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The energy of the spin-glass state of a binary mixture at T = 0 and its variational properties
Physica 100A (1980) 24-44 © North-Holland Publishing Co.
THE ENERGY OF THE SPIN-GLASS STATE OF A BINARY MIXTURE AT T = 0 AND ITS VARIATIONAL PROPERTIES Sakari INAWASHIRO and Shigetoshi KATSURA
Department of Applied Physics, Tohoku University, Sendai, Japan Received 17 Jury 1979
In the random-bond model of Ising spins, the concept of a multiple-bond distribution of effective field was introduced in the pair approximation. The integral equation for a single-bond distribution was derived intuitively. The variational energy at T = 0 is expressed in terms of two parameters /~ and ~ where /~ is the probability of zero effective field in the single-bond distribution and v/ is the magnetization per spin. For 7/= 0, the energy of the spin-glass state corresponds to a local minimum as a function of /z, for an even z (number of the nearest neighbours) and to an inflection point for an odd z. It was shown that the spin-glass state corresponds to a local minimum with respect to /~ and y/ for z = 4, to an inflection point with respect to/~ and a local minimum with respect to 7/for z = 3. It is conjectured that the maximum of the energy of the spin-glass state of Sherrington and Kirkpatrick is attributed not to the replica method, but to the mean field approximation. Stationary properties of the energy as a function of both ~t and ~ were examined in detail.
1. Introduction E d w a r d s a n d A n d e r s o n ~) p r o p o s e d a s p i n - g l a s s m o d e l in w h i c h s p i n s a r e r a n d o m b u t f r o z e n . S h e r r i n g t o n a n d K i r k p a t r i c k 2) ( r e f e r r e d as S K h e r e a f t e r ) s o l v e d a r a n d o m Ising m o d e l w i t h infinitely l o n g - r a n g e d i n t e r a c t i o n b y t h e r e p l i c a m e t h o d s h o w i n g t h e s p i n - g l a s s state. A b i n a r y m i x t u r e o f a r a n d o m I s i n g m o d e l w a s a l s o s h o w n to h a v e t h e s p i n - g l a s s s t a t e ( g l a s s - l i k e p h a s e , G L P ) b y M a t s u b a r a a n d S a k a t a 3) i n d e p e n d e n t l y o f EA~). M o r i t a 4) p r o p o s e d a v a r i a t i o n a l f r e e e n e r g y o f a r a n d o m - b o n d m o d e l in a p a i r a p p r o x i m a t i o n a n d d e r i v e d a n i n t e g r a l e q u a t i o n f o r a d i s t r i b u t i o n f u n c t i o n o f t h e e f f e c t i v e fields. K a t s u r a , I n a w a s h i r o a n d F u j i k i 5) ( r e f e r r e d as K I F h e r e a f t e r ) s h o w e d t h a t t h e i n t e g r a l e q u a t i o n b y M o r i t a 4) is e q u i v a l e n t to t h a t i n t r o d u c e d b y M a t s u b a r a 6) p r e v i o u s l y , a n d t h e y d e r i v e d t h e r e s u l t o f S K 2) f o r a l o n g - r a n g e d i n t e r a c t i o n . F u r t h e r t h e y 5) o b t a i n e d t h e e n e r g y o f the s p i n - g l a s s s t a t e at T = 0 f o r a s h o r t r a n g e d b i n a r y m i x t u r e o f f e r r o - a n d a n t i f e r r o m a g n e t i c b o n d s , b y s o l v i n g the integral e q u a t i o n o f t h e p a i r a p p r o x i m a t i o n e x a c t l y . As for the stability of the spin glass state, much controversies are debated a m o n g a u t h o r s . F i s c h e r 7) s h o w e d t h a t t h e v a r i a t i o n a l f r e e e n e r g y is n o t m i n i m u m b u t m a x i m u m in the s p i n - g l a s s state. T h o u l e s s , A n d e r s o n a n d 24
SPIN-GLASS STATE OF A BINARY MIXTURE AT T = 0
25
Palmer s) modified SK theory so as to have zero entropy at T = 0. Their free energys) of the spin-glass state also has a maximum below the glass transition temperature, and a minimum in the paramagnetic state. The lower critical dimensionality is supposed to be 4 by Fisch and Harris 9) and by Bray, Moore and Reed~°). Southern, Young and Pfeuty H) did not obtain the spin glass transition in two dimensions but did in three dimensions. The purpose of the present paper is to obtain the energy at T = 0 of the spin-glass phase as well as the ferromagnetic phase of a binary mixture and to investigate the stability of these phases. In section 2, we give a simplified derivation of the integral equation making clear the meaning of the pair approximation. In section 3, a variational form of the solution is introduced, and a set of algebraic equations are derived from the integral equation. In section 4, the free energy is expressed in terms of two variational parameters. In section 5 the solutions are classified into several categories, and the stationary properties of these solutions are examined. Section 6 is devoted to conclusions and discussions.
