KFKI-1981-0^4
H ungarian Academy o f Sciences
CENTRAL RESEARCH
INSTITUTE FOR PHYSICS
BUDAPEST
J, SÓL YO M P. PFE UT Y
RENORMALIZATION GROUP STUDY OF THE
ONE-DIMENSIONAL QUANTUM POTTS MODEL
■ t'
RENORMALIZATION GROUP STUDY OF THE ONE-DIMENSIONAL QUANTUM POTTS MODEL
J. Sólyom*
Department of Physics
University of Illinois at Urbana-Champaign Urbana, Illinois, 61801
and P. Pfeuty
Laboratoire de Physique des Solides, Université Paris-Sud, Orsay 91405, France
HU ISSN 0368 5330 ISBN 963 371 780 9
*Permanent address: Central Research Institute for Physics, H-1525 Budapest 114, P.O.B. 49, Hungary
particular the order of the transition as the number of components q in
creases, is studied by constructing renormalization group transformations on the equivalent one-dimensional quantum problem. It is shown that the block transformation with two sites per cell indicates the existence of a critical q separating the small q and large q regions with different critical behav
iours. The physically accessible fixed point for q>q is a discontinuity fixed point where the specific heat exponent a=l andctherefore the transition is of first order.
АННОТАЦИЯ
Изучаются фазовые переходы в классической двухмерной модели Поттса, частично зависимость порядка фазового перехода от числа компонентов q по
средством исследования эквивалентной одномерной квантовой проблемы, применяя метод ренормализационной группы. Показано, что в проведенном преобразовании блоков, содержащих по два узла решетки, появляется критическое значение q , разделяющее две области, в каждой из которых критическое поведение различное.
При q > qc достигается дисконтинуальная фиксированная точка, при которой по
казатель теплоемкости а = 1, и поэтому происходит фазовый переход первого рода.
KIVONAT
A klasszikus kétdimenziós Potts model fázisátalakulásait, pontosabban az átalakulás rendjének a q komponensszámtól való függését tanulmányozzuk az ekvivalens egy dimenziós kvantum probléma renormálási csoportos vizsgálatá
val. Megmutatjuk, hogy a két rácspontot tartalmazó cellákon elvégzett blokk transzformációban megjelenik egy kritikus q^ mely különböző kritikus visel
kedésű két tartományt választ el. q>qc eseten egy diszkontinuitási fix pontot érünk el, ahol a fajhő exponens a = 1 és ezért az átalakulás elsőrendű.
p h a s e t r a n s i t i o n in the q - s t a t e P o t t s m o d e l 1 . This m o d e l is a p o s s i b l e g e n e r a l i z a t i o n of the t w o - s t a t e I s i n g m o d e l to a r b i t r a r y n u m b e r of s tates, b u t m u c h l e s s is k n o w n e x a c t l y . The c r i t i c a l t e m p e r a t u r e or c r i t i c a l c o u p l i n g can be d e t e r m i n e d
2
f r o m a d u a l i t y t r a n s f o r m a t i o n . B a x t e r has s h own t h a t the p h a s e t r a n s i t i o n in the t w o - d i m e n s i o n a l (2-d) P o t t s m o d e l is of s e c o n d o r d e r if the n u m b e r of c o m p o n e n t s q _< 4 , b u t of f i rst o r d e r for q>4. E x c e p t for s p e c i a l c a s e s the c r i t i c a l e x p o n e n t s are not k n o w n e x a c t l y , a l t h o u g h t h e r e are r e c e n t g u e s s e s b o t h for the
3 4
t h e r m a l and m a g n e t i c e x p o n e n t s .
We k n o w e v e n less a b o u t the m o d e l in three d i m e n s i o n s .
T h e r e is e v i d e n c e from h i g h - t e m p e r a t u r e s e r i e s 5 t h a t the t h r e e - e s t a t e P otts m o d e l has a f i r s t - o r d e r t r a n s i t i o n , t h e r e is,
h o w e v e r , still no g e n e r a l a g r e e m e n t (see Refs. 6 a n d 7) w h e t h e r the c r i t i c a l v a l u e q^ w h i c h s e p a r a t e s the r e g i o n s w i t h s e c o n d - - o r d e r and f i r s t - o r d e r t r a n s i t i o n s , r e s p e c t i v e l y , is b e t w e e n q=2 and q=3 or not. Even if the t r a n s i t i o n if of f i r s t - o r d e r , it is, at most, o n l y w e a k l y f i rst o r d e r and t h e r e f o r e a p p r o x i m a t e t r e a t m e n t s m a y e a s i l y fail to p r e d i c t c o r r e c t l y the o r d e r of the t r a n s i t i o n .
In fact m o s t a p p r o x i m a t e t r e a t m e n t s do fail in d e s c r i b i n g the p h a s e t r a n s i t i o n s in the P o t t s model. The m e a n - f i e l d t h e o r y p r e d i c t s f i r s t - o r d e r t r a n s i t i o n for the t h r e e - s t a t e P o t t s m o d e l in a n y d i m e n s i o n s . The r e n o r m a l i z a t i o n g r o u p (RG) r e s u l t s w h i c h s h o u l d im p r o v e u p o n the m e a n - f i e l d t r e a t m e n t are r a t h e r c o n t r o -
8
G i n z b u r g - L a n d a u m o d e l v e r s i o n s of the P o t t s m o d e l ' g i v e first- - o r d e r t r a n s i t i o n n o t o n l y in e - e x p a n s i o n a r o u n d d=4 b u t al s o in d=3 and d=2. On the o t h e r h a n d the r e a l space r e n o r m a l i z a t i o n g r o u p t r a n s f o r m a t i o n in its s t a n d a r d f o r m ' a l w a y s g i v e s
s e c o n d - o r d e r t r a n s i t i o n i r r e s p e c t i v e of the n u m b e r of c o m p o n e n t s . The r e a l - s p a c e RG t r a n s f o r m a t i o n s are not free f r o m a m b i g u i t i e s . The m a p p i n g of the c e l l w i t h s e v e r a l s i t e s to a s i ngle P o t t s s p i n is r a t h e r a r b i t r a r y . It w a s thi s a r b i t r a r i n e s s w h i c h a l l o w e d N i e n h u i s e t al.'*'3 to c h o o s e t h a t m a p p i n g w h i c h in its c o n s e q u e n c e s c o m e s c l o s e s t to the k n o w n r e s u l t s of the 2-d P o t t s m odel. The m a p p i n g can g e n e r a t e v a c a n c i e s in the l a t t i c e of P o t t s - s p i n s if the P o t t s - s p i n s in a c e l l are in d i f f e r e n t states. The g e n e r a t i o n of v a c a n c i e s is m o r e p r o b a b l e for l a r g e q
(large n u m b e r of states) t h a n for s m a l l q and t h i s g i v e s ris e to an a b r u p t c hange in the o r d e r of t r a n s i t i o n at a f i nite q .
c The f i x e d p o i n t o f the RG t r a n s f o r m a t i o n , w h i c h w a s a c c e s s i b l e f r o m the p u r e P o t t s m o d e l for q £ q ^ , is a n n i h i l a t e d by a n o t h e r f i x e d p o i n t and the a c c e s s i b l e fixed p o i n t for q > q is a dis -
c c o n t i n u i t y fixed p o i n t .
