francisco c. alcaraz et al- three-dimensional spin systems without long-range order

8/3/2019 Francisco C. Alcaraz et al- Three-Dimensional Spin Systems Without Long-Range Order

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Nuclear Physics B265 [FS15] (1986) 161-186

T H R E E - D I M E N S I O N A L S P I N S Y S T E M S

W I T H O U T L O N G - R A N G E O R D E R

Francisco C. ALCARAZ

Departamento de Ffs,ca, Unwersidade Federal de S~o Carlos, 13560 S~o Carlos, SP, Brasd

Laurence JACOBS

A T&T Bell Laboratories, Holmdel, NJ 07733 USA

andInstitute for Theoretical Phystcs, Umverstty of Cahforma, Santa Barbara, CA 93106, USA

Robert SAVIT2

Phystcs Department, Unwerstty of Michigan, Ann Arbor, MI 48109, USA

Received 15 April 1985

Several examples of three-dimensional statistical models with multibody interactions and

unusual symmetry properties are studied from a unified point of view. The symmetries of these

theories are neither global nor local, but are something in between In the nomenclature of a

recently proposed classificat ion scheme, these theories have a symmetry index, n ~< 2, and so have

no long-range order when the internal symmetry is continuous Duality properties and topological

excitations of the theories are studied, and their phase diagrams are qualitatively discussed.

Similarities between these three-dimensional semi-local theories and ordinary globally-symmetric

theories in one and two dimensions are d~scussed.

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

R e c e n t l y , a s c h e m e f o r t h e c l a s s i fi c a ti o n o f s ta t is t ic a l s y s t e m s p o s s e s s i n g m u l t i b o d y

i n t e r a c ti o n s w h o s e s y m m e t r ie s a r e m o r e g e n e r a l t h a n t h e u s u a l g l o b a l o r lo c a l

s y m m e t r ie s h a s b e e n p r o p o s e d [ 1] a n d h a s l e d to t h e i d e n t if i ca t io n o f a n e w s y m m e t r y

index, n, f o r t h e s e t h e o r i e s . J u s t a s th e u s u a l g l o b a l l y - s y m m e t r i c s t a ti s ti c a l t h e o r i e s

w i t h c o n t i n u o u s i n t e r n a l s y m m e t r ie s h a v e a l o w e r c r it ic a l d i m e n s i o n , d = 2, s y s t e m s

w i t h m o r e g e n e r a l s y m m e t r i e s h a v e a l o w e r c ri ti c a l v a l u e f o r t h e i r a s s o c i a te d

s y m m e t r y i n d e x . A t h e o r e m w a s r e c e n t l y p r o v e d [ 2] w h i c h s h o w s t h a t , re g a r d le s s

o f th e d i m e n s i o n a l i t y o f a s y s t e m , t h e o r i e s w i t h a U ( 1 ) s y m m e t r y w i t h n ~<2 h a v e

n o l o n g - r a n g e o r d e r . A p a r t f r o m t h e i n t ri n s i c t h e o r e t i c a l in t e r e s t e m b o d i e d i n t h e s e

s y s t e m s , s t a ti s ti c a l m o d e l s w i t h n = 2 ( a n d o t h e r c l o s e l y r e l a t e d m o d e l s ) a r e t h o u g h t

t o b e r e l e v a n t t o t h e d e s c r i p t i o n o f c e r ta i n a s p e c t s o f l iq u i d c r y s t a ls a n d h e l i c a l

m a g n e t s . T h i s p a p e r is c o n c e r n e d w i th a fu l l e r d i s c u s s i o n o f t h e m a r g i n a l c a s e ;

Work supported in part by Conselho Nacional de Desenvolvimiento CienUfico (Brazil).

2 A.P Sloan Foundation Fellow Research supported in part by the National Science Foundation underGrant PH78-0846.

162 F C . Alcaraz e t a l / Sp in sys tems

n = 2. We will make a n umb er of general observations abou t n = 2 theories and

study in some detail severa l n = 2 statistical theories in three d imens ions. As sug-

gested by the theorem of ref. [2], we shall find that these models have critical

properties qualitatively similar to ordinary globally-symmetric statistical models in

two dimensions. It is our intention that the study of these models at their lower

critical value of n be a first step in the systematic study of the critical properties of

the rich set of theories included in our classification scheme.

In subsect. 2.1 we will define what is mean t by an index-n symmetry in its simplest

sense. In subsect. 2.2 we will describe the proof of the theorem which states that

theories with a U(1) symmetry and n <~ 2 have no long-range order. In subsect. 2.3

we will discuss an extension of the definition of index-n symmetries to a wider class

of theories, and we will argue that the theorem of subsect. 2.2 applies to this larger

class as well. Subsect. 2.4 contains some additional, miscellaneous comments. In

sect. 3 we will analyze three closely related three-dimensional models, all of which

have an n = 2 symmetry in the restricted sense of subsect. 2.2 but which differ in

detail. These models are quite simple a nd serve to clarify most of the general features

of n = 2 symmetries. In particular, we shall discuss their properties under duality

transformations and the relations between order and disorder variables. Next we

present the results of Monte Carlo simulations of the ZN version of one of these

models. These simulations lend strong suppor t to the general theoretical expectations

for this class of systems; in particular, they verify the prediction that the phase

diagrams of n = 2 theories with a ZN symmetry are qualitatively quite similar to

those of the d = 2 globally-symmetric, two-dimensional clock models [3]. Lastly,

we shall briefly describe two somewhat more complex n = 2 theories which have

been di scussed in the literature [4-6] and which, in a sense, motivated much of the

present work. These two theories are examples of the generalized n = 2 theories

discussed in ref. [1].

We summarize our results in sect. 4 and conclude with some suggestions for

further work. Finally, we show how to construct the hamiltonian formulation of

one of our models in an appendix.

2 . P re l i m i n a r i e s

2.1. INDEX-n SYMMETRIES

The theories we will discuss in this paper will be statistical theories of a single

field which is associated with the sites of a lattice. In this section we shall consider

those theories which have a U(I) internal symmetry with the site-associated spins

given by s ( r ) = e '~r~. For simplicity we will consider theories defined on a d-

dimensional simple hypercubic lattice. The defining characteristic of our models isthat they are described by hamiltonians which are invariant under the transformation

$ ( x l , . . . , X d ) - . " $ ( X l , . . . , X d ) + A ( X ,, + I . . . . , X d ) . (2.1)

F.C. A lcaraz e t a l / Sp in sys tems 16 3

T h e s e t h e o r i e s w i l l b e s a i d t o p o s s e s s a symmetry index e q u a l t o n . ( A c t u a l l y , m o s t

o f th e r e s u lt s o f t h is p a p e r a p p l y t o a w i d e r c l a s s o f t h e o r ie s . T h i s g e n e r a l i z a t i o n is

d e f i n e d i n s u b s e c t . 2 .3 .) I n f a c t, m o s t o f t h e m o d e l s w e w i ll e n c o u n t e r h a v e a

s o m e w h a t m o r e c o m p l e x s y m m e t r y s t ru c t u re i n w h i c h t h er e m a y b e m o r e t h a n o n e

d i s t i n c t s e m i - l o c a l s y m m e t r y ( s e e , e .g ., t h e t h e o r y d e f i n e d i n eq . ( 3. 4) b e l o w ) . S u c h

t h e o r i e s w il l b e s a id to h a v e a c o m p o u n d s y m m e t r y .

A s w e h a v e d i s c u s s e d e l s e w h e r e [ 1] , t h e h a m i l t o n i a n s o f o u r m o d e l s c a n b e w r i t te n

i n t h e f o r m

w h e r e t h e f p a re f u n c t i o n s o f a l i n e a r c o m b i n a t i o n o f q(p) ~b 's an d the cpj a r e s o m e

s e t o f c o n s t a n t c o e f f i c i e n t s . T h e c o o r d i n a t e o f ~bj i s x + r~. W e w i l l a s s u m e t h a t t h e r ea r e n o e x p l i c i t l o n g - r a n g e i n t e r a c t i o n s , s o t h a t a l l I ~ 1 a r e f i n i t e . F o r t h e s a k e o f

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

( 2. 2) a r e d e f i n e d o n t h e k - d i m e n s i o n a l ( k ~< d ) s i m p l i c e s o f t h e d - d i m e n s i o n a l l a tt ic e .

F o r e x a m p l e , n e a r e s t - n e i g h b o r i n t e ra c t i o n s a l o n g l i n ks , f o u r - b o d y i n te r a c t i o n s

a r o u n d a p l a q u e t t e , e t c . O u r r e s u l t s a p p l y a s w e l l i n m o r e g e n e r a l c a s e s . (S e e re f .

[ 1 ] a n d t h e e x a m p l e s b e l o w . )

2 2 A B S E N C E O F L O N G - R A N G E O R D E R F O R n ~< 2

I t w a s r e c e n t l y s h o w n [ 2 ] t h a t t h e o r i e s w i t h n = 1 o r n = 2 s y m m e t r y i n d i c e s d o

n o t h a v e l o n g - r a n g e o r d e r f o r a n y T > 0 . L e t u s b r ie f ly r e v i e w t h e m a i n f e a t u r e s o f

t h is r es u lt . T h e p r o o f is a n e x t e n s i o n o f t h e M e r m i n - W a g n e r t h e o r e m [ 7] a n d r e li es

o n t h e u s e o f a B o g o l i u b o v in e q u a l i t y [ 8 ] t o b o u n d t h e s q u a r e o f t h e m a g n e t i z a t i o n

f r o m a b o v e .

C o n s i d e r a s t a ti s ti c a l t h e o r y w i t h a h a m i l t o n i a n a s t h a t g i v e n in ( 2 .2 ) d e f i n e d o n

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

t h e o r y w e f o l lo w t h e g e n e r a l p r o c e d u r e u s e d to p r o v e t h e M e r m i n - W a g n e r t h e o r e m

b y m a k i n g u s e o f a B o g o l i u b o v i n e q u al it y ,

½/3({A, A*}) ( [ [C , H ] , C* ] ) /> I ( [C , A] ) [ 2 , ( 2 .3 )

w h e r e a d a g g e r s t a n d s f o r h e r m i t i a n c o n j u g a t i o n , a n d [ , ] , { , } d e n o t e , r e s p e c t i v e l y ,

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

o p e r a t o r O a t i n v e r s e t e m p e r a t u r e / 3 . C a l l in g t h e s t a t e o f t h e s y s t e m d e f in e d b y t h e

s e t { & ( x ) } f o r a l l p o i n t s , x , o n t h e l a t t i c e , 1 4 ) ( x ) ) , w e c a n d e f i n e s u i t a b l e o p e r a t o r s

, 4 a n d C t h r o u g h t h e i r a c t i o n o n t h e s e s t a t e s in s u c h a w a y t h a t ( 2 .3 ) le a d s t o a n

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

C k I • ( X ) ) : I t ~ ( X ) - ~ (~ C OS k ° x ) - [ - [ ~ ( x ) - t ~ ) s i n k . x ) , ( 2 .4 )

Ak[qb(x)) = ~ c o s k . y s i n 4 ~ ( Y ) l ~ , ( x ) ) + Y . s i n k . y s in & ( y ) l ~ 2 ( x ) ) , ( 2. 5)v y

w h e r e

F.C . A Ic araz e t a l. / S p in sy s t e m s

I ~ , ( x ) > ~ I ~ ( x ) + 8 ~ c o s k . x > ,(2 .6)

I q b 2 ( x ) ) - I & ( x ) + 8~b s in k - x ) .

I n t h e a b o v e e x p r e s s i o n s 8~b i s a s m a l l c o n s t a n t f i e ld a n d k i s a m o m e n t u m v e c t o r

i n t h e f i rs t B r i l l o u i n z o n e . C o n s i d e r t h e c a s e n = 2 . T o s t u d y t h e p o s s i b i l i t y o f a

s p o n t a n e o u s b r e a k i n g o f th e g l o b a l s y m m e t r y , w e c o n s i d er a h a m i l t o n i a n H o (~ b),

i n v a r i a n t u n d e r ( 2. 1) w i t h n = 2 a n d a d d a s y m m e t r y - b r e a k i n g t e r m i n t h e f o r m o f

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

H = Ho(~b) - h 3~ co s th (x ) . (2 .7 )x

N o w , i f H o (~ b ) i s i n v a r i a n t u n d e r ( 2 .1 ) , it is e a s y t o p r o v e t h a t i t c a n b e w r i t te n i n

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

s o t h a t

H o ( ~ )

A s s u m i n g t h a t t h e f p ar e e x p a n d a b l e i n a T a y l o r s e r ie s a b o u t th e z e r o o f t h e i r

a r g u m e n t s , u s e o f t h e a b o v e o p e r a t o r s i n ( 2.7 ) l e ad s t o a n u p p e r b o u n d f o r t h e

s q u a r e o f t h e m a g n e t i z a t i o n , m ,

y Y ~ p e p + h m ' (2 .9)

w h e r e N i s t h e n u m b e r o f si te s i n t h e l a tt ic e , y i s a p o s i t i v e n u m b e r w h i c h b o u n d s

t h e t h e r m a l a v e r a g e o f f ~ f r o m a b o v e , a n d

= cp, s in k . r , . (2 .1 0)Y ~ c p , c o s k . r , + =~. I=l I

N o t e t h a t w e a r e s t i l l d e a l i n g w i t h a c l a s s i c a l t h e o r y ; b r a s a n d k e t s a r e r o w a n d

c o l u m n v e c t o r s in x s p a c e a n d o p e r a t o r s a r e m a t r i c e s w h i c h a c t o n th i s s p a c e.

