k n o w n a s

m a g n e t i c b u b b l e s . T h e s e c o n f i g u r a ti o n s a r e p r e d o m i n a n t l y t w o - d i m e n si o n a l, i n th e

s e n s e t h a t s i g n i f i c a n t v a r i a t i o n s o f t h e m a g n e t i z a t i o n o c c u r o n l y a l o n g t h e f d m

d i r e c t i o n s . H o w e v e r t h e m a g n e t i z a t i o n v a r i e s t o s o m e e x t e n t a l s o a l o n g t h e t h i r d

d i r e c t i o n , t h e d i r e c t i o n p e r p e n d i c u l a r t o t h e f i l m , l e a d i n g t o t h e f a m i l i a r b a r r e l -

s h a p e d m a g n e t i c b u b b l e s [ 1 , 2 ].

T h e q u a s i - tw o - d i m e n s i o n a l n a t u r e o f th e m a g n e t i c b u b b l es l e a d s u s t o r e co n -

s i d e r D e r r i c k ' s s c al in g a r g u m e n t f r o m t h e p o i n t o f vi ew o f lo w e r d i m e n s i o n ( d = 1

o r 2 ). B y t h i s w e m e a n t o s e a r c h f o r s t at ic s o l u t i o n s o f t h e f o r m M = M(x~) o r

M = M ( x i, x E ) a n d i g n o r e i n t h e d e f i n i t io n o f e n e r g y t h e t ri v ia l i n t e g r a t i o n s o v e r

t h e r e m a i n i n g 3 - d d i m e n s i o n s . O n e s h o u l d a l so n o t e t h a t t h e m a g n e t o s t a t i c f ie l d

is n o l o n g e r g iv e n b y e q . ( 2.5 ). H e n c e , fo r d - 1, H = ( - 4 r i M 1 , 0 , 0 ) , w h e r e a s f o r

d = 2 t h e m a g n e t o s t a t i c f i el d i s o f t h e f o r m H = ( H ~ , H E, 0) and i s g i ven by an

e q u a t i o n a n a l o g o u s t o ( 2 . 5 ) w i t h t h e k e r n e l r e p l a c e d b y a l o g a r i t h m . W i t h t h e s e

p r o v i s i o n s i n m i n d , t h e d - d i m e n s i o n a l a n a l o g o f eq . ( 2 .8 ) is f o u n d t o b e

( d - 2 )We+d (W~ ,+Wm)=O . ( 2 . 9 )

A pp l i ed f o r d = 1 , eq . ( 2 . 9 ) y i e l ds t he v i r i a l r e l a t i on W = W + Wm w h i c h d o e s n o t

e x c l u d e n o n t r i v i a l s t a t ic s o l u t io n s a n d is i n d e e d v e r i f i e d b y t h e f a m i l ia r d o m a i n

w a l l s . H ow eve r , f o r d = 2 , t he v i r i a l r e l a t i on r eads W + W m = 0 a n d a g a i n e x c l u d e s

n o n t r i v i a l s t a t i c s o l u t io n s . T h e r e f o r e , s tr ic t ly t w o - d i m e n s i o n a l s t a ti c m a g n e t i c b u b -

b l e s a r e n o t p o s s i b l e f o r t h e t y p e o f i n t e r a c t i o n s c o n s i d e r e d s o f a r .

H o w e v e r , s t ri c tl y tw o - d i m e n s i o n a l s t a ti c v o r t i c es m a y e x i s t i f t h e L a n d a u - L i f s h i t z

e q u a t i o n i s m o d i f i e d a p p r o p r i a t e l y . F o r e x a m p l e , i f o n l y t h e e x c h a n g e f ie l d isr e t a i ned i n eq . ( 2 . 1 ) ;

OM= M × A M , M 2 = 1 , ( 2 . 1 0 )

d t

t h e n t h e v i ri a l re l a ti o n ( 2.9 ) r ed u c e s t o ( d - 2 ) W e - 0 w h i ch c o n t r ad i c ts t h e

e x i s t e n c e o f n o n t r i v ia l s t a ti c s o l u ti o n s a t d - 1 a n d d - - 3 b u t n o t a t d = 2 .

A l t hough t he m ode l i n eq . ( 2 . I 0 ) i s t oo s i m p l e t o be use f u l f o r r ea l i s t i c bubb l e

ca l cu l a t i on s , i t is no t en t i r e l y w i t hou t p r ac t i ca l i n t e r e s t b ecau se i t de sc r i b e s t he



430 N. Papanicolaou, T.N. Tomara s / Magn etic vort ices

long-wavelength limit of the isotropic Heisenberg ferromagnet. In other words,

solutions of eq. (2.10) could in principle be of some value in a semiclassical study

of quantum ferromagnet s [13, 17]. At any rate, this simple model will often be used

in the present work for purposes of illustration.

A more transparent form of eq. (2.10) is obtained by explicitly resolving theconstraint M 2 = 1. Thus we may use either spherical variables O and • for the

parametrization of the unit vector M, or the stereographic variable

M t + i M 2 iO = = tan ~O exp (i ~ ). ( 2.11 )

1 + M 3

The cartesian components of the magnetization are then given by

O + 0

M t= _ =s in Oco s1 + O O

B

1 O-g2M2 = i 1 + ~ O = sin O sin ~ ,

1 - 0 0M3 = _ = c o s O (2.12)

" 1 + 0 0

where O is the complex conjugate of 0. In terms of the complex variable O, eq.

(2.10) reads

,90 2 0 ( I70. VO)+ an = _ . ( 2 . 1 3 )

i o t 1 + 0 0

We further restrict our attention to two-dimensional solutions; 0 = O(x~, x 2 , t ) .

Using the complex coordinates z = x t +/ x 2 and 2 =x t - / x 2, eq. (2.13) becomes

i & O 2~0~.~z- ~ + O ~ z = _ , ( 2 . 1 4 )4 0 t 1 + 0 0

where subscripts denote differentiation with respect to z and 2. Thanks to the

conformal invariance of the static sector of eq. (2.14), a large class of static

solutions can be constructed by choosing O to be any function of z or 2 alone; i.e.

O - O ( z ) or O = O(,~). However most of these solutions have infinite energy.

Expressing the exchange energy in current notation we find that

w . = f 4 ( 1 0 z l 2 + 10:12) d2x

( l + . o . o ) "( 2 . 1 5 )



N. Papanicolaou, T.N. Tomaras / Magnetic cortices 43 1

I n o r d e r t o s o r t o u t f i n i t e - e n e r g y s o l u t i o n s a n d u n d e r s t a n d t h e i r s i g n i f i c a n c e , w e

m u s t r e c a l l a t th i s p o i n t t h e i m p o r t a n t n o t i o n o f a w i n d i n g n u m b e r . W e s e a r c h fo r

s o l u t i o n s a p p r o a c h i n g a t l a r g e d i s t a n c e s t h e u n i f o r m c o n f i g u r a t i o n w h i c h p o i n t s i n ,

say , the posi t ive th i rd d i rect ion (g~-- ->0 f o r I z l -- * ~ ) . T h e n t h e t w o - d i m e n s i o n a l

p l an e (x ~, x 2 ) is t o p o lo g ica l ly eq u iv a l en t t o a sp h ere , an d M (x ~, x 2 ) d e f in es a m apb e t w e e n t h e p l a n e a n d t h e s p h e r e M 2= 1 . T h e n u m b e r o f ti m e s t h e u n i t v e c t o r

M(x~, x 2 ) c o v e r s t h e s p h e r e M z = 1 a s x~ a n d x 2 v a ry t h r o u g h t h e p l a n e is g iv e n

b y

1 f iQ = 4~r qd 2x ' q= ~e#,, (O, ,MXO uM)'M, ( 2 . 1 6 )

w h e r e t h e G r e e k i n d ic e s / ,~ , u , . . . t a k e t h e t w o v a lu e s 1 o r 2 a n d eu~ i s the

t w o - d i m e n s i o n a l a n t i s y m m e t r i c t e n s o r . Q is i n t e g e r v a l u e d ( Q = 0 , + 1 , _+ 2 , . . . )

f o r c o n t i n u o u s f i e l d c o n f i g u r a t i o n s a n d w i l l b e r e f e r r e d t o a s t h e w i n d i n g n u m b e ro r t o p o lo g ica l ch arg e , wh i l e q wi l l b e ca l l ed th e t o p o lo g ica l d en s i ty . Fo r fu tu re

r e f e r e n c e , w e a l s o ex p r e s s q a s

4 ( l a ~ l 2 - I g2 :l2 )

q = ( 1 + , ~ g ] ) 2 - - e~,~ s in O 0 ~ O 0 u ~ . ( 2 . 1 7 )

O f c o u r s e , t h e p r e c e d i n g d e f i n i ti o n o f t h e w i n d i n g n u m b e r is n o t s p e c i a l t o th e

m o d e l ( 2 . 1 4 ) b u t a p p l i e s t o a ll v a r i a ti o n s o f t h e L a n d a u - L i f s h i t z e q u a t i o n e x a m -

in ed in t h e p ap er , a s we l l a s t o r e l at i v is t ic n o n l in ea r t r -mo d e l s i n 2 + 1 d ime n -s ions [6 ] .

R e tu rn in g to t h e p ro b lem a t h an d , we n o te t h e i n eq u a l i t y We >_ -4 ~ rlQI wh ich

f o ll o w s f r o m a s im p l e c o m p a r i s o n o f t h e e n e r g y d e n s i t y in e q . ( 2 . 1 5 ) w i t h t h e

to p o lo g ica l d en s i ty (2 .1 7 ) . Fo r s t a t i c co n f ig u ra t io n s o f th e fo rm O (z ) o r .Q(~,) t h e

in eq u a l i t y i s sa tu ra t ed ; W = 4 ~ ' IQI . Hen ce , t o o b ta in so lu t io n s wi th f i n i t e en erg y ,

i t i s su f f i c i en t t o secu re a f i n i t e win d in g n u mb er , wh ich i s ach i ev ed wi th t h e ch o ice

~Q----"/'/'I(Z)/TFE(Z) o r " / ' / ' l ( Z ) / ' / ' / ' 2 ( z ) w h ere 1 r I an d "/T2 are a rb i t r a ry p o ly n o mia l s .

T h e s e a r e t h e w e l l- k n o w n B e l a v i n - P o l y a k o v i n s t a n t o n s [ 1 1 ], v i e w e d h e r e a s s ta t ic

m a g n e t i c v o r t i c e s i n a 2 + 1 - d i m e n s io n a l f e r r o m a g n e t i c c o n t i n u u m .F o r i l l u s t r a t i o n , w e q u o t e t h r e e e l e m e n t a r y e x a m p l e s ;

~ = a / z , ~ / Y . a n d ( ~ / ~ ) 2 , ( 2 . 1 8 )

wi th win d in g n u m b ers Q = - 1, 1 an d 2 , r e sp ec t iv e ly . Here a i s an a rb i t r a ry

c o m p l e x c o n s t a n t r e f le c t in g i n p a r t t h e c o n f o r m a l i n v a r i an c e o f t h e s t a t ic s e c t o r o f

eq . (2 .1 4 ) . Th e co n f ig u ra t io n s (2 .1 8) a re sch em at i ca l ly d ep ic t ed in fi g. 1; t h e

m a g n e t i z a t i o n is s h o w n o n l y a r o u n d t h e c ir c le Iz i = l a l , w h e r e its t h i r d c o m p o n e n t

v an i sh es , an d a t z = (i, wh ere i t p o in t s t o wa rd th e so u th p o le . Th e ma g n e t i za t io n



4 3 2 N. Papanicolaou, T.N. Tomaras / Magnetic cortices

0

O O O O

0 = -1 Q = I 0 = 2

F i g . l . S c h e m a t i c r e p r e s e n t a t i o n o f m a g n e t i c v o rt i ce s w i t h w i n d i n g n u m b e r s Q = - 1 , 1 a n d 2 .

p o i n t s t o w a r d t h e n o r t h p o l e a t s p a t i a l i n f i n i t y , i n a c c o r d a n c e w i t h o u r s t a n d a r d

c o n v e n t i o n . T h e p h a s e o f a w a s c h o s e n e q u a l t o 7 r / 2 .

A l t h o u g h t h e r e s u l t i n g p i c t u r e s a r e e s s e n t i a l l y i d e n t i c a l t o t h o s e u s u a l l y d r a w n

f o r r e a l i s t i c m a g n e t i c b u b b l e s , o n e s h o u l d k e e p i n m i n d t h a t t h e B e l a v i n - P o l y a k o v

i n s t a n t o n s a r e m a r g i n a l e x a m p l e s o f ( m e t a s t a b l e ) m a g n e t i c v o r t ic e s b e c a u s e o f t h e

u n d e r l y i n g c o n f o r m a l i n v a r i a n c e . M o r e " r e a l i s t i c " s t r ic t ly tw o - d i m e n s i o n a l m a g -

n e t i c v o r t i c e s a r e o b t a i n e d b y a d d i n g t o t h e e n e r g y f u n c t i o n a l ( 2 . 6 ) t e r m s t h a t

con t a in h ighe r -o rde r spa t i a l de r i va t i ves , such a s

J w = f A M + O M )4 I • ( 2 . 1 9 )

T he sca l i ng r e l a t i on (2 . 9 ) becomes

( d - 2 ) W + d ( W a + W m) = ( 4 - d ) A W (2.20)

and does n o t con t r a d i c t t he ex i s t ence o f non t r i v i a l s ta t i c so lu t i ons a t d = 2 , no t

e v e n a t d = 3 . S u c h m o d e l s h a v e a t t r a c t e d s o m e a t t e n t i o n i n r e c e n t l i t e r a t u r e [ 1 2 ] .

T h u s w e c o n c l u d e o u r b r i e f s u r ve y o f b a s ic f a ct s a b o u t t h e L a n d a u - L i f s h i t z

e q u a t i o n a n d t u r n t o t h e m a i n t a s k o f t h i s p a p e r .

3 . C o n s e r v a t i o n la w s

W e n o w b e g i n t o a d d r e s s s o m e d y n a m i c a l q u e s t i o n s . A s a f i r s t s t e p w e i n t e r p r e t

t h e L a n d a u - L i f s h i t z e q u a t i o n a s a h a m i l to n i a n s y st e m , w i th h a m i l t o n i an W =

I V ( M ) e n d o w e d w i t h t h e P o i s s o n b r a c k e t s

{ M , ( x ) . M , ( x ' ) } = - M , ( x ) 8 ( x - x ' ) . ( 3 . 1 )



N. Papanicolaou, T.N. Tomaras / Magnetic vortices 433

T h e h a m i l t o n e q u a t i o n is t h e n g i v e n b y

O M O M 6 W

= + M × - - = O ( 3 . 2 )O t + { M ' W } o t '

a n d c o i n c i d e s w i th t h e g e n e r a l f o r m o f t h e L a n d a u - L i f s h i t z e q u a t i o n ( 2 .1 ) w i th t h e

e f fec t iv e f i e ld F g iv en b y eq . (2 .7 ).

