Two Kondo impurity problem: Relevance to heavy fermions

Journal of Magnetism and Magnetic Materials 63 & 64 (1987) 251-253 North-Holland, Amsterdam

251

TWO KONDO IMPURITY PROBLEM: RELEVANCE

TO HEAVY

FERMIONS

B A JONES** and C M VARMA* *AT & T Bell Laboratories, 600 Mountain Ave, Murray Hdl, NJ 07974, USA *Laboratory of Atomtc and Sohd State Phystcs, Clark Hall. Cornell Umverstty, Ithaca, N Y 14853 and Instztute for Theorencal Physzcs, Umvers~ty of Cahfornta-Santa Barbara, Santa Barbara, CA 93106, USA We have used Wflson's numerical renormallzatlon group method to study the problem of two spin-one-half magnetic moments m a Fermi gas Even when the lnmal effect,ve RKKY couphng K~ between the Impurmes ~s much smaller than the Kondo energy Tr., the system scales to strong couphng and the asymptonc behawor ~s that of a correlated Kondo effect

1. Introduction A n i m p o r t a n t q u e s t i o n for a system of two m a g n e t i c i m p u r i t i e s in a F e r m i gas ms the n a t u r e of the e f f e c t i v e l n t e r a c t t o n s at low t e m p e r a t u r e s A b r a h a m s a n d V a r m a [1] find a l o g a r t t h m l c s l n g u l a r t t y in the m t e r a c t l o n b e t w e e n the two m o m e n t s t h e m s e l v e s , in a d d i t i o n to the wellk n o w n K o n d o s l n g u l a r t t y i n v o l v i n g the c o n d u c tion e l e c t r o n s Just as in the K o n d o p r o b l e m , h o w e v e r , t h e i r a p p r o a c h c a n n o t g i v e inform a t i o n on the g r o u n d state a n d the l o w - l y i n g e x c i t a t i o n s W e h a v e u s e d the t e c h n i q u e s of W l l s o n ' s t r e a t m e n t of the o n e - i m p u r i t y p r o b l e m [2] to s t u d y the c a s e of two K o n d o i m p u r i t i e s W e find that the e f f e c t i v e R K K Y a n d K o n d o c o u p hngs both scale to s t r o n g c o u p l i n g as T---~ 0 In the g r o u n d state the s e p a r a t e d i m p u r i t i e s a r e c o r r e l a t e d e v e n t h o u g h b o t h e x p e r i e n c e the K o n d o effect the o v e r a l l g r o u n d state is a singlet T h e m o d e l H a m t l t o n i a n is H =

I

d3k E k a k . a k .

*

-- J [ s c ( R / 2 ) • S, + s o ( - R / 2 ) . S~]

(1)

W e a p p r o x i m a t e the c o n d u c t i o n b a n d with l i n e a r d i s p e r s i o n a n d a c o n s t a n t d e n s i t y of states W e also n e g l e c t t h o s e c o n d u c t i o n o p e r a t o r s n o t c o u p l i n g to the t m p u r l t y T h e resulting H a m l l t o n i a n c a n b e w r i t t e n in t e r m s of c o n d u c t i o n states w h i c h h a v e e v e n (e) a n d o d d (o) p a r i t y with

r e s p e c t to the o r i g i n b e t w e e n the two i m p u r t t l e s F o r s l m p h c l t y we e v a l u a t e two e f f e c t i v e c o u p l m g c o n s t a n t s Je a n d Jo at the F e r m i w a v e v e c t o r T h e n (1) r e d u c e s to t" 1

H/ D

[

dE ~:[aee~a~eg *

3- 1

+ a tcoo. a e o ~ ]

-fl_ld'~lde'[lJea*,e.O'.#a,e

#

+ Joa*~o.O'~. a.,o,. } • (S, + $2) + l(JeJo)l/2{a~e~O'u.u.a.,o# - a*.o~.O'~.wa.,e.,}" ($1 - Sz)],

(2)

where

J°=7

J°=7

T h e r e s u l t i n g e f f e c t i v e R K K Y i n t e r a c t i o n is t h e n f e r r o m a g n e t i c at all R K o l D = 8 In 2(J~ - Jr,,)2 T h e p r o c e s s of l o g a r i t h m i c d i s c r e t i z a t i o n a n d c o n v e r s i o n to an i t e r a t l v e H a m i l t o m a n is t h e n i d e n t i c a l to that p r e s e n t e d by W i l s o n [2], a n d g e n e r a t e s a s e q u e n c e of e f f e c t i v e H a m i l t o n l a n s T h e s e d e s c r i b e the l o w - l y m g m a n y - b o d y states at successively lower temperatures knT(N)/D~x A -~N-l)/2 A t e a c h i t e r a t i o n the H a m l l t o n i a n is d l a g o n a l i z e d a n d the e n e r g y flows as a f u n c t i o n of t e m p e r a t u r e g i v e the a s y m p t o t i c l o w - t e m p e r a t u r e b e h a v i o r A s T 9 0 the levels r e a c h a

