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QUANTUM SINE-GORDON THERMODYNAMICS
X. Zotos
To cite this version:
X. Zotos. QUANTUM SINE-GORDON THERMODYNAMICS. Journal de Physique Colloques, 1983,
44 (C3), pp.C3-477-C3-480. �10.1051/jphyscol:1983395�. �jpa-00222834�
JOURNAL DE PHYSIQUE
Colloque C3, suppl6ment au n06, Tome 44, juin 1983 page C3-477
QUANTUM SINE-GORDON THERMODYNAMICS
X . Z o t o s
P h y s i c s Dept., U n i v e r s i t y of S o u t h e r n C a l i f o r n i a , Los A n g e l e s , CA 90089-0484, U.S.A.
Resume
-
En u t i l i s a n t l a methode du theoreme de Bethe, nous analysons l e s--
p r o p r i e t e s thermiques du modele sine-Gordon quantique. Nous presentons un systeme dl@quations i n t e g r a l e s n o n l i n e a i r e s pour l a d i s t r i b u t i o n thermique des diverses e x c i t a t i o n s , que l ' o n rPsoud numeriquement. La chaleur s p e c i f i - que que nous obtenons a i n s i e s t en bon accord avec l e s experiences recentes dans l e s materiaux unidimensionnels magnetiques CsNiFg e t TMMC, ce q u i i n d i q u e que l e s e f f e t s quantiques sont importants.A b s t r a c t
-
Using t h e Bethe ansatz method we analyze t h e thermodynamics o f t h e quantum sine-Gordon model. We present a f i n i t e s e t o f coupled n o n l i n e a r i n t e g r a l equations f o r t h e thermodynamic e q u i l i b r i u m d e n s i t i e s o f t h e various e x c i t a t i o n s , which we solve numerically. We c a l c u l a t e t h e s p e c i f i c heat and we compare our r e s u l t s w i t h r e c e n t experiments on CsNiF and TMMC. The good accord obtained i n d i c a t e s t h e importance o f quantum e f f s c t s .One dimensional magnetic m a t e r i a l s have r e c e n t l y a t t r a c t e d a l o t o f i n t e r e s t as p o s s i b l e candidates f o r t h e observation o f n o n l i n e a r modes. I n p a r t i c u l a r CsNiF3 and TMMC a r e described by quantum hamiltonians which, i n a c e r t a i n regime of temperature and magnetic f i e l d , were shown t o be approximated by t h e sine-Gordon
(SG) h a m i l t o n i a n /I/. These are chains o f p l a n a r spins coupled ferromagnetical l y ( a n t i f e r r o m a g n e t i c a l l y ) i n t h e presence o f a magnetic f i e l d i n t h e easy plane, where t h e s o l i t o n corresponds t o a 2 a - t w i s t ( a - t w i s t ) o f t h e spins. Experimental evidence f o r magnetic s o l i t o n c o n t r i b u t i o n has been obtained /2,3/ by neutron s c a t t e r i n g experiments, t h e r e s u l t s analyzed i n terms o f c l a s s i c a l sine-Gordon dynamics. This i n t e r p r e t a t i o n however i s c o n t r o v e r s i a l as m u l t i p l e l i n e a r e x c i t a t i o n s and quantum e f f e c t s are important and t h e mapping o f t h e s p i n h a m i l t o n i a n t o t h e SG model ambiguous. Furthermore i n r e c e n t s p e c i f i c heat experiments /4,5/ a magnetic f i e l d dependent peak was observed i n q u a l i t a t i v e agreement w i t h c l a s s i c a l SG
thermodynamics. Thus i t i s e s s e n t i a l t o analyze t h e quantum SG model i n o r d e r t o conclude about the importance o f quantum e f f e c t s and c l a r i f y t h e v a l i d i t y o f t h e SG mapping./6/
