Revista Brasileira de Física, Vol. 7, Nº 1 , 1977
Green Function Theory of Random Ferromagnets with Large
Exchange Anisotropy
J: D. P A T TE R SO N *
Behlen Laboratory o f Ph ysics The University o f Nebraska-Lincoln, Lincoln, Nebraska 68588
Recebido em 9 de Fevereiro de 1977
Spin 1/2 systems which a r e coupled w i t h an s i n g - l i k e H a m i l t o n i a n w i t h
f l u c t u a t i n g exchange a r e discussed by t h e use o f thermodynamic
f u n c t i o n s i n f o u r d i f f e r e n t approximations.
Green
The f i r s t i s e q u i v a l e n t t o
a l o c a l mean f i e l d approxirnation and the second i s an approximation t o
t h e f i r s t i n which t h e l o c a l mean f i e l d i s assumed t o be
t o the overal
"bonds".
proportional
m a g n e t i z a t i o n and t o t h e sum o f the nei ghboring exchange
The t h i r d i s a l s o a s p e c i a l case o f t h e f i r s t , b u t t h e l o c a l
mean f i e l d i s approximated i n such a way as t o be a p p r o p r i a t e
d i s c u s s i o n o f s p i n glasses.
approximation (CPA)
.
The f o u r t h i s a coherent p o t e n t i a l
f o r the
-
like
The second approximation can be shown t o imply no
,
r e d u c t i o n o f the C u r i e temperature ( T ~ )due t o exchange f l u c t u a t i o n s
whi l e the CPA does r e s u l t i n a l o w e r i n g o f Tc (assuming small f l u c t u a t ions about an average pos i t i v e exchange i n b o t h cases)
.
However, t h e
CPA t h a t was used i s a l s o an approximation t o t h e second method. Hence,
even though t h e CPA does r e s u l t i n a l o w e r i n g o f
TC (which i s general ly
conceded t o be c o r r e c t f o r t h e o r i g i n a l model), t h i s
used as an argument f o r the v a l i d i t y o f t h e CPA.
fact
cannot
be
Our c a l c u l a t i o n thus
emphasizes t h e necessi t y o f c r i t i c a l l y examining any CPA-l i k e c a l c u l a t i o n b e f o r e a c c e p t i n g i t s p r e d i c t i o n s as v a l i d .
Sistemas de s p i n 1/2 com acoplamento do t i p o I s i n g , no qual a i n t e r a ção de intercâmbio
é
v a r i á v e l , são estudados através de funções de Green
termodinâmicas em q u a t r o d i f e r e n t e s aproximações.
valente
equi
-
-
a aproximação de campo médio l o c a l , enquanto na segunda
o, campo médio é assumido p r o p o r c i o n a l a mag-
aproximação da a n t e r i o r
*
A primei ra é
-
On s a b b a t i c a l leave from the South Dakota School o f Mines &
logy, Rapid C i t y , South Dakota-57701,
USA.
uma
Techno-
n e t i z a ç ã o t o t a l e à soma das constantes de i n t e r a ç ã o com os v i z i n h o s
.
é também um caso e s p e c i a l da p r i m e i r a , mas aqui o campo méd i o l o c a l é aproximado de maneira a r e p r e s e n t a r um v i d r o de s p i n . A
A terceira
q u a r t a é uma aproximação de p o t e n c i a l coerente (CPA)
.
Resulta a segun-
da aproximação em uma não d i m i n u i ç ã o da temperatura de C u r i e ( T ~ ) dev i d o 2s f 1 utuações da i n t e r a ç ã o de t r o c a enquanto que a CPA r e s u l t a em
uma d i m i n u i ç ã o de Tc (supondo-se pequenas f lutuações em
torno
é também uma aproximação do segundo método.
uso da CPA r e s u l t a r em uma redução de T
de uma
u t i 1 i zada
i n t e r a ç ã o média p o s i t i v a em ambos os casos). Todavia, a CPA
Em consequência, apesar do
(o que é geralmente
reconhe-
c i d o como sendo c o r r e t o para o modelo o r i g i n a l ) , esse f a t o não podeser
usado como argumento p a r a a v a l i d a d e da CPA.
