SOLID 3He : LOCKED-IN MAGNETISM ?

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SOLID 3He : LOCKED-IN MAGNETISM ?

M. Papoular

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque C6, supplPment au

no

8, Tome 39,

aolit 1978, page C6-133

S O L I D

3 ~ e

:

LOCKED-IN MAGNETISM

?

M. Papoular

Centre

de

Recherehes sur les Tr'r8.s Basses Tempdratures,

C. N.R.S., BP 166 X, 38042

Grenoble Ce'dex,

France.

Rhsum5.- Dans une approche basde s u r l a s e l f - c o n s i s t a n c e de l ' o r d r e dans 3 ~ e s o l i d e , on o b t i e n t une c o n s t a n t e d'dchange qui dCpend de T

B

b a s s e tempbrature. Le problhme e s t comparable, e n t r e a u t r e s s i t u a t i o n s physiques analogues,

P

c e l u i de l a t r a n s i t i o n commensurable d'une onde de d e n s i t d de charge.

A b s t r a c t . - The s e l f - c o n s i s t e n t n a t u r e of t h e o r d e r i n g t r a n s i t i o n i s emphasized, y i e l d i n g an exchange c o n s t a n t which i s T-dependent a t low temperature. The p i c t u r e which emerges, f o r e x c i t a t i o n s and tem- p e r a t u r e e v o l u t i o n , i s found t o c l o s e l y p a r a l l e l the lock-in model f o r charge d e n s i t y waves, and t o b e a r s t r o n g a n a l o g i e s w i t h o t h e r p h y s i c a l s i t u a t i o n s .

I . - The low temperature n u c l e a r magnetism of s o l i d 3 ~ e i s an open problem / I

/.

Recently, we proposed a non-linear l a t t i c e - d y n a m i c a l approach / 2 / whereby t h e o r d e r i n g temperature TN (= 1 mK) i s v i e - wed a s an i n s t a b i l i t y , comparable t o a metal-insula-

t o r t r a n s i t i o n . Below TN, long range antiferromagne- t i c o r d e r i s "frozen i n " , due t o t h e hard c o r e cons- t r a i n t : t h e r e l e v a n t e x c i t a t i o n s a r e domain w a l l s coupled t o a l o c a l l a t t i c e d i s t o r t i o n , r a t h e r t h a n c o n v e n t i o n a l s p i n waves. These l o c a l i z e d , d e f e c t - l i k e e x c i t a t i o n s were d i s c u s s e d i n Ref. / 2 / , u s i n g a crude one-dimensional, phenomenological r e p r e s e n - t a t i o n of t h e f u l l exchange o p e r a t o r . I n t h i s approach, t h e low-T m a g n e t o e l a s t i c problem i s h i g h l y s e l f - c o n s i s t e n t . Due t o t h e s e v e r e c o n s t r a i n t p l a - ced on exchange by hard c o r e e x c l u s i o n , t h e r e i s a c o u p l i n g between two competing o r d e r i n g s : magnetic and c r y s t a l l i n e . Hence t h e i n s t a b i l i t y c h a r a c t e r of

TN.

The 1st o r d e r t r a n s i t i o n t h e r e c e r t a i n l y has no- t h i n g t o do w i t h c o n v e n t i o n a l m a g n e t o e l a s t i c cou- p l i n g ( t h e Bean-Rodbell c r i t e r i o n i s o f f by about 3 o r d e r s of magnitude / 3 / ) .

As shown i n Ref. / 2 / , u s i n g a p a t h i n t e g r a l method f i r s t proposed by Krumhansl and S c h r i e f f e r / 4 / , t h e dynamics of t h e problem can be e x t r a c t e d from t h e SchrEdinger e q u a t i o n of a f i c t i t i o u s par-

*

t i c u l e w i t h " y s s " m

,

t u n n e l l i n g i n a t r i p l e - w e l l

*

p o t e n t i a l . m i s T-dependent and i n c r e a s e s when de- c r e a s i n g T a s : m * ( ~ ) % T - ~ . I n a c o r r e l a t e d s i t u a -

t i o n , any elementary exchange i s accompanied by t h e rearrangement of a number of neighboring p a r t i c l e s , which i s t h e l a r g e r , t h e s t r o n g e r t h e c o r r e l a t i o n s : t h e r e a r e fewer w a l l s a t lower t e m p e r a t u r e s , b u t any given w a l l " b u i l d s up" a s T J.. By r e f e r e n c e t o

r o t o n s i n He 11, we may c a l l t h a t a blackflow pro- c e s s . This i s q u a l i t a t i v e l y r e f l e c t e d i n t h e T-de-.

