Decay of magnetization in the one-dimensional XY model

Volume 30A. n u m b e r 8

DECAY

OF

PHYSICS

MAGNETIZATION

IN

LETTERS

THE

15 D e c e m b e r 1969

ONE-DIMENSIONAL

XY

MODEL

M . D . G I R A R D E A U **

Institutt for Teoretisk Fysikk. Norges Tekniske Hdgskole. Trondheim. Norway Received 6 November 1969

The F o u r i e r components Mq of the magnetization approach t h e r m a l equilibrium in the one-dimensional 0 and q : =~-n due to existence of many s i m p l e constants of the motion.

X Y model if 0 < I q ] < ~" Non-ergodic b e h a v i o r occurs both for q

T h e o n e - d i m e n s i o n a l X Y m o d e l c o n s i s t s of a • 1 c h a i n of l o c a l i z e d s p l n - ~ p a r t i c l e s w i t h a n i s o t r o p i c c o u p l i n g d e f i n e d by t h e H a m i l t o n i a n [ 1 . 2 ] n-1 H :-2J=~I[(I+r)S

X S.x" V V j j + l . (1 - r ) ~ j 3 j + l ] .

(1)

The thermal equilibrium properties are exactly s o l u b l e s i n c e (1) c a n b e r e d u c e d to d i a g o n a l f o r m b y a J o r d a n - W i g n e r t r a n s f o r m a t i o n to l o c a l i z e d F e r m i o p e r a t o r s , a t r a n s f o r m a t i o n to p l a n e wave Fermi operators, and a final diagonalizat i o n b y a B o g o l i u b o v - V a l a t i n t r a n s f o r m a t i o n . It h a s b e e n s h o w n b y N i e m e i j e r [3] a n d B a r o u c h a n d D r e s d e n [4] t h a t t h e e x a c t n o n - e q u i l i b r i u m b e h a v i o r of t h e t o t a l m a g n e t i z a t i o n z - c o m p o n e n t c a n a l s o b e e v a l u a t e d in c e r t a i n c a s e s . It i s found that the magnetization exhibits a damped o s c i l l a t o r y d e c a y to a n a s y m p t o t i c v a l u e d i f f e r eat from its thermal equilibrium value. Such p e c u l i a r b e h a v i o r i s e x p e c t e d on t h e b a s i s of M a z u r ' s p r o o f [5] t h a t t h e m a g n e t i z a t i o n i s n o n e r g o d i c in t h i s m o d e l . T h e r e a s o n i s t h a t a n o n n e g l i g i b l e p a r t of t h e m a g n e t i z a t i o n o p e r a t o r c a n b e e x p r e s s e d l i n e a r l y in t e r m s of t h e c o n s t a n t s of t h e m o t i o n . Consider more generally the Fourier comp o n e n t s Mq = ~ Sj z c o s (qj). S u p p o s e t h a t at t = 0 t h e s t a t i s t i c a l a v e r a g e @In(t)} i s o u t of e q u i l i b r i u m b y a p r e s c r i b e d a m o u n t a n d t h e n a l l o w e d to p r o p a g a t e a c c o r d i n g to t h e H a m i l t o n i a n (1). T h e n

(Mq(t)} = Wr[p(0) exp (itH)Mq exp ( - i t H ) ]

a n d ;~ i s to b e d e t e r m i n e d s o a s to g i v e t h e p r e s c r i b e d i n i t i a l v a l u e . (Mo(t)} c a n b e e v a l u a t e d e x a c t l y [6], l e a d i n g to an e x p r e s s i o n d i f f e r i n g o n l y in n o t a t i o n f r o m t h e c a s e b = 0 of eq. (13) of B a r o u c h a n d D r e s d e n [4]. It e x h i b i t s a d a m p e d o s c i l l a t o r y d e c a y to a n o n - z e r o a n d h e n c e n o n e q u i l i b r i u m a s y m p t o t i c v a l u e , a s e x p e c t e d [5]. F o r q ¢ O, ,:Mq(t)} h a s n o t b e e n e v a l u a t e d in c l o s e d f o r m . b u t f o r i n v e s t i g a t i o n of t h e q u e s t i o n of w h e t h e r o r n o t it i s e r g o d i c it i s s u f f i c i e n t to u s e t h e a p p r o x i m a t i o n 1