T o c o m p l e t e t h e c a n o n i c a l d e s c r i p t io n w e m u s t a ls o e x a m i n e t h e a l g e b r a i c

c o n s t r a i n t p r e s e n t i n e q . (2 .1 ). T h u s w e n o t e t h a t t h e f ie l d M 2 c o m m u t e s w i t h th e

h a m i l t o n i a n a n d m a y b e s e t e q u a l t o a n y p r e s c r i b e d f u n c t i o n o f t h e s p a t ia l

c o o r d i n a t e s , t h e s p e ci fi c c h o i c e M 2 = 1 b e i n g t h e o n e o f p h y si c a l in t e r e s t. T h i s

s i t u a t i o n i s r e m i n i s c e n t o f a g a u g e t h e o r y , t h e c o n s t r a i n t p l a y i n g th e r o l e o f, s a y,

t h e G a u s s l a w i n a h a m i l t o n i a n f o r m u l a t i o n o f e l e c tr o d y n a m i c s . I n d e e d t h e

L a n d a u - L i f s h i t z e q u a t i o n m a y b e v i e w e d a s a U ( 1 ) g a u g e t h e o r y [ 1 3 ] . B e c a u s e t h e

c o n s t r a i n t i s a l g e b r a ic , a n d a g a u g e c o n d i t i o n m a y b e c h o s e n t h a t is a l s o a lg e b r a ic ,

i t c a n b e r e s o l v e d e x p l ic it ly t o e x p r e s s th e t h r e e c o m p o n e n t s o f t h e m a g n e t i z a t i o n

i n te r m s o f tw o r e a l v a r i a b l e s t h a t a r e c a n o n i c a ll y c o n ju g a t e . T h e r e a r e m a n y w a y s

o f d o i n g s o , th e s i m p l e st o n e b e i n g t o u se t h e s t a n d a r d s p h e r i ca l p a r a m e t r i z a t io n ,

w h e r e c o s 19 is t h e c a n o n i c a l m o m e n t u m f o r th e a z i m u t h a l a n g l e 4 ):

n = c o s O , { n ( x ) , , t , ( x ' ) } ( 3 . 3 )

T h e H a m i l t o n e q u a t i o n s t h e n a s s u m e t h e u s u a l f o r m

6 W 6 Wq b= ~ H ' H = ~ ' ( 3.4 )

w h e r e t h e d o t d e n o t e s t i m e d e r iv a t iv e , a c o n v e n t i o n t h a t w ill b e a d o p t e d f r o m n o w

o n . An eq u iv a l en t fo rm o f eq . (3 .4 ) i s g iv en b y

6 W 6 W( si n O ) ( b = ( 5 0 ' ( si n 0 ) 0 = 6 - ~ " ( 3 .5 )

H a v i n g b r o u g h t t h e L a n d a u - L i f s h i t z e q u a t i o n t o a s t a n d a r d c a n o n i c a l f o r m , t h ec o n s t r u c t i o n o f th e a s s o c i a t e d c o n s e r v a t i o n l a w s w o u l d a p p e a r s t r a ig h t f o r w a r d . F o r

i n s t a n c e , t h e m o m e n t u m i n t e g r a l w o u l d b e t a k e n t o b e

p , = f ( 1 - co s O ) O i ( ~ d3x, ( 3 . 6 )

w h e r e t h e m o m e n t u m d e n s i t y d i f f e r s f r o m t h e s t a n d a r d d e f i n i t i o n - H Oil=

- c o s t 9 a i ~ o n l y b y a t o ta l d e r i v a ti v e w h i ch i s a d d e d t o e n s u r e p r o p e r b e h a v i o r a t

i n f in i ty w h e re , b y co n v en t io n , 19 - -, 0 . Th e c o r res p o n d in g ex p ress io n fo r t h e an g u la r



4 3 4 N. Papanicolaou, T.N . Tom aras / Magnetic cortices

m o m e n t u m w o u l d t h e n b e g iv e n b y

! , = f t 1 - cos O ) e i i x i 0 , ~ d 3 x . ( 3 . 7 )

A c t u a l ly th e a n g u l a r m o m e n t u m w o u l d n o t b e c o n s e r v e d u n l e s s w e a d d t o i t t h e

t o t a l m a g n e t i z a t i o n

, , , , = f ( M - M , , °, ) a 3 x ( 3 . 8 )

H er e w e sub t r ac t t he t r iv i a l m ag ne t i z a t i on o f t he un i f o r m conf i g u r a t i on M {°} - -

( 0, 0 , 1 ). I f t h e m a g n e t o s t a t i c f i el d i s n e g l e c t e d i n t h e L a n d a u - L i f s h i t z e q u a t i o n ,

t h e n t h e a n g u l a r m o m e n t u m ( 3 .7 ) a n d t h e t o t a l m a g n e t i z a t i o n ( 3 .8 ) w o u l d b e

s e p a r a t e l y c o n s e r v e d . O f c o u r s e , i n t h e c a s e o f a n o n v a n i s h i n g u n i a x i a l a n i s o t r o p y ,

o n l y t h e t h i rd c o m p o n e n t s w o u l d b e c o n s e r v e d .

H o w e v e r , t h e s e c o n s e r v a t i o n l a w s a r e a m b i g u o u s a s t h e y s t a n d , a n d t h e i r u s e f o r

t he ana l ys i s o f t he dynam i cs o f m agne t i c vo r t i c e s i s dub i ous [ 12 ] . A supe r f i c i a l

o b j e c t i o n i s t h a t t h e m o m e n t u m d e n s i t y i n e q . ( 3 . 6 ) i s n o t i n v a r i a n t u n d e r i n t e r n a l

r o t a t io n s , b u t i t t r a n s f o r m s b y a n a d d i t i v e t o t a l d i v e r g e n c e ; s u c h a n i n s t a n c e is

c o m m o n p l a c e i n g a u g e t h e o r i e s a n d w o u l d n o t b y i t se l f c o n s t i t u t e a n a m b i g u i t y .

M o r e s e r io u s i s t h e n o n - d i f f e r e n t i a b i l it y o f th e a z i m u t h a l a n g l e qb a t p o i n t s w h e r e

t h e m a g n e t i z a t i o n i s d i r e c t e d t o w a r d t h e n o r t h o r t h e s o u t h p o l e . W h i l e t h e f a c t o r

1 - c o s O w o u l d s u p p r e s s t h e a m b i g u i t y a t th e n o r t h p o l e , th e p r o b l e m p e r s i s t s a t

t h e s o u t h p o l e . A s a c o n s e q u e n c e , a n o m a l o u s ~ - f u n c t i o n c o n t r i b u t i o n s a r i s e w h i c h

p r e v e n t t h e c o n s er v a ti o n o f t h e l i n e ar m o m e n t u m ( 3.6 ) a n d t h e a n g u l a r m o m e n t u m

( 3 . 7 ) . S i n c e t h e m a g n e t i z a t i o n i n a m a g n e t i c v o r t e x r e a c h e s t h e s o u t h p o l e a t l e a s t

o n c e , t h e p r e c e d i n g o b j e c t i o n s t o t h e c o n s e r v a t i o n l a w s a r e c r u c i a l f o r o u r

p u r p o s e s .

T h e a m b i g u i ti e s i n h e r e n t in th e d e f i n i t i o n o f t h e l i n e a r m o m e n t u m ( 3. 6) w e r e

p r ev i ous l y ana l yzed by H a l dane [ 14 ] and by V o l ov i k [ 15 ] bu t t he o f f e r ed r e so l u t i ons

d o n o t s e e m t o s h e d li g h t o n t h e c u r r e n t p r o b l e m . A c o m p l e t e r e s o l u t i o n o f t h e s e

d i f f i c u l t i e s i s a c h i e v e d h e r e t h r o u g h a d e e p e r a p p r e c i a t i o n o f t h e t o p o l o g i c a l

s t r u c t u r e a s s o c i a t e d w i t h t h e L a n d a u - L i f s h i t z e q u a t i o n . B e c a u s e t h e t o p o l o g i c a l

s t r uc t u r e i s s ens i t i ve t o space d i m ens i on , ou r a r gum en t w i l l be ca r r i ed ou t i n t w o

s t e p s . T h e b u l k o f t h e p a p e r w i l l b e d e v o t e d t o t h e s t u d y o f s t r i c t l y t w o - d i m e n -s i on a l m a g n e t i c v o rt i ce s w h e r e t h e e f f e c t s o f t o p o lo g y a r e s t r o n g e s t . T h e d i s c u s s io n

of t he t h r ee - d i m ens i ona l ca se i s r e l ega t ed t o s ec t . 7 .

T h e t w o - d i m e n s i o n a l r e s t r i c t i o n s o f t h e c a n o n i c a l c o n s e r v a t i o n l a w s d i s c u s s e d

above a r e g i ven by

p u = f ( 1 - - C O S O ) cg u c/)d 2 x , # = l o r 2 ,

l = f ( l - c o s O ) e , , . x , ~ , ., I, d : ' x , = f ( c o s 6 ) - 1 d ' x ( 3 . 9 )



N. Papanicolaou, T.N . Tom aras / Mag netic vortices 435

N o t e t h a t t h e r e is o n ly o n e a n g u l a r m o m e n t u m c o m p o n e n t , w h i c h w e d e n o t e b y l,

w h e r e a s m d e n o t e s t h e th i r d c o m p o n e n t o f t h e t o ta l m a g n e t i z a t i o n . T h e q u a n t it i e s

( 3 . 9 ) a r e a t l e a s t a s a m b i g u o u s a s t h e i r t h r e e - d i m e n s i o n a l a n a l o g s .

I t wou ld be d i f f i cu l t t o convey in a f ew l ines ou r de t a i l ed mo t iva t ion fo r t he

c o n s t r u c t i o n o f t h e u n a m b i g u o u s c o n s e r v a t i o n l a w s g iv e n b e lo w . W e t h u s i n it ia ll yp r o c e e d i n a s o m e w h a t a d h o e m a n n e r , r e s e r v i n g a d e t a i l e d d i sc u s s io n f o r l a t e r

s t ag e s . H e n c e w e c o n s i d e r t h e t im e e v o l u ti o n o f t h e t o p o lo g i c a l d e n s i t y q d e f i n e d

i n e q . ( 2 . 1 6 ) , a s s u m i n g t h a t t h e m a g n e t i z a t i o n s a t i s f i e s t h e L a n d a u - L i f s h i t z e q u a -

t ion in i t s gen era l fo rm (2 .1 ) r es t r i c t ed to two d imens ions . A re l a t ive ly s imp le

c a l c u l a t i o n s h o w s t h a t

O - e~v(a~,F" a ~ M ) = e j,~ a j,( F . a ~ M ) , ( 3 . 1o )

f o r a n y c h o i c e o f th e e f f e c t iv e m a g n e t i c f i el d F . T h e r i g h t - h a n d s i d e o f t hi s

equ a t ion i s a t o t a l d ivergence , so tha t eq . (3 .10 ) is a l r eady in the fo rm o f a l oca l

c o n s e r v a t i o n l aw . A s t r o n g e r v e r s i o n o f th i s c o n s e r v a ti o n l aw i s o b t a i n e d t h r o u g h a

c l o s e r e x a m i n a t i o n o f t h e v e c t o r

L =- - ( F . a ~ M ) = - - ~ . a ~ M = aatr,,a , (3 .11 )

w h e r e t h e t e n s o r tr va m a y b e e x p r e s s e d i n t e r m s o f t h e e n e r g y d e n s it y w a c c o r d in g

to

Owova = w6va O(cg aM i) O~,M . (3 .12 )

Eq . (3 .10 ) i s t hen wr i t t en in the fo rm

4 = (3 .13 )

wh ich wi l l p rove fundamen ta l fo r a l l subsequen t ca l cu la t ions .

A l t h o u g h o n l y s o m e g e n e r a l p r o p e r t i e s o f t h e t e n s o r trv a w il l b e i m p o r t a n t f o r

ou r a rgu m en t , i t i s no t d i f f i cu lt t o i l lu s t r a t e he re i ts expl ic i t fo rm . W e m ay wr i t e

" . , + . . . , (3 .14 )

w h ~ r e t h e f i r s t t h r e e t e r m s c o r r e s p o n d t o t h e e x c h a n g e , a n i s o t r o p y a n d m a g n e t o -

s t a t i c con t r ibu t ions , wh i l e t he do t s s t and fo r poss ib l e add i t iona l t e rms o r ig ina t ing

in in te rac t ion s of the fo rm g iven in eq . (2 .19) . Th e var iou s term s in eq . (3 .14) are

ca lcu la t ed by in se r t ing the exp l i c i t exp ress ion o f t he energy dens i ty f rom eq . (2 .6 )

in eq . (3 .12) . Thus we f ind that

cr,.c^ + tr,~ = ( w~ + w,,)6,, a - (O ,, M 'O ^ M ), ( 3 . 1 5 )



4 3 6 N. Papanicolaou, T.N . Toma ras / Ma gnetic rortices

w h e r e we a n d wa a r e t h e e x c h a n g e a n d a n i s o t r o p y e n e r g y d e n s i t i e s o f e q . ( 2 . 6 )

r e s t r i c t e d t o t w o d i m e n s i o n s :

' ! M 2 )e + W a = ~ ( O ~ , M ' O ~ , M ) + ~ o t ( M ? + • ( 3 . 1 6 )

I t s h o u l d b e n o t e d t h a t t h e p o r t i o n o f t h e t e n s o r t rv a d i s p l a y e d i n e q . (3 . 15 ) is

s y m m e t r i c, b u t t h i s p r o p e r t y is n o t s h a r e d b y t h e m a g n e t o s t a t i c c o n t r i b u t i o n t r y .

F o r s i m p li ci ty , w e d e m o n s t r a t e t h i s p o i n t b y m o d e l i n g t h e n o n l o c a l m a g n e t o s t a t i c

e n e r g y d e n s i t y w i t h t h e l o c a l H u b b a r d - l i k e i n t e r a c t i o n

! 2W m = ~ (O ~ ,M ~ , ) , ( 3 . 1 7 )

s o t h a t

( Tv ~ = W m ~ v a - - (oqt. tM~)cgvMA, ( 3 . 1 8 )

w h i c h i s i n d e e d n o n s y m m e t r i c . F u r t h e r t e r m s i n e q . ( 3 . 1 4 ) c a n b e c o m p u t e d i n a

s i m i l a r f a sh i on .

T h e v e c t o r f , , a p p e a r e d e a r l i e r i n t h e w o r k o f T h i e l e [ 1 6 ] w h e r e i t w a s

i n t e r p r e t e d a s a f o r c e d e n s i ty . A c c o r d i n g l y t h e t e n s o r o ;, A c o u l d b e c a l l e d t h e s t r e ss

t e n s o r . H o w e v e r w e s h a l l n o t u s e t h i s t e r m i n o l o g y f u r t h e r i n t h e p r e s e n t p a p e r , f o r

i t m a y e a si ly l e a d t o o v e r i n t e r p r e t a t i o n . I n s t e a d a n u m b e r o f i m p o r t a n t c o n c l u s i o n s

w i l l be de r i ved d i r ec t l y f r om eq . ( 3 . 13 ) t o w h i ch w e r e t u r n p r om pt l y .

A n i m m e d i a t e c o n s e q u e n c e o f e q . ( 3 . 1 3 ) i s t h a t t h e i n t e g r a t e d t o p o l o g i c a ld e n s i ty , th e w i n d i n g n u m b e r Q o f e q . (2 .1 6 ), is c o n s e r v e d , a s e x p e c t e d . F u r t h e r -

m o r e t h e a p p e a r a n c e o f a d o u b l e d e r i v a t iv e o n t h e r i g h t - h a n d s i de o f eq . (3 .1 3 )

s u g g e s ts th a t s o m e o f t h e l o w m o m e n t s o f th e t o p o l o g i c a l d e n s i t y a r e a l s o

c o n s e r v e d . T h e l o w e s t n o n t r i v i a l m o m e n t s a r e g i v e n b y

t . = f x , , q d Z x , /z = 1 or 2 , (3 .1 9)

a n d t h e i r c o n s e r v a t i o n i s d e m o n s t r a t e d b y a s im p l e a p p l i c a t io n o f eq . (3 .1 3 ):

L = f x , , O d = x = -

= ) + d x. ( 3 . 2 0 )

S i nce bo t h t e r m s i n t he i n t e g r an d o f eq . (3 . 20 ) a r e i n t he f o r m o f a t o t a l

d i v e r g e n c e , a n e l e m e n t a r y a p p l i c a t i o n o f t h e d i v e r g e n c e t h e o r e m s h o w s t h a t

/'~, = 0 , f o r f ie l d con f i gu r a t i ons w i t h r ea so nab l e beh av i o r a t i n fi n it y . By r ea so nab l e



N. Papanicolaou, T.N . Tom aras / Magnetic vortices 437

w e m e a n t h o s e f i e l d c o n f i g u r a t i o n s f o r w h i c h t h e e n e r g y i s f i n i t e , a s i s a s s u m e d

t h r o u g h o u t t h i s p a p e r .

O u r n e x t c o n c e r n i s t o i d e n t i f y t h e p h y s i c a l c o n t e n t o f th e c o n s e r v e d q u a n t i t i e s

( 3 . 1 9 ) . T h i s w e a c h i e v e b y e x a m i n i n g t h e s y m m e t r y t r a n s f o r m a t i o n s t o w h i c h t h e

a b o v e c o n s e r v a t io n l aw s m a y c o r r e sp o n d . H e n c e w e e x a m i n e t h e P o i ss o n b r a c k e ts{ I~ ,, M } w hi ch w e ca l cu l a t e i n two ways . F i r s t by us ing i n eq . ( 3 .19) t he t opo l og i ca l

d e n s i t y q i n t h e f o r m ( 2 . 1 6 ) t o g e t h e r w i t h t h e o r i g i n a l P o i s s o n b r a c k e t s ( 3 . 1 ) ,

p r o v i d e d t h a t t h e c o n s t r a i n t M 2 = 1 i s e n f o r c e d w h e r e v e r p o s s i b l e . S e c o n d b y

reso lv ing t he cons t r a in t exp l i c i t l y , t hus us ing t he t opo log i ca l dens i t y i n t he fo rm

q = % ,, sin 0 a,,Oa~,,t, = e,.,~,.lla,,cl, , ( 3 . 2 1 )

t o g e t h e r w i t h t h e c a n o n i c a l P o i s s o n b r a c k e t ( 3 . 3 ) . I n b o t h c a s e s w e f i n d t h a t

{ I v , M } = - e u , , a , , M . ( 3 . 2 2 )

T h e r e f o r e t h e m o m e n t s I v a r e e s s e n t ia l l y t h e g e n e r a t o r s o f s p a c e t r a n s la t i o n s .

M o r e p r e c i s e l y , t h e q u a n t i t i e s p # = - e v v I ~ , o r

= -e v f xv q d2x, ( 3 . 2 3 )

wi th ~ t = 1 o r 2 , s a t i s fy t he P o i s son b r a ck e t r e l a t i on {p~ ,, M } - -O~, M and m us t

t h u s b e i d e n ti f ie d w i th t h e c o m p o n e n t s o f li n e a r m o m e n t u m .T h i s f o r m o f t h e m o m e n t u m i n t e g ra l is f r e e o f th e a m b i g u i t ie s d i s c u s se d e a r l i e r

i n connec t i on w i th eq . ( 3 . 6 ) o r ( 3 . 9 ) . T he momentum dens i t y i n eq . ( 3 . 23) i s

i n v a r i a n t u n d e r i n t e r n a l r o t a t io n s , a s is a p p a r e n t f r o m t h e d e f i n i ti o n o f t h e

t o p o l o g i c a l d e n s i t y in e q . ( 2 .1 6 ). F u r t h e r m o r e , e v e n i f w e u s e t h e s p h e r i c a l

p a r a m e t r i z a t i o n , t h e n o n - d i f f e r e n t i a b i li t y o f t h e a z i m u t h a l a n g l e • is h a r m l e s s in

e q . ( 3 .2 1 ) b e c a u s e t h e a t t e n d i n g f a c t o r si n O w o u l d s u p p r e s s a m b i g u i t i e s at b o t h

t h e n o r t h a n d t h e s o u t h p o l e .