0 3 0 4 - 8 8 5 3 / 8 7 / $ 0 3 50 O E l s e v i e r S c i e n c e P u b l i s h e r s B V ( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )

B A Jones, C M

252

Varma / Two Kondo tmpunty problem

s t a b l e fixed p o m t By c o m p a r m g the d e g e n e r a c i e s a n d level flows n e a r the fixed p o i n t to systems with k n o w n phystcs, we c a n d e s c r t b e the low t e m p e r a t u r e b e h a v t o r for v a r i o u s lnttial v a l u e s of J a n d Ko

2.

Results

H e r e we dtscuss the flows of the l o w e s t - l y i n g e n e r g y levels as a f u n c t i o n of N T h e levels are d e n o t e d by ( Q , 2S, P) w h e r e Q ts the c h a r g e , 2 S twtce the total spin, a n d P the p a r t t y ( e v e n = 0, a n d o d d = 1) In the figures we s h o w o n l y a l o w e r s u b s e t of the flows W e discuss a n u m b e r of c a s e s l e a d i n g to o u r c o n c l u s i o n s n o n - i n t e r a c t i n g tmpurtttes, the o n e - a n d t w o - c h a n n e l p r o b l e m s wtth the t m p u r i t t e s c o n f i n e d to a trtplet state, a n d cases w h e r e the R K K Y e n e r g y ( f e r r o m a g n e t t c ) is b o t h l a r g e r and s m a l l e r t h a n the K o n d o e n e r g y

A Non-interacting lmpurmes S e t t i n g J~ = J,, d e c o u p l e s the two t m p u r t t t e s I n d e e d , at all t e m p e r a t u r e s the c a l c u l a t e d e n e r g y l e v e l s are just

/ /

"

J

10

'©21)2

p a l r w l s e sums of levels f r o m the o n e - i m p u r i t y case, thus p r o v t d m g a c h e c k on o u r c o m p u t a t i o n Ftg 1 shows the e n e r g y flows for a s u b s e t of the levels T h e s y m m e t r y c h a r a c t e n s n c s of md e p e n d e n t ~mpurtttes are t l l u s t r a t e d by the d e g e n e r a c y of 110 a n d 111 (different p a n t y ) , a n d of 000~ a n d 021 (spm s y m m e t r y s m g l e t to triplet with a p a n t y flip) A s T - - ~ 0 , e a c h s p m is s e p a r a t e l y c o m p e n s a t e d by the K o n d o effect 13 Impurity spin constrained to be untty W e w o r k m the m a n i f o l d in whtch the total t m p u n t y s p m ts f o r c e d to be o n e First for the c a s e of the two i m p u r t t l e s on t o p of e a c h o t h e r ( R = 0) t h e r e ~s o n l y o n e e f f e c t w e e l e c t r o n c h a n n e l O u r c o m p u t a n o n s , c o n f i r m m g the w o r k of C r a g g a n d L l o y d [3], yield a o n e - s t a g e K o n d o effect, wtth c o m p e n s a t i o n tn the e v e n c h a n n e l o n l y , the r e s u l t m g final t m p u r t t y - c o n d u c t ~ o n e l e c t r o n o b j e c t has s p m o n e - h a l f Ftg 2 shows the flows for the m o r e m t e r e s t m g case (S, mp = 1) of s e p a r a t e d i m p u r t t l e s w h e r e t h e r e is b o t h an e v e n a n d an o d d e l e c t r o n c h a n nel N o t e that the s y m m e t r t e s are c o m p l e t e l y dtfferent f r o m the n o n m t e r a c t i n g c a s e 0 0 0 , Is now d e g e n e r a t e wtth 0002 (no spin s y m m e t r y ) , a n d 110 a n d 111 a r e n o n d e g e n e r a t e as well, m d t c a t m g a b s e n c e of p a n t y s y m m e t r y A twos t a g e K o n d o effect, slmtlar to t h a t p r o p o s e d by J a y a p r a k a s h , K r l s h n a - m u r t h y , a n d W t l k m s [4] for

\

[ J LN

\ t

-

> i

[Ji ttl

J

U,

o o 1

5

9 ITERArION

15

17

J 21

J m

LLJ [ J

t t(O00)l ~

NUMBER

Fig 1 Energy flows for case of two non-interacting Kondo impurities The energies m umts of the bandwidth D are scaled by A ~N-1~/2 where N is the iterate number on the honzontal ax~s Increasing N ~mphes decreasing energy or temperature according to kBT(N)~ A ~N-I)/2, so that the ground and exc,ted states at zero temperature are at the far right hand s~de The many-body energy levels are labelled by (Q, 2S, P) where Q is the charge, 2S twice the total spin and P the panty (even = 0 and odd = 1) Note for non-interacting tmpunhes that different parltwS are degenerate, e g (111) and (110)