( Mq(t)}lin = -½ ~ f['/Mq (0)Mq (t+ iris))eq + 0

(4)

(Mq(l - iris) Mq(0)}eq] d s of l i n e a r t r a n s p o r t t h e o r y . H e r e ( }eq d e n o t e s t h e e q u i l i b r i u m a v e r a g e . One f i n d s in t h e t h e r modynamic limit

:Mq(l)i~lin = - ~~v . .~. .*. ( I q+ . I ( -t ) where 7;

C k q ( A k + q ± A k ) -1 [ t a n h ( ~ * A k ÷ q )

(6)

-~

+ t a n h (i3*A k)] c o s [~(Ak+ q + A k ) l * ] d k and

C;q : ½(1 +A]~ 1 A ; l q [ c o s k

c o s (k + q) +

(7)

(2) -

where p(0) = e x p ( - r i H - ~ 4 q ) / T r

exp (-fiH-~14q)

* Supported in part by NORDITA. ** On leave from the University of Oregon, Eugene, Oregon. 442

(3)

(5)

7 2 s i n k s i n (k+ q)]}

w i t h f ~ * : f l J , ~ = 3" ~*, 1" - 4Jl , a n d A k =: -- ( c o s 2 k + ) / 2 s i n 2 k ) , ~ . T h e a s y m p t o t i c b e h a v i o r of t h e i n t e g r a l s (6) f o r t* ~ c a n b e e v a l u a t e d in c l o s e d f o r m . T h e y d e c a y to z e r o in a d a m p e d o s c i l l a t o r y f a s h i o n if 0 ~:, 7 ": 1 a n d 0 ": q ~,. [i

Volume 30A, n u m b e r 8

PHYSICS

q = 0, I - i s c o n s t a n t i n t i m e , s o t h a t t h e d e c a y i s to a n a s y m p t o t i c n o n - e q u i l i b r i u m v a l u e , in agreement with the exact solution. However, s i n c e A k i s p e r i o d i c w i t h p e r i o d n, s i m i l a r n o n e r g o d i c b e h a v i o r a l s 0 o c c u r s if.q = ± ~ , i . e . , t h e s t a g g e r e d m a g n e t i z a t i o n ~;j(-1)J S~ a l s o f a i l s to a p p r o a c h e q u i l i b r i u m . T h i s i s r e I a t e d to t h e n o n - e r g o d i c b e h a v i o r of Mo, s i n c e t h e u n i t a r y transformation S x ~ Sx, Sf 4 (_I)JsY ' S z• ~ ( - )l J S .z c o n vs e r t s sM i n t o t h e s t aJg g e r e d J ) u magnetization and converts the Hamiltonian H ( V , J ) [eq. (1)] i n t o H ( ~ - I , v J ) . T h e c o n s t a n t s of the motion responsible for the non-ergodic beh a v i o r of t h e s t a g g e r e d m a g n e t i z a t i o n d i f f e r

INTERNAL

FIELD

MEASUREMENTS IN ORDERED

LETTERS

15 D e c e m b e r 1969

from those responsible for the non-ergodic beh a v i o r of M o b y t h e s u b s t i t u t i o n y - 7 -1.

References 1. E. Lieb, T. Schultz and D. Mattis, Ann. Phys. (N.Y.) 16 (1961) 407. 2. S. Katsura, Phys. Rev. 127 (1962) 1508. 3. Th. N i e m e i j e r , Physica 36 (1967) 377. 4. E. Barouch and M. Dresden, Phys. Rev• L e t t e r s 23 (1969) 114• 5. P. Mazur, Physica 43 (1969) 533. 6. The details of the calculations and r e s u l t s a r e available as No. 15-1969 of Arkiv for Det Fysiske Seminar i Trendheim.