*

pendence of t h e e f f e c t i v e mass m

.

Obviously, w e a r e f a c e d h e r e w i t h much more c o o p e r a t i r e a mechanism than simple 3 o r 4 - p a r t i c l e /3,5/ exchange. As T i n c r e a s e s above TN, t h e c o r r e l a t i o n s weaken and f i - n a l l y fade away above a c h a r a c t e r i s t i c temperature Tp(= 4 TN / 2 / ) : we a r e then l e f t w i t h an u n c o r r e l a -

t e d paramagnetic s t a t e w i t h t r i v i a l 2 - p a r t i c l e exchange. The problem, a s we s e e i t , i s a s o r t of "dynamic l o c a l i z a t i o n " problem, and we e x p e c t a

&-

c r e a s e i n molar volume a t low T (a few 2 , s e e Ref.

12.1 85).

2.- The lowest e i g e n l e v e l cO(T) of our f i c t i - t i o u s p a r t i c l e motion y i e l d s t h e thermodynamics of t h e problem / 4 , 2 / , i n

articular

t h e e n t r o p y jump a t TN / 2 / ) . E~ i s c a l c u l a t e d , and i t s T-dependence de- p i c t e d , i n Ref. / 2 / f o r a given 3-well shape. Rough- l y , t h e r e a r e t h r e e r e g i o n s : a s t e e p - l i n e a r i n c r e a - s e a t T goes from 0 t o TN, an i n c r e a s e t o J a s T r e a c h e s T and then a s a t u r a t i o n a t J o . J o ( % 1 mK)

P'

i s t h e high-T v a l u e of t h e ( n e a r e s t neighbor) exchange i n t e g r a l . The t r i p l e w e l l s t r u c t u r e i t s e l f i s i n f a c t T-dependent as T approaches T

.

This enhan-

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c e s t h e temperature v a r i a t i o n of E~ between TN and Tp. i s known / 4 / t o be i d e n t i c a l t o t h e lowest eigenvalue of t h e t r a n s f e r i n t e g r a l o p e r a t o r :

-Bf (ui,ui-l

/

dui- l e ~ ~ ( u ~ - ~ ) = e - B E ~ w ~ ( u ~ ) where f i s t h a t p a r t of t h e s e l f - c o n s i s t e n t poten-

t i a l , which depends on t h e (average) p o s i t i o n s ui and ui-, of two neighboring atoms. I n o t h e r words,

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E~ i s a measure of t h e exchange c o n s t a n t J . So, below T J d e c r e a s e s w i t h temperature and i s pro-

-

p'

g r e s s i v e l y quenched p r o p o r t i o n a l l y t o E ~ . Of c o u r s e , because of t h e crudeness of our 3-well model, t h i s important r e s u l t - which should be submitted t o low- T NMR t e s t s

-

must be taken a s simply semi-quantita- t i v d . N e v e r t h e l e s s , i t i s c o n s i s t e n t w i t h t h e obser- ved / 6 , 7 / p r e t r a n s i t i o n a l i n c r e a s e ('L 50-100 % a t T _N) of t h e magnetic s u s c e p t i b i l i t y above t h e Curie- Weiss v a l u e . Moreover, i t a l l o w s us t o ~ r e d i c t a p r e t r a n s i t i o n a l d e c r e a s e of t h e s p i n d i f f u s i o n coef- f i c i e n t (Ds % ~ a ~ ) . Both t h e s e r e s u l t s a r e a t odds

w i t h conventional a n t i f e r r o m a g n e t i c b e h a v i o r .

3.- We a r e l e f t now with two c h a r a c t e r i s t i c e-

I

nergy s c a l e s : a c o r r e l a t i o n e n e r g y E D p ( E D p

,

10 J o , s e e Ref. / 2 / ) , and an exchange energy J = E 0'

.