It w o u l d b e . d e s i r a b l e to have RG t r a n s f o r m a t i o n s w h i c h have l i t t l e a r b i t r a r i n e s s in t h e m and a l l o w for s y s t e m a t i c i m p r o v e m e n t s The z e r o - t e m p e r a t u r e r e n o r m a l i z a t i o n g r o u p s for q u a n t u m sys-
1 4 - 1 9
terns can be f o r m u l a t e d in a way t h a t a c o n s i s t e n t p e r t u r - b a t i o n a l c a l c u l a t i o n can be d o n e in p r i n c i p l e to a n y o rder.
In th i s p a p e r we p r e s e n t a f i r s t - o r d e r c a l c u l a t i o n for the P o tts m o d e l u s i n g q u a n t u m RG t r a n s f o r m a t i o n s .
It is well k n o w n th a t d d i m e n s i o n a l c l a s s i c a l s t a t i s t i c a l m e c h a n i c s p r o b l e m s can be r e l a t e d by the t r a n s f e r m a t r i x to d-1 d i m e n s i o n a l q u a n t u m p r o b l e m s . In the p a r t i c u l a r case of the 2-d I s i n g m o d e l , the 1-d q u a n t u m e q u i v a l e n t is the I s i n g c h a i n in t r a n s v e r s e f ield. The P o t t s m o d e l b e i n g a g e n e r a l i z a t i o n of the I s i n g m o d e l , the q u a n t u m v e r s i o n wi l l s i m i l a r l y be a P o tts c h ain w i t h some k i n d of " t r a n s v e r s e field". S i n c e the q u a n t u m RG c a l c u l a t i o n s do not give u s u a l l y v e r y g o o d n u m b e r s for the c r i t i c a l e x p o n e n t s in the f i rst a p p r o x i m a t i o n , it is n o t e x p e c t e d to get a s e p a r a t i o n of f i r s t - o r d e r a n d s e c o n d - o r d e r t r a n s i t i o n s at q = 4, but at l east we w o u l d lik e to get an i n d i c a t i o n that
c
s o m e t h i n g h a p p e n s at a f i n i t e q ^ . In fact, as we w i l l show, the q u a n t u m RG c a l c u l a t i o n s r e p r o d u c e the fixed p o i n t s t r u c t u r e
o b t a i n e d by N i e n h u i s at al. 13, and a c r o s s o v e r f r o m s e c o n d - o r d e r to f i r s t - o r d e r t r a n s i t i o n is o b t a i n e d at a f i n i t e q .
c
The o u t l i n e of the p a p e r is as follows. The 1-d H a m i l t o n i a n v e r s i o n of the 2 - d Potts m o d e l is p r e s e n t e d in Sec. II. The r e s u l t s of a s e l f - d u a l RG t r a n s f o r m a t i o n are g i v e n in Sec. III.
A n o t h e r t r a n s f o r m a t i o n , a b l o c k t r a n s f o r m a t i o n w i t h t w o sites p e r c e l l is p e r f o r m e d in Sec. IV. T h e c r i t i c a l e x p o n e n t s and the o r d e r of t r a n s i t i o n are s t u d i e d in Sec. V. The d i s c u s s i o n of the r e s u l t s is g i v e n in Sec. VI.
I I . H A M I L T O N I A N V E R S I O N OF THE P O T T S M O D E L
In the s t a t i s t i c a l m e c h a n i c a l f o r m u l a t i o n of the P o t t s m o d e l the P o t t s spin c a n be in q p o s s i b l e s t a t e s at e a c h site of the l a t t i c e . The e n e r g y of the c o n f i g u r a t i o n d e p e n d s on w h e t h e r the P o t t s spins are in the s a m e or d i f f e r e n t s t a t e on n e i g h b o r i n g sites. D e n o t i n g th i s e n e r g y d i f f e r e n c e by e, the e n e r g y of a g i v e n c o n f i g u r a t i o n is g i v e n by
E = - e l 6
s . . s (2.1)
<1D> i' j
w h e r e the P o t t s spin s^ at site i c a n have the v a l u e s s^ = l ,2,... ,q, and the s u m m a t i o n goes o v e r n e a r e s t n e i g h b o r s .
The t r a n s f e r m a t r i x o f the P o t t s m o d e l w a s c o n s t r u c t e d by M i t t a g and S t e p h e n 2 0 . A s s u m i n g d i f f e r e n t e n e r g i e s a n d z^ for the v e r t i c a l a n d h o r i z o n t a l n e a r e s t n e i g h b o r s , they h a v e shown tha t the t r a n s f e r m a t r i x is p r o p o r t i o n a l to
T ъ exp (- — - a ) exp(
К. X
(г?>*B)
(2 .2 )w h e r e (e^/kT) i s the d u a l о f Е д У к Т ,
A =
N q - 1
l l l Í2q "k
“ i + 1 ' (2 .
i=l k = l
and
N q - 1
l l м к
i=l k = l 1 (2.4)
the s u m m a t i o n o v e r i goes a l o n g a r o w of the l a t t i c e , ft is a d i a g o n a l m a t r i x
/1
ш ft =
Шq-i
, ш = exp ( — —2iri ) ( 2 . 5 )
while
/ 0 1 О 0 0 1 . . .
1 0 0
S t e p h e n and M i t t a g 21 h a v e also s h o w n t h a t a simple p s e u d o - - H a m i l t o n i a n , a l i n e a r o p e r a t o r t h a t c o m m u t e s w i t h the t r a n s f e r mat r i x , e x i s t s at the c r i t i c a l p o i n t of the P o t t s m o del w h i c h is at
e . q e 2q
- Й - n ' W n ( 2 -7)
and t h e r e the p s e u d o - H a m i l t o n i a n is p r o p o r t i o n a l to A+B.
A H a m i l t o n i a n f o r m u l a t i o n of the P o t t s m o del is t h e r e f o r e p o s s i b l e , s t r i c t l y s p e a k i n g , at the c r i t i c a l p o i n t only.
(2 .6 )
A s s u m i n g , h o w e v e r , t h a t the l a t t i c e a n i s o t r o p y is i r r e l e v a n t near t h e c r i t i c a l p o int, the d i s c r e t e l a t t i c e c a n be m a d e c o n t i nuous in one d i r e c t i o n . I d e n t i f y i n g t h i s d i r e c t i o n as the time axis of the q u a n t u m model, a t i m e c o n t i n u u m q u a n t u m v e r s i o n of
the P o t t s m o d e l can be d e r i v e d n e a r the c r i t i c a l p o i n t in the for m
H = - - A - h B , (2.8)
q
a n d the c r i t i c a l p o i n t c o r r e s p o n d s to A / q=h.
The f i r s t p a r t of the H a m i l t o n i a n is the u s ual P o t t s
c o u p l i n g b e t w e e n the n e i g h b o r i n g s i t e s (the e n e r g y is lower by -A if the n e i g h b o u r s a r e in the same s t a t e than if t h e y are in d i f f e r e n t s t a t e s ) . The s e c o n d t e r m is the a n a l o g u e of the t r a n s v e r s e field, it r o t a t e s any s t a t e into a l i n e a r c o m b i n a t i o n of all o t h e r s t a t e s .