W e w i l l n o w s h o w t h a t i f / 4 0 is i n v a r ia n t u n d e r t h e t r a n s f o r m a t i o n

4 a ( x , . . . . , x a ) ~ d p ( x , . . . . . x a ) + A ( x 3 , . . . , X d ) , ( 2 . 1 1 )

t h e n t h e s u m o v e r k o n th e r i g h t - h a n d s i d e o f ( 2. 9) d i v e r g e s i n t h e t h e r m o d y n a m i c

l i m i t a s h -~ 0 , t h u s i m p l y i n g t h a t t h e m a g n e t i z a t i o n v a n i s h e s i n t h i s l im i t .

S i n c e e a c h o f t h e f p a re s e p a r a t e l y i n v a r i a n t u n d e r ( 2 .1 0 ) , f o r e a c h p t h e c o e f f ic i e n t s

a p p e a r i n g i n t h e a r g u m e n t s o f th e f p ( s ee ( 2 . 2) ) s a t i sf y

Z cpj = 0 , (2 .12)

F.C. Alcaraz et al / Spin systems 16 5

w h e r e a r e p r e s e n t s t h e s e t o f a l l s p i n l a b e l s w h i c h l i e i n a g i v e n ( x l , x 2) p l a n e . ( I n

c o n t i n u u m n o t a t i o n t h is is e q u i v a l e n t t o t h e s t a t e m e n t t h a t H o d e p e n d s o n ~b o n l y

t h r o u g h 0 1~ b a n d 02~b. T h i s a l l o w s t e r m s i n H o s u c h a s ( 0 30 24 ,) 2, b u t n o t t e r m s s u c h

a s , f o r e x a m p l e , ( 0 3 ~ b ) 2 . )

B e c a u s e ep = 0 f o r k ~ - k 2 = 0 th e f a c t th a t i t h a s a d o u b l e T a y l o r e x p a n s i o n i n

p o w e r s o f k~ a n d k2 i m p l i e s t h a t , i n g e n e r a l ,

e p = k 2 u p ( k ) + k E v p ( k ) + k l k E w p ( k ) , (2 .13 )

w h e r e up, vp a n d w p a r e f i n it e f u n c t i o n s o f k . C o n s i d e r a l a r g e l a tt i c e o f l in e a r

d i m e n s i o n L . D e f i n i n g u ( k ) = y ~ p u p ( k ) , e t c . , a n d c o n v e r t i n g t h e s u m o v e r k i n

( 2 . 9 ) i n t o a n i n t e g r a l l e a d s t o

( 2 ~ ) d / 3m 2 ~ ( 2 . 1 4 a )

I a ( L , h ; m ) '

w h e r e

Id = | d d k [ k E u + k 2 v + k , k E w + h m ] -1 • ( 2 . 1 4 b ).11/L

S i n c e ep >! 0 f o r e a c h p , l o o k i n g a t t h e r e g i o n o f i n t e g r a ti o n w h e r e k l a n d k 2 a r e

c l o s e t o 1 / L , i t i s c l e a r t h a t Id d i v e r g e s a t l e a s t a s f a s t a s I n L i n th e l i m i t a s h - ~ 0 .

S i m i l a r l y , t h i s i n t e g r a l w i l l a l s o d i v e r g e i f t h e l a r g e - L l i m i t is t a k e n f ir s t. T h u s , i n

t h e t h e r m o d y n a m i c l i m i t m 2 - - ~ 0 a s h ~ 0 a n d t h e s y s t e m h as n o l o n g - r a n g e o r d er :

T h e g l o b a l s y m m e t r y c a n n o t b e s p o n t a n e o u s l y b r o k e n . C o n t r a s t in g t h e a b o v e a r g u -

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

t h a t t h e e s s e n t i a l f a c t o r l e a d i n g t o t h e a b s e n c e o f lo n g - r a n g e o r d e r i s n o t t h e

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

w h i c h t h e s y m m e t r y i s g l o b a l . I t is t ri v ia l to a p p l y t h e a b o v e a r g u m e n t s t o p r o v e

t h a t t h e r e i s a l s o n o l o n g - r a n g e o r d e r i n a n n = 1 t h e o r y . T o d o s o , i t i s s u f f ic i e n t

t o n o t i c e t h a t , w h e n n = 1 , ep c a n b e w r i t t e n j u s t a s f o r n = 2 b u t w i t h v p - - w p = 0.

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

N o t o n l y i s t h e g l o b a l s y m m e t r y u n b r o k e n f o r n - 2 , b u t i n f a c t, t h e f u ll s e m i - lo c a l

s y m m e t r y is a ls o u n b r o k e n i n th i s c la s s o f th e o r i e s. T o d e m o n s t r a t e t h is , l et u s a g a i n

c o n c e n t r a t e o n t h e n = 2 c a s e ; i t i s s i m p l e t o c o n s t r u c t a s i m i l a r a r g u m e n t f o r n = 1.

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

a n o p e r a t o r w h i c h w i ll o n l y b r e a k t hi s s y m m e t r y , l e av i n g o t h e r s u b s y m m e t r i e s o f

t h e f u l l n - - 2 s y m m e t r y , p a r t i c u l a r l y t h e g l o b a l o n e , i n t a c t . W i t h H 0 i n v a r i a n t u n d e r

( 2 . 1 1 ) , o n e c a n e a s i l y s e e t h a t t h e r e l e v a n t o b j e c t t o t e s t i s t h e t w o - p o i n t f u n c t i o n

d e f i n e d b y

F M = c o s [ ~ b ( x ) - ~ b ( x+ M ) ] , ( 2.1 5)

w h e r e M is s o m e v e c t o r w i th a n o n - z e r o p r o j e c t i o n o u t o f t h e ( x l , x 2) p l a n e . W e

16 6 F.C . Al c a raz e t a l. / Sp in sy s t e m s

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

H = Ho- h~ Y r~(x)x

a n d c o m p u t e

(2 .16)

w h e r e

AM (~b(y ) ) =- s in (~b (y ) - ~b (y + M ) ) .

I t is n o w s t r a i g h t f o r w a r d t o c o m p u t e ( 2 .1 7 ). W e f in d [ 2 ]

m ~ < (2 7 r)a /3 f d d k ~ 2 ( k ) ] - '4 L J a / L k 2 u ( k ) + k 2 v ( k ) + k ~ k 2 w ( k ) + ~ ( k ) h ~ m ~ '

~ : ~ (k ) = 1 - c o s k - M . ( 2. 19 )

S i n c e M h a s a c o m p o n e n t o u t o f t h e ( X l, x 2) p l a n e , t h e n u m e r a t o r o f t h e i n t e g r a n d

i n ( 2 .1 9 ) d o e s n o t v a n i s h w h e n k l = k 2 - - 0 . B e c a u s e t h e i n t e g r a n d i n ( 2 .1 9 ) i s p o s i t i v e ,

t h e a n a l y s i s p r o c e e d s a s f o r ( 2.1 5 ) a n d t h e d i v e r g e n c e s a r e s im i l a r . T h u s , m 2 - - 0

i n t h e t h e r m o d y n a m i c l i m i t a n d t h e f u ll n = 2 s y m m e t r y i s n o t s p o n t a n e o u s l y b r o k e n .

N o t i c e t h a t i t is e s se n t i a ll y th e s a m e m e c h a n i s m w h i c h is r e s p o n s i b l e f o r t h e

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

t h e s a m e l o n g - w a v e l e n g t h p h o n o n e x c i ta t io n s w h i c h p r e v e n t b o t h t h e l o ca l as w e ll

a s th e s e m i - l o c a l s y m m e t r i e s f ro m b e i n g b r o k e n . O n e c a n a p p l y s i m i la r a r g u m e n t s

t o t es t f o r s y m m e t r y - b r e a k i n g i n o t h e r d i r ec t io n s , a n d s y m m e t r y - b r e a k i n g a s s o c i a te d

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

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

h a s b e e n d i s c u s s e d i n t h e l i t e r a tu r e [ 5 ] a n d w i ll b e b r i e fl y a n a l y z e d i n t h e n e x t

s u b s e c t i o n .

B e f o r e p r o c e e d i n g t o t h e n e x t s e ct io n , w e m a k e o n e i m p o r t a n t e x t e n s io n o f o u r

t h e o r e m . F o r a th e o r y w i t h n = 1, t h e g l o b a l s y m m e t r y c a n n o t b e s p o n t a n e o u s l y

b r o k e n , w h e t h e r t h e i n t e r n a l s y m m e t r y i s c o n t i n u o u s o r d is c re t e . A s w e s h a ll s h o wi n s e ct . 3 , t h i s is b e c a u s e a n n = 1 t h e o r y i n d d i m e n s i o n s c a n b e t r a n s f o r m e d i n t o

a se t o f n o n - i n t e r a c t i n g s y s t e m s in d - 1 d i m e n s i o n s i n a m a n n e r s i m i l a r t o th e w a y

t 2 1 7 ,A s w e d i d f o r t h e c a s e o f g l o b a l s y m m e t r y b r e a k i n g , w e w i ll te s t th e p o s s i b il i ty

o f b r e a k i n g t h e s e m i - l o c a l s y m m e t r y b y c o m p u t i n g ( 2.1 7 ) u s in g ( 2 .1 6 ) a n d t h e n

t a k i n g t h e l im i t as h ~ t e n d s t o z er o . A n o n - v a n i s h i n g v a l u e o f mM i n t h i s l i m i t

w o u l d s ig n a l t he s p o n t a n e o u s b r e a k i n g o f t h e s e m i - lo c a l sy m m e t r y . T o c o n s tr u c t

a n u p p e r b o u n d o n m 2 , w e a g a i n us e a B o g o l i u b o v in e q u a l it y w i th Ck d e f i n e d a s

i n ( 2 . 6 ) b u t w i t h

m k [ t ~ ( X ) ) = ~ , ( CO S k " y - c o s k " ( y - M ) ) A M ~b(y ) ) I~ I ( X ) )

+ E ( s in k - y - s i n k . ( y - - M ) ) A M ( q b ( y ) ) l ~ 2 ( X ) ) , (2 .18)Y

F.C. Alc ara z et al. / Spin systems 167

in w h i c h t h e o n e - d i m e n s i o n a l I si n g m o d e l c a n b e r e w r i tt e n a s a se t o f n o n - i n t e r a c ti n g

s p in s . H e r e a g a i n w e s ee h o w a s y s te m w i t h a s y m m e t r y in d e x n a n d d d i m e n s i o n s

b e h a v e s q u a l i t a ti v e l y li k e a n o r d i n a r y n - d i m e n s i o n a l t h e o r y w i t h g lo b a l s y m m e t ry .

2.3 GENERALIZ ATIONS OF INDEX-n SYMMETRIES

T h e s y m m e t r i e s d e f i n e d i n ( 2 . 1 ) c a n b e g e n e r a l i z e d i n a n a t u r a l a n d i m p o r t a n t

w a y . T h e s p a c e - d e p e n d e n c e o f A i n ( 2. 1) c a n b e e x p r e s s e d b y s a y i n g t h a t A is a

f u n c t i o n o f a ll d c o o r d i n a t e s , b u t is c o n s t r a i n e d t o s a t i s fy

A ~A = A 2A . . . . . A , A = 0 , ( 2 .2 0 )

w h e r e z a, i s a f i n i t e - d i f f e r e n c e o p e r a t o r i n t h e x , d i r e c t i o n . S u c h a s e t o f c o n d i t i o n s

c a n b e g e n e r a l i z e d t o

01A = 02A . . . . . O,,A = 0 , ( 2 . 2 1 )

w h e r e t h e (9, a r e a s et o f n l i n e a r l y i n d e p e n d e n t d i f f e re n c e (o r , in c o n t i n u u m

l a n g u a g e , d i f f e r e n ti a l ) o p e r a t o r s . T h e o r i e s c o n s t r u c t e d c o n s i s t e n t w i t h th e c o n d i t i o n s

g i v e n i n ( 2 .2 1 ) w i ll g e n e r a l l y h a v e i n t e r a c t i o n s d e f i n e d o n l a tt ic e e l e m e n t s o t h e r

t h a n s i m p l e h y p e r c u b e s .