These a r e a s s o c i a t e d , r e s p e c t i v e l y , w i t h two k i n d s of e x c i t a t i o n s : l o c a l i z e d domain w a l l s , a s a l r e a d y d i s c u s s e d

-

and long wavelength o s c i l l a t i o n s of t h e w a l l network (with " l a t t i c e c o n s t a n t "

5,

i n c r e a s i n g e x p o n e n t i a l l y a s T C ) . The l a t t e r a r e simply t h e Goldstone modes of t h e problem, w i t h v e l o c i t y pro- p o r t i o n a l t o J . They would reduce t o conventional s p i n waves i n an u n c o r r e l a t e d problem. I n a d i s p e r - s i o n r e l a t i o n diagram, t h e y a r e r e s t r i c t e d t o t h e long wavelength c o r n e r : q< < 1 , which r a p i d l y f a d e s away a s T -+ 0. For s h o r t e r wavelengths, t h e r e is an

upper branch of w a l l e x c i t a t i o n s , s e p a r a t e d from t h e Goldstone mode branch by a gap EDp. This upper branch i s what would be c a l l e d a "phason branch" 181 i n a charge d e n s i t y wave c o n t e x t . Note t h a t , s t r i c t l y , t h e "lock-in" t r a n s i t i o n , h e r e , i s a t T=O, n o t a t

I t i s a p p a r e n t t h a t t h e r e a r e s t r o n g p a r a l l e l s w i t h o t h e r problems w i t h two competing energy s c a l e s and two t y p e s of e x c i t a t i o n s : r o r o n s i n He 11, va- c a n c i e s i n quantum c r y s t a l s , t h e

XY

model, t h e rou- ghening and s p i n g l a s s t r a n s i t i o n s / 9 / . The analogy w i t h s t r u c t u r a l t r a n s i t i o n s /4/

i s

obvious from t h e s t a r t . One of t h e c h a r a c t e r i s t i c e n e r g i e s : t h e c o r r e l a t i o n energy, o r i g i n a t e s from some " f r u s t r a - t i n g " c o n s t r a i n t : hard c o r e , lower d i m e n s i o n a l i t y , " e p i t a x ~ a l " p o t e n t i a l of a r e f e r e n c e l a t t i c e , epc. From t h a t p o i n t of view, i t i s tempting t o s p e c u l a t e

t h a t t h e magnetism of t h e Wigner l a t t i c e , and a l s o t h a t of t h e "Kondo l a t t i c e " / l o / , could b e a r some resemblence w i t h ' t h e p r e s e n t approach.

Another obviously r e l a t e d problem i s t h a t of t h e r e g i s t r y t r a n s i t i o n i n p h y s i s o r p t i o n . The analogy w i t h CDW has a l r e a d y been p o i n t e d o u t / ] I / . I t i s

amusing t o n o t e t h a t , f o r a p p r o p r i a t e thermodynamic c o n d i t i o n s , t h e a p p a r e n t l y c o n f l i c t i n g Novaco and Venables models might be r e c o n c i l e d a s d e s c r i b i n g ,

r e s p e c t i v e l y , t h e formation of a " d e n s i t y wave" below some temperature

"T

"

and i t s f u r t h e r non-linear

P '

d i s t o r t i o n and l o c k i n g a s T i s reduced through "TN1'. These i d e a s w i l l be developed i n a s e p a r a t e p u b l i c a t i o n .

References

/

1

/

See f o r example, c o n t r i b u t i o n s t o t h e "Interna- t i o n a l Conference on Quantum C r y s t a l s " , F o r t C o l l i n s , Colorado (1977) ; Guyer, R., J . Low Temp. Phys.

30

(1978) 1 ; See a l s o Ref. / 3 / and /5/ below, and : HERITIER, M. and Lederer, P.,

J. Physique L e t t .

2

(1977) L-209 ; Beal-Monod, M.T. Commun Phys.

2

(1977) 199.

/ 2 / Papoular, M . , J. Low Temp. Phys.

2

(1978) 595 and S o l i d S t a t e Commun (1977) 113.

/3/ Roger, M., D e l r i e u , J . M . and Landesman, A . , Phys. L e t t .

62

A. (1977) 449.

/ 4 / Krumhansl, J.A. and S c h r i e f f e r , J . R . , Phys. Rev. Bfi (1975) 3535.

/ 5 / H e t h e r i n g t o n , J.H., C o n t r i b u t i o n t o t h e p r e s e n t Conference.

161 B e r n i e r , M. and D e l r i e u ,

J.M.,

Phys. L e t t .

60

A (1977) 156.

171 Bakalyar, D.M., B r i t t o n , C.V., Adams, E.D. and Hwang, Y.C., Phys. L e t t .

64

A (1977) 208.

/ 8 / Mc M i l l a n , W.L., Phys. Rev. B (1977) 4655. 191 Toulouse, G . , Vannimenus, J . and M a i l l a r d , J.M.

J. Physique L e t t .

2

(1977) L-459.

/ l o / As suggested by Fouquet, J., t o b e p u b l i s h e d . / I l l See Proc. of t h e " I n t e r n a t i o n a l Conference on