I t i s r e a s o n a b l e t o a s k w h e t h e r o r n o t t h e r e e x i s t s a f o r m a l r e l a t i o n s h i p b e t w e e n

t he two de f i n i t i ons o f momentum g iven i n eqs . ( 3 . 9 ) and (3 . 23) . Us ing eq . ( 3 . 21) ,o n e m a y d e r i v e t h e a w k w a r d i d e n t i t y

x , , q = e v a [ ( 1 - c o s 1 9 ) 0 a q ) ] - O u [ x . ( 1 - c o s O ) e u a O , ( b ]

+ x , , ( 1 - cos O)(euaOu ~A~ ) . ( 3 . 2 4 )

I f t h e s e c o n d a n d t h i r d t e r m w e r e a b s e n t i n th i s e q u a t i o n , t h e t w o d e fi n i ti o n s o f

m o m e n t u m w o u l d b e i d e n t i c a l . N o w t h e s e c o n d t e r m c a n b e n e g l e c t e d b e c a u s e i t i s

a t o t a l d i v e r g e n c e a n d w o u l d n o t c o n t r i b u t e t o t h e m o m e n t u m i n t e g r a l . T h e t h i r d



43 8 N. Papanicolaou, T.N. Tomaras / Magnetic cortices

t e r m a l s o a p p e a r s t o v a n i s h b e c a u s e t h e a n t i s y m m e t r i c t e n s o r e ,A i s c o n t r a c t e d

w i t h t h e s y m m e t r i c o n e O ,Oa. H o w e v e r t h i s t e r m n e e d n o t v a n i s h t h a n k s t o t h e

non- d i f f e r en t i ab i l i t y o f q~ . To i l l u s t r a t e t h i s po i n t w e use t he exp l i c i t vo r t ex

c o n f i g u r a t io n s o f e q . ( 2 .1 8 ) f o r w h i c h q ~ - c q~ , w h e r e c i s s o m e ( i n t e g e r ) c o n s t a n t

a n d ~ = a r c t a n ( x 2 / x l ) i s t he po l a r ang l e . I t i s no t d i f f i cu l t t o s ee t ha t e , a O,O a ~ =2 r r c ~ ( x ) ~ 0 . N e v e r t h e l e s s t h e t h i r d t e r m i n e q . ( 3 . 2 4 ) w o u l d s t i l l b e e q u a l t o z e r o

b e c a u s e x ~ 6 ( x ) = 0 . I n g e n e r a l , h o w e v e r , t h e m a g n e t i z a t i o n r e a c h e s t h e s o u t h p o l e

a t x = x 0 ~ 0 , so t ha t e ,^ O , O a ~ = 2 7 r c 6 ( x - x o ) a n d t h e t h i r d t e r m i n e q . ( 3 . 2 4 )

d o e s n o t v an i sh . T h e r e f o r e t h e t w o d e f i n it i o n s o f m o m e n t u m a r e a c t u a ll y n o t

i den t i ca l .

H a v i n g a c c e p t e d e q . ( 3 . 2 3 ) a s t h e p r o p e r d e f i n i t i o n o f t h e l i n e a r m o m e n t u m , a

n u m b e r o f in t e r e s t i n g c o n s e q u e n c e s e m e r g e a l m o s t i m m e d i a t e ly . A t f i rs t s i g h t, th e

m o m e n t u m i n t e g r a l ( 3 . 2 3 ) p o s s e s s e s s o m e r a t h e r s t r a n g e p r o p e r t i e s . A s i m p l e

c a l cu l a ti o n o f t h e P o i s s on b r a c k e t b e t w e e n t h e t w o c o m p o n e n t s o f m o m e n t u m

yi e l ds a nonvan i sh i ng r e su l t :

{ P l , P 2 } = 4 r r Q . (3 . 25 )

T h i s r e s u l t w o u l d a p p e a r t o b e s u r p r i s i n g , h a d i t n o t o c c u r r e d p r e v i o u s l y i n t h e

p r o b l e m o f e l e c t r o n m o t i o n i n a u n i f o r m m a g n e t i c f ie l d . A l t h o u g h t h e l a t t e r

p r o b l e m i s m u c h s i m p l e r th a n t h e L a n d a u - L i f s h i t z e q u a t i o n , i t o f f e rs a u s e f u l

g u i d e f o r a p h y s i c a l i n t e r p r e t a t i o n o f t h e c u r r e n t f i n d i n g s . T h u s a b r i e f s u r v e y o f

t h e c a n o n i c a l s t r u c t u r e a s s o c i a t e d w i t h t h e e l e c t r o n p r o b l e m i s g i v e n i n a p p e n d i xA . T h e r o le o f t h e a p p l i e d m a g n e t i c f i el d B i n t h a t p r o b l e m i s p l a y e d h e r e b y th e

w i n d i n g n u m b e r Q , a s is e v i d e n t f r o m a c o m p a r i s o n o f t h e P o i s s o n b r a c k e t ( 3 . 2 5 )

w i t h t hose o f eqs . ( A . 3 ) and ( A . 5 ) .

F o r f i e l d c o n f i g u r a t i o n s w i t h v a n i s h i n g w i n d i n g n u m b e r t h e P o i s s o n b r a c k e t

( 3 . 2 5 ) v a n i s h e s a n d t h e c o n s e r v e d q u a n t i t y ( 3 . 2 3 ) p l a y s t h e r o l e o f o r d i n a r y

m o m e n t u m . H o w e v e r , f o r Q ~: 0 , t h e m o m e n t s ( 3. 19 ) y i e l d a m e a s u r e o f p o s i t io n

m o r e s o t h a n m o m e n t u m . L e t M = M ( x , t ) b e a f i e l d c o n f i g u r a t i o n w i t h t o p o l o g i -

ca l dens i t y q = q ( x , t ) a n d m o m e n t s I , ca l cu l a t ed w i t h eq . ( 3 . 19 ) . N ow i m ag i ne a

r i g i d t r ans l a t i on o f t h i s con f i gu r a t i on by a co ns t a n t ve c t o r c = (c~, c2 ), so t ha t t hen e w t o p ol o gi ca l d e n si t y is g iv e n b y q ' = q ( x - c , t ) a n d t h e n e w m o m e n t s b y

I'~ = I , + 4 ~ r Q c , . ( 3 . 2 6 )

O n e w o u l d n o r m a l l y e x p e c t t h e m o m e n t u m t o r e m a i n u n c h a n g e d u n d e r r i g i d

t r ans l a t i ons , a p r op e r t y t ha t is c l ea r l y v i o l a t ed w hen eq . (3 . 26 ) is u sed t og e t h e r w i t h

t h e d e f i n i t i o n o f m o m e n t u m ( 3 . 2 3 ) e x c e p t f o r f i e l d c o n f i g u r a t i o n s w i t h v a n i s h i n g

w i n d i n g n u m b e r . T h e r e f o r e , f o r Q , 0 , it is m o r e s e n s i b l e to u s e as c o n s e r v e d



N. Papanicolaou, T.N . Tom aras / M agnetic vortices

quan t i t i e s t he moment s (3 .19) norma l i zed by the wind ing number :

439

f x ~ q d 2 x I v .

R u = ] q d2 x = 4 r r Q , Q ~ 0 " (3 .2 7)

T h e v e c t o r R t r a n s fo r m s u n d e r t r a n s l a ti o n s a s R - - ,R + e a n d m a y t h u s b e

in t e r p re t ed a s t he mean p os i t i on o f t he vort ex . Ye t t he Po i sson b racke t be tween

the two componen t s o f R i s nonvan i sh ing :

{ R , , R 2 } = 1 / 4 " n ' Q , (3 .28)

which i s ana logous to the seco nd equ a t ion o f (A.5 ) i n the e l ec t ron p rob lem and

sugges t s t ha t t he po in t de f ined by the vec to r R be ca l l ed the g u i d i n g c e n t e r o f t h e

vor tex.

Th e im po r t an t f ac t is tha t i n the absence o f ex te rna l f ie ld g rad ien t s t he vec to r R

is t ime in dep end en t . In o the r words , t he vor t ex is spon taneous ly p inned a round i ts

f ixed guiding center , even i f i t s de ta i led s t ruc ture would exhibi t a nont r iv ia l t ime

evo lu t ion . O f course , spon tane ous p inn ing o f t he vor t ex can occur anywhere in the

(x l , x2 ) -p l an e , i n com ple t e ana logy wi th the fami l i a r cyc lo t ron m ot ion , so tha t

t r ans l a t ion inva ri ance i s i nd i rec tly r e s to red . A mo re de t a i l ed d i scuss ion o f t he

impl i ca t ions o f t h i s conse rva t ion l aw fo r t he dynamics o f m agne t i c vor ti ce s will be

given in la te r sec t ions.

Th i s sec t ion i s comple t ed wi th a co r re spond ing d i scuss ion o f t he angu la r

m o m e n t u m . T h u s w e c o n s i d e r t h e s e c o n d m o m e n t o f t h e t o p o lo g i ca l d e n si ty :

l = ½ x 2 q d 2 x , ( 3 . 2 9 )

which may be shown to sa t i s fy the Po i sson b racke t r e l a t ion

{ I , M } = - e ~ , , x ,O , , M . (3 .30)

Th e re fo re the quan t i ty in eq . (3 .29) is t he ge ne ra to r o f space ro t a t ions and may be

iden t i f i ed wi th the angu la r momentum, rep lac ing the na ive de f in i t i on g iven ea r l i e r

in eq. (3.9).

I t shou ld be in t e res t ing to examine the t ime depen dence o f 1 de f ined above

using again the basic equat ion (3 .13) :

, 2 a ~ ( x , o ', , ^ ~,~] d 2 x .¢ ¢ . . , ) + ) + (3 .31)



4 4 0 N. Papanicolaou, T.N . Tom aras / Ma gnetic t'ortices

Th e f i r s t two t e rms in the in t e g ra n d a re in the fo rm o f a to t a l d ive rge nc e a n d do

n o t c o n t r i b u t e t o t h e i n t e g r a l . H o w e v e r t h e l a s t t e r m m a y b e d i f f e r e n t f r o m z e r o ,

un le ss the t e nso r t r ,~ i s symme t r i c so tha t e ,~tr ,~=O. In f a c t , r e c a l l i ng ou r

d iscussion fo l low ing eq . (3 .14) , the ten so r t r ,~ i s sym me tr ic i f we n egle c t the

ma gne tos t a t i c i n t e ra c t ion , i n wh ic h c a se 1 wo u ld be by i t se l f c onse rve d . Oth e rw isd 't he c on se rve d qua n t i ty i s ! + m , w he re m is t he to t a l m a gne t i z a t ion in the th i rd

d i re c t ion , a s e xpe c te d .

A s i n th e c a s e o f l i n e a r m o m e n t u m , t h e q u a n t i ty 1 o f e q . (3 .2 9 ) m a y b e

i n t e r p r e t e d a s o r d i n a r y a n g u l a r m o m e n t u m o n l y f o r f i e l d c o n f i g u r a t i o n s w i t h

va n i sh ing wind ing num be r . Fo r Q ~ 0 , l is r a th e r a me a su r e o f t he vo r t e x s iz e . A

mo re p re c i se de f in i t i on o f t he mean squared radius is given by

/ ( X - - R ) 2 q d 2 x 1r 2 __-- - . R 2

/ ' q d E x 2 r r Q ' ( 3 . 3 2 )

wh e re R i s t he c ons e rve d gu id ing c e n te r ve c to r o f eq . (3 .27 ) . Eq . (3 .32 ) p rov ide s a n

ob je c tive de f in it i on o f t he vo r t e x r a d ius , e spe c ia l ly wh e n the ma gne tos t a t i c i n t e ra c -

t ion i s neglec ted ; in tha t case , r 2 i s t i m e i n d e p e n d e n t i r re s p e ct iv e l y o f th e t i m e

e vo lu t ion o f t he de ta i l e d s t ruc tu re o f t he vo r te x .

A byp rod uc t o f t he p re c e d ing d i sc uss ion i s t he c u r ious f a c t tha t t he " a n gu la r

mome n tum" l i s e xpe c te d to be nonva n i sh ing e ve n fo r a s t a t i c vo r t e x . Fo r

i l l u s t r a t ion , we re tu rn to the Be la v in -Po lya kov in s t a n tons d i sc usse d in se c t . 2 . In

pa r t i c u la r , we c ons ide r t he e xp l i c i t e xa mple s

0 = ( ~ / 2 ) " o r ( a / z ) " , ( 3 . 3 3 )

wh e re n i s a pos i ti ve in t e ge r , wh ic h a re s imp le g e ne ra l i z a t ions o f t he e l e m e n ta ry

examples g iven ear l ie r in eq . (2 .18) , and in f ig . 1 , and correspond to a winding

num be r Q = _+ n . Ca lc u la t ing the topo log ic a l de ns i ty f rom e q . (2 .17 ) we f ind tha t

4,12 la i 2,~2 ~,- l)

q = + (92" + l a l 2 , , ) 2 , (3 .34 )

w he re Q2 = ~,z = x~ + x 2 = x 2 i s t he squa re d po la r r a d iu s .

Be c a use o f t i l e a x ia l symme t ry o f t he topo log ic a l de ns i ty (3 .34 ) the ve c to r R o f

e q . (3 .27 ) va n i she s a nd the gu id ing c e n te r c o inc ide s wi th the po in t whe re the

ma g ne t i z a t ion i s d i r e c t e d " towa rd the so u th po le . H ow e ve r th i s c o inc ide nc e is

me re ly a n a c c ide n t a ssoc ia t e d wi th the s imp le e xa mple s (3 .33 ) . In ge ne ra l , t he

ma gne t i z a t ion a t t he gu id ing c e n te r ne e d no t po in t t owa rd the sou th po le ; i n f a c t ,

for vor t ices wi th IQI > l , t he ma gn e t i z a t ion m a y re a c h the sou th po le a t m ore tha n

one po in t .



N. Papanicolaou, T.N . Tom aras / Magnetic cortices 441

N o w i n s e r t t h e t o p o l o g i c a l d e n s i t y ( 3 .3 4 ) in e q . (3 .2 9 ) a n d p e r f o r m t h e i n t e g ra -

t i o n s t o f i n d

1 = 2 7 r Q [ ( ' n - / Q ) c o s e c ( ~ r / Q ) ] l a l 2 , ( 3 . 3 5 )

w i th Q = + n , w h i c h is in d e e d d i f f er e n t f ro m z e ro . T h e c o r r e s p o n d i n g m e a n

s q u a r e d r a d i u s is c a l c u l a t e d f r o m e q . (3 .3 2 ) u s i n g 1 f r o m e q . ( 3 .3 5 ) a n d R - - 0 :

r 2 = [ ( r r / Q ) c o s e c ( z r / Q ) ] l a 12 . ( 3 . 3 6 )

T h i s r a d i u s i s d i v e r g e n t f o r IQ I = 1 , r e f l e c t i n g t h e s l o w d e c a y o f t h e f u n d a m e n t a l

B e l a v i n - P o l y a k o v i n s ta n t o n , a p r o p e r t y n o t s h a r e d b y m o r e r e al is t ic m a g n e t i c

vo r t i ce s . I n t h e o ppo s i t e l i m i t , I Q I ~ 0o, t he vo r t ex r ad i us ( 3 .36 ) co i nc i des w i th t he

n a i v e d e f i n i t i o n r = [ a l ; t h a t i s , t h e r a d i u s a t w h i c h t h e t h i r d c o m p o n e n t o f th e

m a g n e t i z a t i o n v a n i s h e s . F i n a l l y , f o r f u t u r e r e f e r e n c e , w e c a l c u l a t e t h e t o t a l m a g n e -t i z a t i o n i n t h e t h i r d d i r e c t i o n :

m = f ( M 3 - 1 ) d2 x = - 2 7r[( ~ r / Q ) c o s e c ( r t / Q ) l l a 12 . ( 3 . 3 7 )

I n v i e w o f e q . ( 3 . 3 6 ) w e m a y a l s o w r i t e

m = - 2 r r r 2 . ( 3 . 3 8 )

T h i s f o r m u l a h a s a s i m p l e g e o m e t r i c a l m e a n i n g . I f w e i m a g i n e a c r u d e m o d e l f o rt h e v o r t e x i n w h i c h M 3 t a k e s t h e v a l u e - 1 f o r ra d i a l d i s t a n c e s s m a l l e r t h a n t h e

v o r t e x r a d i u s r , a n d t h e v a l u e + 1 o t h e r w i s e , t h e n M 3 - 1 t a k e s th e v a l u e s - 2 a n d

0 , r e spec t i ve l y , and t he t o t a l m agne t i za t i on i s i ndeed g i ven by eq . ( 3 . 38 ) .