T h e c la s s o f th e o r i e s w i th s u c h a s u i ta b l y d e f i n e d e x t e n d e d i n d e x - n s y m m e t r y is

e x p e c t e d t o s h a r e m a n y o f t h e p r o p e r t ie s o f t h e o ri e s w i th a n o r d i n a r y i n d e x - n

s y m m e t r y . A m o n g t h e m o s t i m p o r t a n t o f th e s e is t h e a b s e n c e o f lo n g - r a n g e o r d e r

f o r n = 1 o r 2 . I t is n o t d i f f ic u l t t o s e e h o w t h i s r e s u l t g e n e r a l i z e s t o i n c l u d e t h e s e

c a s e s as w e l l. R e c a l l t h a t w e a r g u e d t h a t , i n t h e r e s t r ic t e d c a s e o f t r a n s f o r m a t i o n s

o f th e t y p e g i v e n i n ( 2 . 1 1 ), a l l t h e t e r m s i n H o d e p e n d e d o n ~b o n l y t h r o u g h 01~b

a n d 02~b w h i c h i m p l i e d t h a t t h e i n v e rs e p r o p a g a t o r i n m o m e n t u m s p a c e w a s

n e c e s s a r i l y o f th e f o r m g i v e n i n ( 2 . 13 ). I n m u c h t h e s a m e w a y , t h e r e a r e a l a r g e

c la s s o f o p e r a t o r s , O 1 a n d 0 2 s u c h t h a t s y m m e t r y o f / 4 o u n d e r ( 2. 21 ) im p l i es t h a t

i t c a n o n l y d e p e n d o n ~b t h r o u g h O l ~b a n d O 2~ b. T h i s i n t u r n i m p l i e s a s t r u c t u r e f o r

ep s i m i l a r t o ( 2 . 1 3 ) b u t i n w h i c h t h e r o l e o f k~ a n d k~ w i ll b e t a k e n b y s o m e o t h e r

q u a d r a t i c c o m b i n a t i o n o f th e m o m e n t a , r e f le c ti n g t h e f o r m o f t h e o p e r a t o r s O1 a n d

0 2 . T h e r e w i ll th e n b e m a n y c a se s in w h i c h a b o u n d o n t h e s q u a r e o f t h e m a g n e t iz -

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

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

p r e c i s e , g e n e r a l c o n d i t i o n s o n t h e O , r e q u i r e d t o le a d t o t h is c o n c l u s i o n . A n e x a m p l e

o f a n n = 2 t h e o r y i n v a r i a n t u n d e r a t ra n s f o r m a t i o n o f th e f o r m ( 2. 21 ) w h i c h h a s

n o s y m m e t r y b r e a k i n g [ 4 ] w il l b e b r i ef l y d i s c u s s e d b e l o w .

2.4. MISCELLANEOUS COMMENTS

( i) I t is c l e a r l y r e a s o n a b l e t o b e l i e v e t h a t a r e s u l t s i m i la r to t h e t h e o r e m o f s u b s e ct .

2 .2 c a n b e p r o v e n i n th e c a s e o f n o n - a b e l i a n s y m m e t r i e s, s in c e o n e g e n e r a l ly e x p e c t st h a t d i s o r d e r w i ll c o m e a b o u t m o r e e a s il y w h e n t h e n u m b e r o f d e g r e e s o f f r e e d o m

is i n c r e a s e d . I t is a l s o c l e a r t h a t t h e r e m a y b e e x c e p t i o n s to t h is s i m p l e i n t u i t i v e

168 F . C . A l c a r a z e t al . / S p i n s y s t e m s

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

o r 2 i t s h o u l d b e p o s s i b l e to p r o v e a n a n a l o g o u s t h e o r e m .

( ii ) I t s h o u l d b e p o s s i b l e t o c o n s t r u c t t h e o r i e s, b y a v a r i e t y o f m e t h o d s , w i th

m o r e t h a n o n e f ie ld , a n d w i t h s e m i - l o c a l s y m m e t r i e s o f t h e ty p e w e h a v e d i s c u s s e d

h e r e . I n th i s w a y , it s h o u l d b e p o s s i b l e t o d i s cu s s t h e a b s e n c e o f s y m m e t r y - b r e a k i n g

i n a w i d e v a r i e t y o f th e o r i e s r a n g i n g f r o m s i m p l e d = 2 g l o b a l l y - s y m m e t r i c t h e o r i e s

to b o n a - f i d e l o c a l l y - s y m m e t r i c g a u g e t h e o r i e s in a n y d i m e n s i o n . W i th t h is a p p r o a c h

o n e m a y b e a b l e t o d is c u ss b o t h t h e M e r m i n - W a g n e r t h e o r e m a n d E l i t z u r' s t h e o r e m

[ 9] in th e s a m e f r a m e w o r k a n d p r o b e f o r a d e e p e r c o n n e c t i o n b e t w e e n t h e m w h i c h

i s p r e s e n t l y n o t a p p a r e n t .

( ii i ) T h r o u g h o u t t h e a b o v e d i s c u s si o n w e h a v e b e e n s o m e w h a t c a v a l ie r a b o u t

p r o b l e m s w h i c h m i g h t a r i se a s a r e s u lt o f b o u n d a r y c o n d i t io n s . I t is c le a r t h a t th i s

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

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

t h e f o r m ~ x O ~ b ( x ) O 2 ( a ( x ) i s i n v a r i a n t u n d e r ( 2 .1 1 ) w h i l e a t e r m s u c h a s

~ x ~ b (x ) 0 10 2~ b(x ) i s n o t . C l e a r l y , a m o r e p r e c i s e s t a t e m e n t o f o u r t h e o r e m a n d t h e

c o n d i t i o n s u n d e r w h i c h i t is v a l id m u s t a c c o u n t p r o p e r l y f o r b o u n d a r y e f f ec ts .

B o u n d a r y c o n d i t i o n s a r e a l s o c le a r ly r e le v a n t i n t h e c o n t e x t o f t h e r e p r e s e n t a t i o n

o f th e s e t h e o r i e s i n t e r m s o f t h e i r t o p o l o g i c a l e x c i t a t i o n s , a s w e w i ll s e e w h e n w e

d i s c u s s s p e c i f i c m o d e l s i n th e f o l l o w i n g s e c t i o n . I n t h e c a s e o f t h e o r i e s w i t h a n n = 1

o r 2 s y m m e t ry ~ i t f r e q u e n t l y h a p p e n s t h a t t h e t o p o l o g i c a l e x c i t a ti o n s i n t e r a c t t h r o u g h

v e r y l o n g r a n g e p o t e n t i a l s ; l o g a r i t h m i c o r e v e n w o r s e . I n c a s e s s u c h a s t h e se ,

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

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

3 . S p e c i f i c m o d e l s

3.1. THREE SIMPLE MODELS WITH n~<2

M o s t o f th e r e l e v a n t p r o p e r t ie s o f t h e o r ie s w i th u n c o m m o n s y m m e t r ie s o f t h e

t y p e w e h a v e b e e n t a l k i n g a b o u t c a n b e e n c o u n t e r e d i n v er y s i m p l e m o d e l s . I n t hi ss e c t i o n w e s h a l l m e n t i o n t h r e e s u c h t h e o r i e s a n d e n u m e r a t e t h e i r s a l i e n t f e a t u r e s .

F o r c l a r it y , t h e s i m p l e s t o f t h e s e m o d e l s w i ll b e d i s c u s s e d i n s o m e d e t a il .

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

d e f in e a n i n t e r a c t i o n t e r m , I t, a s s o c i a t e d w i t h t h e 2 r s it e s o f a n r - d i m e n s i o n a l

h y p e r c u b e i n a d - d i m e n s i o n a l h y p e r c u b i c l a tt ic e w i t h r<~ d. W i t h t h e s p i n s, s ( r ) ,

d e f i n e d o n t h e l a t t i c e , w e h a v e

L ( i ; x l , x 2 , . . . , x r ) -~ ½ s ( i) s *( i + ~ l ) s * ( i + :c2)

. . . s * ( i + ~ , r ) s ( i + :~l + ~ 2 ) s ( i + ~ l + ~3 )

• - . s ( *) i + ~ + h . c . ,

F.C. Alcaraz et al / Spin systems 16 9

w h e r e i l a b e l s a l a t t ic e s it e a n d ~ i s t h e u n i t v e c t o r in t h e j d i r e c t i o n . T h e l a s t f a c t o r

i n ( 3. 1 ) i s c o m p l e x - c o n j u g a t e d i f r is o d d a n d i s n o t w h e n r is e v e n .

T h e fi rs t a n d s i m p l e s t m o d e l w e s h a ll d i s c u ss i s a n i n d e x - 2 , t h r e e - d i m e n s i o n a l

t h e o r y , t h e s o - c a ll e d 2 / 2 4 m o d e l * . I ts h a m i l t o n i a n is g iv e n b y

H = OL1 ~ I 1 ( i ; Z ) + a 2 ~ I 2 ( t ; X , y ) , ( 3 . 2 )l !

w h e r e I i a n d / 2 a r e d e f i n e d b y ( 3. 1) a n d t h e sp i n s, s ( r ) , a r e U ( 1 ) ( o r Z , ) v a r i ab l e s

o f th e f o r m

s ( r ) = e ' '/ '(r) .

I n t h e Z N c a s e , t h e p h a s e o f s ( r ) t a k e s t h e v a l u e s

2 ~t b (r ) = - ~ - p ( r ) , p ( r ) = 0 , 1 . . . , N - 1 .

T h e i n t e r a c t i o n s a r e t h u s a s im p l e t w o - b o d y t e r m i n t h e z - d i re c t i o n , a n d a f o u r - b o d y

t e r m a l o n g a n e l e m e n t a r y p l a q u e t t e i n t h e x - y p la ne ( s e e f ig . 1 ). I t is e a s y to s e e

t h a t t h i s t h e o r y i s i n v a r i a n t u n d e r

6 ( x , y , z ) ~ ~b (x , y , z ) + Al ( X) + A 2( y ) ,

w h e r e A1 a n d A2 a r e a r b i t r a r y f u n c t i o n s o f t h e i r a r g u m e n t s . T h e r e a r e , t h e r e f o r e ,

t w o d i s t i n c t n = 2 sy m m e t r i e s : H i s i n v a r i a n t u n d e r a u n i f o r m r o t a t i o n o f a l l s p i n sw h i c h l i e i n a n y g i v e n y - z p l a n e o r i n a n y g i v e n x - z p l a n e , b e c a u s e A , ( x ) + A 2 ( y )

is n o t a n a r b i t r a r y f u n c t i o n o f x a n d y , t h e t h e o r y d o e s n o t h a v e a n n = 1 s y m m e t r y .

L e t u s r e w r i t e ( 3 .2 ) e x p l i c i t l y f o r t h e c a s e o f Z N s p i n s :

H = a [ ~ L c o s A 2 p ( r )+ ~. c o s A 4p (r ) ] ( 3 .3a )

w h e r e

A 2 p ( r ) =---~ - [ p ( r + 2 ) - - p ( r ) ] , ( 3 .3b )

+ ( x 2 ~

r + i^

r r + x

F ] g. l . T h e h a m i l t o n ] a n o f e q . ( 3 .3 a )

* T h i s m o d e l i s s o m e t i m e s r e f e r r e d t o as th e February24 m o d e l

170 F.C Alcaraz et al / Spin systems

2~rA 4 p ( r ) - - ~ - [ p ( r ) - p ( r + ~ ) + p ( r - £ + f i ) - p ( r + f i ) ] , ( 3.3 c)

w h e r e ~ , f i a n d £ re p r e s e n t u n i t v e c t o r s , a n dp =

0 , 1 , . . . , N - 1. F o r s im p l i ci ty , w eh a v e s p e c i a l i z e d o u r d i s c u s s i o n t o t h e c a s e , a ~ = a 2 = a . T h e f ir s t s u m i n ( 3 . 2 ) is

o v e r l in k s i n t h e z - d i r e c t i o n ( L ) a n d t h e s e c o n d o v e r a ll e l e m e n t a r y p l a q u e t t e s i n

e a c h x - y p l a n e ( P ) o f t h e l a t ti c e . I n w h a t f o l l o w s w e a s s u m e t h e s y s t e m is d e f i n e d

o n a l a t ti c e w i th p e r i o d i c b o u n d a r y c o n d i t io n s .