It is s o m e t i m e s m o r e c o n v e n i e n t to u s e a d i f f e r e n t r e p r e s e n t a t i o n , n a m e l y one in w h i c h t h e field p a r t is d i a g o n a l . D e n o t i n g the s t a t e s at site i in the r e p r e s e n t a t i o n w h e r e the P o t t s c o u p l i n g is d i a g o n a l by |k>^, k = l , 2 , . . . , q , the s t a t e s
„ U - D ( k - D
Ш k > ± , Ä = 1,2 f q (2.9)
are e i g e n s t a t e s of the transverse* field w i t h e i g e n v a l u e - ( q - l ) h for I 1 * > and h for all t h e o t h e r states. In this r e p r e s e n t a t i o n
H A_
q N
l
i=l q —1
l
k=l
m* Mq'i 1 1+1 - h
N
l
i=l
R.l (2.10)
w i t h
(2.1 1)
1
T h i s form of the H a m i l t o n i a n s h o w s that in this r e p r e s e n t ation the e f f e c t of the P o t t s c o u p l i n g is to r a i s e one spin and to l o w e r the n e i g h b o r i n g spin. T h e Potts s p i n s are, h o w e v e r , not r e a l spins. R a i s i n g the P o t t s sp i n wh e n it is in its h i g h e s t state Iq * > b r i n g s it to its l o w e s t s t a t e |1 * > .
I I I . A S E L F - D U A L R E N O R M A L I Z A T I O N G R O U P T R A N S F O R M A T I O N
T h e v a r i o u s r e a l i z a t i o n s of the q u a n t u m RG t r a n s f o r m a t i o n s
. 2 3
and t h e i r r e l a t i o n s h i p is d i s c u s s e d in the a c c o m p a n y i n g p a p e r Her e we will p r e s e n t the r e s u l t s of a s e l f - d u a l RG t r a n s f o r m a t i o n and t h e n in the n e x t s e c t i o n a n o t h e r t r a n s f o r m a t i o n w i l l be
c o n s i d e r e d .
F e r n a n d e z - P a c h e c o 18 p r o p o s e d an RG t r a n s f o r m a t i o n for the I s i n g m o d e l , w h i c h is s e l f - d u a l and t h e r e f o r e g i v e s the c r i t i c a l c o u p l i n g e x a c t l y . In a d d i t i o n to t h a t it g i v e s also the v e x p o n e n t e x a c t l y for the I s i n g m o del. W h e n u s i n g t h i s m e t h o d for the P o t t s model, the s i t e s are g r o u p p e d i n t o c l u s t e r s l a b e l l e d by £ , e a c h c l u s t e r h a v i n g b sites. The s p i n - c o n f i g u r a t i o n is f i x e d on the f i r s t site of e a c h c l u s t e r and t h e spins' on the r e m a i n i n g sites are e l i m i n a t e d by t a k i n g t h e i r l o w e s t e n e r g y c o n f i g u r a t i o n . The H a m i l t o n i a n g i v e n in Eqs. (2.8), (2.3) and
(2.4) is split as
H = H . + H . t ,
fixed spin intermediate (3.1)
w h e r e
Hfixed spin
N/b
£=1
(3.2)
w i t h
H„ = A V
У
L „к QK p 1 0q-kp p - h 11 q-iУ
L M*“ p л k=l Х" Х k=l У" 1(3.3)
c o n t a i n e s the s i n g l e site t e r m on site ( £ , 1 ) , w h e r e the spin wil l be c o n s i d e r e d fixed, a n d the c o u p l i n g of this spin to its
r i g h t n e i g h b o u r , w h ile
Hintermediate
N/b
£=1
I '
H£ д + i
w i t h
(3.4)
Ha, i+i
b-l q-l
h l I
q-l
4 a=2 k=l £,a A,a+1 4 г,Ъ Ä-+1'1 - $ I ОС
k=l
,q-k
b q-l - h i i M1
a=2 k=l £ ,a
(3.5)
c o n t a i n s the f i e l d a c t i n g on the i n t e r m e d i a t e spins a n d all o t h e r c o u p l i n g s . Th i s s e p a r a t i o n is such t h a t b o t h t e r m s are
s e l f - d u a l a n d t h e r e f o r e the c r i t i c a l c o u p l i n g will a g a i n be o b t a i n e d e x a c t l y .
In the d e c i m a t i o n t r a n s f o r m a t i o n the spins are f i x e d on s ites (£,1), £ = 1,2 ..., N/b. T h e t h i n n i n g of the d e g r e e s of f r e e d o m is a c h i e v e d by k e e p i n g for e a c h c o n f i g u r a t i o n of the f i x e d spins a s i n g l e c o n f i g u r a t i o n of the i n t e r m e d i a t e spins w h i c h has the l o w e s t e nergy. S i n c e H ^ doe s n o t c o n t a i n the c o u p l i n g to the site (£,1), the c l u s t e r d e s c r i b e d b y H^ has b s i tes w i t h o n e end spin f i x e d a n d the e n e r g y e i g e n v a l u e s s h o u l d be c a l c u l a t e d for all p o s s i b l e e n d spin c o n f i g u r a t i o n s . H^ w i l l be t r e a t e d as p e r t u r b a t i o n , it w i l l c o u p l e the c l u s t e r s and w i l l flip the s p i n s w h i c h were u n t i l now fixed. As the s i m p l e s t case, we w i l l do the c a l c u l a t i o n w i t h a s c a l e f a c t o r b=2.
The c l u s t e r p a r t of the H a m i l t o n i a n is then
H£. £+1
\ q_1
= l
4 k=l
Ü ftq-k
£,2 £ +1,1 - h q-l k=l
l
M£, 2 (3.6)
i.e. the c l u s t e r h a s two s p i n s c o u p l e d w i t h a P o t t s c o u p l i n g , the t r a n s v e r s e f i e l d acts on one spin only, the o t h e r is h e l d fixed. F o r any f i x e d state of the spi n at (£+1,1) the c l u s t e r h a s q sta t e s . The e i g e n s t a t e s of H can be e a s i l y f ound.
X/ f X/ • _L
T h e r e is one l o w e s t l y i n g l e v e l wi t h e n e r g y
Eо
1
2
[A
(q-2 ) h ] - / у [A - (q—2) h] 2 + (q-l)h2 (3.7)q-2 d e g e n e r a t e l e v e l s w i t h e n e r g y E^=h, and a n o t h e r l e v e l at
E „ = - [A + ( q - 2 )h] + / T [A - (q-2) h J + (q-l)h' (3.8)
The w a v e f u n c t i o n of the l o w e s t l y ing level, w h e n the s p i n at (£ + 1,1) is in s t ate |i > , is
X/ t X / -L
* i (i" £+1) = {3 X>£ , 2 + a l2>£,2+ -*-+ a li-1>£,2+ li>£,2+ a li+1>£,2+ -
+ a lq > £,2 } I1>£+1,1 Х1 (^Д+1) l1>£+ljl ' (3.9)
w i t h
a = (q-_j;)h { - j [A - (q-2) h] + / j [A - ( q - 2 ) h ] 2 + ( q - l ) h 2 } .