N e e d l e s s t o s ay , t h e m a i n f u n c t i o n o f th e c o n s e r v a t i o n l a w s o b t a i n e d i n t h is

s e c t i o n i s n o t t h e d e s c r i p t i o n o f s t a t i c m a g n e t i c v o r t i c e s , b u t t h e d e r i v a t i o n o f

u s e f u l p r e d i c t i o n s c o n c e r n i n g t h e i r d y n a m i c a l b e h a v i o r ; a t a s k t a k e n u p i n t h e

r e m a i n d e r o f t h i s p a p e r .

4. Steady mo t ion

A s t a t i c vo r t ex i s t he s i m p l e s t , pe r haps con t r i ved , exam pl e o f a vo r t ex i n s t eady

m o t i o n . O u r n e x t ai m is t o a s c e r t a i n t h e p o s s ib l e t y p e s o f t i m e - d e p e n d e n t g e n e r a l -

i z a t i o n s o f t h e n o t i o n o f a s t a t i c v o r t e x t h a t a r e c o m p a t i b l e w i t h t h e c o n s e r v a t i o n

l aw s de r i ved i n s ec t . 3 . Jus t a s s t a t i c so l u t i ons m ay be v i ew ed a s s t a t i ona r y po i n t s o f

t h e e n e r g y f u n c t i o n a l , m o r e g e n e r a l s t e a d y - s t a t e s o l u t i o n s w i l l b e v i e w e d a s

s t a t io n a r y p o i n t s o f t h e e n e r g y u n d e r t h e f u r t h e r c o n d i t io n t h a t o n e o r m o r e o f t h e

a d d i t i o n a l c o n s e r v e d q u a n t i t i e s b e a s s i g n e d a d e f i n i t e v a l u e . S u c h a p r o c e d u r e w a s

a l r e a d y u s e d i n th e e a r l y d is c u s s i o n s o f o n e - d i m e n s i o n a l m a g n e t i c s o l it o n s [1 7].



442 N. Papanicolaou, T.N. Tom aras / Ma gnetic vortices

I t s h o u l d b e s t r e s s e d f r o m t h e o u t s e t t h a t t h e a c t u a l c o n s t r u c t i o n o f s t e a d y s t a t e

s o l u t i o n s s t i l l r e s t s w i t h t h e e x p l i c i t s o l u t i o n o f h i g h l y n o n l i n e a r e q u a t i o n s , w h i c h i s

i n g e n e r a l d i f f i c u l t t o a c h i e v e a n a l y t i c a l l y . T h e r e f o r e , f o r t h e m o s t p a r t , t h e r e s u l t s

o f th i s s e c t io n m a y b e t h o u g h t o f a s a q u a l it a t iv e t h e o r y o f s t e a d y m o t i o n w h i c h

m a y s e r v e a s a g u i d e f o r n u m e r i c a l s o l u t i o n s .T h e s i m p l e s t q u e s t i o n a l o n g t h e s e l in e s is w h e t h e r o r n o t t h e r e e x is t v o r t i c e s in

l i n e a r m o t i o n w i t h c o n s t a n t v e l o c i t y . O n e w o u l d n o r m a l l y a n s w e r s u c h a q u e s t i o n

b y a p p l y i n g a g a l i l e a n b o o s t t o a s t a t i c v o r t e x . H o w e v e r a g a l i l e a n s y m m e t r y i s n o t

k n o w n t o e x is t f o r th e L a n d a u - L i f s h i t z e q u a t i o n , e x c e p t f o r v e r y s p e c i a l o n e -

d i m e n s i o n a l m o d e l s [1 3]. N e v e r t h e l e s s t h e q u e s t i o n c a n b e a s k e d a n y w a y b y

e x p l o it in g t h e m o m e n t u m i n t e g ra l . T h u s w e s e e k a f t e r s t a ti o n a r y p o i n t s o f th e

e x t e n d e d e n e r g y f u n c t i o n a l

U = W - v r , p ~ , , ( 4 . 1 )

w h e r e t h e c o n s t a n t s v ~ , w i t h /~ = 1 a n d 2 , p l a y t h e r o l e o f L a g r a n g e m u l t i p l ie r s

e n f o r c i n g a d e f i n i t e v a l u e f o r t h e m o m e n t u m . T a k i n g t h e v a r i a t i o n s o f U w i t h

r e s p e c t t o t h e i n d e p e n d e n t v a r i a b l e s ( 9 a n d ~ , u s i n g t h e e x p l i c i t e x p r e s s i o n f o r t h e

m o m e n t u m f r o m e q s . ( 3 . 2 3 ) a n d ( 3 . 2 1 ) , w e f i n d t h a t

6 W 6 Ws i n O ( v , O ~ ) = ~ 0 ' - s i n O ( v ~ O , O ) = ~ . ( 4 . 2 )

G i v e n a s o l u t i o n O = O o ( X ; V ) a n d ~ = ~ 0 ( x ; v ) o f t h e t i m e - i n d e p e n d e n t e q u a-t io n s (4 .2 ), a s o l u ti o n o f t h e o r i g i n a l t i m e - d e p e n d e n t e q u a t i o n s ( 3 .5 ) w o u l d b e

o b t a i n e d f r o m O ( x , t ) = O o ( X - v t ; v ) a n d ~ ( x , t) = ~ 0 ( x - v t ; v ) a n d w o u l d d e -

s c r i b e a c o n f i g u r a t i o n i n r i g i d l i n e a r m o t i o n w i t h v e l o c i t y v .

B u t a r e t h e r e a n y i n t e r e s t i n g s o l u t i o n s o f e q s . (4 . 2 ), s u c h a s s o l u t i o n s w i t h f i n i t e

e n e r g y a n d n o n v a n i s h i n g w i n d i n g n u m b e r ? T h e g e n e r a l d i s cu s s io n o f s e ct . 3

a l r e a d y i m p l i e s t h a t s u c h s o l u t i o n s d o n o t e x i s t, f o r t h e y w o u l d y i e l d a t o p o l o g i c a l

d e n s i t y o f t h e f o r m q = q ( x - v t ; v ) a n d a t i m e - d e p e n d e n t g u i d i n g c e n t e r , R ( t ) =

R ( O ) + v t , w h i c h w o u l d c o n t r a d i c t t h e c o n s e r v a t i o n o f R u n l e s s t h e v e l o c it y v a n i s h .

N o t e t h a t t h e a s s u m p t i o n Q ~ 0 i s c r u c i a l in t h i s a r g u m e n t .L e a v i n g a s i d e f o r t h e m o m e n t t h e p o s s i b i l i t y o f s o l u t i o n s w i t h Q = 0 , w e w o u l d

l i k e t o c o n f i r m t h e a b o v e g e n e r a l c o n c l u s i o n d i r e c t l y f r o m e q s . ( 4 . 2 ) . U s i n g b o t h o f

t h e s e e q u a t i o n s w e m a y w r it e

( S W ~SW )v ~, s i n O ( O ,O O, cl) - O~O O q ~ ) = - - ~ 0 ~0 + --6-~O l ) . ( 4 . 3 )

T h e l e f t- h a n d s i d e m a y b e e x p r e s s e d in t e r m s o f t h e t o p o l o g i c a l d e n s i ty q , w h e r e a s

t h e r i g h t - h a n d s id e is b u t a n o t h e r f o r m f o r t h e v e c t o r - f , , i n t r o d u c e d e a r l i e r i n e q .



N. Papanicolaou, T.N . Tom aras / Magnetic vortices 443

( 3 . 1 1 ) . U s i n g t h e r e l a t i o n f~ = 0 ~ f r o m e q . ( 3 . 1 1 ) , e q . ( 4 . 3 ) m a y b e w r i t t e n a s

qev~,v# = -a,~tr,, ,~. (4 .4 )

I n t e g r a t i n g b o t h s i d e s o v e r a ll s p a c e , t h e r i g h t - h a n d s i d e y i e l d s z e r o , f o r c o n f i g u r a -

t i o n s w i t h f i n i te e n e r g y a n d t h u s r e a s o n a b l e b e h a v i o r a t i n f in i ty , w h i l e t h e l e f t - h a n d

s i d e i s e x p r e s s e d i n te r m s o f th e w i n d i n g n u m b e r Q :

4 ~ r Q e v , t ' ~ = 0 . ( 4 . 5 )

H e n c e , f o r Q ~ 0 , f in i t e - e n e r g y s o l u ti o n s w i t h n o n v a n i s h i n g v e l o c it y d o n o t e x i st , i n

a g r e e m e n t w i th o u r e a r l ie r c o n c lu s io n .

N e v e r t h e l e s s f i n i t e - e n e r g y s o lu t i o n s w i t h n o n v a n i s h i n g v e l o c i ty a n d Q - 0 a r e

n o t c o n t r a d i c t e d b y e i t h e r e q . ( 4 . 5 ) o r t h e g e n e r a l d i s c u s s i o n o f s e c t . 3 . O n t h e

o t h e r h a n d , e q . (4 . 4) d o e s i m p l y c e r t a i n c o n d i t i o n s f o r t h e e x i s t e n c e o f s u c h

s o l u t i o n s . W e c o n t r a c t b o t h s i d es o f t h i s e q u a t i o n w i t h x ~ a n d t h e n i n t e g r a t e o v e r

a l l space . The l e f t - hand s i de y i e l ds e ~ c ~ [ x ~ q d 2 x = v ' p , w h e r e w e u s e d t h e

d e f i n i t i o n o f t h e m o m e n t u m f r o m e q . ( 3 .2 3 ), a n d t h e r i g h t - h a n d s i d e g iv e s

E m p l o y i n g t h e u s u a l n o t a t i o n t r , ,V - t r t r f o r t he t r ace o f t he t enso r , t he r e su l t i ng

r e l a t i o n i s

v - p = f t r t r d 2 x , ( 4 . 6 )

which i s a v i r i a l re la t ion sa t i s f ied by a l l so lu t ions of eqs . (4 .2) wi th Q = 0 , f in i t e

ene r gy , and f i n i t e m om en t um . A c t ua l l y eq . ( 4 . 6 ) i s equ i va l en t t o t he v i r i a l r e l a t i on

o b t a i n e d b y a p p ly i n g D e r r i c k ' s s c a l in g a r g u m e n t t o t h e f u n c t i o n a l U o f e q . (4 .1 ).

T o o b t a i n a m o r e e x p l ic i t r e l a t io n o n e m u s t s p e c if y t h e t y p e s o f i n t e r a c t i o n

i nc l ud ed i n t he t en so r o -~a o f eq . (3 . 14 ) . I f w e i nc l ude on l y t he exch ange and

an i so t r opy con t r i bu t i ons , t he t enso r i s g i ven by eqs . ( 3 . 15 ) and ( 3 . 16 ) and i t s t r ace

by t r t r = a ( M ~2 + M 2 ) = a s in 2 0 . T h e a b s e n c e o f a c o n t r i b u t i o n t o t h e t r a c e f r o m

t h e e x c h a n g e i n t e r a c t i o n is a r e f l e c ti o n o f t h e c o n f o r m a l i n v a r ia n c e o f t h e s t a ti c

s e c t o r o f t h e p u r e e x c h a n g e m o d e l . T h e n t h e v i r i a l r e l a t i o n ( 4 . 6 ) t a k e s t h e e x p l i c i t

f o r m

v p - a f s in 2 19 d2x . ( 4 . 7 )

I n t e r e s t i ng l y t h i s r e l a t i on con t r ad i c t s t he ex i s t ence o f non t r i v i a i s t a t i c so l u t i ons

( v = 0 ) bu t l e aves op en t he p os s i b il i ty o f so l u t i ons w i th t , , 0 , and Q = 0 .



444 N. Papanicolaou, T.N. Tom aras / Magnetic cortices

The construction of explicit solutions is still too difficult a problem to tackle

analytically. Nevertheless some progress can be made in the case of the pure

exchange model of eq. (2.10) for which eqs. (4.2) are equivalent to

- c . O . M = M × A M , M 2= 1, (4.8)

where we have returned to vector notation and A is the two-dimensional laplacian.

The nonlinear system (4.8) was shown by one of the present authors to be

completely integrable [18], in the sense that it possesses a Lax pair and an

associated dual symmetry. It was then hoped that one could analytically obtain

examples of Belavin-Polyakov instantons propagating with nonvanishing velocity.

However such solutions were never accomplished and the program was eventually

abandoned. The Lax pair of ref. [18] was apparently rediscovered in ref. [19] where

an explicit one-soliton solution was also given. Although the details of that solutionare essentially two-dimensional, its overall structure is that of a one-dimensional

domain wail, and its energy is infinite.

The present analysis makes it clear that finite-energy solutions of eq. (4.8) do not

exist, except possibly for Q -- 0. Even in the latter case, a severe restriction results

from the virial relation (4.7), which now takes the degenerate form v . p = 0, while

angular-momentum conservation yields e,~,t', p~ = 0. Therefore the linear momen-

tum must vanish for any value of the velocity. This is surely a stringent condition

from the physical point of view, but it still does not seem to mathematically

contradict the existence of solutions of eq. (4.8) with finite energy and Q = 0.Returning to the general problem, we have already established that in the

absence of external field gradients the guiding center of a nontrivial (Q :~ 0) vortex

is spontaneously p inned at some a rbitrary point in the (x l, x2)-plane. There for e

the only allowed time-dependent generalizations of a static vortex are those that

preserve the location of the guiding center and are consistent with the remaining

conservation laws; namely, the angular momentum 1 of eq. (3.29) and the total

magnetization m of eq. (3.9). We may thus envisage a steady state in which the

vortex rotates as a whole around the fixed guiding center, while the magnetization

at each point precesses, at some specified rate. Such a state would be the analog of

the familiar cyclotron motion of an electron in a uniform magnetic field.

One may again consider an extended energy functional of the form

U = W - tol - to'm, (4.9)

where the constants to and to' are the frequencies of rotation and precession,

respectively. If the magnetostatic interaction is included, so that only the sum

l + m is conserved, the above frequencies must be chosen equal (to = to'). As

before, we search for stationary points of U, which can be shown to satisfy the



N. Papanicolaou, T.N. Tomaras / Magnetic cortices

d i f f e r e n t i a l e q u a t i o n s

445

sin 19(toe~,,,x~,0,,q~ - t o ') = 8 W / ~ 1 9 ,

- to sin 1 9 e ~ , ~ x ~ , O v O = $ W / 6 q ~ .

( 4 . 1 0 )

Giv en a so lu t ion 19 = 190(x ; to, to ') an d q~ = ~ 0 (x ; to, to ') o f the se equ at ion s , a

s o l u ti o n o f th e o r i g in a l t i m e - d e p e n d e n t e q . ( 3 . 5 ) w o u l d b e o b t a i n e d f r o m

o ( x , t ) = O o ( ~ ' ( x - Z o ) ; , ,, , " 3 ,

q , ( x , t ) = ' I ' 0 ( ~ ( x - X o ) ; , o , , 0 3 + o , ' t , ( 4 . 1 1 )

w h e r e ~ d e n o t e s s y m b o li c al ly t h e t i m e - d e p e n d e n t r o t a ti o n m a t ri x

~ q ~ = ( c o s t o t s i n t o t ) ( 4 .1 2 )- s i n to t c o s t ot '

a c t i n g o n t h e c o l u m n v e c t o r x - x o . H e r e x o is a n a r b i t r a ry c o n s t a n t v e c t o r

r e f l e c t i n g t h e t r a n s l a t i o n i n v a r ia n c e o f t h e o r i g in a l e q u a t io n s . H o w e v e r t h e r e -

d u c e d e q u a t i o n s ( 4 .1 0 ) a r e n o t t r a n s l a t io n i n v a r ia n t , s o th e o r i g in o f c o o r d i n a t e s i n

t h e s e e q u a t i o n s h a s a n o b j e c t i v e m e a n i n g ; i . e . i t i s t h e p o i n t a r o u n d w h i c h t h e

v o r t ex ro t a t e s w i th an g u la r f r eq u en cy to . As we sh a l l s ee sh o rt l y , t h e p o in t x = 0 in

eq . (4 .1 0 ) , o r t h e p o in t x - x 0 in eq . (4 .1 1 ), co in c id es wi th t h e g u id in g cen te r o f t h e

v o r t e x , a s e x p e c t e d o n c o n s i s t e n c y g r o u n d s .

F i r s t we d e r iv e t h e an a lo g o f eq . (4 .4 ) s t a r t i n g f ro m eq . (4 .1 0 ) :

t o x , , q - to 'O~ (cos 1 9 - 1 ) = -Oao '~ a . ( 4 . 1 3 )

A n i m m e d i a t e c o n s e q u e n c e o f t h is r e la t i o n is t h a t

to f xvq d2x = O, ( 4 . 1 4 )

w h e r e q = q ( x ; to , a ; ) . T h e r e f o r e , f o r s o l u ti o n s w i th n o n v a n i s h i n g a n g u l a r f r e -

q u e n c y , t h e f i r s t m o m e n t s o f t h e t o p o l o g i c a l d e n s i t y m u s t v a n i s h . T h e p h y s i c a l

i n t e r p r e t a t i o n o f t h is r e su l t d e p e n d s o n w h e t h e r o r n o t t h e w i n d i n g n u m b e r i s

d i f f e re n t f ro m ze ro . F o r Q = 0 , eq . (4 .14 ) is eq u iv a l en t t o s ta t i n g th a t t h e l i n ea r

m o m en tu m v an i sh es . Fo r Q 4= 0 , eq . (4 .1 4 ) co n f i rm s th a t t h e v o r t ex ro t a t es a ro u n d

i ts f ix e d g u i d i n g c e n t e r . T h e l a t t e r m a y b e p i n n e d a n y w h e r e i n t h e ( X l , x 2 ) ' p l a n e ,

a s is e v i d e n t i n e q . (4 .1 1 ) th r o u g h t h e a p p e a r a n c e o f th e a r b i t ra r y p o s i t io n v e c t o r

x 0 . In a l l cases , t h e m ag n e t i za t io n p recesses wi th co n s t an t f r eq u e n cy to ' wh ich d o es

n o t ex p l i c i t l y en t e r eq . (4 .1 4 ) .