T h e t h e o r y d e f i n e d b y ( 3 .3 ) is s e l f- d u a l i n t h e s e n s e t h a t i t s p e r i o d i c g a u s s i a n o r

v i l la i n - li k e f o r m u l a t i o n i s se l f -d u a l . F u r t h e r , i n t e r a c t i o n s o f th e d u a l f o r m o f th e

f u l l h a m i l t o n i a n ( 3 .3 ) a r e o f t h e s a m e f o r m a s t h o s e a p p e a r i n g i n ( 3 .3 ) . T h e r e a r e ,

i n a d d i t i o n , h i g h e r h a r m o n i c s o f t h o s e i n t e r a c t i o n s [ 1 0 ]. A g e n e r a l i z a t i o n o f t h is

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

T o d i s c u s s t h e d u a l i t y p r o p e r t i e s o f (3 . 3) m o r e c o m p l e t e l y , i t is u s e f u l t o s t u d y a

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

g a u s s i a n v e r s i o n o f ( 3 . 3 ) i s g i v e n b y

{p(r)=--oo} {Kp=--oo} {KL=--~O}

w h e r e w e h a v e i n t r o d u c e d t h e i n te g e r fi el ds K L a n d K e a t t a c h e d t o l i n k s i n t h e

z - d i r e c t io n a n d t o p l a q u e t t e s i n x - y p l a n e s , r e s p e c ti v e ly . F o r c o n v e n i e n c e , w e h a v e

e x t e n d e d t h e s u m o v e r p ( r ) t o g o f r o m - c o t o o o. T h i s in t r o d u c e s a u n i f o r m a n d

h a r m l e s s o v e r c o u n t i n g [ 1 1 ] w h i c h w i l l n o t c o n c e r n u s h e r e .

T o c o n s t r u c t t h e d u a l o f ( 3 . 4 ) 'w e u s e t h e P o i s s o n i d e n ti ti e s

t o o b t a i n

F [ p ( r ) ] = ~ I f o o d ~ ? 8 ( ~ ? - / ) F ( ~ 7 )p(r ) =--oo l=--oo ,

I {d ( ) e x p{fc(r)=-oo} {KL, Kp=--oo} -oo

w h e r e t h e l a s t s u m ( o v e r S ) i s a s u m o v e r al l s it e s o f t h e l a tt ic e . I n t h e a b o v ee q u a t i o n w e h a v e i n t r o d u c e d a n i n t e g e r - v a l u e d f i e ld , K , l o c a t e d a t th e l a tt ic e s it es .

W e n o w i n t r o d u c e c o n t i n u o u s z - l i n k a n d x - y pl aq ue t te f i e lds , ~'L an d ~-~, In w ha t

F.C . Alcar az e t a l. / Sp in sys tem s 171

f o l l o w s t h e s e w i l l b e a s s o c i a t e d w i t h t h e s it es o f t h e d u a l l a t ti c e . D r o p p i n g i r re l e v a n t

o v e r a l l m u l t i p l i c a t i v e ( f l - d e p e n d e n t b u t f i e l d - i n d e p e n d e n t ) f a c t o r s , w e o b t a i n

+ ~ p [ - ~ , 2 + i r p ( a 4 7 7 - 2 r r K p ) ] + ~ 2 r r i K ( r ) T l ( r ) ] . (3.6)

T o d o t h e i n t e g r a l o v e r 77, w e f i r s t i n t e g r a t e ( s u m ) b y p a r t s t h e t e r m s

E r ,. & r l = - E ~ ( r )& ~ ' L ( r- ½ ) ,L L

Y~ ~ 4 , 7 = Y~ ~ ( r ) a , ~ ( r - ½ ( ~ + ~ ) ) .P P

T h e i n t e g r a l o v e r 77 c a n n o w b e d o n e , l e a d i n g t o a p r o d u c t o f & f u n c t i o n c o n s t r a i n t s

o f t h e f o r m

] 1 8 12 ¢r K ( r ) - a 2 r L ( r - ½ £ ) + a 4 r p ( r - ½(a~+ J~))] .L

T o s o l v e t h e s e c o n s t r a i n t s , w e d e f i n e a d u a l f i e ld , ~ /,, a t t h e s i te s o f th e d u a l l a t ti c e ,1 ^ A A

F = r + ~ ( x + y + z ) , i n s u c h a w a y t h a t

1A N A A

~-,.(r+ ~z ) = 2 -7 a 4 ~ , ( ~ - x - y ) + & N ,

Nr p ( r + ½ ( ~ + f i ) ) = ~ A 2 O ( ~ - e ) + 7 L N ,

w h e r e -IL a n d J p a r e i n t e g e r f i e l d s l o c a t e d o n t h e z - l i n k s a n d x - y p l a q u e t t e s o f t h e

d u a l ( s h i f t e d ) l a tt ic e . W e c a n r e p l a c e t h e s u m s o v e r r L a n d re b y s u m s o v e r $ , a ; t

a n d J e [ 1 1 ] . T h e r e s u l t , n e g l e c t i n g o v e r a l l c o n s t a n t s , i s

+ ~ , - ( a E O + 2 ~ r j L ) 2 - i N K ~ E 4 , ( r - £ ) (3 .7)~, ~ '

w h e r e t h e s u m s o v e r z - l i n k s ( L ) a n d x - y p l a q u e t t e s ( P ) r e f e r t o t h e o r i g i n a l l a tt i ce .

172 F.C . A l c araz e t al . / Sp in sy s t e m s

H o w e v e r , i t i s e a s y t o s e e t h a t

. 2 ~ - .

' ~ K L A 4 ~ b ( r - . I ~ - I~) -t- E K r , ~ 2 t ~ ( / 7 - e ) - - ~ & ( r ) - ~ r n ( r ) ,

m ( ~ ) =- - A 2 K p + A 4 K L . (3 .8)

T h i s a l l o w s u s t o r e f e r a ll s u m s i n ( 3 .7 ) t o t h e d u a l l a t t i c e , w i t h t h e r e s u l t

Z = _ D~b ex p,,(~) =-~ yL, p= - ~

+ ; ( A 2~ -- 21r'JL)2+ ; 2 ~'im( r)~b(r) ] } (3 .9)

w h e r e w e h a v e d e f i n e d a n e w c o u p l i n g , / 3 -= N 2 / 8 ~ 2 /3 .

U s i n g ( 3 .9 ) , o n e c a n p r o v e t h e s e l f - d u a l i t y o f t h is m o d e l b y p e r f o r m i n g t h e

s u m m a t i o n s o v e r t h e m ( ~ ). A l t e r n a ti v e l y , i n t e g r a t i n g o v e r ~b l e a d s t o a r e p r e s e n t a t i o n

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

t h e i n t e g e r - v a l u e d f ie l d m . L e t u s fi rs t d i s c u s s t h e s e l f - d u a l n a t u r e o f th e m o d e l . W e

s h a l l t h e n d i s c u s s i ts t o p o l o g i c a l e x c i t a t i o n s ,

U s i n g t h e P o i s s o n s u m m a t i o n f o r m u l a i n (3 . 9) it i s s t r a i g h t f o r w a r d t o p r o v e t h a t

t h is v e r s i o n o f t h e t h e o r y i s s e lf - d u a l. T h e r e s u l t o f t h is c a l c u l a t i o n is j u s t ( 3 .4 ) w i t h

t h e d i ff e r e n c e t h a t / 3 h a s b e e n r e p l a c e d b y / 3 a n d t h a t t h e d e g r e es o f f r e e d o m a r e

n o w d e f i n e d o n t h e d u a l l at ti ce . A s w e h a v e m e n t i o n e d b e f o r e , w e h a v e c h o s e n t o

d e a l w i th t h e p e r i o d i c g a u s s i an a p p r o x i m a t i o n t o o u r t h e o r y f o r s im p l i ci ty . I n d e e d ,

f o l l o w i n g e s s e n t i a l ly t h e s a m e s t e p s u s e d t o d e r i v e ( 3 .9 ) f r o m ( 3 . 4 ) , o n e c a n m a n i p u -

l a te ( 3 . 3) t o p r o v e t h a t t h a t m o d e l i s p r e c i s e l y s el f -d u a l . ( T h a t i s, n o e x t r a h a r m o n i c s

o f t h e i n t e r a c t i o n s i n H a r e g e n e r a t e d b y th e d u a l i t y t r a n s f o r m a t i o n f o r N = 2 , 3

a n d 4 .) T h i s is e n t i re l y a n a l o g o u s t o t h e s t r u c t u r e f o u n d i n t h e 2 - d i m e n s i o n a l Z N

c l o c k m o d e l s . A s i s t h e c a s e f o r t h o s e m o d e l s , b e c a u s e o f s e l f- d u a l i t y , i f t h e t h e o r y

h a s a s i n g l e p h a s e t r a n s i t i o n , t h e c r i t i c a l t e m p e r a t u r e m u s t s a t i s f y / 3 = / 3 . T h a t t h i s

is tr u e i s d e m o n s t r a t e d b y o u r n u m e r i c a l a n a l y s i s o f t h e m o d e l f o r N <~ 4 d i s c u s s e d

i n t h e n e x t s u b s e c t i o n . F o r N > 4 t h e m o d e l i s n o t p r e c i s e l y s e l f - d u a l , i n t h a t t h e

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

a p p r o x i m a t i o n , ( 3 .4 ) , is s e l f- d u a l f o r a ll v a l u e s o f N . I n t h i s c a s e , t h e s e l f - d u a l p o i n t

i s g i ven b y 13 = / ~ = N / 2 z r . A s o u r n u m e r i c a l a n a l y s i s i n d i c a t e s , f o r N / > 5 , t h e m o d e l

( 3. 3) h a s t w o p h a s e t r a n s i ti o n s * , o n e o f w h i c h b e c o m e s e s s e n ti a ll y i n d e p e n d e n t o f

N a s N g r o w s , a n d t h e o t h e r o c c u r ri n g a t a t e m p e r a t u r e w h i c h a p p r o a c h e s z e r o

* I n f a ct , o n e c a n r i g o r o u s l y s h o w t h a t t h i s m u s t h a p p e n : A s i s t r u e f o r t h e d = 2 Z N c l o c k m o d e l s , it

i s p o s s i b l e t o d e r i v e a s e t o f G n t t i th s - l i k e i n e q u a l i ti e s f o r c e r t a i n n - p o i n t c o r r e l a t i o n f u n c t i o n s m o u r

m o d e l ( s e e r e f. [ 5 ]) . U s i n g t h e s e i n e q u a h t i e s , o n e c a n d e d u c e t h e e x i s t e n c e o f t h r e e p h a s e s f o r t h e

p e r i o & c g a u s s l a n a p p r o x i m a t i o n o f t h e m o d e l I t a l so f o ll o w s f r o m t h e se n g o r o u s i n e q u a h t ie s t h at

t h e i n t er m e d i a t e p h a s e w h i c h a p p e a r s b e t w e e n o r d e r e d a n d d i s o r d e r e d f e r r o m a g n e t i c p h a s e s h a s n o

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

F.C. Alcaraz et al. / Spin systems 173

l ik e N - 2. T h i s i s c o n s i s t e n t w i t h t h e b e h a v i o r e x p e c t e d f o r t h e s e l f -d u a l g a u s s i a n

v e r s i o n o f t h e m o d e l ( 3 . 4 ), a n d is q u a l it a t i v e l y s i m i l ar to t h e b e h a v i o r f o u n d f o r t h e

d = 2 c l o c k m o d e l s . I t i s i m p o r t a n t t o s t r e s s t h e f a c t t h a t s e l f - d u a l i t y o f ( 3 . 4 ) i s n o t

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

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

m o d e l ; i n d e e d , t h e r e a r e t h r e e - d i m e n s i o n a l n = 2 m o d e l s w h i c h a r e n o t s e lf - du a l

( a n e x a m p l e w i ll b e b r i e f ly d e s c r ib e d b e l o w ) . T h e s e lf - d u a li ty o f o u r m o d e l f o l lo w s

f r o m t h e f a c t t h a t t h e t h e o r y h a s t w o d i s t in c t t y p e s o f in t e r a c t i o n s f o r a s i n g le f ie ld :

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

( A g e n e r a l d i s c u s s i o n o f th i s p o i n t c a n b e f o u n d i n r e f . [ 1 2 ]. )