Nosso c á l c u l o , assim, en-
f a t i z a a necessidade de se r e a l i z a r um exame c r i t i c o de qualquer
cãl-
c u l o do t i p o CPA, antes de se a c e i t a r suas predições.
1. INTRODUCTION
Since the c l a s s i c paper o f ~ u b a r e v ' , the use o f thermodynamic
Green
f u n c t i o n s i n s o l i d s t a t e physics has i n c r e a s i n g l y grown. I n p a r t i c u l a r ,
the magneti c p r o p e r t i e s
Green f u n c t i o n s .
o f s o l i ds have been e x t e n s i vel y
s t u d i e d by
I n recent years t h e study o f magnetism i n amorphous
m a t e r i a l s has been spurred by the l a r g e t e c h n o l o g i c a l progress t h a t h a s
been made i n connection w i t h metal1 i c g l a s s e s 2 - 5 . At the same time the
t h e o r e t i c a l techniques f o r h a n d l i n g n o n - c r y s t a l l i n e s o l i d s
have
been
im p rovin g 6 , and i t has been found t h a t thermodynamic Green functions i n
connection w i t h 'coherent p o t e n t i a l a p p r o x i m a t i o n s ' can be very useful
f o r c a l c u l a t i n g the p r o p e r t i e s o f s o l i d s w i t h some degree of randomness.
The e x i s t e n c e o f a new type o f phase ( t h e s p i n g l a s s ) has even apparent l y been found (both e x p e r i m e n t a l l y and t h e ~ r e t i c a l l ~
i n) c e r t a i n k i n d s
o f random systems a t s u f f i c i e n t l y low tem p eratures 7 .
this
paper
I n Section 2 of
we d e f i n e o u r model and s e t up t h e b a s i c equations f o r t h e
thermodynamic Green f u n c t i o n s which we w i l l use.
I n Section 3,wesol-
ve these equations i n a simple random phase approximation,
and
t h a t o u r r e s u l t s a r e e q u i v a l e n t t o a l o c a l mean f i e l d theory.
show
In
Sec-
t i o n 4, we make two f u r t h e r approximations, one o f which w i l l be
used
again i n S e c t i o n
5 and the o t h e r w i l l a l l o w us t o make a s h o r t discus-
s i o n o f s p i n glasses.
I n S e c t i o n 5, we s o l v e f o r the m a g n e t i z a t i o n o f
o u r system w i t h i n a random phase coherent p o t e n t i a l - l i k e approximation.
Despite t h e p l a u s i b i l i t y o f t h i s CPA method and the reasonableness
i t s r e s u l t s , o u r use o f a coherent p o t e n t i a l
certain difficulties.
- like
approximation
of
has
I t i s suggested t h a t t h í s c a l c u l a t i o n can serve
as a warning a g a i n s t the u n c r i t i c a l acceptance o f the
predictions
of
such approximations.
2. MODEL AND GREEN FUNCTION FORMALISM
L e t GE(A,B) be t h e energy (E) dêpendent F o u r i e r time t r a n s f o r m
-
doubl e time
o f the
temperature dependent thermodynami c Green f u n c t i o n
(re-
t a r d e d i f I m(E)>O, advanced i f Im(E)<O) a s s o c i a t e d wi t h o p e r a t o r s A and
B, Refs.
1,8.
where <.
. . .>
Then, as i s wel l known, the Green f u n c t i o n s 5 a t i S f y
refers
t o a quantum s t a t i s t i c a l average, H i s the
m i l t o n i a n , a n d [ ~ , ~ ] = A B - BA. Acommonway o f s o l v i n g E q . ( l )
Hais
to
w r i t e GE( E , ~ , B ) i n terms o f G (A,B) by some decoupl i n g aproximationl>?
E
Assuming Eq. ( I ) can be solved f o r GE(A,B), from the general t h e o r y o f
Green f u n c t i o n s we have f o r equal time c o r r e l a t i o n f ~ n c t i o n s ' ~ ~ ,
where the
E*
6 = l/(kT),
l i m i t i s t o be taken a f t e r the i n t e g r a l i s performed and
where k
i s BOItzmann's constant and T i s t h e temperature.
Eqs. ( I ) and (2) a r e t h e b a s i c Green f u n c t i o n s equations
t h a t we wi I 1
need.