446 N. Papanicolaou, T.N . Tom aras / Magn etic t 'oru'ces

Eq . (4 .1 3 ) a l so imp l i es an an a lo g o f t h e v i r i a l r e l a t i o n (4 .6 ) , o b t a in ed b y

co n t rac t in g b o th s id es wi th x ~ , an d b y in t eg ra t in g o v er a l l sp ace . Ap p ly in g p a r t i a l

i n te g r a ti o n w h e r e a p p r o p r i a t e a n d r e c a ll in g t h e d e f i n i ti o n s o f t h e a n g u l a r m o m e n -

t u m 1 f r o m e q . ( 3 .2 9 ) a n d t h e t o t a l m a g n e t i z a t i o n m f r o m e q . (3 .9 ) , w e f i n d t h a t

to l + to 'm = ½ t r o - d2 x , ( 4 . 1 5 )

w h i c h i s a g a i n e q u i v a l e n t t o t h e r e l a t i o n o b t a i n e d b y a p p l y i n g D e r r i c k ' s s c a l i n g

a r g u m e n t t o t h e f u n c t i o n a l U o f e q . ( 4 .9 ) . T h i s v i ri a l r e l a t i o n d o e s n o t c o n t r a d i c t

t h e e x i s te n c e o f s o l u t io n s w i th f in i te e n e r g y , a n g u l a r m o m e n t u m , a n d t o t a l m a g n e -

t iz a t io n , a s w e ll a s Q ~ 0 , f o r e s s e n ti a l ly a ll v a r ia t i o n s o f t h e L a n d a u - L i f s h i t z

e q u a t i o n c o n s i d e r e d i n t h i s p a p e r .

T o c o m p l e t e t h e q u a l i ta t i v e d e s c r i p t io n o f ro t a t i n g s o l u t io n s o n e m u s t r e c a l l t h e

o b s e r v a t i o n s m a d e a t t h e e n d o f s e c t. 3 . F i r st , th e f i x e d g u i d i n g c e n t e r n e e d n o tc o i n c i d e w i t h t h e p o i n t w h e r e t h e m a g n e t i z a t i o n r e a c h e s t h e s o u t h p o l e , j u s t a s i n

t h e o r d i n a r y c y c l o t r o n m o t i o n t h e g u i d i n g c e n t e r d o e s n o t c o i n c i d e w i t h t h e a c t u a l

p o s i t io n o f t h e e l e c t ro n . S e c o n d , f o r Q ~ : 0 , t h e a n g u l a r m o m e n t u m is a m e a s u r e o f

t h e v o r t e x r a d i u s m o r e s o t h a n a m e a s u r e o f h o w fa s t t h e v o r t e x r o ta t e s . A s a

co n seq u en c e , t h e d i sp e r s io n ! = l ( to ) is ex p ec t ed to ex h ib it u n u su a l b eh av io r , su ch

as 1 (0) ~ 0 .

T h e s e o b s e r v a t i o n s c o n c l u d e o u r a n a l y s is o f s t e a d y m o t i o n , h a v i n g g o n e a s f a r a s

p o ss ib l e wi th o u t ac tu a l ly so lv in g th e n o n l in e a r d i f f e ren t i a l eq u a t io n s (4 .1 0 ). A l -

t h o u g h a n a l y t i c a l s o l u t i o n s d o n o t s e e m f e a s i b l e , n o t e v e n f o r t h e s i m p l e e x c h a n g em o d e l , f u r t h e r p r o g r e s s m a y r e l y o n n u m e r i c a l s o l u t i o n s b a s e d o n t h e q u a l i t a t i v e

p ic tu re p resen ted in t h i s sec t io n .

5 . M o t i o n i n a f i e ld g r a d i e n t

T h e m o s t c h a r a c te r i s ti c f e a t u r e s o f v o r t e x d y n a m i c s b e c o m e a p p a r e n t i n t h e

p r e s e n c e o f a n e x t e r n a l m a g n e t i c f ie l d g r a d i e n t . S u c h a f i e ld p l a y s t h e r o l e o f a n

e l e c t r i c f i e l d i n t h e a n a l o g o u s e l e c t r o n p r o b l e m , w h i l e t h e w i n d i n g n u m b e r Q

c o n t i n u e s t o p l a y t h e r o l e o f a n a p p l i e d u n i f o r m m a g n e t i c f i e l d . A s a r e s u l t , t h em o t i o n o f a v o r t e x r e s e m b l e s t h e f a m i l i a r H a l l m o t i o n .

Sp ec i fi ca lly we co n s id e r t h e fo l lo win g p h y s i ca l s i t u a t io n . A v o r t ex in i ti a ll y p in n ed

a t so me p o in t i s su b jec t t o an ex t e rn a l mag n e t i c f i e ld o f t h e fo rm

Hext=[O,O,h(x,t)]. ( 5 . 1 )

In wo rd s , t h e ap p l i ed f i e ld p o in t s i n t h e t h i rd d i r ec t io n an d i ts m ag n i tu d e h m ay

d e p e n d o n p o s i ti o n a s w e l l a s o n t i m e . T h e q u e s t i o n is t h e n t o p r e d i c t t h e b e h a v i o r

o f th e v o r t ex a f t e r t h e f i e ld is imp o sed . W e sh a l l f i n d th a t i n t h e ab sen c e o f



N. Papanicolaou, T.N. Tom aras / Magnetic vortices 447

d i s s i p a t io n t h e v o r t e x w o u l d d r i f t in a d i re c t i o n p e r p e n d i c u l a r t o t h e f ie l d g r a d i e n t

Vh, a s i t u a t i o n m a d e m o r e p r e c i s e b e l o w .

O u r s t a r t i n g p o i n t i s a g a i n t h e f u n d a m e n t a l r e l a t io n ( 3 .1 3 ) e x t e n d e d t o in c l u d e

th e ap p l i e d f i e ld (5 .1) . To ca l cu l a t e t h e e f f ec t o f t h i s fi e ld o n eq . (3 .1 3 ) , we ju s t

sub st i tu te F ~ F + Hex in eq . (3 .10) to ob ta in

i l = e ~ a g ( F . O ~ M ) + et,~ a~ ( H e x t " O v M )

= + ( 5 . 2 )

H e r e t r ~ i s t h e t e n s o r c o n s t r u c t e d i n s e ct . 3 , i n t h e a b s e n c e o f t h e a p p l i e d f ie l d ,

a n d M 3 is th e t h i r d c o m p o n e n t o f t h e m a g n e t i z a t i o n . T h e r i g h t - h a n d s id e o f e q .

(5 .2 ) is st i ll in t h e fo rm o f a t o t a l d iv e rg en ce , so t h a t t h e i n t eg ra t ed to p o lo g ica l

d e n s i ty , t h e w i n d i n g n u m b e r Q , i s s ti ll c o n s e r v e d . H o w e v e r th e m o m e n t s o f th et o p o l o g i c a l d e n s i t y a r e n o l o n g e r c o n s e r v e d . Y e t s t u d y i n g t h e d e g r e e t o w h i c h

t h e s e c o n s e r v a t i o n l a w s a r e v i o l a t e d b y t h e a p p l i e d f i e l d y i e l d s d i r e c t i n f o r m a t i o n

o n t h e b e h a v i o r o f th e v o r te x .

T h u s w e c o n s i d e r t h e t i m e e v o l u t i o n o f t h e m o m e n t s I~, o f e q . ( 3 .1 9 ):

z . = f x .O f x .a , ( ha M ) d x , ( 5 . 3 )

w h e r e w e u s e t h e f a c t t h a t t h e f ir s t t e r m i n t h e r i g h t - h a n d s i d e o f e q . (5 .2 ) d o e s n o tc o n t r i b u t e i n e q . ( 5.3 ), r e f le c t i n g t h e c o n s e r v a t i o n o f t h e m o m e n t s i n t h e a b s e n c e

o f t h e a p p l i ed f i e ld . By p a r t i a l i n t eg ra t io n , eq . (5 .3 ) may b e wr i t t en as

[ . = - e . f h a M 3 d 2 x = - e . f h O ( M 3 - 1)d2x. ( 5 . 4 )

In t h e sec o n d s t ep o f t h i s eq u a t io n we in s e r t ed th e t ri v ia l i d en t it y O M 3 = 0 v (M3 - 1)

i n o r d e r t o p r e p a r e a s e c o n d p a r t i a l i n t e g r a t i o n ;

1)d x , ( 5 . 5 )

w h i c h i s n o w l e g i t i m a t e b e c a u s e M 3 ~ 1 , a t l a rg e d i s t an ces , i n acco rd an ce wi th o u r

s t a n d a r d c o n v e n t i o n .

F o r a g e n e r a l a p p l i e d f i e ld h - - h ( x , t ) t h e r i g h t - h a n d s id e o f e q . ( 5 .5 ) is n o t

su f f i c i en t ly ex p l i c it t o p ro v id e a d e t a i l ed d esc r ip t io n o f th e t ime ev o lu t io n o f th e

m o m e n t s , b u t i t a l r e a d y i n d i c a t e s t h a t t h e m o m e n t s w o u l d t e n d t o v a r y m o s t l y i n

t h e d i r e c t i o n p e r p e n d i c u l a r t o t h e f i e l d g r a d i e n t . T h i s s i t u a t i o n b e c o m e s c o r n -



448 N. Papanicolaou, T.N. Tomaras / Magnetic vortices

p l e t e ly t r a n s p a r e n t i n t h e c a s e o f th e s i m p l e f i e ld h =gx~, o r

H ex t = ( 0 , 0 , g x l ) , g = g ( t ) , ( 5 . 6 )

w h e r e t h e g r a d i e n t g is s p a c e i n d e p e n d e n t b u t m a y st il l d e p e n d o n t im e , c o n s t a n t

g b e i n g a n a d m i s s i b l e s p e c i a l c a s e . T h e n e q . ( 5 . 5 ) y i e l d s

i , = o , i2= -gf(M 3 - 1 )d 2 x = - g m , ( 5 . 7 )

w h e r e m is t h e t o t a l m a g n e t i z a t i o n i n t h e t h i r d d i r e c ti o n .

T h e f i rs t i m p o r t a n t c o n s e q u e n c e o f e q . ( 5 .7 ) is t h a t o n l y t h e c o m p o n e n t o f t h e

m o m e n t i n t h e d i r e c t i o n p e r p e n d i c u l a r t o t h e f i e l d g r a d i e n t v a r i e s w i t h t i m e .

F u r t h e r m o r e t h e r i g h t - h a n d s i de o f t h e s e c o n d e q u a t i o n i n e q . ( 5 .7 ) is e x p r e s s e d

e n t i r e ly i n t e r m s o f th e f i e l d g r a d i e n t a n d t h e t o t a l m a g n e t i z a t i o n m o f t h e v o r t e xi n q u e s t i o n . I f w e n e g l e c t t h e m a g n e t o s t a t i c i n t e r a c t io n , m w o u l d b e c o n s e r v e d

e v e n i n t h e p r e s e n c e o f a n e x t e r n a l f i e ld o f t h e f o r m ( 5 . 6 ) b e c a u s e t h e l a t t e r p o i n t s

i n t h e t h i r d d i r e c t i o n . I n t h a t c a s e , m c a n b e c o m p u t e d f r o m t h e i n i t i a l c o n f i g u r a -

t i o n o f t h e v o r t ex an d eq . (5 .7 ) p ro v id es a co m p le t e ly ex p l i c it d esc r ip t io n o f t h e

t im e e v o l u t i o n o f t h e m o m e n t s . I n t h e p r e s e n c e o f t h e m a g n e t o s t a t i c i n t e r a c t i o n ,

o n l y t h e s u m l + m is c o n s e r v e d , s o t h a t m w o u l d p ic k u p s o m e t i m e d e p e n d e n c e

d u r i n g t h e e v o l u ti o n o f th e v o r t e x u n d e r t h e i n f lu e n c e o f t h e f ie l d g r a d i e n t . I n s u c h

a c a se , a c o m p l e t e l y ex p li ci t c a l c u l a t io n o f t h e t i m e d e p e n d e n c e o f t h e r i g h t - h a n d

s id e o f eq . (5 .7 ) is n o t p o ss ib l e wi th o u t a d e t a i l e d so lu t io n o f t h e i n i ti a l v a lu e

p r o b l e m .

F o r a p h y si ca l i n t e r p r e t a t i o n o f th e p r e c e d i n g r e s u lt s , w e m u s t r e c a l l t h e p h y s i c a l

co n ten t o f t h e m o m en t s I~, d i scu ssed in sec t. 3 . W e sh a l l d i s t in g u i sh tw o cases

d e p e n d i n g o n w h e t h e r o r n o t t h e w i n d in g n u m b e r Q v a n i s h e s . W e d i sc u s s f ir s t t h e

c a s e Q 4 : 0 w h e r e t h e m o m e n t s p r o v i d e a m e a s u r e o f th e m e a n p o s i ti o n o f

t h e v o r t e x t h r o u g h t h e g u i d i n g c e n t e r c o o r d i n a t e s d e f i n e d i n e q . ( 3 . 2 7 ) . S i n c e t h e

w i n d i n g n u m b e r r e m a i n s c o n s e r v e d in t h e p r e s e n c e o f th e a p p l i e d f i e ld , e q s . (5 .7 )

m a y b e i m m e d i a t e l y t r a n s l a t e d i n t e r m s o f t h e g u i d i n g c e n t e r c o o r d i n a t e s :

/~ l = 0 , /~ 2 = - g m / 4 ~ r Q . ( 5 . 8 )

Th ese r e l a t i o n s s t i p u l a t e t h a t t h e v o r t ex wi l l d r i f t a s a wh o le i n t h e d i r ec t io n

p e r p e n d i c u l a r t o t h e f i e l d g r a d i e n t , w i t h " H a l l v e l o c i t y " VH = / ~ 2 , o r

V H = -g m/ 4~ rQ . ( 5 . 9 )

I t s h o u l d b e e m p h a s i z e d t h a t t h e a b o v e g e n e r a l r e s u l t d o e s n o t d e p e n d o n t h e

f in e r d e t a i ls o f t h e t ime ev o lu t io n o f t h e v o r t ex co n f ig u ra t io n , wh ich sh o u ld s t i ll b e

d e t e rm in ed th ro u g h an ex p l ic i t so lu t io n o f t h e i n it ia l v a lu e p ro b lem . In p a r t i cu l a r ,



N. Papanicolaou, T.N . Tomaras / Magnetic cortices 449

t h i s r e su l t i s v a l id ev en i f t h e v o r t ex d o e s n o t r eac h a s t ead y s t a t e . In f ac t , a s t ead y

s t a t e i s n o t e x p e c t e d t o o c c u r w i t h o u t d i s s i p a t i o n w h i c h i s a s s u m e d t o b e a b s e n t i n

t h e c a l c u l a t i o n s o f th i s s e ct io n . T h u s , i n t h e a b s e n c e o f d is s i p at io n , t h e g u i d i n g

c e n t e r o f t h e v o r t e x w o u l d d r i f t i n a d i r e c t i o n p e r p e n d i c u l a r t o t h e f i e l d g r a d i e n t ,

w i t h v e l o c i t y g i v e n b y e q . ( 5 .9 ), e v e n i f t h e d e t a i l s o f t h e v o r t e x u n d e r g o ac o m p l i c a t e d t i m e e v o lu t io n . T h e a b o v e re s u l t al s o d o e s n o t d e p e n d o n w h e t h e r t h e

in i ti a l co n f ig u ra t io n i s a s t a t ic v o r t ex o r a v o r t ex in a s t ead y s t a t e o f t h e t y p e

s tu d ied ea r l i e r i n sec t . 4 .

F o r a n e x p l i c i t i l l u s t r a t i o n , w e r e t u r n a g a i n t o t h e B e l a v i n - P o l y a k o v i n s t a n t o n s

o f e q . ( 3 . 3 3 ) w h i c h a r e s t a t ic s o l u t io n s o f t h e s i m p l e e x c h a n g e m o d e l o f e q . (2 .1 4 ).

I f a m a g n e t i c f i e l d o f t h e f o r m ( 5 . 6 ) i s a p p l i e d t o t h e s e v o r t i c e s , t h e i r g u i d i n g

c e n t e r w i l l m o v e a t a r i g h t a n g l e t o t h e f i e l d g r a d i e n t w i t h v e l o c i t y g i v e n b y e q .