L e t u s n o w d i sc u s s t h e t o p o l o g i c a l e x c i t a ti o n s o f th e m o d e l . T h e r e a r e tw o t y p e s

o f i n t e g e r - v a lu e d o b j e c ts i n t h is t h e o r y w h i c h m a y l o o s e l y b e t h o u g h t o f a s to p o l o g i c a l

i n o r i g in . O n e is a s s o c i a t e d w i t h th e d i s c r e t e n e s s o f t h e Z N s y m m e t r y a n d t h e o t h e r

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

w e w i l l t a k e t h e U ( 1 ) l i m i t o f ( 3 . 9 ) . W e c a n w r i t e

Z = ~ [ D ~/, e x p ( - H [ t p ] ) , ( 3.1 0 ){ r e ( r ) } . /

w h e r e

T r a n s f o r m i n g H t o m o m e n t u m s p a c e a n d i n te g r a ti n g o v e r ¢ l e a d s to a r e p r e s e n t a t i o n

o f t h e m o d e l i n te r m s o f p o i n t - li k e e x c i t a ti o n s r e p r e s e n t e d b y t h e f i e ld m :

d 3 k 2 1

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

e x p l i c i t ly a n d t h e i n v e r s e p r o p a g a t o r , D ( k ) , i s g i v e n b y

D ( k ) = 6 - 4 c os k x - 4 c o s k y + 4 c o s k x c o s k y - 2 c o s k z . ( 3 .1 3 )

D ( k ) h a s z e r o s i n s i d e t h e B r i l lo u i n z o n e , a n d s o o n l y t h o s e c o n f i g u r a t i o n s in ( 3 .1 2 )f o r w h i c h D -1 i s f in i te w il l c o n t r i b u t e t o Z . T h e z e r o s o f D o c c u r a l o n g t h e l i n e s

k x = k z = 0 f o r a ll ky e B ,

k v = k z = 0 f o r a ll kx e B , ( 3.1 4 )

w h e r e B is t h e f ir st B r i l l o u i n z o n e , [ - ~ , ~ '] . E x c l u d i n g c o n f i g u r a t i o n s f o r w h i c h D

v a n i s h e s t r a n s l a t e s i n t o a s e t o f n e u t r a l i t y c o n d i t i o n s f o r t h e c h a r g e s re(r) . T h e s e

c a n e a s i l y b e s e e n t o b e g i v e n b y

~.e 'Yk , m (r )= O f o r a n y k x ,r

e'xk~m(r) = 0 f o r a n y k y . ( 3 .1 5 )r

174 F C A l c a r a z e t a l . / S p i n s y s t e m s

T h e s e c o n d i t i o n s i d e n t i fy th e c o n f i g u ra t io n s w h i c h c o n t r i b u t e t o th e p a r t it i o n f u n c -

t i o n a s t h o s e f o r w h i c h t h e s u m o f to p o l o g i c a l c h a r g e s , ~ , m ( r ) , v a n i s h e s o n e a c h

o f th e p r i n c i p a l x - z a n d y - z p l a n e s o f t h e la t ti c e . T h i s i s a d i r e c t c o n s e q u e n c e o f

t h e c o m p o u n d n = 2 sy m m e t r y o f o u r m o d e l . O f co u r s e , s i nc e t h is s y m m e t r y c o n t a i n s

a g l o b a l U ( 1 ) s y m m e t r y , t h e a b o v e c o n d i t i o n s i m p l y , a s t h e y s h o u l d , g l o b a l n e u t r a l i ty

o f t h e m f i e l d a s w e l l.

T h e c o m p l e t e p a r t i t io n f u n c t i o n c a n t h e r e f o r e b e w r i t te n a s

Z = Z o Y, e x p ( - ~ m ( r ) G ( r , r ' ) m ( r ' ) ) , (3 . 16 ){re(r)}

w h e r e t h e s u m o v e r { m ( r ) } i n c l u d e s o n l y t h o s e c o n f i g u r a t i o n s f o r w h i c h ( 3. 15 ) is

v a l i d , a n d G(r , r ') is th e c o n f i g u r a t io n s p a c e p r o p a g a t o r . T h e e x p l ic i t f o r m o f th ep r o p a g a t o r c a n b e o b t a i n e d a p p r o x i m a t e l y b y c o n s i d er in g t h e d o m i n a n t c o n tr ib u -

t i o n s o f t h e F o u r i e r t r a n s f o r m o f D -1 c l o s e t o t h e s i n g u l a r l i n e s g i v e n b y ( 3 . 14 ) .

T h e r e s u l t i s

( z - z ' ) 2G(r , r ' ) - l n I x - x ' l In [ Y - Y ' I 2 ( x _ x , ) 2 ( y _ y , ) 2 , (3 . 17 )

w h e r e s - s ' is t h e c o m p o n e n t o f r - r ' i n t h e s d i r ec t io n .

T h e s i m i l a r it y b e t w e e n ( 3 .1 6 ) a n d t h e l o g a r i t h m i c g a s r e p r e s e n t a t i o n o f t h e

t w o - d i m e n s i o n a l x - y m o d e l s u g g e s t s t h a t , j u s t a s t h e p o i n t c h a r g e s i n t h e d = 2 x - ym o d e l a r e t h e t o p o l o g i c a l e x c i ta t io n s o f t h e m o d e l , s o t h a t m ( r ) a r e t o p o l o g i c a l

e x c i t a t i o n s o f t h e p r e s e n t t h e o r y . T h e i n t e r p r e t a t i o n o f t h e p o i n t c h a r g e s i n th e

g l o b a l l y - s y m m e t r i c x - y m o d e l a s r e p r e s e n t in g t h e m u l t ip l e w i n d i n g s o f t h e U ( 1 )

s p in s a r o u n d s o m e c l o s e d c o n t o u r is s t r ai g h t fo r w a r d . T h e s a m e is u n f o r t u n a t e l y n o t

t r u e i n t h e p r e s e n t c a s e . R e c a l l , f r o m ( 3 . 8 ), t h a t t h e m ( r ) a r e p a r t ic u l a r c o m b i n a t i o n s

o f t h e i n t e g e r f ie ld s K p a n d K L w h i c h w e i n t r o d u c e d t o r e ta i n t h e p e r i o d i c i ty o f

t h e h a m i l t o n i a n i n g o i n g f r o m ( 3 .3 ) t o ( 3 .4 ) . T h i s i s st il l i d e n t i c a l t o th e c o r r e s p o n d i n g

o r i g in o f t h e p o i n t c h a r g e s i n th e x - y m o d e l . H o w e v e r , h e r e i t i s d i f f ic u l t t o t e l l

w h a t a r e th e c o n f i g u r a t i o n s f o r th e t h e o r y d e f i n e d t h r o u g h ( 3 . 3 ) w h i c h l e a d ton o n - z e r o v a l u e s f o r m . In t h e c a s e o f n o n - z e r o KL, t h e i n t e r p r e t a t i o n i s l ik e t h a t o f

t h e x - y m o d e l : K L r e p r e s e n ts t h e n u m b e r o f c o m p l e t e r o ta t i o n s u n d e r g o n e b y a

s p i n a s w e m o v e f r o m a s i t e a t r t o t h e n e x t i n t h e z - d i r e c t i o n . T h e i n t e r p r e t a t i o n

o f K p is n o t q u i t e s o s i m p l e , s o t h a t a g e o m e t r i c a l i n t e r p r e t a t i o n o f th e m ' s i s n o t

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

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

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

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

q u i t e p o s s i b l e t h a t t h e m ( r ) a r e r e l a t e d t o s o m e q u a l i t y o f a d i f f e r e n t o b j e c t ( e .g .t h e e n d s o f a s t ri n g ), s o t h a t a c o m p l e t e u n d e r s t a n d i n g o f t h e t o p o l o g i c a l e x c i t a ti o n s

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

F . C A l c a r a z e t a l. / S p i n s y s t e m s

r+l~r r+l~

F i g . 2 . T h e h a m i l t o m a n o f e q . (3 1 8 ).

A s e c o n d s i m p l e m o d e l w h i c h w e w o u l d l ik e t o d e s c r i b e h e r e is d e f i n e d b y t h e

h a m i l t o n i a n

H = Y . [ a ~ I2 ( i; x , z ) + a 2 1 2 ( i ; y , z ) ] , ( 3 . 1 8 )

w h e r e t h e n o t a t i o n i s t h e s a m e a s in ( 3 .1 ) ; t h a t i s, t h e t h e o r y d e s c r i b e s f o u r - b o d y

i n t e r a c t io n s a r o u n d e l e m e n t a r y p l a q u e t t e s in t h e x - z a n d y - z p l a n e s ( s e e f ig . 2 ) .

T h i s t h e o r y i s i n v a r i a n t u n d e r

~ ( x , y , z ) ~ & ( x , y , z ) + A , ( x , y ) + A 2 ( z ) ,

w h e r e t h e A , a r e a r b i t r a r y f u n c t i o n s o f t h e i r a r g u m e n t s . T h e t h e o r y t h u s h a s a

c o m p o u n d n = 2 p lu s n - - 1 s y m m e t r y .

T h e a p p e a r a n c e o f a n n = 1 s y m m e t r y h a s r a d i c a l c o n s e q u e n c e s : I t i s n o t d i ff ic u lt

t o s e e t h a t , b e c a u s e o f th i s s y m m e t r y , t h e t h e o r y d e c o u p l e s i n to a s e t o f n o n -

i n t e r a c ti n g t w o - d i m e n s i o n a l m o d e l s . I n f a c t, fo r th e c a s e o f Z N ( U ( 1 ) ) s p i n s , (3 .1 8 )

c a n b e t r a n s f o r m e d i n t o a s et o f d e c o u p l e d d -- 2 N - s t a t e c l o c k ( x - y ) m o d e l s . T h i s

is a g e n e r a l f e a t u r e o f t h e o r i e s w i t h n = 1 s y m m e t r i e s . A n y s u c h t h e o r y ( w h e n d e f i n e d

o n t h e e l e m e n t a r y s i m p l i c e s o f t h e l at ti c e) d e c o u p l e s a l o n g t h e d i r e c t io n d e f i n e d b y

t h e n = 1 a x i s . T o s e e t h i s , n o t i c e t h a t t h e n - - 1 s y m m e t r y o f t h i s m o d e l a l l o w s u s

t o p e r f o r m g l o b a l r o t a t i o n s o f a ll sp i n s w h i c h l ie o n a n y l in e a l o n g t h e z - d i re c t io n .

T h e r e f o r e , b y p e r f o r m i n g t h e s e l i n e ar t r a n s fo r m a t i o n s , a n y p a r t i c u la r x - y p l a n e c a n

b e t r a n s f o r m e d s o t h a t a ll it s s p i n s h a v e t h e s a m e p h a s e . H a v i n g f i x e d t h e g a u g e i n

t h is m a n n e r , t h e p a r t i t i o n f u n c t i o n f o r t h e m o d e l w i ll fa c t o r i z e b y t h e u s e o f a

J o r d a n - W i g n e r - l i k e t r a n s f o r m a t i o n w h i c h e x p l ic i tl y d e c o u p l e s t h e m o d e l .

T h e a r g u m e n t c a n b e a p p l i e d t o a n y n = 1 t h e o r y i n d d i m e n s i o n s t o s h o w t h a t

i t w i ll d e c o u p l e i n to a s e t o f n o n - i n t e r a c t i n g t h e o r i e s i n d - 1 d i m e n s i o n s . F o r t h e

p u r p o s e o f i ll u s t ra t io n , i t is in s t ru c t iv e t o g o t h r o u g h t h e a r g u m e n t f o r t h e o n e -

d i m e n s i o n a l I s i n g m o d e l , a p a r t i c u l a r ly s i m p l e n = 1 t h e o r y .

