Revisa Brasileira de Flsica, Vol. 6, N? 3, 1976
Ising Chain with Two Random Components in Zero Field
L.C. DE MENEZES*and G.M. OBERMAIR
FB. Physik, Universitat Regensburg, Germany
Recebido em 28 de Junho de 1976.
An exact, simple, c l o s e d form,
r e s u l t f o r the f r e e energy o f a two com-
ponent random I s i n g chain, w i t h nearest neighbour i n t e r a c t i o n s , i n
the
two p h y s i c a l l y i n t e r e s t i n g cases o f the completely quenched and complet e l y annealing system f o r any c o n c e n t r a t i o n o f the components,
sented.
pre-
is
Systems w i t h d i l u t e magnetic components a r e p a r t i c u l a r
cases
o f our r e s u l t s .
Apresenta-se um r e s u l t a exato, simples, em form fechada, para a energia
l i v r e de uma cadeia de I s i n g a l e a t ó r i a , a duas componentes, com i n t e r a çÕes e n t r e v i z i n h o s mais próximos.
Isso
é f e i t o nos d o i s casos f i s i c a fi-
mente i n t e r e s s a n t e s , v a l e d i z e r , em o caso de impurezas totalmente
xas ou totalmente móveis, para qualquer concentração dos
componentes
.
Sistemas com componentes magnéticos d i l u i d o s aparecem como casos p a r t i c u l a r e s de nossos resultados.
1. INTRODUCTION
Because o f the growing i n t e r e s t i n t h e e f f e c t s o f random d i s o r d e r o n t h e
model
p r o p e r t i e s o f condensed matter, we consider once more the simple
of a l i n e a r I s i n g c h a i n o f two p h y s i c a l l y d i s t i n c t s p i n 1/2 components,
A and B, randomly
d i s t r i b u t e d over the c h a i n s i t e s ,
according t o neighbourhood,
t h r e e d i f f e r e n t exchange constants,
JBB
and JAB
, between nearest neighbours.
*
which take
on,
JAA ,
Present address: I n s t i t u t o de ~ f s i c a , Universidade de São Paul0,C.P.
20516, 01000
-
São Paulo SP
I n t h i s paper, o n l y t h e case o f z e r o rnagnetic f i e l d w i l l be c o n s i d e r e d .
The r e s u l t s o f p r e v i o u s a u t h o r s (Refs. 1-5) a r e sirnpl i f i e d and g e n e r a l i zed t o a r b i t r a r y c o n c e n t r a t i o n s , cA, c B = 1
- cA '
to yield
the
free
energy i n t h e two cases:
( i ) f i x e d ( o r "f rozen")
,
t e m p e r a t u r e independent, d i s t r i b u t i o n
of
the
in
the
components, c o r r e s p o n d i n g t o a quenched a l l o y .
( i i ) Completely m o b i l e cornponents, f r e e t o a r r a n g e thernselves
rninimurn f r e e energy c o n f i g u r a t i o n s , a t any ternperature,
corresponding t o
f r e e anneal i n g .
I n what f o l lows,
t h e two cases w i I 1 be r e f e r r e d t o as
the
"quenched"
and " anneal ing" cases, r e s p e c t i v e l y .
2. FORMULATION OF THE PROBLEM, PARTITION FUNCTION OF A GIVEN
CONFIGURATION
L e t us assume an open c h a i n o f N I s i n g s p i n s , a
i'
o f which
m
are
of
and c B = k i n d A, n = N - rn 5 m o f k i n d B, so t h a t c A =
In this
N '
one d i m e n s i o n a l rnodel, t h e word "sequence" can and w i l l be used t o den o t e a g i v e n component c o n f i g u r a t i o n ( s ) ,
( AAA
...
ABB
n o t t o be confused w i t h t h e s p i n c o n f i g u r a t i o n s (o) ,
+H
. . .) , w h i c h e x i s t s
...
(44
BBA
...
...