(3.10) K e e p i n g on l y t h i s l o w e s t e n e r g y s tate, the r e d u c e d s p a c e of s t a t e s wil l be
11>£ , 1 4 . (1,2)|i2>
2,1 *'1£>£ ,lXi
£+1
(£,£+1) |i£+1>£+1/1---
(3.11)
a n d it w i l l be m a p p e d onto the state
1 >
1 1 cell 12>2 cell 1г>£ cell 1íí+l>í.+l cell (3.12)
w h ere |i > is the i. s t ate o f the Ä.th cell. T h e P o t t s c o u p l i n g
X» C6 1JL X/
b e t w e e n t h e r e n o r m a l i z e d s t a t e s is o b t a i n e d by c a l c u l a t i n g the e n e r g y d i f f e r e n c e w h e n the n e i g h b o r i n g s p i n s are in the same or d i f f e r e n t s tates a n d we get
X . = A cell
1 -a 1+ (q-1) a'
(3.13)
The r e n o r m a l i z e d v a l u e of the t r a n s v e r s e f i eld is g i v e n by the m a t r i x e l e m e n t b e t w e e n two s t a t e s d i f f e r i n g by a spin flip.
h cell11 h 2 a + ( g - 2 )a 2 1 + ( q - 1 ) a 2
(3.14)
R e p e a t i n g this RG t r a n s f o r m a t i o n n times, the r e c u r s i o n r e l a t i o n for the r a t i o A / h is
n + 1
*n + l
1 -a 2 a + (q- 2 )a
n n
(3.15)
w h ere a is o b t a i n e d fr o m Eq. (3.10) by u s i n g A and h in it.
n n n
The r a t i o is use d b e c a u s e o n e of the c o u p l i n g s c a n be u s e d to set the e n e r g y scale.
*
A p a r t from the t r i v i a l f i x e d p o i n t s (A/h) = 0 (A = 0, h finite)
★
and (A/h) = “ (A finite, h = 0 ) , t here is one n o n - t r i v i a l f i x e d p o i n t at ( A / h ) c = q, w h i c h is the e x a c t r e s u l t for the c r i t i c a l c o u p l i n g . L i n e a r i z i n g a r o u n d the fixed p o i n t , the " t h e r m a l " e i g e n value is
Уt
q + 2 /q + 2
^q + 2
(3.16)
a n d the c r i t i c a l e x p o n e n t v is o b t a i n e d f r o m
l o g 2
iog y fc (3.17)
An e x p o n e n t can a l s o be d e f i n e d f r o m the b e h a v i o r of the e n e r g y gap. In the w e a k c o u p l i n g case, w h e n A / h < ( A / h ) = q,
c
A r e n o r m a l i z e s to zero, b u t h g o e s to a c o n s t a n t v a l u e , p r o p o r t i o n a l to the e n e r g y gap. N e a r the c r i t i c a l c o u p l i n g , the gap Д b e h a v e s as
(3.18)
The e x p o n e n t can be o b t a i n e d by f o l l o w i n g the r e n o r m a l i - z a t i o n of h or u s i n g the s c a l i n g a r g u m e n t s of F r a d k i n and R a b y 19 We get
log Z ^ ( (^) ) v = --- c—
Д log у (3.19)
w h e r e Z, is the m u l t i p l i c a t i v e f a c t o r in the r e n o r m a l i z a t i o n o f h, h
Zh
2 a + ( g - 2 )a 2 1 + ( q - 1 ) a 2
(3.20)
W h e n e v a l u a t e d at the c r i t i c a l c o u p l i n g , it g i v e s
í 1 •
(3.21)As was s h o w n by J u l i i é n et al. the d y n a m i c a l e x p o n e n t z can be o b t a i n e d fr o m the r e n o r m a l i z a t i o n of the a b s o l u t e e n e r g y s p a c i n g , n a m e l y
b-z A n + 1 An
(3.22)
w h e n t h ese r a t i o s are e v a l u a t e d at the f i x e d p o i n t v a l u e of А/h. Th i s m e a n s th a t for b=2
z =
log Z h 1 ( (-R-) c )
log 2 (3.23)
C o m p a r i s o n of Eqs. (3.17), (3.19) and (3.23) g i v e s v . = z v
Д (3.24)
We h a v e al s o c a l c u l a t e d the e i g e n v a l u e for a m a g n e t i c p e r t u r b a t i o n of the f o r m
Hmagn
The r e c u r s i o n r e l a t i o n for the p e r t u r b a t i o n у is
(3.25)
Уn+1
2 + ( q - 2 ) a^
l+(q-l) a 2n
a n d the e i g e n v a l u e at the f i x e d p o i n t is
(3.26)
_ 3 /q + 4 Ут ~ 2/q + 2
The 3 m a g n e t i c e x p o n e n t has b e e n c a l c u l a t e d fro m
3/v = d
log log
(3.27)
(3.28)
F i n a l l y the s p e c i f i c he a t e x p o n e n t has b e e n o b t a i n e d u s i n g the s c a l i n g l a w
2 - a = (1+z)v . (3.29)
U s u a l l y the c o r r e s p o n d i n g s c a l i n g law for the s t a t i c c r i t i c a l e x p o n e n t s is w r i t t e n in the f o r m 2 - a = dv . In the p r e s e n t
q u a n t u m cas e the d i m e n s i o n a l i t y is 1+1, b u t the ti m e or e n e r g y d i m e n s i o n is s c a l e d d i f f e r e n t l y f r o m the s p a t i a l d i m e n s i o n . T h i s l eads to the f a c t o r 1+z i n s t e a d of d.
The n u m e r i c a l v a l u e s for the c r i t i c a l e x p o n e n t s are g i v e n in T a b l e I. for some v a l u e s of q. S i n c e v e r y few e x a c t r e s u l t s a r e k n o w n for the P o t t s model, t h e s e n u m b e r s s h o u l d be c o m p a r e d w i t h the c o n j e c t u r e d v a l u e s and w i l l be c o n t r a s t e d to the r e s u l t s of the n e x t s e c t i o n s . T h e t e n d e n c y in the v a r i a t i o n of the e i g e n v a l u e s and e x p o n e n t s w i t h q is c o r r e c t for small q. The e x a c t v a l u e of v for q=2 s e e m s , h o w e v e r , to be an a c c i d e n t . No o t h e r e x p o n e n t s are o b t a i n e d e x a c t l y a n d t h e r e is no i n d i c a t i o n t h a t the o r d e r of t r a n s i t i o n c h a n g e s as q i n c r e a s e s . The t r a n s i t i o n o b t a i n e d by t h i s m e t h o d is a l w a y s a s e c o n d o r d e r p h a s e t r a n s i t i o n .
This r e s u l t is n o t v e r y s u r p r i s i n g a f t e r all. The m o d e l has o n l y one r e l e v a n t c o u p l i n g , Л/h, the o r d e r e d and d i s o r d e r e d
p h a s e s are s e p a r a t e d b y a u s ual f i x e d p o i n t . A f i r s t - o r d e r t r a n s i t i o n c a n be e x p e c t e d to o c c u r o n l y in a model, w h e r e the s p a c e of c o u p l i n g s is e n l a r g e d , as in the P o t t s l a t t i c e gas v e r s i o n 13 of the P o t t s model. We wi l l s h o w in the n e x t S e c t i o n t h a t the g e n e r a t i o n of n e w c o u p l i n g s n a t u r a l l y h a p p e n s in the o t h e r v e r s i o n s of the q u a n t u m RG t r a n s f o r m a t i o n .