C o n s i d e r a o n e - d i m e n s i o n a l I s in g m o d e l i n a f in it e l a tt ic e o f N s p i n s w i th f r ee

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

Z = Y. 3 ~ . . . ~ e -~E,' ,s ,+ , , (3.1 9)S 1 s 2 S N

176 F C. Alcaraz et al. / Spin systems

w h e r e s , = ± 1 i s a n I s i n g s p i n a t s i t e i. D e f i n e n e w I s i n g s p i n s tr, s u c h t h a t

sM = l-I o ' , . (3 .2 0)

I t fo l l o w s t h a t % = s js j_ l f o r j > 1 , a n d o'1 = s~ . I n t e r m s o f o r,, ( 3 . 19 ) c a n b e w r i t t e n

Z = Y~ Z " "" Y~ e -~2'~=2o-' = 2(2 co sh f l ) N - , . (3 .21)o- 1 o- 2 o-N

I t i s s t r a i g h t f o r w a r d t o a p p l y a n a n a l o g o u s t r a n s f o r m a t i o n t o ( 3 .1 8 ) a l o n g t h e

z - d i r e c t i o n t o s h o w t h a t i t is e q u i v a l e n t to a s t a c k o f t w o - d i m e n s i o n a l Z N c l o c k

m o d e l s . F u r t h e r m o r e , a g e n e r a l i z a t i o n o f t h e t h e o r e m o f se c t. 2 c a n b e s e e n t o fo l l o w

f r o m th i s n o n - l o c a l t r a n s f o r m a t i o n : B e c a u s e t h e ( d - 1 ) - d i m e n s i o n a l t h e o r i e s i n t ow h i c h t h e d - d i m e n s i o n a l n = 1 t h e o r y w a s t r a n s f o r m e d a r e n o n - i n t e r a c ti n g , t h e

e x p e c t a t i o n v a l u e o f th e s p i n , ( s ) = 0. T h e r e f o r e , a s w e m e n t i o n e d i n s u b s ec t . 2.2 ,

t h e o r i e s w i t h s y m m e t r y i n d e x , n = 1 c a n n o t b r e a k t h e i r g l o b a l s y m m e t r y s p o n -

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

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

H = Y . [ a , I 2 ( i ; x , z ) + t~212(i ; y, z ) + ot312(i; x , y ) ] , ( 3 . 2 2 )I

w h e r e w e h a v e u s e d t h e n o t a t i o n d e s c r i b e d i n ( 3 .1 ). T h e m o d e l t h u s d e s c r ib e s

f o u r - b o d y i n t e r a c t i o n s o f s p in s w h i c h l ie a t t h e c o r n e r s o f e l e m e n t a r y p l a q u e t t e s i n

t h e x - z , y - z a n d x - y p l a n e s ( s e e f i g . 3 ) . T h i s t h e o r y i s i n v a r i a n t u n d e r t h e t r a n s f o r -

m a t i o n

~b(x, y , z) --> ~b(x, y , z ) + Al(X ) + A 2(y) + A 3( z) ,

a n d s o , h a s a t ri p l e n = 2 s y m m e t r y . B e c a u s e o f th e l a s t t e r m i n ( 3 .2 2 ) t h o u g h , i t

d o e s n o t h a v e t h e n = 1 s y m m e t r y o f (3 .1 8 ).

U s i n g t h e t e c h n i q u e s o f r ef . [1 3 ], i t i s s t r a i g h t f o r w a r d , i f t e d i o u s , t o p e r f o r m t h e

s a m e a n a l y s i s f o r t h is m o d e l a s w a s d o n e f o r th e 2 / 2 4 m o d e l . I t i s n o t d i f fi cu l t t o

s e e , h o w e v e r , i n w h a t w a y s t h e se t w o m o d e l s d i ff e r i n th e i r p r o p e r t i e s u n d e r a

d u a l i ty t r a n s f o r m a t i o n . T h e f u n d a m e n t a l i n t e r a c ti o n s i n ( 3 .2 2 ) a re d e f in e d o n t h e

~ [ O~ 1r

r r + ~ c

Fig. 3. The ham iltonian of eq. (3.22).

r r + k

F C. Alcaraz et al. / Sp in systems 177

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

a l in k , u~, i n t h e d u a l t h e o r y , p a s s i n g t h r o u g h t h e c e n t e r o f th e o r i g i n a l p l a q u e t t e .

T h e d e l t a - f u n c t i o n c o n s t r a i n ts w h i c h f o l l o w f r o m t h e i n t e g r a t io n o v e r a n g le s r e q u i r e

t h a t th e s u m o f t h e 12 u , ' s w h i c h f o r m t h e e d g e s o f t h e c u b e s u r r o u n d i n g t h e o r i g in a l

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

s u c h t h a t u ~ = V x ( V y V y - V z v z ) a n d s o o n f o r t h e o t h e r t w o d i r e c t i o n s . T h e v ~ ' s

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

D e f i n i n g 8 z- -- V xV y , e t c. , t h e d u a l h a m i l t o n i a n c a n b e w r i t t e n i n th e f o r m

H = Z F 2 o ,~O

F,,v = a, ,vv - avv ,, .

T h e s y m m e t r y u n d e r t r a n s f o r m a t i o n s o f t h e v~ is o f t h e f o r m

v, , -- , v~ + a~A ( r ) .

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

p r e s e n t i n g t h i s e x a m p l e i s t h a t , a l t h o u g h t h i s t h r e e - p l a q u e t t e m o d e l s e e m s t o b e

f u n d a m e n t a l l y d i ff e re n t f ro m t h e 2 / 2 4 m o d e l , a n d c e r t a in l y h as a d i f fe r e n t d u a l i ty

s t r u c t u r e , t h e p h a s e s t r u c t u r e o f b o t h t h e o r i e s i s q u a l i t a t iv e l y th e s a m e . T h e r e f o r e ,

t h e r e is n o s i m p l e c o r r e s p o n d e n c e b e t w e e n t h e d u a l i t y p r o p e r t ie s o f th e s e m o d e l s

a n d t h e i r p h a s e s t ru c t u re .

3 .2 . MO NT E CARLO SIMULATIONS

W e h a v e p e r f o r m e d m o r e o r l es s e x te n s i v e n u m e r i c a l a n a ly s is o f all th e a b o v e

m o d e l s . D e t a il s o f t h e p r o c e d u r e s w h i c h w e r e u s e d h a v e b e e n d i s c u ss e d i n th e

l i t e ra t u r e [ 1 3 ] . H e r e w e w i ll b r ie f l y r e c a p i t u l a t e t h e g e n e r a l s t r a t e g y e m p l o y e d i n

o u r a n a l y s i s a n d w i l l s t a t e t h e r e s u l t s .

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

t h e r m a l c y c l e s . S t a r t in g f r o m a n o r d e r e d c o n f i g u r a t i o n a t a la r g e v a l u e o f t h e i n v e r s e

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

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

n e x t i t e r a t i o n . T h i s p r o c e d u r e i s r e p e a t e d u n t i l f l = 0 a n d t h e n r e v e r s e d u n t i l fl = flo .

S e v e r a l q u a n t i t ie s o f i n t e r e s t ar e m e a s u r e d f o r e a c h i t e ra t i o n . A l t h o u g h i n s u c h a

n u m e r i c a l e x p e r i m e n t t h e s y s t e m i s n e v e r r e a ll y in e q u i l i b r iu m , i t is g e n e r a l l y c lo s e

t o e q u i l i b r i u m a s l o n g a s t h e s y s t e m d o e s n o t g o t h r o u g h a p h a s e t r a n s i t i o n . I n t h e

n e i g h b o r h o o d o f a tr a n s i ti o n , h o w e v e r , t h e d i s t a n c e f r o m e q u i l ib r i u m i n c re a s e s, a n d

t h e m e a s u r e d q u a n t i t ie s w i l l s h o w h y s t e r e s is - l ik e lo o p s . F o r t h e Z N v e r s i o n o f t h e s e

m o d e l s , t h i s t y p e o f a n a l y s is i s d o n e f o r se v e r a l v a l u e s o f N . E x a m p l e s o f t h e i n t e r n a l

e n e r g y o f t h e 2 / 2 4 m o d e l f o r N = 4 , 5 , 6 , 7 , 8 a n d 9 a r e s h o w n i n f ig s. 4 - 9 . W h e r e a st h e m o d e l f o r N = 4 d i s p l a y s a s in g l e p r o n o u n c e d l o o p , as N i n c r e a s e s t h r o u g h 5 ,

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

178 E C. Alcaraz et ml. / Spin systems

o,",,0 6

0 0 I0 . 0 0 5 1 .0

I1 .5 2 .0

Fig. 4. The internal energy in a thermal cycle for the model defined in (3.4) for N =4. A single,

pro nou nced l oop is seen to occur at the model 's self-dual point/3 = In (1 + 2 L/2) = 0 .8 8. .. Furthe r analysis

shows that this is a first-order transiUon. The lattice size for this simulation measured 15 x 15 x 15 with

periodic bo und ary conditions. The step size m/3 was 8/3 = 0.001. All simulations were done using a heat

bath updating algorithm

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

t y p e o f s i m u l a t i o n s h o u l d b e u s e d t o d e t e r m i n e t h e t y p e o f t r a n si t io n ( in t h e

i n f i n i t e - v o l u m e l im i t ) w h i c h c a n b e a s s o c i a t e d w i t h a g iv e n h y s t e r e s is l o o p . T h e

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

t r an s i ti o n s f o r N t> N o. T h e h i g h - t e m p e r a t u r e t r a n s it i o n s o o n b e c o m e s i n d e p e n d e n t

10 0 0 5 1 .0 1 5 2 0 2.5

Fig. 5 Same as fig. 4 for N = 5. A less pronou nced l oop than tha t seen for N = 4. Furt her analysis points

to two cont inuo us nearby trans itions at inverse tem pera tur es// ~ ~ 0.9 and // 2 ~ 1.1. For thts simulation,

as well as for those pic tured in figs. 6-9, we used cubi c lattices o f linear dtmens ion L = 20 with periodK

bound ary conditions. The step size in/3 was 8 // = 0 002.

F.C. Alc ara z et al. / Spin systems 179

0 . 8 '

0 . 0 0 5 1 0 1 .5 2 0 2 , 5 3 . 0

Fig. 6. Same as fig 5 for N = 6. Two clearly separated loops and the appe aranc e of an intermediate

phase. Furth er analysis points to two cont inuous transit ions at inverse temper atures fll ~ 1.0 and l/2 ~ 1.5.

o f N a s N g r o w s , in a g r e e m e n t w i th g e n e r a l a r g u m e n t s . T h e l o w t e m p e r a t u r e

t r a n s i t i o n s c a l e s w i t h N a s f l~ --- 3 "/[ 1 - c o s ( 2 ¢ r / N ) ] , w i t h a v a l u e f o r t h e c o n s t a n t ,

% w h i c h d e p e n d s o n t h e d e t a il s o f th e m o d e l . F o r t h e 2 / 2 4 m o d e l , 3' = 0 .7 5.

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

c r it ic a l t e m p e r a t u r e . T h i s p e r m i t s a v e r y p r e c i s e d e t e r m i n a t i o n o f th e c r i ti c al t e m -

p e r a t u r e i n t h e s e c a s e s . B y p r e p a r i n g a n i n i t ia l s t a t e in w h i c h h a l f o f th e l a t t ic e is

O 0 0 5 1 0 1 5 2 .0 2 5 3 .0

Fig. 7. Same as fig. 5 for N = 7. Analyses performed in the critical regions points to two continuoustransitions at inverse te mpera tures 131 = 1.0 and/3 2 = 2.0. In accordance with general principles and our

discussi on of sect 3, the second tr ansition is seen to move towa rd/ 3 ~ o% as N grows The lengthening

intermediate phase can be seen to be dominated by massless, spin-wave excitations.

180 F.C. Alcaraz et al / Spin systems

0 .00 0

1 0 2 0 3 0

Fig. 8. Same as fig. 5 for N = 8.

ordered and half of it is completely disordered, simulations are performed at a set

of temperat ures in the region of hysteresis and the time-evolu tion of the initial state

is observed. At the critical temperature in a first-order transition, each half of the

lattice will evolve into a different stable state, and the value of, say, the energy will

appea r to be rou ghly constant in time. At temperatures away from the critical point,

however, the system rapidly evolves into a unique stable state. Once a good estimate

of the critial temperatu re has been obtai ned in this manner, it is possible to distinguish

fairly easily between first-order and continuous transitions by observing the time

evolution of two extreme initial states, one completely disordered and one ordered,

0 00 0

Ftg. 9

1 0 2 . 0 3 0 4 0

Same as fig. 5 for N = 9.