The SG model i s known t o be completely i n t e g r a b l e i n i t s c l a s s i c a l v e r s i o n and d i a g o n a l i z a b l e i n i t s quantum v e r s i o n u s i n g t h e Bethe ansatz method. To develop t h e thermodynamics we use t h e equivalence between t h r e e models: t h e XYZ s p i n 1/2 chain, t h e massive T h i r r i n g model (MTM) and t h e quantum SG model. I t i s e s t a b l i s h e d i n t h e l i t e r a t u r e t h a t the basic e x c i t a t i o n s o f the MTM ( a r e l a t i v i s t i c fermion model) a r e holes i n t h e D i r a c sea and s t r i n g s ( i n momentum space) o f p a r t i c l e s forming bound s t a t e s corresponding t o t h e s o l i t o n s and quantized breathers r e s p e c t i v e l y . The e x c i t a t i o n spectrum i s given by t h e DHN expression Ei=2Ec s i n j a ( r - ~ ) where Es i s
A
t h e s o l i t o n energy and u t h e MTM c o u p l i n g ( p i s r e l a i e d t o
the'^^
c o u p i i n g gL by:~ = a - ~ ~ / 8 ) The thermodynamics o f t h e XYZ model was analyzed by Takahashi and Suzuki;
by t a k i n g t h e a p p r o p r i a t e continuum l i m i t t o t h e MTM we recover t h e thermodynamics of t h e quantum SG model. This a n a l y s i s /7/ leads t o a s e t o f coupled n o n l i n e a r i n t e g r a l equations d e s c r i b i n g the thermodynamic e q u i l i b r i u m d e n s i t i e s o f t h e various
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983395
C3-478 JOURNAL. DE PHYSIQUE
e x c i t a t i o n s . F o r p a r t i c u l a r v a l u e s o f t h e c o u p l i n g of t h e f o r m U = ~ K ,n an
i n t e g e r , t h i s s e t i s f i n i t e : n
where
*
denotes c o n v o l u t i o n s , B.k a r e known dressed phase s h i f t s between e x c i t a t i o n s o f o r d e r j and k and h . d e s c r i b g t h e e q u i l i b r i u m d e n s i t i e s . S o l v i n g t h e s e e q u a t i o n s n u m e r i c a l l y by i t e r a t i a n , we o b t a i n t h e d e n s i t i e s h . and t h e n t h e f r e e energy i sg i v e n by: J
The t h e o r e t i c a a n a l y s i s i s expressed i n terms o f two d i m e n s i o n l e s s parameters,
&
t h e c o u p l i n g g and t h e t e m p e r a t u r e t o s o l i t o n mass r a t i o T/Es.
The t h e o r e t i c a l c u r v e i s r e l a t e d t o e x p e r i m e n t b y a p r o p o r t i o n a l i t y f a c t o r ;
where a i s t h e l a t t i c e spacing and C t h e v e l o c i t y i n t h e SG system. The c l a s s i c a l s o l i t o n energy E* i s p r o p o r t i o n a l t o o n f o r CsNiF3 and t o H f o r TMMC and r e l a t e d t o ES by a quantam r e n o r m a l i z a t i o n f a c t o r , ES=rE;. So p l o t t i n g t h e e x p e r i m e n t a l r e s u l t s as C e x p m
hCo
f o r t h e d i f f e r e n t magnetic f i e l d v a l u e s ( c o r r e s p o n d i n gR o Eza
- - -
t o d i f f e r e n t s o l i t o n e n e r g i e s ) as a f u n c t i o n o f T/E
,
a l l p o i n t s s h o u l d s c a l e on one u n i v e r s a l c u r v e CBA depending on g2. The v a l u e of g2(a19 f o r CsNiF and 3.19 f o r TMMC) i s known f r o m t h e s p i n h a m i l t o n i a n s so t h e o n l y a r b i t r a r y p a r a m h e r e n t e r i n g t h e comparison i s t h e r e n o r m a l i z a t i o n f a c t o r r.Furthermore Ramirez and W o l f p r e s e n t t h e s p e c i f i c h e a t d i f f e r e n c e w i t h and w i t h o u t m a g n e t i c f i e l d . As t h e s i m p l e s t a p p r o x i m a t i o n t o t h e z e r o m a g n e t i c f i e l d s p e c i f i c h e a t we s u b s t r a c t a t e r m l i n e a r l y dependent on temperature, c o r r e s p o n d i n g t o a phonon gas. F o r TMMC, Borsa e t a1/8/ f u r t h e r s u b s t r a c t t h e s p e c i f i c h e a t due t o l i n e a r s p i n waves. I n t h i s case we c o n s i d e r a p p r o p r i a t e t o s u b s t r a c t t h e s p e c i f i c h e a t o f a massive phonon gas o f mass ml=2sin
SG phonons. The r e s u l t s a r e p r e s e n t e d i n F i g u r e s 1. and 2.
