THE KONDO LATTICE : A THEORETICAL MODEL FOR Tm Se ?

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THE KONDO LATTICE : A THEORETICAL MODEL

FOR Tm Se ?

R. Jullien, P. Pfeuty, A. Bhattacharjee, B. Coqblin

To cite this version:

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JOURNAL DE PHYSIQUE

Colloque

C5,

supplkment

au

n

O 6 ,

Tome

41,

juin

1980, page C5-33 1

THE

KONDO

LATTICE

: A THEORETICAL MODEL FOR Tin

Se

?

R. Jullien, P. Pfeuty, A.K. Bhattacharjee, B. Coqblin

L a b o r a t o i r e d e P h y s i q u e d e s S o l i d e s B d t i r n e n t 5 2 0 U n i v e r s i t & de P a r i s - S u d

9 1 4 0 5 O r s a y ( F r a n c e )

R6sum6

-

En utilisant une mgthode du groupe de renormalisation dans l'es~ace

reel,

nous montrons que le fondamental de ltHamiltonien

du r6seau Kondo

a

une dimension

est isolant quand il y a juste un 6lectron de conduction par site. Nous discutons

ensuite la possibilit6 d'appliquer ce modsle

5 Tm

Se

a

basses temp6ratures.

Abstract

-

By use of a real space renornalization group method, we show that the

ground state of the one dimension Kondo lattice hamiltonian is insulation whex there

is just one conduction electron per site. Then we discuss the possibility to ap?ly

the model to Tm Se at low temperatures.

The "Kondo lattice" is the extension

of the Kondo impurity model (i.e. a single

impurity spin interacting antiferro-magne-

tically with the conduction electron spins)

to the concentrated Qse of one spin per

site of the lattice.

The theoretical investigation of the

Kondo lattice has been firstly developed

/1/ to explain the low temperature proper-

ties of Ce A12 and Ce A13 by emphasizing the

competition between magnetic order (due to

R.K.K.Y. interactions) and Kondo effect.

But, another interesting property of the

Kondo lattice is that, when the conduction

band is just half-filled, one can obtain an

insulating ground state.

This will be shown here by a real spa-

ce renormalization group study of the one

dimension (1 D) Kondo lattice hamiltonian

:

nerate and characteristic of a metal, while,

in the atomic limit

J/FT-+~,

the conduction

electrons must localize to form a s ~ i n

sin-

glet state on each site leading to a singlet

ground state for the d o l e chain characte-

ristic of an insulator. So, we expect a

change in the ground state by increasing

J/VJ

from a degenerate band state to a singlet

insulating state.

We firstly transform the Hamiltonian

(1) into a three spin chain Hamiltonian

through the Wigner-Jordan transformation

:

which introduces two sets of pseudo 1/2

-

+

spins

Zi+

and

zi4.

SO, Hamiltonian

(1)

trans-

%

=J

r

( 2

(c:+c~+s;

+

c;+ci+si)

+

(ci+cit-

_{forns into}

_:

i

which considers a set of 1/2 spins regular-

W

z

+

-

Z

+

+ 7

-

ly spaced on an infinite chain and interac-

₊

₃

_~ ₍ _T _~ ₊ _~ _~ ₊ _~ _~ ₊ _~ ₊ ₊ _T _~ ₊ _~ _~ ₊ _~ _~ ₊ _~ _* _T _~ _~ _T _~ ₊ _~ ₊ _T _~ ₊ _~ ₊ ₊

ting antiferromagnetically, on each site,

-

z

+

with the co'nduction electrons spins.

J

is

Ti+Ti+l+Ti+l+)

the strengh of the Kondo coupling,

W

the

half-width of the band (when there is no

Kondo coupling) and we restrict ourselves

to the case of a half-filled conduction

band

( E = ~ 0)

.

When J

= 0,

the ground state is dege-

where the band part is transformed into two

coupled X-Y chains. Different mean-field

calculations have been described elsewhere

/2/

/3/

and they all yield a metal-insulator

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JOURNAL DE PHYSIQUE

t r a n s i t i o n a t a f i n i t e J / W v a l u e ; however, t h e s e c a l c u l a t i o n s a p p e a r t o b e d o u b t f u l s i n c e t h e o b t a i n e d r e s u l t s a r e n o t e x a c t l y t h e same f o r t h e d i f f e r e n t used methods / 3 / .