W
i n d e p e n d e n t l y and s irnul taneousl y w i t h any
)
,
. ..
given
cornponent c o n f i g u r a t i o n .
The energy,
f o r a g i v e n sequence ( s ) , i s
t a k e s on t h e v a l u e r J ~ J~~
~ and
, J
= +Iand J 's)
i
i,i+l
t
o
t
h
e
o
c
c
u
p
a
t
i
o
n
o
f
s
i
t
e
s
i
and
i
+
l
i
n
(
s
)
ding
where a
.
The p a r t i t i o n f u n c t i o n f o r t h i s sequence i s
3 98
c c o,r ~ a~
transfer
which, i n standard fashion, can be expressed i n terms of the
m a t r i x , Pi
J
i+,
(see Note
with
where
=
1is
6r,
K i,
i +1
,
as
so t h a t
the u n i t m a t r i x , and
x
0
the Paul i m a t r i x .
From ( h ) ,
is
it
e v i d e n t t h a t the t h r e e d i f f e r e n t P ' s t h a t can occur i n t h e product
( 3 ) , PAA, PBB and PAB = PBA, a l 1 commute, and can t h e r e f o r e
taneous 1 y d i agona 1 i r e d t o g i ve fhe e i genva luer
and
~j:i+~
= 2 ~inhI$;+~.
\z ,(;i,
= 2
be
C O S ~
in
simul-
Ki,i +
1
I n o t h e r words, Z'S) does n o t depend on any o t h e r f e a t u r e o f ( s ) b u t rat h e r on the number o f times i t c o n t a i n s neighbour p a i r s of type AA, AB,
BA and BB, i .e.,
on t h e number r ( s ) o f AB boundaries i n t h e
sequence
.
-
r
T h i s y i e l d s r f a c t o r s PAB, P f a c t o r s P B A (see Note i ' ) ,l e a v i n g
f a c t o r s PAA and
Osr<nm,
-
n
r
factors
PBB,f o r any such r-sequence.
so t h a t f o r r = O one has a sing3e B-domain a t the
s i n g l e A-domain ( .
f rom each o t h e r ,
rn
Of
left
of
a
. . .BBBB/AAA.. . .), w h i l e f o r r = n a l l B ' s a r e i s o l a t e d
.
. . . . . .A/B/AA.. .. . . . .AA/B/A.. .. . .. .) .This
(. .AA/B/AAA..
independence o f the d e t a i l s , o f an r- sequence, f o r o b t a i n i n g the
t i t i o n f u n c t i o n i s the key f e a t u r e o f the l i n e a r chain
f r e e case.
course
With t h a t
in
the
parfield
= 2'
coshnrr
(compare Note 7 t o Eq. (3).
KA coshn->. K
B
cosh2'
K~~
)
3. QUENCHED RANDOM SYSTEM
Now, knowing the p a r t i t i o n f u n c t i o n of any g i v e n sequence,
f i r s t the quenched case.
we
What c h a r a c t e r i z e s quenched randorn
treat
disorder
i s t h a t t h e f r e e energy o f the i n f i n i t e system i s t h e a r i t h m e t i c
mean
o f the f r e e energies o f a11 p o s s i b l e , d i s t i n g u i s h a b l e , sequences (s) o f
length
N, N
-+ w ,
I n o t h e r words,
i t s p a r t i t i o n function i s thegemetric
mean o f the p a r t i t i o n f u n c t i o n s o f these sequences.
I f we c a l 1 M(s) t h e number o f a1 1 those d i f f e r e n t sequences,
we
have,
therefore,
There a r e as many d i f f e r e n t sequences as d i f f e r e n t permutations o f
m i n d i s t i n g u i s h a b l e A's,
the
and t h e n í n d i s t í n g u í s h a b l e B's, so t h a t
According t o ( 5 ) , a1 1 sequences ( s ) , wi t h t h e sarne r ( s ) , a r e degenerate
w i t h respect t o t h e p a r t i t i o n f u n c t i o n .