J

'-

k ~ 10)

t.

e.._

(111) ~ 1 ~UZ ) 5

~ ITERATION

1 ~,

17

"1

NbMBER

Fig 2 Energy flows versus ~teranon number N for the case where only t m p u n n e s triplet states are permitted, l e S,.w = 1 The order of the levels is qmte different from those m fig 1 In particular analysis of the flows reveals a two-stage Kondo screemng of the impurity spins

B A Jones, C M Varma I Two Kondo ampunty problem

full problem (see p o m t C ), occurs tn which the spin-1 Is c o m p e n s a t e d to spin-½ and then to spin-0 T h e r e are then two charactertstlc K o n d o temperatures, one for the even channel and one for the odd C Full manifold of Impurity spins, and Ko >> TK T h e spms ahgn, and there follows a two-stage K o n d o effect, wtth two characteristic K o n d o temperatures [4] D Full mamfold as m C, for Ko ~ TK Both impurities are fully compensated, but (fig 3) the flows are unhke the n o n m t e r a c t m g case (see A ) at any fimte T T h e degeneracies are asymptotically that of the S,mp--- 1 problem Thts means that the R K K Y couphng as well as the K o n d o effect determine the low-temperature behavtor even for Ko ~ TK In addmon, the system shows coherence, as odd and even parity channels approach the zero t e m p e r a t u r e fixed point at dtfferent rates T o analyze the flows near the T = 0 fixed point, we expand m effective Fermi llqmd operators In general, there are 6 Irrelevant operators to leadmg order In A T h e operators can be expressed m terms of excltattons about the K o n d o c o m p e n s a t e d state Oa = o s , ( f ta~ef2.e + f 2t . e f a . e ) 02

1 = o~2(f,1-uof2~,o +[2~,o[,,,o)

O~ = ~ O ' ~ ( . f I ~ f , ~ e - 1) 2 04

~"

t04(/*~o/,~o- 1) 2

o~ = ,o~(f, oo-f,O ( f , o ~ , o ) 06 = ~6[(flu.eflt.,.et

t

1)(fl.of,.ot

1)

1

+ 2 ( f ' t e f l ~ f l + o f ' t o + fltof,~ofl+~flte)] T h e operator 0 6 at the T = 0 fixed point m the S, mp = 1 mamfold has only zero expectation values, to6 = 0 H o w e v e r m the case where full tmpunty spm manifold is used (S, mp= 1 and 0), we nonetheless find to6 = 0 even for K o ' ~ TK (wtthm our numerical accuracy) We have e x a m m e d our results for the relatmnshlps between the ton's, these wdl be pubhshed elsewhere We have also examined the case of an antlferr o m a g n e t R K K Y couphng by expltcitly addmg such a term - K ' S I S2 to (2) and setting J~ = Jo (to a v m d generating a ferromagnettc R K K Y )

253

/

•

~

(110)2

,0

r~

000)2 ..(11o)~'%.

n

<-

,

" ~

C) ½1

@ (3o

1

(111) .~ 5

9 ITERATION

15

1 17

21

~

25

NUIvlBER

Fag 3 Energy flows for the general case of two l m p u r m e s All parameters are the same as an fig 1 wath the exceptaon that JdJ,, = 4 (m contrast to value of umty for fig 1) Some features mclude the development of the S, mp= 1 state as as evidenced by the rise of (000)t to meet (000)2 at tterate N = 7, and the same collection of levels an the zero-temperature ground state as m fig 2, illustrating a c o m m o n ground state for both cases

For K ' small and negattve, we get a K o n d o effect, but w~th antlferromagnet~c correlations ( ~ K'---~-oo j - + + ~ ) For K ' large and negative, the ground state ~s just a smglet, and there ts no K o n d o effect T h e ~mportant Imphcatmns for the heavy f e r m m n problem are that the low temperature behavior of the heavy fermton latttce may be dtscussed in terms of a sum of pmr-w~se Hamtltontans constructed from the operators above At low temperatures th~s results m coherence (since ~01 ¢ w2) and magnetm correlattons due to interactions 03 through 06 This work was supported in part (B A J) by NSF Grant D M R - 8 3 1 4 7 6 4 We wish to thank E Abrahams, F D Haldane and J W Wllkms for extensive and useful dtscusslons

References [1] E A b r a h a m s and C M Varma, Phys Rev Lett (submatted) [2] K G Wdson, Rev Mod Phys 47 (1975) 773 [3] D M C r a g g and P Lloyd, J Phys C12 (1979) L215 [4] C Jayaprakash, H R K r l s h n a - m u r t h y and J W Wdklns, Phys Rev Lett 47 (1981) 737