F C. Alcaraz et al. / Spin systems 181

a t t h e e s t i m a t e d c r i ti c a l p o i n t . I n a c o n t i n u o u s t r a n s i t i o n , a f t e r a p e r i o d o f r e l a x a t i o n ,

t h e v a l u e s o f th e e n e r g y f o r b o t h s t a t e s w i ll c o i n c i d e . A t a f i rs t - o r d e r t r a n s i t i o n , o n

t h e o t h e r h a n d , t h e e n e r g y o f t h e t w o s t at e s w il l ev o l v e to t w o d i f fe r e n t v al u e s . U s i n g

t h is k i n d o f s i m u l a t i o n , w e d e t e r m i n e d t h a t f o r al l m o d e l s , w i t h th e e x c e p t i o n o f

t h a t g i v e n b y ( 3 . 18 ) , t h e t r a n s i t i o n s o b s e r v e d f o r N ~ < 4 a r e f ir s t o r d e r . A l l o t h e r

t r a n s i ti o n s o b s e r v e d a r e c o n t i n u o u s . N o e f fo r t w a s m a d e t o d e t e r m i n e t h e o r d e r o f

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

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

N ~ N c a l o n g th e l i ne s d i s c u s s e d i n r ef . [ 5 ] w a s p e r f o r m e d f o r t h e s e m o d e l s . T h e

r e s u lt s a r e c o n s i s te n t w i t h th e a b s e n c e o f l o n g - r a n g e o r d e r i n t h e i n t e r m e d i a t e p h a s e

a n d t h e r e f o r e w i t h th e a b s e n c e o f l o n g - r a n g e o r d e r i n t h e U ( 1 ) l im i t , i n a c c o r d a n c e

w i t h th e p r e d i c t i o n o f t h e t h e o r e m o f s e ct . 2 , a n d t h e b e h a v i o r o f t w o - d i m e n s i o n a l

Z N c l o c k m o d e l s .

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

s t u d i e d i n th i s s e c ti o n a n d c o i n c i d e s w i t h t h e v a l u e f o u n d i n t w o - d i m e n s i o n a l c l o c k

m o d e l s . A l l t h e s e m o d e l s s h a r e t h e s a m e t y p e o f i n d e x - n s y m m e t r y ; t h a t is , t h e

c o n s t r a i n t s o n A a r e a l l o f t h e t y p e s h o w n i n (2 . 20 ) . I n th e n e x t s e c t i o n w e w i ll

d e s c r i b e a n n = 2 t h e o r y w h o s e g a u g e f u n c t i o n s a t is f ie s c o n s t r a i n t s o f a d i ff e r e n t

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

t h a t o f t h o s e s a t i s f y i n g ( 2 . 2 0) , b u t w i t h a d i f f e r e n t v a l u e o f N o .

3 3 M O RE GENERAL M ODELS

T h e m o d e l s w e h a v e d i s c u s s e d s o f a r s h a r e t w o i m p o r t a n t c h a r a c t e r is t ic s : T h e y

a r e a ll d e f i n e d o n c u b i c l a tt i c e s a n d t h e y a l l h a v e i n d e x - n s y m m e t r i e s in t h e r e s t r i c t e d

s e n s e o f ( 2 .2 0 ) . I n t h i s s e c t i o n w e w i ll b r i e fl y d i s c u s s t w o t h e o r i e s w h i c h d i f fe r i n

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

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

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

T h e f ir s t m o d e l i s a s i m p l e b u t i n t e r e s t i n g s t a t i s t ic a l t h e o r y o f Z N s p i n s d e f i n e d

o n a t h r e e - d i m e n s i o n a l F C C l a tt ic e [6 ]. T h e m o d e l i s d e s c r i b e d b y th e h a m i l t o n i a n

H = ce ~ co s A~ ( t~) . (3 .24)

w h e r e A ~ ( q5 )= ( 2 7 r / N ) ( ~ b ~ + qb 2 - ~ b 3 - ~b4). T h e s u m i n ( 3 . 2 4) i s o v e r a ll e l e m e n t a r y

t e t r a h e d r a o f t h e l a t t ic e , a n d t h e i n d i c e s o n ~b r e f e r to t h e f o u r v e r t i c e s o f a g i v e n

t e t r a h e d r o n . H h a s c o m p o u n d n = 2 s y m m e t r i e s o f t h e t y p e ( 2 .2 0 ) a n d i s s e l f -d u a l

i n t h e s e n s e w e h a v e e m p l o y e d a b o v e . A s N ~ oo , t h e m o d e l h a s p o i n t - l i k e to p o l o g i c a l

e x c i t a t i o n s w i t h i n t e r a c t i o n s w h i c h a r e b a s i c a l l y l o g a r i t h m i c . F o r f in i te N t h e r e a r e

a l so d o m a i n - w a l l - l i k e e x c i t a ti o n s . T h e p h a s e s t r u c t u r e o f t h is m o d e l i s q u i t e s i m i l a rt o t h a t o f t h e t w o - d i m e n s i o n a l c l o c k m o d e l s [ 3 ], t h e l o c a l l y - s y m m e t r i c Z N g a u g e

t h e o r i e s [ 1 4 ] , a n d t h e m o d e l s d e s c r i b e d i n s u b s e c t s 3 .1 a n d 3 .2 a b o v e . F o r N s m a l l e r

1 8 2 F . C . A l c a r a z e t a l / S p i n s y s t e m s

t h a n a c r it i c a l v a l u e , N o , t h e m o d e l h a s a s i n g le , f i r s t- o r d e r t r a n s i t i o n s e p a r a t i n g a n

o r d e r e d f r o m a d i s o r d e r e d p h a s e ( th e c o r r e s p o n d i n g t r a n s it i o n s in t h e Z N g a u g e

t h e o r y i s a l s o f i rs t o r d e r , w h e r e a s t h a t o f th e d = 2 c l o c k m o d e l s i s s e c o n d o r d e r ) .

F o r N t> N ~ t h is t r a n s i ti o n b i f u r c a t e s i n to t w o s o f t e r o n e s d e m a r c a t i n g a n i n t e r m e d i -

a t e p h a s e w i t h n o l o n g - r a n g e o r d e r . I n t h e l im i t a s N ~ oo, w h e r e t h e m o d e l h a s a

U ( 1 ) s y m m e t r y , t h e l o w - t e m p e r a t u r e t ra n s i ti o n m o v e s t o w a r d T = 0 as N - 2, a n d

t h e s y m m e t r y - b r e a k i n g p h a s e d i s a p p e a r s, c o n s i st e n t w i th t h e t h e o r e m o f t h e p a s t

s u b s e c t i o n . F o r th i s m o d e l , a s f o r t h e c lo c k m o d e l i n d = 2 a n d t h e Z N g a u g e t h e o r y

i n d = 4 , a s w e l l a s t h e m o d e l s d e s c r i b e d i n s u b s e c t . 3. 1, N ~ a p p e a r s t o b e c l o s e t o

5. A n o t h e r e x a m p l e i n t hi s c a t e g o r y w h i c h h a s b e e n d i s c u s s e d i n th e l i t er a t u re i s

d e s c r i b e d b y a h a m i l t o n i a n s i m i l a r t o ( 3. 24 ) b u t d e f i n e d o n a n H C P l a t ti c e. T h e

g e n e r a l f e a t u r e s o f t h is m o d e l a r e a g a i n e s s e n t ia l ly t h e s a m e a s th o s e o f t h e a b o v e

m o d e l s .

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

t h e m o r e g e n e r a l t y p e o f s y m m e t r y o f t h e t y p e ( 2 .2 1 ). T h e t h e o r y is d e s c r i b e d b y

t h e h a m i l t o n i a n

H = R e Y , [as( t )s*(r+~,)+f l s*(r- .~)s*(r+.~)s*(r-~)s*(r+~)s4(r)] . ( 3 . 2 5 )r

I n i ts p e r i o d i c g a u s s i a n f o r m , t h e p a r t i ti o n f u n c t i o n o f t h e m o d e l t a k e s t h e f o r m [ 4 ]

N - 1 o o

z = E E{ tO =0 } {m,n =--oo}

(2"/r2 2 }e x p / - - ~ - ~ [ - f l ( V p ~ b - Nn , ) 2 - a ( V z< - N m ,) 21 , ( 3 .26 )

w h e r e j l a b e l s a l a t t ic e s i t e a n d

2Vp~bj = ~ j _ ~ . + t b , - , - + t h j+ . ~ + t h , - . ~ - 4 ~ , ,

V,& ~ = ~b, - ~b~_e, (3 .2 7)

w h e r e £ , fi, £ r e p r e s e n t u n i t v e c t o r s o n a c u b i c l a t ti c e . A n nj i s a s s o c i a t e d w i t h a n

e l e m e n t a r y c r o s s i n t h e ( x , y ) p l a n e , a n d a n m j i s a s s o c i a t e d w i t h a li n k i n t h e

z - d i r e c t i o n .T h e h a m i l t o n i a n o f t h i s m o d e l , c o n s i d e r e d a s a f u n c t i o n o f t h e 4 ~ 's ( t h a t is , a f t e r

s u m m i n g o v e r t h e n ' s a n d r e ' s ) is i n v a r i a n t u n d e r ( 2 .2 1 ) w i t h n = 2 w h e r e O ~ = V ~

a n d O 2 = V z . T h e p h a s e d i a g r a m o f t h is m o d e l i s s i m i l a r t o t h a t o f t h e m o d e l s

d e s c r i b e d a b o v e , b u t w i t h t w o i m p o r t a n t d i f f e r e n c e s : F i r s t, f o r N < N o, t h e s in g l e

t r a n s i t i o n i n t h i s c a s e a p p e a r s t o b e s e c o n d o r d e r , r a t h e r t h a n f ir s t, a n d f o r N > i N o ,

t h e t w o t r a n s i t i o n s a r e k n o w n t o b e i n fi n it e o r d e r [ 4 ] . S e c o n d , t h e v a l u e o f N ~ i s

n o t 5 , b u t , a s a p r e l i m i n a r y M o n t e C a r l o a n a l y s i s in d i c a t es , s e e m s t o b e c lo s e t o

1 1. It i s s i g n i f i c a n t t h a t a l l t h e k n o w n m o d e l s w i t h s y m m e t r y i n d e x n = 2 o f t h e t y p e

( 2 .2 0 ) h a v e t h e s a m e v a l u e o f N ~ , w h i l e t h e o n e t h a t d o e s n o t f it i n t o th i s c a t e g o r yb u t r a t h e r h a s a m o r e g e n e r a l s y m m e t r y o f t h e t y p e ( 2. 21 ) h a s a d i f fe r e n t v a l u e f o r

F..C. Alcaraz et al. / Spin sy stem s 183

4 . Summary

I n th i s p a p e r w e h a v e p r e s e n t e d t h e r e s u l t s o f a fi rs t s t e p in a s y s t e m a t i c s t u d y

o f t h e o r i e s w i t h u n u s u a l s y m m e t r i e s a s d e s c r ib e d i n r e f. [1 ]. W e h a v e d i s c u s s e d a

n u m b e r o f th e o r i e s w i t h s y m m e t r y i n d e x n < ~2, a n d h a v e c o n c e n t r a t e d o n t h r e e

d i m e n s i o n s . I n t h e s a m e s e n s e i n w h i c h d = 2 i s t h e l o w e r c ri t ic a l d i m e n s i o n f o r

o r d i n a r y s p i n t h e o r i e s w i t h a c o n t i n u o u s g l o b a l s y m m e t r y , n = 2 i s t h e l o w e r c r i ti c a l

v a l u e o f th e s y m m e t r y i n d e x f o r o u r w i d e r c l a ss o f t h e o ri e s . I n o u r s t u d y w e f o u n d

r e m a r k a b l e s im i l ar it ie s b e t w e e n t h e p h a s e d i a g r a m s a n d o t h e r p r o p e r t ie s o f o u r

n = 2 t h r e e - d i m e n s i o n a l m o d e l s a n d g l o b a l l y - s y m m e t r i c m o d e l s i n tw o d i m e n s i o n s.

W e a l so d i s c u s s e d a m o d e l w h o s e n = 2 s y m m e t r y i s o f a s o m e w h a t d i f f er e n t c h a r a c t e r

t h a n t h e n = 2 s y m m e t r i e s o f o u r o t h e r e x a m p l e s . S i g n i fi c an t ly , t h e Z N v e r s i o n o f

t h is m o d e l h a s a v a l u e f o r N c w h i c h i s v e r y d i ff e re n t f r o m t h a t s h a r e d b y a ll o u ro t h e r m o d e l s a n d t h e d = 2 c lo c k m o d e l s . I t t h u s a p p e a r s t h a t t he p r e c i s e f o r m o f

t h e c o n s t r a i n t s o n t h e g a u g e f u n c t i o n h a v e a s t r o n g i n f lu e n c e o f t h e v a l u e o f N o .