( 6 ) i s t h e the nurnber N ( r )
me number r o f
AB boundaries,
evaluate
t o obtain:
N ( r ) i s e a s i 1y found as f o l lows.
- i n s e r t s ) between t h e m A's;
Thus, a11 we need t o
o f d i f f e r e n t sequences t h a t c o n t a i n t h e sa-
F i r s t one i n s e r t s r AB boundaries (B-
t h i s can be done i n
different
For any such i n s e r t i o n , one uses r B's, t h e remaining n
-r
ways
B i s can
.
be
attached t o any o f t h e r bound B's w i t h o u t a l t e r i n g t h e number
boundaries;
t h i s , again, can be done i n
(r)
of
AB
d i s t i n g u i s h a b l e ways. There-
fore,
O f course, as i t should be,
I n s e r t i n g (7) and (9) i n t o
(8), we
obtain
Using S t i r l i n g ' s formula, we f i n d f o r t h e f r e e energy per s p i n
-
,.
I
--
exp 1-2rlog r
. I og (n-r)
+
-
-
(m-r) iog(m-r)
1og 1og Z ( r )
(n- r)
.
I.
Converting the sum i n t o an i n t e g r a l , one f i n d s t h e saddle p o i n t r
obey t h e e q u a t i o n
2 l o g ro
-
-
l o g (m-ro)
l o g (n-ro) = O
The term i n l o g l o g ~ ( rc o
) n t r i b u t e s o n l y O(N-l)
.
t o the p o s i t i o n o f
saddle pol i n t , and t h e r e f o r e t h e s o l u t i o n r,, does n o t depend
cons t a n t s o r temperature b u t o n l y on
the concentrations:
the
on c o u p l i n g
From (12), we
find
r
o
=N
CACB
;m-r
o
=N
CA2
; n-r
o
=N
CB2
(1 3)
I n o t h e r words, the above procedure amounts t o t h e usual replacements o f
40 1
such surns by t h e i r l a r g e s t member, where upon t h e b i n o m i a l f a c t o r s
can-
c e l o u t and, u s i n g ( 5 ) , one g e t s sirnply, up t o i r r e l e v a n t a d d i t i v e constants,
+
c 2 I o g cosh
B
= c;fA
+
c2c
B B
KB +
+
2~
2 c c l o g cosh K A B )
A B
C
A B
(1 4)
fA B '
a r e t h e f r e e e n e r g i e s of p u r e
where fA , fg and f A B
I sing chains
wi t h
c o u p l i n g c o n s t a n t s JA, JB and JAB, r e s p e c t i v e l y .
O f c o u r s e i n t e r n a 1 energy and s p e c i f i c h e a t have s i m i l a r s t r u c t u r e
as
(14) :
+
u
=
C
= c c
qu
2
qu
A A
c;uB
+
+ c;cB +
2c c u
A B A B '
2cAcBcAB
,
where
4. ANNEALING RANDOM SYSTEM
I n t h i s case, a11 t h e p o s s i b l e d i s t i n g u i s h a b l e sequences a r e r e a l ,
rea-
l i z a b l e component c o n f i g u r a t i o n s o f t h e systern, cornpet i n g thermodynarnic a l l y i n t h e energy b a l a n c e o f t h e p a r t i t i o n f u n c t i o n .
The t o t a l
t i t i o n f u n c t i o n i s , t h e r e f o r e , a c t u a l l y t h e surn o v e r a 1 1 component
spin configurations
parand
where, again,
Inserting
the degeneracy o f a l l sequences o f equal r has been used.
(5) and (9) i n t o (161, we o b t a i n
whe r e
cosh 2 KAB
x
=
cosh K cosh K
A
B
Applying once more S t i r l i n g ' s formula, we can w r i t e
'ann
= const.(m cos
m
KA)
n
1
n
( n c o s h KB)
exp [ r l o g x
-
r=O
where we w r i t e f o r t h e sum, as before,
o n l y t h a t now the saddle p o i n t equation,
O
' .[
(m-r
.;: ]
1 (n-r.
