A new renormalization approach to the coherent state in the Kondo lattice

~

Solid State Communications, Vol.61,No.12, pp.775-777, 1987. Printed in Great Britain.

0038-1098/87 $3.00 + .00 ©1987 Pergamon Journals Ltd.

A New R e n o r m a l i z a t i o n Approach to the Coherent S t a t e in the Rondo L a t t i c e

Chen Chang-feng and Department of Physics,

Zhang Li-yuan Peking University,

( R e c e i v e d 18 December 1986 by V.Y.

Beijing,

China

Ruan)

We develop a new theoretical approach to study the coherent state in the Rondo lattice. A r e n o r m a l i z a t i o n operator is introduced to reflect the many-body interactions. A characteristic temperature, below which the system enters a coherent state (i.e., h e a v y - f e r m l o n state), is obtained, and this is in agreement with the recent theoretical result derived from a different method.

The study of h e a v y - f e r m i o n system has recently been an evolving and promising field which attracts much attention in both experimental [I] and theoretical [2,3] activities. An essential problem is concerned with the formation of the h e a v y - f e r m i o n state in these systems, many of which can be called Rondo lattices. We have proposed [4-6] a model to discuss the formation of the heavy q u a s i p a r t i c l e s from conduction and f electrons, the effective mass enhancement and the s u p e r c o n d u c t i v i t y in the h e a v y - f e r m i o n systems, where the periodic Anderson Hamiltonian is treated in a mean-field theory and the renormalization effect is taken into account in an a p p r o x i m a t e way [ 5 - 7 ] . In t h i s comm u n i c a t i o n we d e v e l o p a new r e n o r m a I i z a t i o n approach to deal w i t h the format i o n of the c o h e r e n t s t a t e in the Rondo lattice and o b t a i n a characteristic t e m p e r a t u r e , T$, below which the system is in a c o h e r e n t s t a t e , i . e . the heavyfermion s t a t e . The r e l a t i o n of T* w i t h s i n g l e i m p u r i t y Rondo t e m p e r a t u r e and the p r o p e r t i e s of the system below T* is b r i e f l y discussed. Our r e s u l t s are in agreement w i t h those o b t a i n e d by Tesanovic and V a l l s ( T V ) [ 8 ] , which are developed in a d i f f e r e n t approach.

After a s t r a i g h t f o r w a r d performance [4,5] we obtain the Hamiltonian in the mean-field theory with renormalization effect taken into account

H=~e(k)C~¢ C~

(2)

where E 3 : ~ 3 -÷ T and T : ~ < n t ~ > . The par a m e t e r s are r e n o r m a l i z e d in the f o l l o w i n g way: E ~E /R, h=h/R , where R= [(I-~Z/~E)5= o ]" , Z is the self-energy of f electrons in the single-sure approximation [7]. I is, in general, a function of temperature[5]. We can introduce an operator ~ to represent the r e n o r m a l i z a t i o n process in the Rondo lattices with the understanding that means thermal average. For an operator Amwhich contains pf and qf (p and q are the numbers of f and f+ , and p*q=n) operator(s), we define A

:

(3)

Eq. (3) i m p l i e s t h a t e v e r y f or f+ oper a t o r associates with a factor R- ~ . This is easy to understand because the renormaIization effect results from conduction-f or c o n d u c t i o n - e l e c t r o n - i n duced f-f interactions and hence can be naturally associated with f-operators in the Hamiltonian. It ls~more direct and convenient to rewrite R as

As suggested in p r e v i o u s papers [ 4 - 6 ] we d e s c r i b e the h e a v y - f e r m i o n systems (Rondo l a t t i c e s ) with the periodic Anderson U a m i l t o n i a n

: where 6(k) a n d . ~ are one-electron energy of conduction and f electrons respectively, both measured from the Fermi energy, U is the Coulomb c o r r e l a t i o n energy, h the h y b r i d i z a t i o n potential. C~ and ~ are the c r e a t i o n opera~ors f o r c o n d u c t i o n and f e l e c t r e n s , i , k, are the s i t e , w a v e v e c t o r and spin i n d exes r e s p e c t i v e l y . The o r b i t a l degeneracy is neglected for s i m p l i c i t y .

n*n

(4)

and i n t r o d u c e the f o l l o w i n g mapping f

.......

> nf

(5) f*

....... > n , f *

to represent the renormalized Hamiltoalan and meantime m a i n t a i n the paramet e r s n o n - r e n o r m a l i z e d . In t h i s way Eq.

775

776

A NEWRENORMALIZATIONAPPROACH TO THE COHERENT STATE

(2) can be rewritten

r.%

Ho = ~E (k)

C*

H, = h~(n*Ci f" io" i~¢

*

as

T* = 1.13Dexp[-'l~fl/p]

f.

f,.