IV. S C A L I N G E Q U A T I O N S OF THE B L O C K T R A N S F O R M A T I O N
• 1 4 -17
We w i l l c o n s i d e r n o w the u s u a l b l o c k t r a n s f o r m a t i o n
w i t h two s i t e s p e r cell. The H a m i l t o n i a n is s p lit in t o i n t r a c e l l and i n t e r c e l l p a r t s
H = H . + H , intra inter w h e r e the i n t r a c e l l p a r t is
H intra “£
N/2
I H. .
(4.1)
(4.2)
wi th
H£
q-i k=l
q-
n, - h 2
l
a=l q-i
l
k=l
4.3)
w h i l e the i n t e r c e l l c o u p l i n g is
w i t h
H. *- inter
N/2
= I
£=1
H£, £+1
H£ ,£+1
_A q
q-i
l
k=l
q-k
£ +1,1 ‘
(4.4)
(4.5)
As a first ste p of the RG t r a n s f o r m a t i o n , the e i g e n f u n c t i o n s a n d e i g e n v a l u e s of a s i n g l e c e l l have to be d e t e r m i n e d . A ce l l of two spins has q 2 st a t e s . T h i s e i g e n v a l u e p r o b l e m can be c o n v e n i e n t l y s o l v e d in the r e p r e s e n t a t i o n g i v e n in Eq. (2.10) w h e r e the e i g e n s t a t e s of the t r a n s v e r s e f i e l d are u s e d as a b a s i s . In t h i s r e p r e s e n t a t i o n the H a m i l t o n i a n of a cell is
H£
, q-1 - I q k=i
мк Mq_k - h
£, 1 £, 2
2
l
a=l
R£,a (4.6)
Since the Potts coupling c o r r e s p o n d s now to raising and lowering, respectively, the neig h b o r i n g spins, the q 2 states of the cell fall into q subgroups which are not mixed by H . The state
I 1 ’ 1 * > , when both spins are in the state |l'> (see Eq. (2.9)), is mixed to |2 * q '> , |3' (q — 1) '> ... and |q'2'> . Looking for the e i g e n v a l u e s of H in the form
ф = a 1 11 * 1 1 > + a 2 |2'q' > + ... + a ^ | q ' 2 ' > (4.7)
a s t r a i g h t f o r w a r d d i a g o n a l i z a t i o n gives the f o l l o w i n g wave functions and energies:
Фл =
1 / l+(q-l)a2
{ 11'1'> + a|2 'q'> + a|3'(q-1)'> + ... + a|q'2'>} , (4.8)
where
a = (q-1)A { ~qh + A + / (qh - A) 2 + f
2q (4.9)
is a n o n - d e g e n e r a t e lowest lying level with energy
E ± = - (q --2 ) h (q-2) , / , , q-2 ,,2 . q-1 ,2 - A - / (qh - ^ A) + ^ A
q
(4.10)
There is a (q-2)-fold d e g enerate level at a higher energy E_ = 2h + — ,
2 q (4.11)
with wave functions
Ф (1) = 7 =
2 */cp^l= { I 2 *q * > + eI 3 * (q-1)'> + e2 I 4'(q-2)'> +...+ e4"2 |q'2'> ,
(2)
^4.
== {|2'q’>+ e2 |3'(q-1)’>+e4 |4'(q-2)’>+...+e2(q 2 ) |q'2' >} ,
Ф2д“2)= y = = { I 2 * q * >+ eq_2|3' (q-1)'>+e2(q_2) I 4 ' (q-2)'>+...+£ (q_2) |q’2 f >}
(4.12)
V- , 2lri \
where e = exp ( — — ) . q-i
F i n a l l y there is an o t h e r non-degenerate level
*3 J~. „ 2 2, 7
(q-1) a +q-l
{ - (q-l)a|l'l'>+ I 2 ’q* > + I 3' (q-1)'> +...+|q'2'>
with e n e r g y
E - - (q-2)h - / ( q h --Í3IÍLX)2 q
A nother set of states can be o b t a i n e d by looking at the state I 1 * 2 * > and the states mixed to it. By symmetry similar sets can be o b t a i n e d by starting from |l'3'>, ...» |l'q’> and the energy spectrum will be the same. S eeking the eige n s t a t e s in the f o r m
ф = b. 11'2' > + b0 I 2'1'> + b |3'q'> + . . . + b |q'3’>
the H a m i l t o n i a n can a g ain be d i a g o n a l i z e d exactly and we for the eigenfunctions of the lowest lying levels:
(1) . 1
{ b|l'2’>+ b 1 2 '
1 ^q-2+2b2
(2) _ 1 { b|1'3’> +12'2
1 Уа-2+2Ъг
+ |3'q’>+|4'(q-1)' > +...+|q'3' >}
+ bI 3'1'> +|4'q'> +... + |q'4'>} ,
-- --- { b|l'q'> +|2' (q-1) *>+| 3* (q-2)'>+...+ b|q'l’> } ,
*V2+2b^
with
Ь = 2Л { 2 h ' ^ X + ^ (2 h “ ^ X)2 + 2'("q' 2 ~ X2 } ' q
(4.16)
(4.17)
and the energy of this (q-l)-fold degenerate level is
E I
3zi2 h (4There is a higher lying level at
E'2 = 2h + I , (4
w h ich is (q-1)(q-3)-fold degenerate. The wave functions are
Ф
2(1,1) 1/q-2
Ф
2(1,2)(1,q~3)_ l 2 " q ^
Ф
2(2,1) 1/q-2
(2,2)
2 /5=2
é (2'q -3)=
{ I 3'q '> + n I 4' ( q - 1 ) '> + n 2 |5 * (q-2)’> + . . . + n4 “3 (q ' 3 ’> } ,
{ I 3' q * > + Л2 I 4 * (q-1) '> + л4 | 5 * (q—2) * > + ...+ 3 ^|q'3'>}
{ I 3' q' > + nq-3 |4' (q-1)'> + n2 (q_3) | 51 (q_2) » > +...+ п (<1~3) |
{ I 2'2 * > + п I 4'q' > + n2 I 5' (q-1) '> +...+ n4 “ 3 | q ’ 4'>} ,
{I 2 '2 '> + л2 I 4 ’q ’>+ n4 |5'(q-l)'> +...+ П2 (4_3) |q'4 »>} ,
2 { I 2'2'> + л4 ”3 I4'q'> + n2(q_2) I5 »(q-1)<> +...+ n (q_3) |q'4'
(4,
and similar functions, where n = exp (2 iri/(q-2 ) ) Finally, the highest l ying level at
E3 ■ - ¥ h - ^ /
q
(4
19) 18)
q ' 3' >},
>}
2 0)
.2 1)
is again (q-1 )-times d e g e n e r a t e and the wave functions a r e ;
/
lT7.ilZI
{ 11
'2 * > +I
2'1' > -- b I 3 ' q' >q-2+2b2 q-2
2
q-2 Ь I q ' 3 ’ > }
ф(2) = / (q-2)/2 I I! » 3 I> ---- b|2 * 2 ’> +|з'1 '>
3 7 ' q-2 1 '
q-2+2b
bIq14'>}
/ (q,-2)/2 { 11 1 q' > - b|2'(q-l)'> - b | 3 ’ ( q - 2 ) | q'1 •> } .
q-2+2b2 q_" q”2
(4.22)
The energy spectrum is such that is the lowest lying
level, the next is the (q-1)-times deg e n e r a t e Ej level. E^ and E' lie higher and are again degenerate. E^ and E^ lie even higher.