= l o g z m
,
does c o n t a i n t h e coupl i n g constants and temperature through x ( T ) .
i s i n t u i t i v e l y clear:
when t h e components .are f r e e t o d i f f u s e
minimum f r e e energy c o n f i g u r a t i o n ,
the optimal o r "typical"
to
number
Thi s
the
r,,
o f AB boundaries must depend on temperature.
I n t roduci ng t h e concentra t i o n o f AB boundar i e s , cAB(T) =
can r e w r i t e (17) t o read
I
3 r. (T) ,
one
cosh 2 KAB
c2
AB
=
x(T) =
J
cosh KA cosh KB
(cA-cAB) b B - c A B )
whose o n l y p h y s i c a l s o l u t i o n i s
A t low temperatures, when
so t h a t cAB = O,
i.e.,
form two pure phases.
as T
-t
O
,
t h e A's and B1s "repel"
to
On the o t h e r hand, f o r
t h e m i x i n g w i l l be maximal,
ween two A's,
each o t h e r , tending
i.e.,
every B i s
inserted bet-
the remaining A's ( i f any) placed anywhere, g i v i n g
t o a h i g h l y degenerate ground s t a t e .
rise
For
i n t h e low temperature l i m i t , o r f o r any
IJABIi n
the h i g h temperature
l i m i t , x + l , s o t h a t ~ ~ ~A =B e' e l i k e t h e d i s t r i b u t i ~ n(13)
quenched c h a i n o f the preceeding paragraph.
of
the
One can c o n s i s t e n t l y
de-
f i n e the degree o f m i x i n g i n the system as cAB/eB
,
a kind of
"compo-
nent d i s o r d e r parameter" t h a t has i t s range between O and 1 , according
t o the temperature i n the d i f f e r e n t cases discussed above.
I n s e r t i n g now (22) and (19) i n t o (18), and t a k i n g the l o g of (181, wher e o n l y the e x t e n s i v e terms a r e k e p t , one gets f o r the f ree energy the
express i o n
where fX =
-k
!
I
'
l o g cosh K X
,
as before, and
l n a s i m i l a r form one can w r i t e t h e i n t e r n a 1 energy and t h e
heat.
specific
For t h e l a t t e r , we g e t
whe r e
*
CX
=
k
-
T~
and, as before,
CX =
k ~ /i
cosh 2 K X
,
X = A, B.
I n (25) and (26), i t i s
i n t e r e s t i n g t o note t h a t one has separated the pure magnetic c o n t r i b u t i o n t o the s p e c i f i c heat frorn the pure annealing
c o n t r i b u t i o n . The
f i r s t o f them, already present i n the quenched case (151, i s
s p i n f l i p s and, the l a t t e r , due t o the m o b i l i t y o f
the
due
to
components
,
appears i n t h e form o f temperature d e r i v a t i v e s o f the AB-boundary conc e n t r a t i o n i n Eq. (26).
5. RESULTS AND CONCLUSIONS
The r e s u l t o f the preceding paragraphs,
f o r the c o n c e n t r a t i o n
of
AB-
-boundaries, was :
In the quenched case, %B, is constant; the r e s u l t represents the most pro-
405
FIG.3
FIG. 3
-pure
-.-
.....e
..
.
annealing cg = . 3
JA=JBs 7K°K
quenched cg * . 3
J ~ e'
J~
The
bable d i s t r i b u t i o n o f independent A's and B's on the s i t e s .
cha-
r a c t e r i s t i c temperature dependence o f ( c ~ ~ ) ~ on
~ ~
the/ oct h~e r , hand
,
i s a s i l l u s t r a t e d i n Fig. 4 and Fig. l c : a t h i g h temperatures,
,
when
i . e.