Thus, i t i s n e c e s s a r y t o go beyond mean-field c a l c u l a t i o n s and we p r e s e n t h e r e a r e n o r m a l i z a t i o n group s t u d y o f h a m i l t o n i a n ( 3 ) by a p p l y i n g a r e c e n t method / 4 / . L e t u s f i r s t r e w r i t e t h e h a m i l t o n i a n i n t o t h e e i g h t s t a t e b a s e which d i a g o n a l i z e s t h e Kondo i n t e r a c t i o n on e a c h s i t e . The e i g h t s t a t e s can be c l a s s i f i e d a c c o r d i n g t o t h e good quantum numbers : t h e number o f c o n d u c t i o n e l e c t r o n Ni and t h e z - p r o j e c t i o n o f t h e t o t a l s p i n I:, on s i t e i : z l z z 0 ;

zi

=

SX

+

2

( T ~ + - T . ) 2 1) ( 4 We f i n d a s i n g l e t - t r i p l e t system formed by t h e one p a r t i c l e s t a t e s and a q u a d r u p l e t formed w i t h i n t h e z e r o and two p a r t i c l e s t a - t e s . Using t h i s b a s e , h a m i l t o n i a n ( 3 ) t a k e s t h e g e n e r a l form :

( 5 )

where Di i s a 8 X 8 d i a g o n a l m a t r i x and where Air, and BiIo a r e 8 X 8 m a t r i c e s r e - p r e s e n t i n g r e s p e c t i v e l y

g

T z

T t

and

,+

,

i n t h e 8 s t a t e baze

s i % g

i ( t h e

2 iis

t i l d a s i n ( 5 ) d e n o t e t h e t r a n s p o s e d m a t r i c e s ) The method i s an i t e r a t i v e and appro- a m a t e c o n s t r u c t i o n o f t h e ground s t a t e of t h e c h a i n . We p r o c e e d a s f o l l o w s : ( i ) We f i r s t c u t t h e c h a i n i n t o a d j a c e n t b l o c k s o f ns s i t e s (ns

w i l l

be t a k e n e q u a l t o 2 , 3 , 4 i n t h e f o l l o w i n g ) and we d i a g o n a l i z e e x a c t l y a b l o c k h a m i l t o n i a n by u s i n g t h e good quantum numbers :

(4)

would o c c u r i n 3D.

S i n c e t h e Kondo l a t t i c e Hamiltonian shows a n i n s u l a t i n g phase a t T = 0 f o r 1D

( a n d a l s o f o r 3D a t l e a s t f o r a s u f f i c i e n t - l y l a r g e Kondo c o u p l i n g c o n s t a n t ) and s i n c e t h e "normal" one i m p u r i t y Kondo e f f e c t h a s been a l r e a d y invoked t o e x p l a i n t h e r e s i s - t i v i t y of Tm Se i n t h e paramagnetic m e t a l l i c phase / 8 / , i t i s t e m p t i n g t o a p p l y t h e p r e s e n t model t o Tm Se a t v e r y low tempera- t u r e s i n t h e a n t i f e r r o m a g n e t i c phase where t h e r e s i s t i v i t y i n c r e a s e s d r a s t i c a l l y w i t h d e c r e a s i n g t e m p e r a t u r e . I n s u p p o r t o f t h i s , t h e i n s u l a t i n g phase i s t h e o r e t i c a l l y o b t a i - ned when t h e c o n d u c t i o n band i s j u s t h a l f - f i l l e d , which c o r r e s p o n d s t o t h e s t r o n g e x p e r i m e n t a l dependence of t h e r e s i s t i v i t y o f Tm Se on s t o i c h i o m e t r y /9/. Moreover, e x t e n s i o n a t f i n i t e t e m p e r a t u r e s o f t h e p r e s e n t c a l c u l a t i o n would c e r t a i n l y y i e l d an i n s u l a t o r - m e t a l t r a n s i t i o n , a s a p p a r e n t - l y o b s e r v e d a t t h e Nee1 t e m p e r a t u r e . However, t h e i n s u l a t i n g phase i s t h e o r e t i c a l l y found h e r e t o be non m a g n e t i c , w h i l e Tm Se i s e x p e r i m e n t a l l y o r d e r e d below 35.K. But we c o u l d c e r t a i n l y o b t a i n a magne- t i c i n s u l a t i n g phase by c o n s i d e r i n g t h e c a s e o f l a r g e r s p i n s (S > 1 / 2 ) . Thus, we c o u l d q u a l i t a t i v e l y a c c o u n t f o r t h e i n s u l a t i n g phase o b t a i n e d f o r a p e r f e c t l y s t o i c h i o m e t r i c Tm Se compound i n t h e framework of t h e Kondo l a t t i c e Hamilto- n i a n which d e a l s w i t h w e l l l o c a l i z e d 4f e l e c t r o n s . W e can however a r g u e t h a t t h i s H a m i l t o n i a n i s c e r t a i n l y l e s s a p p r o p r i a t e t h a n t h e Anderson one t o d e s c r i b e t h e mixed v a l e n c e compound Tm Se. But, when t h e t o t a l number of c o n d u c t i o n and 4f e l e c t r o n s i s an i n t e g e r , t h e Anderson l a t t i c e Hamiltonian y i e l d s a l s o a gap which has c e r t a i n l y t h e same p h y s i c a l , o r i g i n a s t h a t d e s c r i b e d h e r e . An a l t e r n a t i v e model o f Tm Se b a s e d on a Hartree-Fock t r e a t m e n t of t h e more g e n e r a l Anderson l a t t i c e Hamiltonian i s d e s c r i b e d e l s e w h e r e /lo/. R e f e r e n c e s /1/

-

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Phys. Rev.

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(1978) 4889 and r e f e r e n c e s t h e r e i n . / 2 /

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