The H a m i l t o n i a n
S = 1/2,
describing
i s assumed t o be
the
i n t e r a c t i n g spins,
each
with
spin
where t h e i and
unless
j l a b e l d i f f e r e n t l a t t i c e s i tes and J;
zero
s i tes.
The
wi l 1
p a i r t o n.n. p a i r and except f o r t h e
d i s c u s s i o n o f s p i n glasses we w i l l assume
(
-
7 >O
a n d f o r i and
1- J~I [ J ) <~< 1,~ where 7 r e f e r r t o t h e average o f
(which a r e n o t z e r o ) .
n.n.'s
Jji i s
i and j r e f e r t o n e a r e s t neighbors (n.n.)
be allowed t o f l u c t u a t e from n.n.
pairs
=
For s i m p l i c i t y , we w i l l assume
the
j n.n.
a1 1
Jij
number
of
(2) i s 6 ( c o n s i s t e n t w i t h a simple c u b i c s t r u c t u r e ) . The second
t e m on the r i g h t o f E q . ( 3 ) i s the Zeernan term, where B,, i s the magnet i c f i e l d i n suitable units.
Hamiltonian
(3)
.
The
which
re-
We w i l l a l s o use u n i t s w i t h R = 1
can be viewed as an I s i n g - 1 i k e Hamil t o n i a n
s u l t s from a Heisenberg Hami l t o n i a n i n an a p p r o p r i a t e 1 i m i t o f extreme
exchange a n i s o t r o p y .
I f we l e t
Eq. (1) becomes
where
<F.> i s t h e l o c a l m o l e c u l a r f i e l d associated w i t h
3
i s a l s o w o r t h n o t i c i n g f o r S = 1/2 and i = j t h a t
Note t h a t
It
3. RANDOM PHASE APPROXIMATION
The random phase approximation c o n s i s t s s i m p l y i n assuming
1 O0
site
j.
G (F .S,'SÍ
E 3 3
Using Eqs.
) =
?-
<F .> G (s,s' :)
3
E 3
z
.
(7) and (4) we then f i n d
where E = E
-
B,. Using t h a t the D i r a c d e l t a f u n c t i o n 6 ( x ) can
be
presented as
we then o b t a i n by Eqs. ( 2 ) ,
( 6 ) , (8) and (9) w i t h i = j
Eq. (10) can be r e c a s t i n t o the more f a m i l i a r form
I f Jij
d i d not fluctuate,
t h i s would be e x a c t l y the r e s u l t of Weiss nean
f i e l d t h e o r y f o r a c r y s t a l l i n e s p i n 1/2 ferromagnet.
does,
t h e <S
Zi>
<Fi>.
However s i n c e
it
v a r y from s i t e t o s i t e and so does the l o c a l mean f i e l d
Thus, we see t h a t the random phase approximation j u s t
gives
us
the same r e s u l t as would be o b t a i n e d by a l o c a l mean f i e l d t h e o r y i n the
case o f an I s i n g - l i k e i n t e r a c t i o n .
It i s quite possible that Eq.(ll) i s
n o t a very good r e p r e s e n t a t i o n o f t h e system.
ve computer s t u d i e s on Eq.
that T
C
T h i s i s because i t e r a t i -
(11) w i t h B , = O have l e d t o t h e p r e d i c t i o n g
i s increased by f l u c t u a t i o n s i n J,ji
whereas
more
real i s t i c
as wel 1 as r i g o r o u s r e s u l t s
studies
Jji
1o
suggest t h a t a f l u c t u a t i o n
in
w i l l decrease Tc wi t h respect t o the corresponding c r y s t a l l i n e
system i n which
TC(')
i s determined by J")
=
i
j
7for
iand
A s i m p l e c a l c u l a t i o n based on E q . ( l l ) a n d o n t h e c o r r e l a t i o n
Richards"
,
a l s o r e s u l t e d i n an increase i n 2'
c'
(c)
j n.n.'s.
idea
of
Ref. 12.