(6)

* ZRlf/~ fie i# w h e r e "Ho i s t h e H a m i l t o n i a n for a "noninteracting lattice" ---a conduction band plus renormalized periodically arranged f-levels, ~i t h e i n t e r a c t i o n s in the lattice. Physically, one has nf=fn and n~f~.=f~n~. This is the direct result of the fact that.the operator n*(n) will give a factor R-~only if the elgenvalue of f~(f) is nonzero, but no matter f*(f) or n~(n) acts on the eigenstate first. Thus both n~ and n are boson operators. It should be noted that each of them contributes a factor R under thermal average. To determine R we introduce the restriction that the interaction associated with H~ has a stationary point, i.e., ~ n / ~ =0, where

+HC)

is the time variable picture,

and

in

Heisenberg

this yields

R z l(T)O= _hTFZ=nA~(~', ,~. )

(7)

w h e r e b3~ i s the Matsubara frequency, °, w h i c h i s o f o r d e r u n i t y in our model [9], is the f electron occupancy number of the nonrenormallzed system, and A e i s t h e a n o m a l o u s p r o p a g a t o r which i s g i v e n by

and A* is the Hermit conjugation It is=easy to find

From Eqs,

(7)

and

(9)

~-

~_

of A6.

one has

/ I(T)'=h=T

Vol. 61, No. 12

.

.

'_~= (I0)

It has been found that heavy-fermion systems undergo a transition to a state in which they have a v e r y large effective mass enhancement [1]. This can be understood [4,8] that these systems enter a coherent state where E~ is very small. ~e introduce a characteristic temperature T~ at which one has I= -el. By neglecting relatively small h = term in Eq. (10) and taking the sum in the usual way one can obtain

(11)

where r = P ( O ) h =, O is the width of f e a t u r e l e s s conduction band, and )0(0) is the density of states of conduction electrons at the Fermi energy. Eq.(11) gives a characteristic tempeEature which is, in the limiting case '=l, the same as TMF obtained by TV [8] with a different method.

It is interesting to note, as also mentioned by TV [8], that T* takes the same form (assuming " = 1 in our case) as that of impurity Kondo temperature TK, except an unimportant numerical coefficient. This is in agreement with the experimental observation that K o n d o - l a t t i c e systems undergo a transition into h e a v y - f e r m i o n state at temperatures near ~ [I]. It has been shown [10] that the low-temperature electrical resistivity of the Kondo lattices can be understood in such a picture. It is also s t r a i g h t f o r w a r d to find that the Fermi veolocity is greatly reduced from its conduction electron value v~as vF =v~ /(m~/m)<>1. This has been well investigated in experiments

[113.

In conclusion, we have shown in this communication a new theoretical approach to study the coherent state in the Kondo lattice, and consequently derived an explicit expression for the coherent temperature below which the system enters the h e a v y - f e r m i o n state. The key point in this approach is the introdu~tlon of a r e n o r m a l i z a t i o n operator, ~, which raises the bare f elec tron levels towards the Fermi energy and meantime leads the on-site Coulomb interaction and f electron occupancy number behave in such a manner that the system has a very large effective mass enhancement [12] and enters a coherent state below a c h a r a c t e r i s t i c temperature. It is worthy to further investigatethe properties of this coherent state and compare with the experiments of h e a v y - f e r m i o n materials. Also it requires more effort to get a better understanding about the nature of the coherent state.

REFERENCES

[1]

See, G.R. Stewart, Rev. Hod. Phys. 5__6, 755 (1984). This field is growing very rapidly, however.

[2]

C.H. Phys.

[3]

P.A. L.J.

Varma, Comments 11, 221 (1986).

on Sol.

ments on Solid State Phys.

99

[4]

Chen Chang-feng and Zhang Li-yuan, J. Phys. Chem. Solids 47, 547 (1986).

[5]

Zhang L i - y u a n , ICTP, T r i e s t e ,

St.

Lee, T.H. Rice, J.W. Serene, Sham, and J.W. Wilkins, Com-

12,

(1986).

preprint, 1986.

IC/86/177

Vol. 61, No. 12

A NEW RENORMALIZATION APPROACH TO THE COHERENT STATE

[6]

Chen C h a n g - f e n s and Zhang L i - y u a n , ACTA PHYSICA SINICA ( i n p r e s s ) .

[7]

A.

Is] [9]

Y o s h i m o r i and H. K a s a i , J . M a g n . Hash. M a t e r . 3 1 , 475 ( 1 9 8 3 ) ; N. T a c h i k i and S. H a e k a v a , P h y s . Rev. B29, 2497 ( 1 9 8 4 ) . Z. T e s a n o v i c and O.T. V a l l s , Rev. B34, 1918 ( 1 9 8 6 ) .

[10]

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and to

appear.

[11]

Phys.

Chen C h a n g - f e n s =,,d Z h a n s L i - y u a n , C h i n e s e P h y s . L e t t . ~ , 177 ( 1 9 8 5 ) ; Chen C h a n g - f e n s , unpublished.

777

[12]

U. R a u c h s e h w a l b e , W. L i e k e , C. O. B r e d l , F. $ t e s l l c h , J. Aarts, K. M. M a r t i n i , and A.C. Mota, Phys. Rev. L e t t . 49, 1448 ( 1 9 8 2 ) . These points in R e f . [ 5 ] .

have also

been argued