kT >> max(1JI) , i t approaches ( c ~ ~ ) =~ CA,
~ / i .e.,
c ~ the
tical distribution;
statis-
f o r T-tO, we g e t e i t h e r complete s e p a r a t i o n o f the
pure phases, cAB = O, o r maximum mixing, cAB = cB, depending on whether
2
I J ~ ~i sI smal l e r
o r l a r g e r than
r a t u r e we g e t a gradual "melting"
I J ~ I+ I J ;~ w ~i t h
i n c r e a s i n g tempe-
o f the pure phase o r the AB
-
supers-
t r u c t u r e , r e s p e c t i v e l y ; t h i s i s q u i t e as i n t u i t i v e l y expected. The temperature, a t which cAB changes m s t r a p i d l y w i t h temperature, l i e s somewhere i n the middle, between the d i f f e r e n t values o f J
.
Let us remark here t h a t i n t h i s one-dimensional zero f i e l d case,
a11
r e s u l t s , both f o r cAB and f o r the f r e e energy and i t s d e r i v a t e s below,
do n o t depend on the signs, b u t o n l y on the absolute values
IJI
o f the
exchange constants; t h i s i s f o r obvious reasons: i n v e r t i n g the s i g n of,
-
l e t us say, J from + t o
, we f i n d , f o r any s p i n s t a t e , a t JA + O , a
A
s p i n s t a t e o f the same energy, a t JA< O , by r e v e r t i n g a l l spins b e t ween any two successive AA-bounds along the e n t i r e chain; t h i s one-to-one correspondence o f degenerate s t a t e s g i ves r i se t o una1t e r e d t h e r mdynami c r e s u l t s .
For t h e s p e c i f i c heat we had:
whith C
*
from (26).
For the s p e c i a l case J = JAB= 0 , these r e s u l t s reproduce
B
Ref .I f o r the " d i l u t e I s i n g chain" .
We n e x t consider the s p e c i f i c heats i n some d e t a i 1.
those
of
Obviously,
(C)qu i s j u s t a s u p e r p o s i t i o n o f t h e s p e c i f i c h e a t s of I s i n g
c h a i n s w i t h c o u p l i n g c o n s t a n t s JA, JBand JAB,whereby t h e b r o a d maximum o f t h e c h a i n s p e c i f i c h e a t i s i n any case
largely differing
broadened,
and,
for
J, even s p l i t s up i n t o severa1 maxima, as i l l u s t r a -
ted i n Fig. la.
F i g . 2 shows how t h e h e i g h t o f t h e s e p a r a t e maxima depends on i m p u r i t y
c o n c e n t r a t i on c
B'
F i g . 2a s h o u l d be compared w i t h t h e c o r r e s p o n d i n g c u r v e s i n Ref.5,which
a r e q u i t e s i m i l a r , e v i d e n t l y due t o t h e r a p i d convergence,
at
zero
f i e l d , o f t h e s e r i e s expansion g i v e n t h e r e f o r t h e quenched case. Comp l e t e l y d i f f e r e n t i s t h e b e h a v i o u r o f Cann, as can be seen by c o n t r a s t i n g Fig.
l a w i t h I b , and 2a w i t h 2b:
the dominating f e a t u r e
is
s t r u c t u r e , a t l e a s t a t 50% i m p u r i t y c o n c e n t r a t i o n .
than two t i m e s h i g h e r ,
i n a1 1 t h e cases o f Fig.1,
T h i s peak i s
than
those
of
quenched case ( o r o f a p u r e c h a i n ) w i t h i t s c e n t e r r i g h t between
superimposed peaks o f t h e quenched case.
Moreover,
more
the
the
comparison o f F i gs.
l b and l c shows t h a t t h e a n n e a l i n g s p e c i f i c h e a t maxima
coincide w i t h
those temperatures a t w h i c h t h e "mel t i n g i ' o f t h e low t e m p e r a t u r e
- s u p e r s t r u c t u r e goes most r a p i d l y w i t h temperature.