4. SPIN GLASSES AND OTHER DECOUPLING APPROXIMATIONS
We now make an even c r u d e r decoupling approximation which i s c o n s i s t e n t
w i t h the type o f decoupling t h a t has been used by T a h i r - K h e l i i n a d i s cussion o f a random bond Hei senberg ferromagnet13.
where M =
- is
We assume
the o v e r a l l m a g n e t i z a t i o n f o r a quenched system
The b a r r e f e r s t o averaging o v e r the Jijis
where each
Jij
( from
I*.
n.n.
p a i r s ) i s randomly determined by some p r o b a b i l i t y d i s t r i b u t i o n which i s
Going through t h e same s o r t o f manipulation which
the same f o r each Jij.
led t o Eq.(ll),
we now c l e a r l y o b t a i n
To f i n d the C u r i e temperature, we assume M i s small (and B , = O ) so
<S.>='LZ
@c
MC
4
l f Z = 6, we f i n d s i n c e
7
i rJ.,.,
3 3
M-tO
.
- J, M = <Sjlz>,
j 'i-
I n t h i s approximation t h e r e i s no l o w e r i n g o f C u r i e temperature due
f l u c t u a t i n g the J . eis i n t h e random system.
J?:"'
23
=
7 f o r T$,
(14)
where T:~)
That i s ,
to
(R) = T ~ ( ' ) ( w i f h
Tc
and yLC) r e f e r t o t h e Curie temperature
o f the random and c r y s t a l l i n e systems, r e s p e c t i v e l y .
Note
that
after
the approximation o f Eq. ( 1 2 ) , we have made no f u r t h e r approximations i n
a r r i v i n g a t t h e r e s u l t o f Eq. (15).
We w i l l r e t u r n t o t h i s l a t e r .
L e t us now a p p l y o u r r e s u l t s t o s i t u a t i o n s where s p i n glass behaviormay
occur.
We assume Jij
can now be e i t h e r p o s i t i v e o r n e g a t i v e b u t 7 2 0 .
We a l s o assume t h e Jijls
a r e d i s t r i b u t e d i n such a f a s h i o n t h a t
magnetism i s t h e o n l y p o s s i b i 1 i t y f o r long range o r d e r (LRO).
s i t u a t i o n , we can then d e f i n e s p i n glasses by r e q u i r i n g two
ferroFor
our
conditions
( w i t h Bo'O):
Eq.
(16a) i m p l i e s t h e r e i s no LRO.
The parameter q
parameter, b u t i s measures l o c a l o r d e r .
For example,
i s a l s o an
order
i f each s p i n were
' f r o z e n ' i n p o s i t i o n a t low temperature i n such a w a y t h a t
it
had
a
p r e f e r r e d d i r e c t i o n which was randomly d i s t r i b u t e d a l o n g e i t h e r * z from
t e t o s i te, then M = O, q # O and we have a s p i n g ass s t a t e .
e a r l y t h e decoup l i n g represented by ~ q . ( 1 2 ) i s n o t a p p r o p r i a t e f o r the
scussion o f s p i n glasses, f o r t h i s decoupling i m p l i es i f B =O, M = O ,
then <S
Zi>
= O and hence
q = O.
We need somehow t o i n c o r p o r a t e the i dea
<F.>
can be l o c a l l y non zero even thoughM= 0.
L
'
I n p a r t i c u l a r we might expect t h a t i f q # 0, then <F.> # 0,
since q ,
L
'
t h a t the molecular f i e l d
i n some sense, measures l o c a l o r d e r .
To f o r m a l i z e t h i s , we use
coupl i n g 1 i k e Eq. ( 7 ) , b u t we assume t h a t the
a
de-
<F .> i n Eq. (7) can be
re-
p l aced by
3
where
1
J . = - 1 J.,.
3
Z i ' 3 3
.
The f i r s t term i s e x a c t l y t h e m l e c u l a r f i e l d t h a t we would have f o r
c r y s t a l l i n e s y s t e m w i t h Jij
-
J f o r a l l n.n.
=
pairs.
a
term
The second
6
as
vanishes i f J = 5 , as i t should, and i t i s a l s o p r o p o r t i o n a l t o
j
we would expect on dimensional c o n s i d e r a t i o n s alone ( q = M~ f o r a c r y s The Z f a c t o r comes i n because
t a l l i n e f e r r o m a g n e t i c system).