Ke e p i n g the two lowest lying levels with q states, these states c o u l d be mapped onto the states of a cell spin with the i d e n t i f i c a tion .
1' >
cell
(i)
’l -*■ (i+1) ' >
cell l — 1,2,...,q—1 (4.23)
The effective transverse field acting on the cell spin can be o b t a i n e d from the e nergy splitting of the cell spin states
qhcell = e; (4.24)
The Potts coupling between the cell spins can be calculated fro m the matrix elements between the cell states I i * > _ ,
1 i-cell1 £+lcell and I (i+1) ' > £ ceii I (j —1) ' >£+iceii * Since the wave function of the lowest lying cell state 11'> ,, is d i f f erent in structure from
■ cell
the higher lying (q-1) degenerate states, the m atrix element will be different whether all the states I i ’> ,, , |j'> ,, , |(i+l)'>
1 cell 1 cell 1 cell
and |(j-l)' »с е ц are among the degenerate states, or |l’> appears
once or twice. This indicates that the RG trans f o r m a t i o n generates new couplings. For a consistent calculation we will enlarge the space of c o u p l i n g s .,Instead of having one Potts coupling, X, we will introduce three, X^, X^ and X3 . They are most c o nveniently defined by the matrix elements:
— <1 * „<111 H 2'> q ’> , = <1* „<2' H l ’> 2' > . ,
£+1 1 £ 11 £ 141 £+1 £+1 1 £ 11 £ 1 £+1 '
£+1 <(i-l)’1lt<2'|H| . H* II Ы 1,4',...,q' , (4.25)
£+1 <(j-D ’
1
, i'= 2'',...,(q-1)', j1 II Ы • • лThe energy spectrum of a single cell can be c a l c u l a t e d for this general model in the same way. The par a m e t e r s a and b in the wave functions are modified
a = (q-i)x. { _qh + Si f '*3 + /(qh " S r x3) 2+ 5 r Ai } '
\ “ (q-3)x
2q
(4.26)
Ь ( 2 h + 2q / '2 " ■ 2q
and the energies of the two i n t erésting levels are shifted to
E . = - ( q - 2 ) h - X 3
- / (qh v 2 + a=i xi
(4.27)
E i
azi h .\1 + (q-3)X,
2q
- / ( f h + ^ f f l )2 + ^ 4
The effective field acting on the cell spins is still given by
the e n ergy splitting of the cell states, as in Eq. (4.24).
The A^, and A^ type couplings between the cells are given by the appropriate matrix elements be t w e e n the cell states.
1 Cel1 [l+(q-l)a2][ q-2+2b*]
2 {A1 (l+a)2b2 + 2(q-2)A2 (l+a)ab + (q-2)2 A3 a2} ,
2 cel1 /l+(q-l)a2 (q-2+2b2)
Y --- 2~ *3/2 { 2A1 (l+a)fc>2 + A2[l+a)b(q-3+b2) +
+ 2 (q-2) ab ] + (q-2) A3 a(q-3+b ) } , (4.28)
A_ (4A.b2 + 4 A_ b(q-3+b2) + A,(q-3+b2)2 } .
3 cell , „ - , 2 , 2 1 2 3
(q-2+2b )
Further i t e r ations of these r e c u rsion relations do not lead to new couplings. Since in the original Potts model A^ = A2 = A 3 , it is easy to show that the choice
*2 = X *1 ' * 3 = X A 1 (4.30)
reproduces itself. Starting from an arbitrary A^ and x, which o r i g i n a l l y is equal to unity, the renorm a l i z e d values for a cell are
1 Cel1 [l+(q-l)a2] [q-2+2b2 ]
[ (1+a)b +■x(q-2)a] ,
(4.31)
Á
cell
l+(g-l)a
ГТ-
q - 2 + 2 b[2 b + x (q-3+b )]
[ (l + a)b + x(q-2)a]
As mentioned earlier, the quantum d e c imation transformation is dual to this block transformation. Instead of i n t roducing new
19
nearest neighbor interactions, there t h r ee-spin couplings would appear. The effect of the transverse field would depend on the states of the n e i g h b o r i n g spins.
These recursion relations have two trivial fixed points.
Starting from the weak field case, h << A^ = A^ = A^ , the field r e normalizes to smaller values, A^, A^ and A^ renormalizes to different values (x gets different from unity), but at the fixed
•k * * *
p o int x=l again and h =0, while A^ = A^ = A^ can be arbitrary.
On the other hand, the r e n o r m a l i z a t i o n of the strong field
*
case, h >> A^ = = A^ leads to a fixed point, where h can be
* * *
arbitrary and the Potts couplings vanish, A^ = A^ = A^ = О , but
* A^ and A^ vanish faster than A^ and x = O.
The two regions are separated by the critical value of the couplings, which from duality t r a n s f o r m a t i o n is known exactly and should be at h = A/q. The critical couplings obtained in the p resent app r o x i m a t i o n are given in Table II for a few values of q. As it is seen from there, the numbers are better for larger q values and there they give the critical coupling to about 10 I accuracy. Since this is a first-order calculation, it is hoped that next corrections can give quite accurate result.
Starting from the critical coupling and from x = 1, a non- -trivial fixed point is reached. The c o u p lings at this fixed
p o int are also given in Table II. The p o s i t i o n of the fixed p o i n t moves continuously wit h increasing q up to q^ = 6.81, where it jumps to h/A = “ , x = 00
t
к
i
is no other fixed point, for q _> 4 + / 3 , however, two new fixed points appear. A con v e n i e n t r e p r e s e n t a t i o n is to plo t the fixed point values of h/Лх 2 and 1/x for various values of q (see Fig.l) One set of fixed p o i n t s is always on the line 1/x = О with
__ h_ = (g-2)2 (g - 3)
2 2 2
A1 x q (q - 2 q - l )
(4.32)
The other set of fixed p oints moves away to finite x values and merges with the p h y s i c a l l y accessible fixed points of the Potts model at q^ ^ 6.81 and the two fixed p o i n t s annihilate each other For larger q values the fixed point at x = °° becomes ph y s i c a l l y acce s s i b l e .
This fixed p o i n t structure is very similar to that obtained by Nienhuis et a l . 3"3 for their vacancy model. Their q is closer
c
to the q^ = 4 exactly known number, but again this may be due to our first approximation. In the next section we will analyze whether the transition for large q is in fact a first-order transition or not.
V. ANALYSIS OF THE CRITICAL BEHAVIOR
Second - o r d e r phase transitions are usually descr i b e d in the framework of the r e n o r m a l i z a t i o n group c a l c u l a t i o n s in terms of fixed points which are accessible on the critical surface. The situation is not so clear for firs t - o r d e r transi-
2 4 2 5
tions. There is a large class of magnetic systems ' , where the first-order transition has been attributed to the absence of stable fixed points w ithin the domain of s t a b i l i t y of the model. This does not apply here, since there is always an accessible fixed point for any value of q.