a
other
h i g h and r e l a t i v e l y narrow low t e m p e r a t u r e peak, w i p i n g o u t a l l
AB-
A I 1 t h i s makes i t
o b v i o u s t h a t t h e anneal i n g case s p e c i f i c h e a t i s dominated
by t h e en-
t r o p y o f compos i t i o n a l d i s o r d e r s u p e r s e d i n g t h e e n t r o p y connected w i t h
s p i n d i s o r d e r w h i c h causes t h e s t r u c t u r e i n t h e quenched case. F o r t h i s
reason, t h e a r e a under C, i .e.,
u(T=m)
-
u(T=O),
i s always
t h e a n n e a l i n g than i n t h e quenched case (compare, e.g.,
larger
also
in
Fig.3
,
where C
and Cann a r e p l o t t e d i n one graph f o r a n o t h e r r e p r e s e n t a t i v e
qu
case, t o g e t h e r w i t h C f o r t h e p u r e AA- and t h e p u r e AB-chain).
Another way t o l o o k a t t h e r o l e o f c o m p o s i t i o n a l e n t r o p y i s
by
dis-
c u s s i n g t h e f r e e energy i n t h e anneal i n g case, i n p a r t i c u l a r t h e terms
eAfA*
+ eBfB* i n (23) and ( 2 4 ) .
I n t h e h i gh temperature 1 i m i t, where, as d i scussed b e f o r e , ( c ~ ~ ) ~ ~, ~ + c ~
t h e s e two terms go i n t o
w h i c h r e p r e s e n t s j u s t t h e usual e n t r o p y o f a m i x t u r e .
Thus, t h e
re-
s u l t s o f t h i s paper can be s t a t e d as f o l lows:
For z e r o magnetic f i e l d , t h e f r e e energy and i t s d e r i v a t i v e s ,
two component l i n e a r I s i n g c h a i n , can be c a l c u l a t e d e x a c t l y
f r o z e n and f o r m o b i l e components
change c o n s t a n t s .
for
a
both f o r
f o r a r b i t r a r y c o n c e n t r a t i o n s and ex-
T h i s i s due t o a degeneracy o f t h e energy f o r a11 w n -
f i g u r a t i o n s w i t h t h e same number o f AB- boundaries.
A
magnetic
field
l i f t s t h i s degeneracy and t h e degeneracy w i t h r e s p e c t t o t h e s i g n s
of
the coupling constants.
We w i s h t o acknowledge t h e s t i m u l a t i o n we g o t f o r t h i s work fromA.Rauh
and L. K a l o k a t o u r i n s t i t u t i o n .
REFERENCES and NOTES
1. S . K a t s u r a and B. Tsujiyama,
Misc.Pub1.
2. D. W.
273
r e p r i n t e d f r o m Critica2 Phenomena, NBS
(1965).
Hoffmann, M e t a l l u r g i c a l T r a n s a c t i o n s , V o l . 3, Dec. 1972.
3. F. Matsubara, K. Yoshimura and S . Katsura, Canadian J o u r n a l o f Physics,
E,
No. 10 (1973).
4. M. Ya. A z b e l ' ,
Phys.Rev.Letters,
2, No.
9 (1973).
5. F. T. Lee, E. W. M o n t r o l l and Lee-po Yu, J. o f S t a t . Phys. 8, No.4,
(1973).
6. A c t u a l l y , E q . ( 3 ) i s t h e s o l u t i o n o f t h e c l o s e d c h a i n problem.
For
t h e open c h a i n , t h e s o l u t i o n i s
where M i s t h e m a t r i x
[[ [)
whi ch san e a s i 1y be shown t o p r o j e c t
t h e e i g e n v a l u e s coshK, such t h a t Eq. (5) i s e x a c t f o r
and,
the
open
i n t h e thermodynamic l i m i t , a l s o t r u e f o r c l o s e d c h a i n s .
out
chain
7. As a m a t t e r o f f a c t , t h e r e a r e r +- 1 P B A - f a c t o r s f o r open chains
For long chains, however, r I1 = r.
/C ) i s the short- range o r d e r c o e f f i c i e n t o f nearest
AB B
bours, as def ined i n Ref .Z.
8. 1-(c
.
neigh-