ZM
expect the
t i p l i e s (J
i
we would
i n the f i r s t term t o be analogous t o t h e f a c t o r whi ch mul-
- J) i n
t h e second.
l n f a c t , a rnethod formal l y
t o assumption (17) has been developed by ~ l e i n " .
equivalent
Thus f o r the d e t a i l s
o f developing o u r assumptions i n a more mathematical way, r e f e r e n c e can
be made t o K l e i n ' s paper.
t h a t l e d t o Eq.
Using Eqs.
(7) and ( 1 7 ) , and the same
ideas
( I I ) , we now o b t a i n ( w i t h B , = O),
I f we assume t h e J
ij
' s a r e independently d i s t r i b u t e d by a Gaussian d i s -
t r i b u t i o n w i t h w i d t h AJ, then so a r e t h e J . ' s b u t w i t h w i d t h
3
we use f o r a p r o b a b i l i t y d e n s i t y
AJ/>/Z.
If
then
and
We do n o t c a r r y t h i s c a l c u l a t i o n any f u r t h e r , because Eqs. (21) and (22)
(wi t h Eqs. (19) and (20)) a r e formal l y t h e same Eqs.
as
d e r i ved
by
S h e r r i n g t o n and K i r k p a t r i c k 1 6 ' " and i n a r i g o r o u s f a s h i o n ( f o r Z-im)
by
M o r i t a and H o r i g u c h i l * .
of
K l e i n shows, however, t h a t the same
i g h t l y d i f f e r e n t s p e c i f i c heat
procedure leads t o a SI
S h e r r i n g t o n and K i r k p a t r i c k .
f rom
For low enough ternperatures and
type
that
of
AJ /(Cf )
l a r g e enough Eqs. (21) and (22) p r e d i c t
spin glass state.
l 6 ' I=7
O and
~
q # O,
i.e.,
A c t u a l l y t h e r e a r e s t i l l some d i f f i c u l t i e s i n
sent t h e o r i e s o f the s p i n glass s t a t e when they a r e compared
a v a i l a b l e experimental r e , s u l t s l g . Eqs.
( 2 1 ) , and (22) a r e
the
pre-
with
surely
a11
not
the f i n a l and b e s t s o l u t i o n f o r M and q. Mention here should a l s o bymade
'
o f the random s i t e s p i n glass c a l c u l a t i o n o f ~ u t t i n ~ e ,r ~and
of
the
-
random s i t e Green f u n c t i o n CPA c a l c u l a t i o n o f ~ a h i r - K h e l i 2 !
random bond
Both authors o b t a i n the s p i n g l a s s s t a t e f o r a p p r o p r i a t e values of t h e
pararneters governing t h e i r systems.
I t i s also worthwhile t o note t h a t
annealed systems ( i n which the thermodynamics i s determined by averaging
( o v e r the J
ij
' s ) t h e p a r t i t i o n f u n c t i o n r a t h e r than averaging the t h e r -
modynamic f u n c t i o n s d i r e c t l y ) a r e n o t expected t o show s p i n g l a s s behav i o r a t a1 l Z 2 .
5. A COHERENT POTENTIAL APPROXIMATION
We now r e w r i t e Eq. (8) wi t h
i=j
SI)
and use the decoupl i n g
approximation
C.
(12) t o o b t a i n w i t h G ~ ( s ~ ,
2'
where
We now e v a l u a t e
zi (and hence M = <-
by Eqs. (2) and (6)) f o l lowing a
coherent p o t e n t i a l - l i k e approximation which i s somewhat s i m i l a r
c a l c u l a t i o n o f Tahi r-Khel i 13. L e t
A. =
2
Then, we can w r i t e Eq.
(23) as
71
Giz'
.
to
a
-
We i n t r o d u c e a
$
&
,
independent C
t o be d e t e r m i n e d l a t e r , and
define
-
= E- C
g-l
(27a)
and
(26) can be w r i t t e n as
Then Eq.
G.=gA
J
We i n t r o d u c e a Ti d e f i n e d s o
i
+ g V G
ii'
,
V.G = T . g A
3 j
3
j J
and t h u s Eq.
(28) becomes
We d e t e r m i n e C so t h a t
-
T.A. = 0
'L 2.