F i g u r e 1. CsNiF s p e c i f i c heat. ?he s o l i d c u r v e i s t h e Bethe a n s a t z r e s u l t f o r n=28.
F o r CsNiF we use a r e n o r m a l i z a t i o n f a c t o r r=0.7 as suggested i n r e f e r e n c e / 4 / . The r e s u l t s far f i e l d v a l u e s between 4 and 1 0 kGauss s c a l e w e l l , t h e a m p l i t u d e and p o s i t i o n o f t h e peak i n a c c o r d w i t h t h e BA curve. Q u a n t i t a t i v e l y t h e agreement i s g r e a t l y improved, an i n d i c a t i o n o f t h e i m p o r t a n c e o f quantum e f f e c t s . The cause o f t h e d e v i a t i o n s observed m i g h t be t h e crude a p p r o x i m a t i o n f o r t h e z e r o f i e l d
s p e c i f i c h e a t , e x p e r i m e n t a l u n c e r t a i n t i e s o r c o r r e c t i o n s due t o i m p e r f e c t mapping t o t h e SG model.
F i g u r e 2. TMMC s p e c i f i c heat. The s o l i d c u r v e i s t h e Bethe a n s a t z r e s u l t f o r n=8.
F o r TMMC t h e SG a p p r o x i m a t i o n i s expected t o be v a l i d 1 9 1 a t l o w f i e l d s , Hs5 Tesla, where t h e o u t o f p l a n e f l u c t u a t i o n s a r e small. A t p r e s e n t we have o n l y one s e t of d a t a i n t h i s regime so we cannot check t h e s c a l i n g p r o p e r t y (we c o n f i r m however t h a t t h e h i g h e r f i e l d d a t a , H=7.6, 10 T e s l a d o n ' t s c a l e p r o p e r l y ) . We v a r y t h e
r e n o r m a l i z a t i o n f a c t o r r and we o b t a i n an o p t i m a l f i t f o r r-0.54. T h i s v a l u e i s n o t i n a c c o r d w i t h t h e one g i v e n i n reference/3/(r=0.79). To c l a r i f y t h i s d i s c r e p a n c y an a n a l y s i s o f t h e n e u t r o n s c a t t e r i n g r e s u l t s i n terms o f t h e quantum SG dynamics seems necessary, k e e p i n g i n mind t h e a m b i g u i t i e s p r e s e n t i n t h e comparison w i t h t h e s p e c i f i c h e a t experiment.
I n c o n c l u s i o n t h e s e p r e l i m i n a r y r e s u l t s i n d i c a t e t h e importance o f quantum e f f e c t s ; f u r t h e r experiments and t h e o r e t i c a l work i s necessary t o a r r i v e a t d e f i n i t e c o n c l u s i o n s .
Acknowledgements
We would l i k e t o t h a n k Dr. A.P. Ramirez and P r o f . F. Borsa f o r sending us d e t a i l s on t h e i r experiments, K. Maki f o r u s e f u l d i s c u s s i o n s and N. W r i g h t f o r h e l p w i t h t h e numerical work, and CNRS f o r f i n a n c i a l a i d t o a t t e n d Les Arcs conference. T h i s work has been s u p p o r t e d by NSF g r a n t no. DMR 79-16703
JOURNAL DE PHYSIQUE
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