AND Fe-Si

SECCND ALLOYS

NEIGHBOR

EFFECTS

M. B. S T E A R N S , L . A . F E L D K A M P a n d J . F. U L L R I C H Ford Scientific Laboratory, Dearborn, Michigan 48121, USA Received 3 November 1969

The i n t e r n a l field shifts due to second n e a r e s t neighbor Fe atoms of the D s i t e s (8-NN Fe) in o r d e r e d Fe-Si alloys have been well r e s o l v e d in pulsed NMR m e a s u r e m e n t s . The frequency dependence of the Si s i t e s is attributed to the volume dependence of the magnetization.

T h e i n t e r n a l f i e l d s of t h e v a r i o u s s i t e s in o r d e r e d a l l o y s w i t h b c c DO 3 s t r u c t u r e ( F e 3 S i ) h a v e b e e n m e a s u r e d b y M ~ i s s b a u e r [1] a n d p u l s e d N M R t e c h n i q u e s [2]. T h e s p e c t r a of t h e D s i t e s ( 8 - N N F e , 6 - 2 N N Si, 1 2 - 3 N N F e i n F e 3 S i ) s h o w e d s t r u c t u r e in t h e i n h e r e n t l y b e t t e r r e s o l v e d p u l s e d N M R s p e c t r a . It w a s p r o p o s e d [2] t h a t t h i s s t r u c t u r e w a s d u e to a c h a n g e in e l e c t r o n i c s t r u c t u r e of t h e a l l o y s n e a r 23 a t . % Si, b u t no o t h e r t y p e m e a s u r e m e n t s i n d i c a t e a n y such structural change. Our measurements s h o w t h a t t h e s t r u c t u r e i s d u e to s e c o n d NN F e atoms. Some measured and calculated spectra are s h o w n in fig. 1; t h e s m a l l p u l s e l e n g t h s i n d i c a t e d w e r e u s e d to o b t a i n r e a s o n a b l e s i g n a l s t r e n g t h since the alloys have very large enhancement f a c t o r s (10 000 - 50 000). T h e c a l c u l a t e d c u r v e s are obtained by assuming that the excess Fe a t o m s go r a n d o m l y i n t o t h e Si s i t e s a n d t h a t e a c h component has the same shift and width. The latter assumption is not strictly true; a second

F e c a u s e s a s l i g h t l y s m a l l e r s h i f t (0.8 ± 0 . 2 ) t h a n t h e f i r s t . A t h i r d F e a t o m a p p e a r s to c a u s e about the same shift as the second. Considering t h e i n h e r e n t d i f f i c u l t i e s in o b t a i n i n g a c c u r a t e l i n e s h a p e s , w e s e e f r o m fig. 1 t h a t t h e r e i s l i t t l e d o u b t t h a t t h e s t r u c t u r e i s d u e to s e c o n d neighbor effects. In fig. 2 w e p l o t t h e f r e q u e n c i e s f o r e a c h t y p e s i t e a s a f u n c t i o n of Si c o n t e n t . T h e D s u b s c r i p t i n d i c a t e s t h e n u m b e r of s e c o n d NN F e ' s w h i l e f o r A (4NN of e a c h F e a n d Si, 6 - 2 N N F e a n d 1 2 - 3 N N F e in F e 3 S i ) s i t e s i t i n d i c a t e s t h e n u m b e r of f i r s t NN F e ' s . R a t h e r s u r p r i s i n g l y t h e measured shifts agree well with those observed f o r d i l u t e F e - S i a l l o y s [3]. F o r t h e s e a l l o y s t h e m a i n c o n t r i b u t i o n s to t h e i n t e r n a l f i e l d c o m e from the core and conduction s electrons (c.e.). It i s u s u a l to a s s u m e t h a t e a c h of t h e s e c o n t r i b u t i o n s i s p r o p o r t i o n a l to t h e m o m e n t ( i . e . , u = = THin t =Act). F o r t h e D s i t e s t h e v a r i a t i o n i n n u m b e r of 2NN h a s a s m a l l e f f e c t o n t h e m o m e n t a n d t h e o b s e r v e d s h i f t of a b o u t - 0 . 0 2 5 H F e p e r

443