( 5 . 9 ) . B e c a u s e t h e m a g n e t o s t a t i c i n t e r a c t i o n i s a b s e n t i n t h i s m o d e l , t h e t o t a l

m a g n e t i z a t i o n m is c o n s e r v e d a n d m a y b e c o m p u t e d f r o m t h e i ni ti a l c o n f i g u r a t io n ,

a s w a s a l r e a d y d o n e i n e q s . ( 3 . 3 7 ) a n d ( 3 . 3 8 ) . H e n c e w e f i n d t h a t

V H = 2 - -Q ' r 2 = [ ( r r / Q ) c o s e c ( r t / Q ) l l a l 2 , ( 5 . 1 0 )

wh ich i s a co mp le t e ly ex p l i ci t r e su l t fo r th e Ha l l v e lo c i ty ex p ress ed in t e rm s o f t h e

f i e ld g r a d i e n t g , t h e v o r t e x ra d i u s r 0 , a n d t h e w i n d i n g n u m b e r Q . N o t e t h a t w e

c u r r e n t l y u s e t h e s y m b o l r 0 , i n s t e a d o f r , f o r t h e v o r t ex r a d i u s i n o r d e r t o

e m p h a s i z e t h a t r 0 is t h e i n it ia l v a l u e o f t h e r a d i u s w h i c h m a y s u b s e q u e n t l y c h a n g e

u n d e r t h e i n f l u e n c e o f th e f i e ld g r a d i e n t . T h i s i s j u s t o n e e x a m p l e o f th e f i n e rd e t a i l s o f t h e v o r t e x t h a t m i g h t u n d e r g o a n o n t r i v i a l t i m e e v o l u t i o n , e s p e c i a l l y

w h e n d i s s i p a t i o n i s a b s e n t .

T o g e t a g l i m p s e o f t h e t i m e e v o l u t i o n o f t h e v o r t e x r a d i u s r , w e m a y r e c a l l i t s

d e f i n i t io n ( 3 .3 2 ) in t e r m s o f t h e a n g u l a r m o m e n t u m l o f eq . (3 .2 9 ). T h e t i m e

d e r i v a t iv e o f l i n th e p r e s e n c e o f th e f i e ld g r a d i e n t m a y ag a i n b e c o m p u t e d f r o m

eq . (5 .2 ) . Neg lec t in g th e mag n e to s t a t i c i n t e rac t io n , we f in d th a t

[ = f ( e ~ , ,x ~ , O , , h ) ( M 3 - 1 ) d 2 x , (5 .1 l a )

wh ich i s t h e a n a lo g o f eq . (5 .5 ). I f we ag a in r es t r i c t o u r a t t en t io n to t h e s imp le

g r a d i e n t o f e q . (5 .6 ), th e t i m e d e r i v a ti v e o f t h e m e a n s q u a r e d r a d i u s r 2 d e f i n e d i n

e q . (3 . 3 2 ) is in f e r r e d b y u s in g e q . ( 5 . 1 1 a ) w i t h h - g x ~ . W e t h u s f i n d t h a t

citd (r E ) = 2 1 r Q g f ( R z - x 2 ) (M 3 - . 1 ) d 2 x , ( 5 . 1 1 b )

w h e r e R 2 = R 2 ( t ) is th e s e c o n d c o m p o n e n t o f t h e g u i d in g c e n t e r w h o s e t im e

ev o lu t io n is g o v ern ed b y eq . (5 .8 ). Un fo r tu n a t e ly t h e r i g h t -h an d s id e o f eq . (5 .1 1 b )

i s n o t su f f i c i en t ly ex p l i c i t t o a l l o w su b s t an t i a l an a ly t i ca l p ro g ress , b u t su ch a



450 N. Papanicolaou, T.N. Tom aras / Magn etic cortices

r e l a t i o n m a y p r o v e u s e f u l f o r a p p r o x i m a t e e s t i m a t e s o f t h e t i m e e v o l u t i o n o f t h e

vor t ex r ad i us .

T o c o n c l u d e o u r d i s c u s s io n o f v o r ti c e s w i t h Q 4= 0 , w e m u s t c o m m e n t o n a t a c i t

a s s u m p t i o n m a d e t h r o u g h o u t t h i s s e c t i o n ; n a m e l y , t h a t t h e t r a j e c t o r y o f t h e

g u i d i n g c e n t e r i s a f a i t h f u l d e s c r i p t i o n o f t h e m o t i o n o f t h e v o r t e x a s a w h o l e . O fc o u r s e , t h e f a c t t h a t t h e g u i d i n g c e n t e r i s a s e n s i b le d e f i n i t i o n o f t h e m e a n p o s i t i o n

o f t h e v o r t e x w a s a l r e a d y e s t a b l i s h e d i n s e ct . 3 fo l lo w i n g e q . (3 .2 6 ). F u r t h e r m o r e a

m e a s u r e o f th e s p r e a d o f t h e v o r te x a r o u n d i ts g u id i n g c e n t e r is p ro v i d e d b y t h e

m e a n s q u a r e d r a d i u s o f e q . ( 3 .3 2 ). H o w e v e r i t i s d i f fi c u lt to d e c i d e a p r i o r i w h e t h e r

o r n o t t h e v o r te x r a d i u s is s m a l l, c o m p a r e d t o s o m e f u n d a m e n t a l s c a l e in e a c h

m o d e l , w i t h o u t w o r k i n g o u t e x p l i c i t s o l u t i o n s . I n t h i s r e s p e c t , t h e s i m p l e e x c h a n g e

m ode l , o f t en used i n t h i s pape r f o r exp l i c i t i l l u s t r a t i ons , i s a t yp i ca l because i t l a cks

a n i n t r i n s i c s c a l e ; a f a c t m a d e a p p a r e n t i n t h e e x p r e s s i o n f o r t h e v o r t e x r a d i u s i n

eq . (5 . 10 ) by t he a r b i t r a r y s ca l e l a l . N eve r t he l e s s t h i s r e su l t sugges t s t ha t t he

v o r t e x r a d i u s d e c r e a s e s w i t h i n c r e a s i n g Q .

I n a g e n e r i c m o d e l , o n e w o u l d e x p e c t t h a t a p a r t i c le l i k e d e s c r i p t i o n o f v o r t i c e s

i n t e r m s o f t h e g u i d i n g c e n t e r w o u l d b e c o m e i n c r e a s i n g l y f a i t h f u l w i t h i n c r e a s i n g

w i n d i n g n u m b e r , j u s t a s t h e g u i d i n g c e n t e r t r a j e c t o r y i n t h e H a l l m o t i o n b e c o m e s

a n a c c u r a t e d e s c r i p t i o n o f t h e a c t u a l e l e c t r o n t r a j e c t o r y in t h e l im i t o f a l a r g e

a p p l i e d m a g n e t i c f i e l d - t h e s o - c a l l e d a d i a b a t i c l i m i t . O n e m a y p u s h t h i s a n a l o g y

f u r t h e r t o c o n j e c t u r e t h a t v o r t i c e s w i t h l a r g e Q , w h i c h m a y b e c a l l e d h a r d v o r t i c e s

[ 1 , 2 ] , sub j ec t t o a gene r a l f i e l d h = h ( x ) w o u l d d r i f t a l o n g l i n e s o f c o n s t a n t h , b y

ana l ogy w i t h H a l l e l ec t r ons w h i ch , in t he ad i aba t i c l i m i t, d r i f t a long l i ne s o f

c o n s t a n t e l e c t r o s t a t i c p o t e n t i a l . T h e c h o i c e o f t h e m a g n e t i c f i e l d m a d e i n e q . ( 5 . 6 )i s spec i a l in t ha t t he g u i d i ng cen t e r f o l l ow s a li ne o f con s t an t h , na m e l y t he xE-ax is ,

e v e n f o r sm a l l n o n v a n i s h i n g Q ; a l s o in a n a l o g y w i t h t h e e l e c t r o n m o t i o n i n a

u n i f o r m e l e c t r i c f i e l d a n d a n a r b i t r a r i l y w e a k u n i f o r m m a g n e t i c f i e l d .

T h e r e r e m a i n s t o e x a m i n e t h e e x t r e m e n o n a d i a b a t i c l i m i t Q = 0 . A v o r t e x w i t h

van i sh i ng w i nd i ng num ber i s show n i n f i g . 2 . I n t h i s c a se , t he m om en t s I ~ , f a i l t o

p r ov i de an e s t i m a t e o f t he m ean pos i t i on o f t he vo r t ex , a s i s ev i den t f r om eq . ( 3 . 26 )

app l i ed f o r Q = 0 . A r e l a t e d f ac t i s t ha t t he t opo l og i ca l d ens i t y q i s no t po s i t i ve

( nega t i ve ) de f i n i te , e spec i a l l y f o r Q = 0 w h en i t i n t eg r a t e s t o ze r o . O n t h e o t he r

han d , f o r Q 4: 0 , t he pos i t i ve (nega t i ve ) va l ues o f t he t opo l o g i ca l de ns i t y bec om ed o m i n a n t , i n c r e a si n g ly s o f o r i n c r e a si n g Q , a n d q b e c o m e s a b o n a f i d e m e a s u r e o f

t he vo r t ex pos i t i on . Such a p r ope r t y i s com pl e t e l y l o s t f o r Q = 0 .

I n s t e a d t h e m o m e n t s I u a t Q = 0 m a y b e u s e d t o c a l c u l a t e t h e l i n e a r m o m e n t u m

according to eq. (3.23) , or p~, = - e u ~ l V. T h e r e f o r e t h e t i m e e v o l u t i o n o f t h e l i n e a r

m om en t u m o f a Q = 0 vo r t ex sub j ec t t o an ex t e r na l f i e l d o f t he f o r m ( 5 .1 ) m ay

aga i n be i n f e r r ed f r om eq . ( 5 . 5 ) :

= f ( o , h )( g 3 - 1 ) d 2 x , ( 5 . 1 2 )



N. Papanicolaou, T.N. Tomaras / Magnetic cortices 451

®

O--O

Fig. 2. Schem atic represen tation of a magn etic vortex with vanishing winding number.

w h e r e t h e r i g h t - h a n d s i d e m a y b e i n t e r p r e t e d a s t h e f o r c e a c t in g o n t h e v o rt e x d u et o t h e f i e ld g r a d i e n t . I f w e f u r t h e r r e s tr i c t o u r a t t e n t i o n t o t h e s i m p l e g r a d i e n t (5 .6 )

w e f i n d t h a t

/~! = g m , / ~ 2 = 0 . ( 5 . 1 3 )

H e n c e , i n t h is c a s e , t h e m o m e n t u m o f a Q = 0 v o rt e x w o u l d v a r y o n ly in t h e

d i r e c t i o n o f t h e f i e l d g r a d i e n t , i n a n a l o g y w i th t h e m o m e n t u m o f a n e l e c t r o n w h i ch

w o u l d v a r y o n l y in t h e d i r e c t io n o f t h e e l e c t r ic f i e ld i f t h e m a g n e t i c f ie l d w e r e

ab sen t . Ho wev er , s t r i c t l y sp eak in g , t h i s r e su l t d o es n o t n ecessa r i l y imp ly th a t a

Q = 0 v o r t ex wo u ld ac tu a l ly mo v e in t h e d i r ec t io n o f t h e f i e ld g rad ien t ; fo r , u n l ik et h e c a s e o f o r d i n a r y p a r t i c le d y n a m i c s , t h e r e l a t io n b e t w e e n m o m e n t u m a n d

v e l o c i t y i s n o t a p r i o r i k n o w n i n t h e p r e s e n t p r o b l e m .

N e v e r t h e l e s s t h e a b o v e p i c t u r e d o e s n o t c o n t r a d i c t t h e e x p e r i m e n t a l f a c t t h a t

Q = 0 m a g n e t i c b u b b l e s m o v e i n t h e d i r e c t i o n o f t h e f ie l d g ra d i e n t , a s it u a t io n t h a t

wi l l b e c l a r i f i ed a f t e r i n c lu d in g d i s s ip a t io n , i n sec t . 6 . Fu r th e rmo re , fo r mag n e t i c

b u b b les w i th Q ~ : 0 , d i s s ip a t io n wi l l b e sh o w n to b e r esp o n s ib l e fo r t h e ex p er im en -

t a ll y o b s e r v e d Q - d e p e n d e n t d e v i a t io n s o f t h e d e f l e c t io n a n g l e f ro m t h e t h e o r e t i c a l

v a lu e o f 9 0 ° d e r iv ed in t h i s sec t io n .

6 . T h e e f fe c t o f d i s s ip a t i o n

Th e d ef l e c t io n o f mag n e t i c v o r t i ces wi th Q ~ : 0 in a d i r ec t io n p e rp en d icu la r t o

t h e f i e l d g r a d i e n t p r e d i c t e d i n t h e p r e v i o u s s e c t i o n i s a n a l o g o u s t o t h e a b s e n c e o f

l o n g i t u d i n a l H a l l c o n d u c t iv i ty i n t h e a b s e n c e o f d i s si p a ti o n . O n t h e o t h e r h a n d ,

j u s t a s t h e H a l l c o n d u c t i v i t y a c q u i r e s a l o n g i t u d i n a l c o m p o n e n t i n t h e p r e s e n c e o f

d i s s ip a t i o n , t h e v e l o c it y o f t h e v o r t e x w o u l d d e v e l o p a n o n v a n i s h in ~ c o m p o n e n t in

th e d i r ec t io n o f t h e f i e ld g rad ien t . Hen ce th e d e f l ec t io n an g le fo r r ea l i s t i c mag -



452 N. Papanicolaou, T.N. Tom aras / Magnetic cortices

ne t i c vo r t i c e s is exp ec t e d t o be d i f f e r e n t f r om 90 °, a s i s d i s cus s ed m o r e p r ec i s e l y i n

t h e r e m a i n d e r o f t h i s s e c t i o n .

I n a p h e n o m e n o l o g i c a l d e s c r ip t i o n o f d i ss ip a ti o n , t h e L a n d a u - L i f s h i t z e q u a t i o n

( 2 . 1 ) i s m od i f i ed acco r d i ng t o

OM OM

Ot T M × Ot -M×F, M " 1 ( 6 .1 )

w h e r e y i s c a l l ed t he d i s s i pa t i on con s t an t . Eq . ( 6 .1 ) m a y a l so be w r i t t en a s

OM= M x G , M 2 = 1 ,

Ot

G = T ~ F + T 2 ( M X F ) ,

1 T, , T2 = ~ ( 6 . 2 )

Y l = I + T - I + T 2 '

and i s s i m i l a r i n f o r m t o t he o r i g i na l equa t i on ( 2 . 1 ) w i t h t he f i e l d G p l ay i ng t he

r o l e o f t h e e f f e ct i v e m a g n e t i c f ie l d F . T o a v o i d i n t r o d u c i n g a c o m p l i c a t e d n o t a -

t io n , t h e f ie l d F i n e q . ( 6 . 2 ) w i l l b e a s s u m e d t o c o n t a i n a n e x t e r n a l m a g n e t i c f i e ld

o f t he f o r m ( 5 . 1 ) .

Bec ause o f t he f o r m a l ana l og y o f eq . ( 6 .2 ) w i t h eq . ( 2 .1 ) , t he t i m e de r i va t i ve o f

t he t opo l og i ca l dens i t y q m ay be i n f e r r ed f r om eq . ( 3 .10 ) w i t h t he s i m p l e subs t i t u -

ti o n F ---> G :

q = ~u~au( G • J~ M ) . ( 6 . 3 )

T h e w i n d i n g n u m b e r Q c o n t i n u e s t o b e c o n s e r v e d i n t h e p r e s e n c e o f d i s s i p a t io n ,

b u t t h e m o m e n t s I , a r e n o t c o n s e r v e d . T h u s w e f i n d t h a t

t = f x q d x - d2x, ( 6 . 4 )

w he r e w e m ay i n se r t t he exp l i c i t exp r e s s i on o f G f r om eq . ( 6 . 2 ) t o ob t a i n

c . - - f ( F . a . M ) d 2 x , D . = - - f [ ( M x F ) . a . M ] d 2 x . ( 6 . 5 )

T h i s e q u a t i o n h a s b e e n a r r a n g e d s o th a t t h e l e f t - h a n d s i d e c o in c i d e s w i t h t h e t i m e

d e r iv a t iv e o f t h e l i n e a r m o m e n t u m . T h e v e c t o r C ~, w o u l d v a n i sh i n t h e a b s e n c e o f

an ex t e r na l f i e l d g r ad i en t , bu t t he f i e l d F i n eq . ( 6 . 5 ) i s a s sum ed t o con t a i n an



N. Papanicolaou, T.N . Toma ras / Mag netic cortices

e x t e r n a l f i e l d o f t h e f o r m ( 5 . 1 ) . H e n c e w e f i n d t h a t

453

C z = - f ( H e x t ' a ~ M ) d 2 x = f ( O , h ) ( M 3 - 1 ) d 2 x , ( 6 . 6 )

w h i ch i s t he f o r ce due t o t he ex t e r na l f i e l d g r ad i en t d i s cus sed ea r l i e r i n eq . ( 5 . 12 ) .

I f w e f u r t h e r r e s t ri c t th e e x t e r n a l f i e ld t o th e f o r m ( 5 . 6 ) w e f i n d th a t

C ! = g m , C 2 = 0 . ( 6.7 )

T h e f o r c e C~, c o n s t r u c t e d a b o v e is r e s p o n s i b l e f o r t h e 9 0 ° d e f l e c t i o n o f m a g n e t i c

v o r t i c e s d i s c u s s e d i n s e c t. 5 , a p r e d i c t i o n t h a t i s n o w m o d i f i e d b y t h e a p p e a r a n c e o f

t he vec t o r D u i n eq . ( 6 . 5 ) .