2 6
N i e n h u i s and N a u e nberg have introduced an important new concept, the d i scontinuity fixed point. There are systems
where the first-order transition can be described in terms of the b e h a v i o r around a special fixed point. A suf f i c i e n t c o n d i tion to get a discont i n u i t y in the order p a r a m e t e r or other d e r i v a t i v e s of the free e nergy is to have a fixed point, where the thermal and magnetic eigenvalues of the RG t r a n s f ormation are equal to b ^ , where b is the scale factor in the RG transfor mation and d is the dimensionality. A fixed p o i n t where the e i g envalues are equal to b^ is called d i s c o n t i n u i t y fixed p o i n t
The calculated values of the eigenvalues are given in Table II. As one can see both the thermal and magn e t i c eigen- values tend to 2 as q “ but they are not equal to b d for any finite value of q. Nevert h e l e s s one can argue, that the fixed points, which become accessible for q>q^ » are d i scontinuity
fixed p o int and describe a first-order transition.
f
i
in the same way as in Sec. III., while has been determined numerically from the behavior of the gap. For q = 7, 8, 9 we could not calculate the gap exponent for c omputational reasons.
We could not reach the trajectories w h i c h would come close enough to the fixed point, to see the asymptotic behavior. The values of the critical exponents are given in Table III.
It is seen that the n u m erically c a l c u l a t e d v. satisfies A
the scaling law given in E q . (3.24). F u r t h e r m o r e 2ß/v = z is also satisfied.
For q > q.
y t = 2
the thermal and m a g n e t i c eigenvalues ( q -2)2
(q-3)2
are (5.1)
„ q-3
y m = 2 m „ 0q - 2 (5.2)
and the dynamical expon e n t z is o b t a i n e d from
(n + 1) , - z
A U ) l
(n + 1) (n)
<T>
< T >(q-3) (q-2)
(5.3)
It follows from these expressions and from Eq. (3.17) that (1 + z ) v = 1
and therefore E q . (3.25) gives a = 1 . We argue that a = 1 is the indication that the fixed points for q > q^ are d i s c o n t i n u i t y fixed points, they describe first-order transition.
When q -+ q^ , the specific heat e x p o n e n t a -* i . This means that the critical exponent for the en t r o p y goes to zero, i.e.
the entropy gets a step function like temperature dependence, a discontinuity, when q^ is reached. This dis c o n t i n u i t y persists for any q > q
VI. DISCUSSION
In this paper we have p r e s e n t e d a RG treatment of the
qua n t u m version of.the 2-d Potts model. Our RG c a l c u l a t i o n gives a crossover from second-order to first-order tr a n s i t i o n as the number of components increases, in agreement with the known exact results. The fixed p o int structure of our RG transfor- mation is very similar to that obtained by Nienhuis et a l .13 The r e n ormalized H a m iltonian has two couplings, h/A and x, the latter one being generated d u r i n g the r e n o r m a l i z a t i o n only.
Below a certain critical value of the n umber of components, q , c there are three fixed points, two of them at finite values of the couplings, one at infinity. The two fixed p oints at finite values merge at q^ and annihilate each other and th e r e b y the infinite fixed p o int becomes p h y s i c a l l y accessible. At this fixed p o i n t the specific heat exponent a = 1 and we argued that this is an indication of the first-order nature of the transition.
We want to emphasize that our RG tran s f o r m a t i o n is a standard one and no ambiguity was built in, contrary to the m o d i f i e d major i t y rule of the vacancy model of Nienhuis et al.'*'^.
Our r e n o r m a l i z a t i o n group t r a n s f ormation g e n e r a t e d a new coupling x / 1. The coupling was introduced in the r e p r e s e n tation defined by the states of E q . (2.9). In terms of the
original Potts states the new coupling will flip s i m u l t a n e o u s l y two neighb o u r i n g spins. One can try to construct a classical 2-d model whose transfer m a t r i x would c o r respond to our r e n o r m a lized Hamiltonian. This classical model will contain four-spin
coupling on a plaquette, but the strengths of the possible four-spin terms are all re l a t e d to each other. It seems that the i n t roduction of the four-spin c o u p l i n g s has the same effect as the i n t r o d u c t i o n of v a c a n c y variables and allows to describe the crossover fr o m second-order to f i r st-order transition.
The value o b t a i n e d for 4 C ' 9 C = 6.81, is somewhat far from the exact r e s u l t = 4. This is the consequence of taking two sites per cell and k e e p i n g for eac h cell the q lowest lying states only. Improvements c o u l d be o b t a i n e d either by taking larger cells or by taking into account the higher lying states
1 6 in a p e r t u r b a t i o n a l way as p r o p o s e d for the Ising model by Urn
, . 17
and H i r s c h and M a z e n k o . In the first case d i a g o n a l i z a t i o n of large matrices w o u l d be needed, while in the second case an extremely large n umber of n e w couplings would be generated.
We feel that both methods w o u l d give a slight impr o v e m e n t of q c , leaving our conclusion on the c r o s s o v e r from second-order transition to first-order intact.
ACK N O W L E D G E M E N T S
One of us (J.S.) is grateful to E. Fradkin for many useful d i s cussions and to R. Pandit and M. Ma for their help in the numerical calculations.
REFERENCES
1. R.B. Potts, Proc. Cambridge Philos. Soc. 4 8 , 106 (1952).
2. R.J. Baxter, J. Phys. C6, L445 (1973).
3. M.P.M. den Nijs, Physica 95A , 449 (1979).
4. R.B. Pearson, to be published. В. Nienhuis, to be p u b l i s h e d 5. R.V. Ditzian and J. Oitmaa, J. Phys. A7^, L61 (1974) г
D. Kim and R.I. Joseph, J. Phys. A8^, 891 (1975) . 6. M. Yamashita, Progr. Theoret. Phys. 6_1, 1287 (1979) . 7. V.M. Zaprudskii, Zh. Eksp. Teor. Fiz. 73 , 1174 (1977) ;
/Sov. Phys. JETP 4£, 621 (1977)/.
8. L . M ittag and M.J. Stephen, J. Phys. A7_, L109 (1974) . 9. D.J. Amit and A. Shcherbakov, J. Phys. CT_, L96 (1974) ?
R.K.P. Zia and D.J. Wallace, J. Phys. A8, 1495 (1975);
J. Rudnick, J. Phys. A8_, 1125 (1975) . 10. G.R. G o l n e r , Phys. Rev. B£, 3419 (1973) . 11. C. Dasgupta, Phys. Rev. B]J^, 3460 (1977) .
12. T.W. Burkhardt, H.J.F. Knops and M.P.M. den Nijs, J. Phys.
A£, L I 79 (1976) ; B.W. Southern, J. Phys. A10, L253 (1977) . 13. B. Nienhuis, A .N. Berker, E.K. Riedel and M. Schick,
Phys. Rev. Lett. 4_3 , 737 (1979) .
14. S. Jafarey, R. Pearson, D.J. Scalapino and B. Stoeckly
( u n p u b l i s h e d ) : S.D. Drell, M. W e i n s t e i n and' S. Yankielowicz Phys. Rev. Dl£, 1769 (1977) .