,
and hence t h e d e s i r e d average Green f u n c t i o n i s
The p r o b l e m i s s o l v e d , once we e v a l u a t e C f r o m Eq.
te13
by
Note t h a t
Ti
Ãi and t h u s we d e t e r m i n e C tiy
TiAi = T . A . i s t h e o n l y f u r t h e r ? i p p r o x i m a t i o n we have
' L ? ,
Thus any d e v i a t i o n o f o u r r e s u l t s f r o m Eq.
f o l l o w s e x a c t l y from Eq.
T
..
'L
approxima-
- - -
beyond Eq, ( 1 2 ) .
tion.
( 3 1 ) . We
( 1 2 ) , must be due t o t h i s f u r t h e r
I n order t o apply Eq.
By E q .
( 3 0 ) , we have
made
(ls),which
approxima-
( 3 3 ) , we need c c o n v e n i e n t e x p r e s s i o n
for
V.G. = g V A
3 3
3 j
Us i ng Eq.
(27b), Eq. (33) becomes
where g =
(2-
C)-'.
+
V. g TA. g
3
3 3
Eqs. (32) and (34) a r e t h e b a s i c equation
of
our
coherent p o t e n t i a l approxirnation.
To go f u r t h e r , we must assume a p r o b a b i l i t y d i s t r i b u t i o n f o r the Jijls.
We assume, f o r
Z
=
6 and f o r n.n.
J
ij
=?+c5
and
Zi
where
p i s the probabi 1 i t y .
wi 1 1 be assumed.
--
= J
interactions,
that
w i t h p = 1/2,
6 withp
= 1/2
(35a)
,
(3%)
Also, we have i n mind t h a t
Equat i o n (34)
7>0
and l a t e r
then becomes
Although t h e d e t a i l s a r e n o t p a r t i c u l a r l y e d i f y i n g ,
i t turns
out
that
1O 7
i s n o t too d i f f i c u l t t o s o l v e f o r g (where C = E
Eq.(37)
then be o b t a i ned f rom Eq.
.
g-l);
Then u s i ng Eqs. ( 2 ) , (6) , and
G3.
(9)
can
and
we f i n a l l y o b t a i n ( w i t h B o = 0 ) :
solving f o r M =
Assuming
(32)
-
M i s smal 1 (T near TLR)) and A
6/7 << 1,
from Eq.
we o b t a i n
where
and where
Eq. (40) should be an approximation t o Eq. (15). l t i s n o t ,
i n the sense
t h a t i t gives t h e a p p a r e n t l y p l e a s i n g r e s u l t t h a t small A > 0 i m p l i e s t h e
Curie temperature i s lowered.
Thus o n l y i f we b e l i e v e t h a t the
taneous appl i c a t i o n s o f the approximation o f Eq. (12) and the
-
mation T A
i i
E
-T.A.
2 2
simulapproxi-
i s somehow b e t t e r than t h e approximation o f Eq.
alone, can we accept Eqs. (39) and ( 4 0 ) as r e l i a b l e
1 i k e l y , t h e p l e a s i n g r e s u l t o f Eq.
cellation o f errors.
predi ctions.
(40) r e s u l t s from a f o r t u i t o u s
I n f a c t , a p p r o x i m a t i ~ n ss~i m
~ ilar to
those
(12)
More
canwe
have made (and which a l s o r e s u l t i n a l o w e r i n g o f the C u r i e temperature)
a r e known t o be u n s a t i s f a c t o r y i n o t h e r respects
24.
Our demonstration
o f i n c o n s i s t e n c y appears, however, t o be p a r t i c u l a r l y c l e a r . We are n o t
c l a i m i n g the CPA i n general i s n o t u s e f u l .
We a r e o n l y n o t i n g t h a t i t s
s t r a i g h t f o r w a r d appl i c a t i o n can l e a d t o i n c o n s i s t e n t p r e d i c t i o n s . To i m prove o u r c a l c u l a t i o n we would have t o improve on o u r decoupling approx i m a t i o n and/or t r y t o improve on t h e approximation (Eq. (33)) used
d e t e r m i n i n g C.
T h i s may l e a d t o v e r y deep waters,
when one looks c r i t i c a l l y ,
but
for
unfortunately
t h e s i m p l e s t , most s t r a i g h t f o r w a r d
calcula-
t i o n , even when i t leads t o reasonable r e s u l t s , may n o t be a c c u r a t e
to
t h e o r d e r necessary t o p r e d i c t these r e s u l t s w i t h confidence.
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