U n f o r t u n a t e l y t h e e x p r e s s i o n f o r t h e v e c t o r D , is n o t s u f f ic i e n tl y e x p li ci t t o

a l l o w f o r f u r t h e r a n a l y t i c a l p r o g r e s s , m a t c h i n g t h a t o f s e c t . 5 , w i t h o u t f u r t h e r

a s s u m p t i o n s . B u t t h e e s s e n t i a l p o i n t o f t h i s a r g u m e n t c a n b e m a d e p l a u s i b l e b y

a s s u m i n g t h a t i n t h e p r e s e n c e o f a f i e l d g r a d i e n t a s w e l l a s d i s s i p a t i o n a s t e a d y

s t a t e i s e v e n t u a l l y r e a c h e d i n w h i c h t h e v o r t e x m o v e s r ig i d ly w i t h c o n s t a n t v e l o ci ty

v . W e s h a l l n o t e x a m i n e h e r e c o n d i t i o n s o n t h e f o r m o f t h e e x t e r n a l f i el d t h a t

w o u l d g u a r a n t e e t h e d e v e l o p m e n t o f a s t e a d y s t a te , i n p a r t i c u l a r , it is n o t k n o w n

w h e t h e r o r n o t a s i m p l e g r a d i e n t o f t h e f o r m ( 5 . 6 ) w o u l d a c t u a l l y l e a d t o a s t e a d y

s t a t e . I n f a c t , T h i e l e [ 1 6 ] s e e m s t o s u g g e s t t h a t t h e f i e l d g r a d i e n t m a y h a v e t o

e x h i b i t a c a r e f u l l y s e l e c t e d t i m e d e p e n d e n c e .

W i t h t h e s e u n s e t t l i n g c o m m e n t s i n m i n d , w e r e t u r n t o e q . ( 6 . 1 ) a n d a s s u m e t h a tt h e s t e a d y - s t a t e m a g n e t i z a t i o n is o f t h e f o r m M - M ( x - v t; v) , so t ha t

- v ~ , ( O , , M - T M X O ~ , M ) = M X F . ( 6 . 8 )

C o n t r a c t i n g b o t h s i d e s o f t h is e q u a t i o n w i t h - O . M a n d i n t e g r a t i n g o v e r a ll s p a c e

w e f i n d t h a t

D u = d u ,,t.' ,, - y ( 4 7 r Q ) e ~ , , t : ~ , ( 6 . 9 )

w h e r e

=- f ( O . M . O . M ) d 2 x = f ( o . o 0 . O + s in 2 0 o J ) o ~ . ~ ) d 2 x ( 6 . 1 0 )

i s c a l l ed t he d i s s i pa t i on t enso r [ 16 ] . Fo r a s t eady- s t a t e con f i gu r a t i on o f t he above

f o r m w e m ay f u r t he r s e t [ . = 4~- Q c . w h i ch i s u sed i n eq . ( 6 .5 ) t og e t h e r w i th eq .

( 6 . 9 ) t o ob t a i n t he a l geb r a i c r e l a t i on

4 z r Q e . , ,r , , + y d . , , r ,, * C . = 0 . ( 6 . 1 1 )



4 54 N Papanicolaou, T.N . Tom aras / M agnetic cortices

A p p l y i n g t h is r e l a ti o n f o r / z = 1 a n d 2 y i e l d s a s y s t e m o f l i n e a r e q u a t i o n s f o r t h e

co m p on en t s o f t he ve l oc i t y t, t and v 2 .

Fo r t he spec i a l cho i ce o f t he ex t e r na l m agne t i c f i e l d ( 5 . 6 ) , t he vec t o r C~ , i s g i ven

by eq . ( 6 . 7 ) and t he l i nea r sys t em ( 6 . 11 ) i s so l ved t o y i e l d

y d 2 2 4 r r Q - - Y d l 2

t '! = y 2 d e t d + ( 4 7 r Q ) 2 ( - g i n ) , v 2 = y 2 d e t d + ( 4 ~ r Q ) 2 ( - g i n ) . ( 6 . 1 2 )

A s i m p l e r v e r s i o n o f t h e s e r e l a t i o n s i s o b t a i n e d b y i n v o k i n g s o m e f u r t h e r a s s u m p -

t i o n s a b o u t t h e s t e a d y s t a t e p r o f i l e o f t h e v o r t e x , w h i c h m a y b e v i e w e d h e r e a s

r e a s o n a b l e a p p r o x i m a ti o n s. T h u s w e a s s u m e t h a t t h e v o r t e x r e ta i n s a p p r o x i m a t e l y

i t s ax i a l sym m et r y , i . e . i t i s de sc r i bed i n i t s r e s t f r am e by sphe r i ca l va r i ab l e s o f t he

f o r m O = O (~} ) an d t p = cons t , x cp, w he r e ~ and cp a r e t he u sua l po l a r va r i ab l e s

de f i ned f r om x~ =~} cos ~p an d x 2 = ~ s i n ~p. T he n t he e l e m en t s o f t h e d i s s i pa t i on

t e n s o r ( 6 . 1 0 ) r e d u c e t o

d i2 = O = d 2 ! ,

d l I = d 2 2 1~( d l l + d22 ) - - W e , ( 6 . 1 3 )

w h e r e W is t h e e x c h a n g e e n e r g y o f t h e v o r te x , a n d e q . (6 . 12 ) b e c o m e s

y W 4 r r Q

C l = y 2 W e2 + ( 4 ~ ' Q ) 2 ( - g m ) , v 2 = y 2W e 2 + ( 4 7 r Q ) 2 ( - g i n ) . ( 6 . 1 4 )

T h e s p e e d o f t h e v o r t e x i s t h e n g i v e n b y

V = (u2 1 - l - v 2 ) 1 / 2 =Igml

[ y2 W e 2 + ( 4 r r Q ) 2 ] 1 / 2 '( 6 . 1 5 )

a n d t h e d e f l e c t i o n a n g l e b y

V 2 4 ~ ' Q V- - = . ( 6 . 1 6 )sin ~ = V g m

H e r e 8 i s t h e a n g l e b e t w e e n t h e t r a j e c t o r y o f t h e v o r t e x a n d t h e d i r e c t i o n o f t h e

f i e l d g r ad i en t . Re l a t i on ( 6 . 16 ) m ay a l so be w r i t t en a s

g m

Q = 4 r r V s in 8 , ( 6 . 1 7 )

w h e r e w e m a y f u r t h e r s u b s t i t u t e t h e t o t a l m a g n e t i z a t i o n t n = - 2 ~ r 2, a s is



N. Papanicolaou, T.N. Tomaras / Magnetic vortices 455

s u g g e s t e d b y e q . ( 3 . 3 8 ) a n d t h e e n s u i n g d i s c u s s i o n , t o o b t a i n t h e g o l d e n r u l e o f

b u b b l e d y n a m i c s [ 1 , 2 ]

g r 2

Q = 2---Vs in 6 , ( 6 . 1 8 )

w h i c h e x p r e s s e s t h e i n t e g e r v a l u e d w i n d i n g n u m b e r Q i n t e r m s o f d ir e c tl y

m e a s u r a b l e q u a n t i t i e s ; n a m e l y , t h e m a g n e t i c f i e l d g r a d i e n t g , t h e v o r t e x r a d i u s r ,

t h e v o r t e x s p e e d V , a n d t h e d e f l e c t i o n a n g l e 6 . T h u s w e s e e t h a t t h e v o r t e x m o v e s

i n t h e d i r e c t i o n o f t h e f ie l d g r a d i e n t f o r Q - 0 , b u t i t u n d e r g o e s s k e w d e f l e c t io n

fo r Q ~ 0 . In t h e l a t t e r case , an eq u iv a l en t fo rm o f eq . (6 .1 7 ) i s g iv en b y

V = Vn s in 6 , (6 .1 9 )

w h e r e VH i s t h e Ha l l v e lo c i ty d e f in ed ea r l i e r i n eq . (5 .9 ) . Th ere fo re , i n t h e

p r e s e n c e o f d i s s ip a t io n , t h e s p e e d o f t h e v o rt e x is al w a y s s m a l l e r t h a n t h e a b s o l u t e

v a lu e o f i t s ch a rac t e r i s t i c Ha l l v e lo c i ty .

I n c o n c l u s i o n , w e m u s t e m p h a s i z e t h a t s o m e o f t h e a r g u m e n t s o f t h is s e c ti o n a r e

b a s e d o n p l a u s i b l e a s s u m p t i o n s w h o s e v a l id i ty r e m a i n s t o b e ju s t if i ed . A c t u a l ly it is

m o r e o r l e s s e v i d e n t t h a t t h e s e a s s u m p t i o n s a r e e x p e c t e d t o b e v a l i d o n l y i n s o m e

a p p r o x i m a t e s e n s e . H e n c e a r i g o r o u s j u s ti f ic a t io n o f t h e g o l d e n r u l e ( 6 .1 8 ) m i g h t

p r o v e p o s s i b l e o n l y i n t h e a d i a b a t i c ( l a r g e - Q ) l i m i t . O n e w o u l d t h e n e n v i s a g e a n

a d i a b a t i c p e r t u r b a t i o n t h e o r y w h e r e t h e c a l c u l a t io n o f n o n a d i a b a t i c c o r r e c t io n s t o

eq . (6 .1 8 ) mig h t a l so p ro v e f eas ib l e . So me s t ep s i n t h i s d i r ec t io n were t ak en

r e c e n t l y b y M a s l o v a n d C h e t v e r i k o v [ 2 0 ], b u t a c o m p l e t e a d i a b a t i c t h e o r y is s til l

l ack in g .

7 . C o n c l u d i n g r e m a r k s

W e h a v e t h u s c o m p l e t e d o u r d is c u ss io n o f t h e t w o - d i m e n s i o n a l f e r r o m a g n e t i c

c o n t i n u u m a n d n o w t u r n o u r a t t e n t i o n t o s o m e m o r e g e n e r a l is su e s. I n p a r t ic u l a r ,

w e a d d r e s s t h e q u e s t i o n o f c o n s t r u c t i n g c o n s e r v a t i o n l aw s f o r t h e t h r e e - d i m e n -

s i o n a l t h e o r y w h i c h a r e n a t u r a l g e n e r a l i z a t i o n s o f t h e i r tw o - d i m e n s i o n a l a n a l o g s

d er iv ed in sec t. 3 . As id e f ro m i ts o b v io u s acad em ic in t e res t , t h e a n sw er o f t h i s

q u e s t i o n c o u l d p r o v e t o b e o f s o m e p r a c ti c a l s i g n if ic a n c e in t h e s t u d y o f t h r e e -

d i m e n s i o n a l t o p o l o g ic a l s o li to n s w h i c h a r e c h a r a c t e r i z e d b y a t o p o l o g i c al c h a r g e o f

a d i f f e r e n t n a t u r e , n a m e l y t h e H o p f i n v a r i a n t . S u c h s o l i t o n s m a y o c c u r u n d e r

ce r t a in co n d i t i o n s , a t l eas t i n p r in c ip l e [2 1 ] .

In d e ed a n a tu ra l ex t en s io n o f th e co n se rv a t io n l aws o f sec t. 3 ex i st s in th ree

d i m e n s i o n s a n d is b r ie f ly d e s c r i b e d in t h e f o l lo w i n g . T h u s w e c o n s i d e r t h e c u r r e n t

d e n s i t y

' ,~ j M ) " M , ( 7 . 1 ), - ~ e , j k ( O k M ×



4 5 6 N. Papanicolaou, T.N . Tomara s / Magn etic vortices

w h e r e L a t i n i n d i c e s i , j , . . . t a k e o v e r t h r e e d i s t i n c t v a l u e s a n d e i j k i s t h e u su a l

a n t is y m m e t r ic t e n s o r . T h e c u r r e n t d e n s i t y J~ is a s i m p l e g e n e r a l i z a t i o n o f t h e

to p o lo g ica l d en s i ty q o f eq . (2 .1 6 ). I t is t h en n o t d i f fi cu l t to sh o w th a t t h e an a lo g o f

eq . (3 .13) reads

~ . = - - e ~ s k a S a l ~ r k l ( 7 . 2 )

w h e r e t rk t is a s t ra i g h t f o r w a r d t h r e e - d i m e n s i o n a l e x t e n s i o n o f t h e t e n s o r c o n -

s t ru c t ed ea r l i e r i n eq . (3 .1 2 ) .

A n i m m e d i a t e c o n s e q u e n c e o f e q . (7 . 2) is t h a t t h e i n t e g r a t e d c u r r e n t d e n s i t y ,

n a m e l y

1 f J i d 3 x ( 7 . 3 )Q i = 4 ~ r

l e a d s to t h r e e c o n s e r v e d c h a r g e s , w i t h i = l , 2 o r 3 , w h i c h m a y b e t h o u g h t o f a s th e

a n a l o g s o f t h e w i n d i n g n u m b e r Q o f e q. (2 .1 6 ). H o w e v e r , u n l i k e t h e w i n d i n g

n u m b e r i n t w o d i m e n s i o n s , t h e Q i a r e t r i v i a l i n t h e t h r e e - d i m e n s i o n a l f e r r o m a g -

n e t i c c o n t i n u u m w i t h o u t b o u n d a r i e s , i n t h e s e n s e t h a t t h e y a r e e i t h e r d i v e r g e n t o r

v an i sh in g . In p a r t i cu l a r , t h e Q i v an i sh fo r a l l f i e ld co n f ig u ra t io n s wi th f i n i t e

e x c h a n g e e n e r g y . T h i s i s a l r e a d y a n i n d i c a t i o n t h a t n o t h i n g s p e c i a l h a p p e n s i n t h e

d y n a m i c s o f a g e n e r i c t h r e e - d i m e n s i o n a l s i tu a t io n .

N e v e r t h e l e s s w e m a y p r o c e e d e x a m i n i n g f u r t h e r c o n s e q u e n c e s o f e q . ( 7. 2) . T h u s

w e f i n d t h a t t i l e m o m e n t s

f ,J d3 x ( 7 . 4 )

are co n se rv ed , wh i l e t h ey sa t i s fy t h e Po i s so n b rack e t r e l a t i o n

{ I ~ j , M } = % , O , M , ( 7 . 5 )

w h i c h m a k e s i t a p p a r e n t t h a t t h e t e n s o r l i ~ i s an t i sy mmet r i c ; l i j = - l y i . T h e s a m e

r e l a ti o n i m p l ie s th a t t h e t h r e e i n d e p e n d e n t e l e m e n t s o f I i j m a y b e i d e n t if i e d w i t ht h e t h r e e c o m p o n e n t s o f t h e l in e a r m o m e n t u m a c c o r d in g t o

P i = - - 1 2 3 , P 2 = - - 1 3 1 , P3 = - - I12" (7 .6 )

S i m il a rl y t h e t h r e e c o m p o n e n t s o f t h e a n g u l a r m o m e n t u m c a n b e s h o w n t o b e

g iven by

l i = ~ f x 2 J i d 3 x , i = 1 , 2 o r 3 . ( 7 . 7 )



N. Papanicolaou, T.N . Tom aras / M agnetic cortices 45 7

W e h a v e t h u s o b t a i n e d n a t u r a l t h r e e - d i m e n s i o n a l e x t e n si o n s o f t h e c o n s e r v a ti o n

laws o f sec t . 3 and now commen t on the i r poss ib l e imp l i ca t ions fo r dynamics .

A s m e n t i o n e d a l r e a d y , t h e v a n i sh i n g o f c h a r g e s Q i of eq . (7 .3 ) imp l i es t ha t t he

above conserva t ion l aws do no t d i sp l ay unusua l behav io r . Fo r in s t ance , t he

m o m e n t s ( 7 . 4 ) a r e i n v a r i a n t u n d e r r i g i d t r a n s l a t i o n s a n d t h e P o i s s o n b r a c k e t sa m o n g t h e c o m p o n e n t s o f t h e li n e ar m o m e n t u m ( 7 . 6 ) v a n i s h . T h e r e f o r e t h e

t h r e e - d i m e n s i o n a l d y n a m i c s d o e s n o t a p p e a r s p e c ia l i n a n y w a y . T h i s o b s e r v a ti o n

app l i es , i n par t i cu la r , t o t opo log ica l so l i t ons charac t e r i zed by a non t r iv i a l va lue o f

t h e H o p f i n v a ri a n t. T h e l a t t e r m a y a ls o b e e x p r e s s e d i n t e r m s o f t h e c u r r e n t

den si ty J i a cco rd ing to [22 , 23]

1 f ( X - x ' ) j J k ( X ' ) d 3 x d 3 x ' ( 7 . 8 )Q H = 16,lr--------~e i j k J i ( x ) i x _ x , [ 3

H o w e v e r t h e t o p o l o g i c a l c h a r g e Q H d o e s n o t e n t e r d i r e c t l y t h e c o n s t r u c t i o n a n d

i m m e d i a t e p r o p e r t i e s o f t h e c o n s e r v a t io n l aw s , w h i l e th e c h a r g e s Q i v a n i sh f o r a

H o p f c o n f i g u r a t i o n w i t h f i n i t e e x c h a n g e e n e r g y . H e n c e t h e r e i s n o e v i d e n c e t h a t

such topo log ica l so l i t ons wou ld exh ib i t unusua l dynamica l behav io r . One shou ld

a l so no te tha t Der r i ck ' s sca l ing a rgumen t y i e ld s severe r es t r i c t ions on the ex i s t ence

o f f in i t e -energy s t a t i c so lu t ions in th ree d imens ions , bu t such so lu t ions cou ld be

s t ab i l i zed by p recess ion o f t he m agne t i za t ion [21 ].