15. R. Juliién, P. Pfeuty, J.N. Fields and S. Doniach, Phys. Rev. В П Ь 3568 (1978) .
16. G. Urn, Phys. Rev. B 1 5 , 2736 (1977) .
17. J.E. Hirsch and G.F. Mazenko, Phys. Rev. Bl_9, 2656 (1979) . 18. A. F e r n a n d e z - P a c h e c o , Phys. Rev. В^Э, 3173 (1979) .
19. E. Fradkin and S. Raby, Phys. Rev. D2C), 2566 (1979).
20. L. Mittag and M.J. Stephen, J. Math. Phys. 1^, 441 (1971).
21. M.J. Stephen and L. Mittag, Phys.Lett. 4 1 A , 357 (1972). t 22. J. Kogut, Rev. Mod. Phys. 51_, 659 (1979) .
23. J. Sólyom, Phys. Rev. following paper. •
24. S .A . Brazovskii and I.E. D z y a l o s h i n s k i i , Pisma Zh. Eksp.
Teor. Fiz. 2_1, 360 (1975) ;
/Sov. Phys. JETP Lett. 21^, 164 (1975)/; S.A. Brazovskii, I.E. Dzyaloshinskii and B.G. Kukharenko, Zh. Eksp. Teor.
Fiz. 70, 2257 (1976)
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25. D. -Mukamel and S. Krinsky, Phys. Rev. ВЗ^, 5056, 5078 (1976) ; P. Bak and D. Mukamel, ibid. 1_3, 5086 (1976) •
26. B. Nienhuis and M. Nauenberg, Phys. Rev. Lett. 35 , 477 (1975) .
FIGURE CAPTION
Fig. 1. Locations of the fixed p o i n t s on the h/Ax 2 versus 1/x plot. q is a p a r a m e t e r of the curves. Fixed points on the straight lines can be reached from the x = 1 Potts line. The fixed points on the dotted lines are accessible for a more general model only.
CAPTION OF TABLES
Table I. The thermal and magnetic e i g e n v a l u e s of the RG transformation and the critical e x p o n e n t s of the q-state Potts model, obta i n e d by a self-dual RG t r a n s f o r m a t i o n .
Table II. The critical and fixed-point values of the couplings obtained with a block tran s f o r m a t i o n for several q values and the thermal and magnetic eige n v a l u e s of the RG transformation.
Table III. The critical e x p o nents obta i n e d with a block t r a n s formation with two sites per block.
eigenvalue s
q thermal
At
magnetic Am
V z
2 2 1.707 1 0.5
3 2.268 1.683 0.846 0.550
4 2.500 1.667
( 0.756 0.585
5 2.708 1.655 0.696 0.612
6 2.899 1.645 0.651 0.633
7 3.076 1.637 0.617 0.650
8 3.243 1.631 0.589 0.665
9 3.400 1.625 0.566 0.678
10 3.550 1.620 0.547 0.689
00 00 1.500 0 1
critical exponents zv = v .
A 6 ß/v a
0.5 0.228 0.228 0.5
0.465 0.211 0.249 0.689
0.442 0.199 0.263 0.802
0.426 0.190 0.274 0.878
0.412 0.184 0.282 0.937
0.401 0.178 0.289 0.982
0.392 0.174 0.295 1.019
0.384 0.170 0.300 1.050
0.377 0.166 0.304 1.076
0 0 0.415 2
q
c r i t i c a l c o u p l i n g
h A
f i x e d p о i n t e i o e n v a l u e я
h
“ ’ 4
X
P"T h
t h e r m a l X t
ma gne tic Am
x2 1 2
2 0 . 6 3 8 0 . 6 3 8 1 . 0 0 0 0 . 6 3 8 1.5 9 6 1.652
3 0 . 3 7 3 0 . 3 9 2 1. 1 1 5 0 . 3 1 5 1.7 7 8 1 . 6 3 5
4 0 . 2 6 0 0 . 2 9 7 1 . 1 9 3 0 . 2 0 9 1.966 1 . 6 2 1
5 0 . 1 9 9 0 . 2 5 8 1 . 3 0 7 0 . 1 5 1 2 . 1 7 7 1.6 0 7
6 0 . 1 6 1 0 . 2 5 7 1 . 5 1 3 0 . 1 1 3 2 . 4 5 5 1.5 9 2
6 . 8 1 0 . 1 3 9 0 . 4 0 7 2 . 2 5 3 0 . 0 8 0 3 . 0 2 0 1 . 5 6 7
6.82 0 . 1 3 9 00 oo 0 . 0 6 0 3.184 1 . 5 8 5
7 0 . 1 3 5 0 0 o o 0 . 0 6 0 3.1 2 5 1 . 6 0 0
8 0 . 1 1 6 0 0 0 0 0 . 0 6 0 2 . 8 8 0 1.666
9 0 . 1 0 2 00 0 0 0 . 0 5 9 2 . 7 2 2 1.7 1 4
10 0 . 0 9 1 0 0 0 0 0 . 0 5 7 2 . 6 1 2 1 . 7 5 0
СО 0 0 o o
0 2 2
q V -1
у = V z z V
2 1 . 4 8 3 0 . 6 7 4 0 . 5 5 1 0 . 8 1 8
3 1 . 2 0 5 0 . 8 3 0 0 . 5 8 1 0 . 7 0 0
4 1.025 0 . 9 7 5 0 . 6 0 7 0 . 6 2 2
5 0 . 8 9 1 1.122 0 . 6 3 1 0 . 5 6 3
6 0 . 7 7 2 1.295 0 . 6 5 9 0 . 5 0 8
6 .81 0 . 6 2 7 1.594 0 . 7 0 5 0 . 4 4 2
6 .82 0 . 5 9 8 1 . 6 7 1 0 . 6 7 1 0 . 4 0 1
7 0 . 6 0 8 1.6 4 5 0 . 6 4 4 0 . 3 9 2
8 0 . 6 5 5 1 . 5 2 7 0 . 5 2 6 0 . 3 4 4
9 0 . 6 9 2 1.4 4 5 0 . 4 4 5 0 . 3 0 8
10 0 . 7 2 2 1.3 8 5 0 . 3 8 5 0 . 2 7 8
oo 1 1 0 0
V A ß/v ß a
0 . 8 2 0 . 2 7 6 0 . 4 0 9 - 0 . 3 0 1
0 . 7 0 0 . 2 9 1 0 . 3 5 0 0 . 0 9 5
0 . 6 2 0. 303 0 . 3 1 1 0 . 3 5 3
0 . 5 6 0 . 3 1 6 0 . 2 8 1 0 . 5 4 6
0 . 5 1 0 . 3 2 9 0 . 2 5 4 0 . 7 2 0
0 . 4 7 0 . 3 5 2 0 . 2 2 1 0 . 9 3 1
0. 336 0 . 2 0 1 1
0 . 3 2 2 0 . 1 9 6 1
0 . 2 6 4 0 . 1 7 2 1
0 . 2 2 3 0 . 1 5 4 1
0.2 6 0 . 1 9 3 0 . 1 3 9 1
0 0 1
Kiadja a Központi Fizikai Kutató Intézet Felelős kiadó: Krén Emil
Szakmai lektor: Zawadowski Alfréd Nyelvi lektor: Zawadowski Alfréd Példányszám: 520 Törzsszám: 81-14 Készült a KFKI sokszorosító üzemében Felelős vezető: Nagy Károly
Budapest, 1981. január hó