O u r s t u d y h a s l e f t a n u m b e r o f i n t ri g u i n g q u e s t i o n s u n a n s w e r e d . I n "a d di ti on to

t h o s e m e n t i o n e d i n s u b s e c t. 2 .4 , i t w o u l d b e i n t e re s t in g t o k n o w h o w N c d e p e n d s

o n t h e o p e r a t o r s i n (2 .2 1 ) f o r t h e c a s e n = 2 . I t w o u l d a l so b e i n t e r e s t i n g ' to k n o w

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

c o m p a r e w i t h t h o s e o f th e d = 2 c l o c k m o d e l s . S e r ie s e x p a n s i o n a n d f in i te -s iz e

s c a l i n g t e c h n i q u e s m i g h t b e u s e f u l h e r e . F i n a l l y , f o r t h o s e t h e o r i e s w i t h n > 2, i t is

i m p o r t a n t t o e x a m i n e t h e p a t te r n s o f s y m m e t r y b r e a k i n g t h a t o c cu r . I n s u c h t h e o ri e st h e r e a r e a v a r i e ty o f s e m i - l o c a l s y m m e t r i e s w h i c h m a y o r m a y n o t b e b r o k e n a t

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

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

a n d w il l d e e p e n o u r u n d e r s t a n d i n g o f cr it ic a l p h e n o m e n a i n ge n e ra l .

W e a r e h a p p y t o a c k n o w l e d g e t h e h o s p i t a l i ty o f t h e I n s t it u t e f o r T h e o r e t i c a l

P h y s i c s a t S a n t a B a r b a r a , w h e r e t h i s a n d r e l a t e d w o r k w a s b e g u n . T h e r e s e a r c h w a s

s u p p o r t e d , i n p a r t , b y t h e N a t i o n a l S c i e n c e F o u n d a t i o n u n d e r g r a n t P H Y 7 7 - 2 7 0 8 4 .

Appendix

C o n s i d e r t h e c l a s s i c a l h a m i l t o n i a n i n (3 .4 ) a n d t h i n k o f t h e z a x is as d e f in i n g a

( e u c l i d e a n ) t i m e d i r e ct io n . I n t h is a p p e n d i x w e w il l c o n s t r u c t a n o p e r a t o r , / q , w h i c h

w i ll d e s c r i b e t h e e v o l u t i o n o f a s p i n c o n f i g u r a t i o n a t z i n to o n e a t z ' . T h r o u g h o u t

t h i s a p p e n d i x , w e w i l l l a b e l t h e z d i r e c t i o n w i t h a n v a r i a b l e , t ( w i t h a l a t ti c e s p a c i n g

r i n t h e z d i r e c t i o n ) a n d c o n s i d e r a c u b i c la t t ic e o f l i n e a r d i m e n s i o n L . W h e n t h e

s p i n s a r e Z N v a r i a b l e s o f t h e f o r m

s ( r , t ) = e '( 2 ~ / N)n ( r ' O , n = O , 1 , . . . , n - - 1 , ( A . 1 )

w h e r e r = ( x , y ) r e f e r s o n l y t o t h e s p a t i a l c o o r d i n a t e s , t h e s y s t e m c a n e x i s t in a n y

1 8 4 F . C . A l c a r a z e t al. / S p m s y s t e m s

o f M s t a te s f o r e a c h t , w h e r e

M = N L2 .

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

H = Y'. h , ( t ) + h 2 ( t ) ] ,t

w h e r e

(A .2 )

h2 ( t ) = ~ , - ½ a 2 [ s ( r , * A ~ ^) s ( r + x , t ) s ( r + x + y , t ) s * ( r + ~ , t ) + c . c . ] , ( A . 3 )r

h a ( t ) = Y . - ½ c q [ s ( r , t ) s * ( r , t + z ) + c . c . ] . ( A . 4 )r

W e w a n t t o c o n s t r u c t a m a t r i x , t h e t r a n s f e r m a t r i x , 7", s u c h t h a t t h e p a r t i t io n

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

z = T r [ ~ L ] . ( A . 5 )

C a l l i n g I n ) a c o n f i g u r a t i o n o f th e s y s t e m a t t im e t, a n d I n ') a n o t h e r a t t i m e t ', w e

c a n w r i t e

( n ' l 7 " 1 n ) = ( n ' l L I n ) ( n l 2~21n ) , ( A . 6 )

w h e r e

( h i ~ 2 1 n > - e - ~ h 2 < ' ) , ( A . 7 )

< n ' l T l l n > = e - o h , < ' ) , ( A . 8 )

w i t h ha a n d h 2 g i v e n ab o v e . W e c a n t h i n k o f s ( r , t ) a s t h e e i g e n v a l u e o f a n o p e r a t o r

S ( r , t ) w h i c h a c t s o n t h e s p a c e o f c o n f i g u r a t io n s ,

S ( r , t ) l n ) = e ' ( 2 = / N ) " ( " ' ) l n ) . (A .9 )

B e c a u s e T 2 i s d i a g o n a l , i t c a n s i m p l y b e w r i t t e n i n t e r m s o f S . T o c o n s t r u c t t h e

o f f - d i a g o n a l p a r t o f T , 7"1, n o t i c e t h a t , i n t e r m s o f t h e i n t e g e r p h a s e o f s ( r , t ) , n ( r , t ) ,

w e c a n w r i t e

( n ' 1 2 r l l n ) = e x p ( / 3 o t 2 ) S L . _ . , i . o + e x p / 3 a 2 c o s - ~ 8 1 . _ . , i . 1

+ . . . + e x p •o/2 c os ~ .] 8 1 n _ n , i , N _ 1 . (A .10 )

B e c a u s e t h e s e t o f s t a t e s , { I n ) } , i s c o m p l e t e , ( A . 1 0 ) c a n b e w r i t t e n i n t e r m s o f a n

o p e r a t o r R ( r ) w h i c h i s t h e Z N c o n j u g a t e o f S :

( n ' J R J n ) = 6 . , . + ~ ( m o d N ) . ( A . I I )

F.C. Alcaraz et al / Spin systems 185

I t f o l l o w s t h a t

S ( r ) R ( r ) = e " 2 " ~ / ~ ) R ( r) S ( r ) , ( A . 1 2 )

R N = l , ( A . 1 3 )

w i th [ S ( r ) , R ( r ' ) ] = 0 f o r r # r '. T h e r e f o r e ,

^ N - -1 I 1T l = l - I ~,, g g ( r ) e x p f l a l c o s 2 - - ~ k ( A . 1 4 ), k ~ 0 N J '

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

w i t h r t h e s p a c i n g i n t h e t d i re c t i o n . T h e l i m i t m u s t b e t a k e n w i t h A f ix e d , w h e r e

f l a2

I t is e a s y t o s e e f r o m ( A . 1 4 ) t h a t o n l y t h e f i rs t t w o t e r m s i n t h e s u m c o n t r i b u t e , s ot h a t w e c a n i d e n t i f y

1 2 1= - A ~ , ½ [ S ( r ) S * ( r + ~ ) S ( r + ~ + ~ ) S * ( r + f i ) + c . c . l - l y , [ R ( r ) + R * ( r ) ] . (A . 17)r r

T h e d u a l o p e r a t o r c o r r e s p o n d i n g t o /~r c a n b e o b t a i n e d q u i t e e a s i ly . D e f i n e t h e

o p e r a t o r s

I ~( r) = S ( r ) S * ( r + ~ ) S ( r + ~ + ~ ) S * ( r + f i) , ( A . 1 8 )

P ( r ) = ~ I ~ R ( x - nx, y - n y ) , ( A . 1 9 )n x = O n y = O

w h e r e ~ ~ ^ ^r + ~ ( x + y ) . I t is a s i m p l e e x e r c i s e to s h o w t h a t

p(~)/.* (~ ) = e ' ( 2 " / N ) / z ( ~ ) p ( I ) ( A . 2 0 )

a n d t h a t

[ p ( ~ ) , / z (~ :) ] - - 0 . (A . 21)

F u r t h e r m o r e , p N ( ~ ) = / , N ( ~ ) = 1, s o t h e d u a l o p e r a t o r s h a v e th e s a m e a l g e b r a as

t h e o r i g i n a l o p e r a t o r s S a n d R . A l s o ,

p ( r ) p * ( r + x ) p ( r + . ~ + f i )p * ( r & f i ) = g ( x + ~ + fi ) . ( A . 2 2 )

Therefore,

F. .C. Alc ara z e t a l. / Spin s yste m s

+ f ) + h c . ]

Y , [ ~ ( i ) + / ~ * ( ~ ) ] / • ( A .2 3 )J

F r o m ( A . 1 7 ) a n d ( A . 2 3 ) i t f o l l o w s t h a t

/ 4 ( A ) = A / q ( 1 / A ) , ( A .2 4 )

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

g i v e n b y A = l / A , i m p l y i n g t h a t t h e s e l f - d u a l p o i n t o c c u r s a t A = 1 f o r a ll v a l u e s

o f N .

R e f e r e n c e s

[ 1 ] F .C . A l c a r a z , L J a c o b s a n d R . Sa v i t , N u c l Ph y s B2 5 7 [ FS1 4 ] ( 19 8 5 ) 3 4 0

[ 2 ] FC . A l c a r a z , L . J a c o b s a n d R . Sa v i t , Ph y s Re v . L e t t 5 1 ( 1 9 8 3 ) 4 3 5

[ 3 ] S E l i t z u r , R Pe a r s o n a n d J. Sh i g e m i t s u , Ph y s Re v . D 1 9 ( 1 9 7 9 ) 3 6 9 8 ;

A . U k a w a , P . W i n d e y a n d A G u t h , Ph y s . Re v D 2 1 ( 1 9 8 0 ) 1 0 13 ;

E . F r a d k m a n d L K a d a n o f f , N u c l . P h ys B 1 7 0 [ F S 1 ] ( 19 8 0 ) 1,

M . E l n h o r n , E . Ra b m o v i c l a n d R Sa v i t , N u c l Ph y s . B1 7 0 [ FS1 ] ( 1 9 8 0 ) 1 6 ,

L . K a d a n o f f , J . Ph y s A l l ( 1 9 7 8 ) 1 3 99[ 4 ] D A m i t , S . E l i t z u r , E Ra b m o v l c l a n d R . Sa v it , N u c l Ph y s B2 1 0 [ FS6 ] ( 1 9 8 2 ) 6 9

[ 5 ] F .C A l c a r a z , L J a c o b s a n d R Sa v l t , J Ph y s . A 1 6 ( 1 9 8 2 ) 1 7 5

[ 6 ] F C . A l c a r a z , L . J a c o b s a n d R . Sa v i t , Ph y s. L e t t 8 9 A ( 1 9 8 2 ) 4 9 ,

R L l e b m a n n , Ph y s . L e t t 8 5 A (1 9 8 1 ) 59 , Z . Ph y s . B4 5 (1 9 8 2 ) 2 4 3 ;

P . Pe a r c e a n d R . Ba x t e r , Ph y s . Re v . B2 4 ( 1 9 8 1 ) 5 2 9 5 ;

L J a c o b s a n d R S a v it , A n n . o f t h e N e w Y o r k A c a d e m y o f S c ie n c e s , 4 1 0 ( 1 98 3 ) 2 81

[ 7 ] N .D . M e r m i n a n d H . Wa g n e r , Ph y s Re v . L e t t. 1 7 ( 1 9 6 6 ) 11 3 3;

F . We g n e r , Z . Ph y s . 2 0 6 ( 1 9 6 7 ) 4 6 5 ,

N D . M e r m m , Ph y s Re v . 1 7 6 ( 1 9 6 8 ) 2 5 0

[ 8 ] N N Bo g o l i u b o v , Ph y s l k A b h a n d l So w j e t u m o n 6 ( 1 9 6 2 ) 1 , 1 1 3, 2 2 9

H . Wa g n e r , Z . Ph y s . 1 9 5 ( 1 9 6 6 ) 2 7 3

[ 9 ] S E h t z u r , P h y s R e v D 1 2 ( 1 9 7 5 ) 3 9 7 8[ 1 0 ] F .C . A l c a r a z a n d L . J a c o b s , J . Ph y s A 1 5 (1 9 8 2 ) L 35 7

[ 1 1 ] R Sa v l t , Re v . M o d Ph y s . 5 2 ( 1 9 8 0 ) 4 5 3

[ 1 2 ] R . Sa v l t , N u c l Ph y s B2 0 0 [ FS4 ] ( 1 9 8 2 ) 2 3 3

[ 1 3 ] S e e , e .g ., M Cr e u t z , L J a c o b s a n d C . Re b b i , Ph y s Re p o r t s 9 5 ( 1 9 8 3 ) 2 0 1

[ 1 4 ] M Cr e u t z , L J a c o b s a n d C . Re b bx , Ph y s Re v D 2 0 (1 9 7 9 ) 1 9 15

francisco c. alcaraz et al- three-dimensional spin systems without long-range order

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