T h e p i c t u r e c h a n g e s d r a s ti c a ll y in t h e c a s e o f t h e q u a s i - tw o - d i m e n s i o n a l f e r r o -

m a g n e t i c c o n t i n u u m , a c a s e o f sp e c ia l p r a c ti c a l i n t e r e s t b e c a u s e o f t h e m a g n e t i c

b u b b l e s k n o w n t o o c c u r i n t h i n m a g n e t i c f i l m s . T h e n t h e c h a r g e s Q i n e e d n o t

van i sh fo r f i e ld con f igu ra t ions wi th f in i t e energy . As an example , cons ider t he

c h a r g e Q 3 w h i c h w e m a y w r i te a s

[ 1 1 7 9 ,., dx 3 ~ ! ,3

where the x3 -ax i s i s t aken to be perpend icu la r t o the f i lm . Al though the dens i ty J3

de pe nd s in genera l on x 3, i t can s t il l be in t e rp re t ed as t he tw o-d imens iona l

topo log ica l dens i ty q o f eq . (2 .16 ), x 3 p l ay ing the ro l e o f a par am ete r charac t e r i z -

ing the speci f ic f ie ld conf igurat ion . Thus the double in tegral in eq . (7 .9) i s s t i l l

i n t eg er va lued , t ak ing a va lue Q = 0 , +_ 1 , . . . wh ich does no t de pe nd on x3 , and eq .( 7 .9 ) y i e ld s Q 3 - Q L w her e L i s t he th i ckness o f t he f iim . O ne can fu r the r show

tha t Q3 i s bou nde d by the exchange energy acco rd ing to W > /47 r lQal , wh ich is a

s imp le genera l i za t ion o f a s imi l a r i nequa l i ty d i scussed in sec t . 2 . There fo re , as l ong

as the f i lm th i ckness r emains f in i t e , f i n i t e -energy so lu t ions may ex i s t i n wh ich

Q 3 " - Q L :/: O. As a r esu l t , t he g ro ss f ea tu res o f t he dynamics o f quas i - two-d imen-

s iona l m agne t i c bubb les a re s imi l a r to those o f s tr ic t ly two-d im ens iona l magn e t i c

v o r t ic e s . O f c o u rs e , o n e s h o u l d n o t c o n c l u d e f r o m t h e a b o v e g e n e r a l a r g u m e n t t h a t

the th i rd d i r ec t ion do es no t p l ay any ro le in the dynam ics o f magne t i c bubb les , bu t

a detai led d iscuss ion of these f iner po in ts wi l l no t be g iven in th is paper .



458 N. Papanicolaou, T.N. Tomaras / Magnetic cortices

T h e d i sc u s si o n o f c o n s e r v a t i o n l aw s w o u l d n o t h a v e b e e n c o m p l e t e w i t h o u t

c o m m e n t i n g o n t h e s i t u a t i o n i n t h e o n e - d i m e n s i o n a l f e r r o m a g n e t i c c o n t i n u u m .

T h i s c a s e a p p e a r s s i n g u l a r b e c a u s e t h e c u r r e n t d e n s i t y Ji o f eq . ( 7. 1) , o r t h e d e n s i t y

q o f e q. (2 .1 6 ), v a n i s h e s w h e n r e s t r i c t e d t o o n e d i m e n s i o n . W e t h u s r e t u r n t o t h e

c a n o n i ca l d e f in i ti o n o f th e l i n e a r m o m e n t u m

p = _ ( l - c o s O ) ~ x d X ( 7.1 0 )

a n d d i s c u s s f ir s t t h e c a s e o f p u l s e s o l i to n s w h e r e t h e m a g n e t i z a t i o n r e a c h e s t h e

n o r t h p o l e a t b o t h e n d s o f t h e l i n e . T h e n t h e i n t e g r a l (7 .1 0 ) w o u l d b e a m b i g u o u s i f

t h e m a g n e t i z a t i o n h a p p e n s t o r e a c h t h e s o u t h p o l e a t s o m e i n t e r m e d i a t e p o i n t .

H o w e v e r , n o t i n g t h a t t h e i n t e g r a l ( 7 . 1 0 ) i s t h e f a m i l i a r e x p r e s s i o n f o r t h e B e r r y

p h a s e , i t s e e m s n a t u r a l t o d e f i n e t h e l i n e a r m o m e n t u m a s t h e g e n e r i c v a l u e o f t h i s

i n t eg r a l ; n a m e l y , t h e s o li d a n g l e t r a c e d b y t h e m a g n e t i z a t i o n a s x v a r i e s f r o m -

t o + ~ . A d e t a i l e d d i s c u s si o n o f th i s c a s e m a y b e f o u n d i n t h e a r t i c le o f H a l d a n e

[ 1 4]. T h e s i tu a t i o n i s l e ss c l e a r i n t h e c a s e o f d o m a i n w a l ls w h e r e t h e m a g n e t i z a t i o n

r e a c h e s t h e n o r t h ( s o u t h ) p o l e a t x = + oo a n d t h e s o u t h ( n o r t h ) p o l e a t x = - 0 o .

W e s h a ll b e c o n t e n t t o n o t c h e r e t h a t t h e d y n a m i c s o f d o m a i n w a l l s is c o m p l e t e l y

u n d e r s t o o d t h r o u g h a n e x p l ic i t s o l u t i o n o f t h e e q u a t i o n o f m o t i o n [9 ].

I n c o n c l u s i o n , w e b e l i e v e t o h a v e p r e s e n t e d a m o r e o r l e s s c o m p l e t e f r a m e w o r k

f o r t h e s t u d y o f t h e d y n a m i c s i n t h e f e r r o m a g n e t i c c o n t i n u u m . T h e u n u s u a l

d y n a m i c a l f e a t u r e s o f m a g n e t i c v o r t i c e s a r e u n d e r s t o o d a s a re s u l t o f a r a d i c a l

c h a n g e i n t h e c a n o n i c a l s t r u c t u r e o f t h e t h e o r y d u e t o t h e u n d e r l y i n g t o p o l o g y ,r a t h e r t h a n a s a r e s u l t o f th e d e t a i l e d f o r m o f th e e q u a t i o n o f m o t i o n . W e t h u s

e x p e c t t h a t t h e m e t h o d o l o g y o f th i s p a p e r w i ll p r o v e u s e f u l a ls o i n t h e d i s c u s si o n o f

t he r e l a t e d t op i c s m en t i on ed i n s ec t . 1 , t he c l o se s t exam pl e b e i ng t ha t o f vo r t i c e s in

a s u p e r f l u i d o r i n a s u p e r c o n d u c t o r [ 3 ] .

W e a r e g r a t e f u l t o V i c t o r B a r ' y a k h t a r a n d B o r i s l v a n o v f o r c o n v e r s a t i o n s

e m p h a s i z i n g th e p h e n o m e n o l o g i c a l a n a l o g y o f t h is p r o b l e m w i t h t h e H a l l e ff e c t ,

a n d t o M i c h a e l M a r d e r f o r a n u m b e r o f r e l a t e d d i s c u s s io n s . T h i s w o r k w a s

s u p p o r t e d in p a rt b y t w o g ra n t s f r o m t h e E E C ( S c ie n c e S C 1 " - 0 2 5 0 - C a n d E s p r i t

3 0 4 1) a n d b y a N A T O t r av e l g r a n t ( N o . 8 7 0 0 09 ).

Appendix A

TWO-DIMENSIONAL ELECTRON MOTION IN A UNIFORM MAGNETIC FIELD

I n o r de r t o i l l u s t r a t e t he ana l ogy w i t h t he H a l l e f f ec t m or e exp l i c i t l y , w e b r i e f l y

d e s c r i b e h e r e t h e c a n o n i c a l s t r u c t u r e a s s o c i a t e d w i t h t w o - d i m e n s i o n a l e l e c t r o n

m ot i on i n a un i f o r m m agne t i c f i e l d B pe r pe r ~d i cu l a r t o t he p l ane and an i n - p l ane



N. Papanicolaou, T.N. Tomaras / Magnetic vortices 4 5 9

e l ec t r i c f i e ld E = - V U , w h e r e U - U ( x ~ , x 2 ) i s t h e e l ec t ro s t a t i c p o t e n t i a l . W e u se

t h e s y m b o l B f o r t h e m a g n e t i c f i e l d t o d i s t i n g u i s h i t f r o m t h e m a g n e t i c f i e l d s

o c c u r r i n g i n t h e m a i n t e x t , r a t h e r t h a n t o d e n o t e m a g n e t i c i n d u c t i o n . I n f a c t , t h e

m a g n e t i c f ie l d B in t h e e le c t r o n p r o b l e m c o r r e s p o n d s t o th e w i n d in g n u m b e r Q o f

a v o r t e x , w h i l e t h e e l e c t r o s t a t i c p o t e n t i a l U c o r r e s p o n d s t o t h e m a g n e t i c f i e l d h o fe q . (5 .1 ) . T h e m a s s a n d t h e c h a r g e o f t h e e l e c t r o n a s w e l l a s th e s p e e d o f l ig h t a r e

s e t e q u a l t o u n i t y .

T h e e l e c t r o n m o t i o n i s t h e n g o v e r n e d b y t h e e q u a t i o n s

2 1 - - T r 1 , 2 2 - - 7 7 2 ,

~'1 = B ' t r 2 - t J l U , 7 1"2 = - B ' r r I - 0 2 U , ( A . 1 )

where ~-~ and 772 a r e th e c o m p o n e n t s o f t h e m e c h a n i c a l m o m e n t u m . T h e s e

e q u a t i o n s m a y b e u n d e r s t o o d a s a h a m i l to n i a n s y s t em w i t h h a m i it o n i an

' '( A . 2 )

e n d o w e d w i t h t h e P o i s s o n b r a c k e t r e l a t i o n s

{ ' r r l , X l } = 1 = { 'r / '2 , x 2 } , { - r /' 1 ,7 7 2 } = - B , ( A . 3 )

w h e r e w e d i s p l a y o n ly t h e n o n v a n i s h i n g P o i s s o n b r a c k e t s . A m o r e t r a n s p a r e n t s e t

o f v a r i a b l e s i s o b t a i n e d b y in t r o d u c i n g t h e g u i d i n g c e n t e r c o o r d i n a t e s

77 2 'IT 1

R 1 - - X l + B ' R 2 = x 2 B ' ( A . 4 )

wh i l e r e t a in in g r r~ an d 7 72 a s in d e p e n d e n t v a r i a b le s . T h e c o r r e s p o n d i n g P o i s s o n

b r a c k e t r e l a t i o n s r e a d a s

{ ' g ' l ,' T r 2 } = - B ,

1

{ R , , R 2 } = ~ - , {.a-u, R v} = 0 , ( A . 5 )

a n d t h e H a m i l t o n e q u a t i o n s a s

O U O U

(r l = B T r 2 - OR---~l' +2 = - B ' n ' ! - 0R--~2

! O U I ~ U- , - , ( A . 6 )

l~l B O R 2 I~2 B OR l

w h e r eU --- U ( x ) , x 2 ) " - U ( R t - " r r : /B , R , + " r r t / B ) .



4 6 0 N. Papanicolaou, T.N . Tom aras / Ma gnetic vortices

In t he absence o f an e l ec t r i c f i e l d (U = cons t . ) t he l a s t two equa t i ons i n eq . (A . 6 )

b e c o m e / ~ = 0 = / ~ 2 , i m p l y i n g t h a t t h e g u i d i n g c e n t e r r e m a i n s f i x ed , w h i c h is t h e

c o n s e r v a t i o n l a w a s s o c i a t e d w i t h i n v a r i a n c e u n d e r s p a c e t r a n s l a t i o n s . T h e c o n -

s e r v e d q u a n t i t y a s s o c i a t e d w i t h i n v a r i a n c e u n d e r s p a c e r o t a t i o n s , w h i c h r e q u i r e s

t h a t U b e a x i al ly s y m m e t ri c , is g i v e n b y th e a n g u l a r m o m e n t u m

B B 1l = ( X i T F 2 - - X 2 7 " I ' I ) -~- " ~ ( X I + X 2 ) = -~- (R~ + R 2 ) - -~-B-(zr2 + ' r r 2 ) , ( A . 7 )

whi ch i s a l so s een t o be d i f f e r e n t f r om i t s s t an da rd me cha n i ca l v a lue x F r 2 -x2"/'/" 1.

N o w i n t h e p r e s e n c e o f a g e n e r a l e l e c t r o s t a t i c p o t e n t i a l U = U(x~,x 2 ) n e i t h e r

t h e g u i d in g c e n t e r r e m a i n s f ix e d n o r t h e a n g u l a r m o m e n t u m is c o n s e r v e d . N e v e r -

t h e l e s s s o m e g e n e r a l p r o p e r t i e s o f t h e e l e c t r o n m o t i o n c a n b e d e r i v e d w i t h o u t a

d e t a i l e d s o l u t io n o f t h e e q u a t i o n s o f m o t i o n . F o r i n s t a n c e , i n t h e c a s e o f a u n i f o r me l e c tr i c f ie l d p o i n t i n g i n th e x r d i r e c t i o n , U = -Ex~, t h e l a s t t w o e q u a t i o n s i n e q .

(A . 6 ) r ead /~ = 0 and /~2 = - E / B , a n d i m p l y t h a t t h e g u i d i n g c e n t e r d r i f t s i n a

d i r e c t i o n p e r p e n d i c u l a r t o t h e e l e c t r i c f i e l d w i t h c o n s t a n t H a l l v e l o c i t y VH =

- E / B . F o r a m o r e g e n e r a l p o t e n t i a l U , t h e l a s t t w o e q u a t i o n s i n ( A . 6 ) c a n n o t b e

d e c o u p l e d f r o m t h e r e s t , e x c e p t i n t h e a d i a b a t i c ( l a r g e - B ) l i m i t w h e r e

1 0 U 1 0 U

/ ~ 1 - - J~ 2 U = U ( R I , R 2 ) , ( A . 8 )B 0 R 2 B o a r I

w h i c h m a y b e v ie w e d a s a h a m i l t o n i a n s y s te m f o r t h e p a i r o f c a n o n i c al l y c o n j u g a t e

va r i ab l e s R~ and R 2 wi th ham i l t on i an U(R ~, R 2) . As a r e su l t , t he gu id i ng ce n t e r

aga in d r i f t s a l ong equ ipo t en t i a l l i nes , i . e . i n a d i r ec t i on pe rpend i cu l a r t o t he

e l ec t r i c fi e ld , w i th a Ha l l ve loc i t y wh ose m agn i t ud e i s g i ven by I V U I l B .

F i n a l ly w e c o n s i d e r t h e e f f e c t o f d i s si p a t io n m o d e l e d h e r e b y a d d i n g a t e r m

-3 , z r t o the L o ren t z fo r ce , wh ere 3 ' i s t he d i s s i pa t i on co ns t an t . Fo r s impl i c i t y we

a s s u m e t h a t t h e e l e c t r i c f i e l d i s u n i f o r m . T h e n t h e s y s t e m o f e q u a t i o n s ( A . 6 )

b e c o m e s

7 :r I = B r r 2 + E - yr r ~, ~2 = -B rr ~ - 3 ,7T2,

3, E 3,g l = - - n ' ~ 2 , / ~ 2 = n "l- ~ - T r I . ( A . 9 )

T h i s sys t em can be so lved exp l i c i t l y , bu t t he e s sen t i a l r e su l t may be ob t a ined

s imply by assum ing tha t a s tea dy s ta te i s eve ntua l ly re ac he d wh ere "/r~ = 0 = "h-2.

T h en t he f i rs t two equa t i ons beco me a lgebra i c fo r t he un kn ow ns "n'~ and "rr2 which

a re i nse r t ed i n t he l a st two equ a t i on s t o y i e ld t he t e rm i na l ve l oc i t y o f t he gu i d in g



N . P a p a n i c o l a o u , T . N . T o m a r a s / M a g n e t i c c o r t ic e s 4 6 1

c e n t e r :

yE BE

V ! = g l = ,~ 2 - .I- B 2 ' V 2 = 1~ 2 = 7 2 + E 2 "

v = ( v , " +E

( y 2 + B 2 ) l / 2 "( A . 1 0 )

T h u s w e s e e t h a t th e g u i d i n g c e n t e r u n d e r g o e s s k e w d e f l e c t i o n a t a n a n g l e 8 w i t h

r e s p e c t t o t h e d i r e c t io n o f t h e e l e c t r i c f i e ld ;

I/2 Bs i n ~ = V E V" ( A . 1 1 )

T h i s r e l a t i o n i s a n a l o g o u s t o e q . ( 6 . 1 6 ) g i v e n i n t h e m a i n t e x t . R e c a l l i n g a l s o t h a t

t h e H a l l v e l o c it y i n t h e a b s e n c e o f d i s si p a t i o n w o u l d n o w b e g i v e n b y V n = - E / B ,

e q . ( A . 1 1 ) m a y b e w r i t t e n a s

V= VH Sin S, ( A . 1 2 )

w h i c h i s t h e a n a l o g o f e q . ( 6 .1 9 ) . T h e s a m e r e s u l t s a r e o b t a i n e d b y e x p li c it ly s o lv i n g

t h e i n i t i a l v a l u e p r o b l e m f o r e q . ( A . 9 ) a n d t h e n t a k i n g t h e l im i t t ~ ~ .

[1]

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d y n a m i c s o f m a g n e t i c v o r t i c e s

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