2. Distribution of effective fields
We consider a spin in an Ising model with random exchange-interactions between nearest neighbours. The number of the nearest neighbours of a lattice site is denoted by z. In an effective field approximation, the spin is ~onsidered to be placed in an effective field, i.e., a z-bond field which is ~onsisted of z contributions from the nearest neighbours. The contribution through a single bond is called a single-bond field. Then the z-bond field is equal to the sum of z single-bond fields. Then an effective Hamiltonian of the spin is given by HI = -trjh ~ --trl
~ . hlk,
(2.1)
k=l
where h denotes the z-bond field and hlk the single-bond field from the kth neighbour. Next we consider a pair of neighbouring spins with an exchange integral J'. Influence upon the pair of spins from the other spins of the system is represented by a pair of (z - 1)-bond fields acting upon the two spins each. An effective Hamiltonian for the pair is given by HI2 = - J ' oqo'2 - o'~ht ~-~>- or2h [z-l),
where
(2.2)
26
S. INAWASHIRO AND S. KATSURA Z--I
h~-'~ = ~--I hik, (i = 1, 2),
(2.3)
which is a sum of ( z - 1) single-bond fields from the outside of the pair upon the ith spin of the pair. N o w we consider a spin of the pair. The effect of the other spin through the exchange bond Y' is represented by a reduced field, i.e.,
h = (1//3) tanh-| [tanh /3J' tanh(/3 ~=th2k) ].
(2.4)
Due to a random distribution of the exchange integral, the single-bond field has a distribution denoted by g(h~k). A pair approximation requires a selfconsistent condition that the distribution of the reduced field is equal to that of the single-bond field, i.e., g(h)=
f 3[h-(l//3)tanh-'Itanh /3J' tanh(/3 ~
h2k)}]
Z--I
× P(J') dJ' l-I g(hEk)dhzk,
(2.5)
k=l
where we have assumed that the distribution of the single-bond field is independent of the location of spins and bonds. A distribution of M-bond field is defined by
g(M'(h) = f 6 ( h - k~,= l h,k ) kfi= l g(h,D dh,k,
(2.6)
Then the integral equation (2.5) is rewritten as
g(h) = f 8[h -
(1//3) tanh-~{tanh/3J' tanh/3h'}]P(J')
dJ'gtZ-~)(h') dh'.
(2.7)
It was shown by Morita 4) that the integral equation (2.5) is derived through the variational principle from a free energy of the random system, which is given by f = f, + f2,
(2.8)
where
Is = (- 11/3)f
ln[Tr{exp(-
z-1
z-I
k=l
m=l
/3HI2)}]P(J') dJ'
× 1-I g(h,k) dhnk FI g(h2m)dh2m,
(2.9)
and .f2 = (2//3)(1 -
l/z) f
ln[Tr{exp(-/3H1)}1 l~I g(h~k) dhlk. k=l
(2.10)
SPIN-GLASS STATE OF A BINARY MIXTURE AT T--0
27
In this paper, we consider a binary mixture, where the distribution of the exchange interaction is given by
P ( J ' ) = p6(J' - J) + (1 - p ) 8 ( J ' × J),
(J > 0),
(2.1 I)
where p and ( l - p ) represent the concentrations of the ferromagnetic and antiferromagnetic bonds, respectively. The integral equation (2.7) is written as
g ( h ) = p f 6[h - (fiB) tanh-I{tanh OJ tanh flh ,}]g(Z-I)( h ,) dh' + (1 - p ) f 6 [h + ( 1/B) tanh-I{tanh flJ tanh flh'}]gtZ-~)(h') dh'. (2.12) Using the distributions of ( z - 1)-bond and z-bond fields and the distribution of the exchange interaction, fl and [2 are expressed as /~ = -(1/fl) f [p ln{exp(flJ)2 cosh fl(hl + h2) + exp(-/3J)2 cosh fl(hl - h2)} + (1 - p) ln{exp(-/3J)2 cosh fl(hl + hz) + exp(flJ)2 cosh fl(hl - hz)}]
× g~-l)(hOg~Z-l)(h2) dhl dh2,
(2.13)
and f2 = (2/fl)(1 - I/z) f In(2 cosh flh)gtZ)(h) dh.
(2.14)
(2.13) and (2.14) are the free energy used in KIF.
3. The integral equation at T = 0
At absolute zero temperature, the distribution of the single-bond field is assumed to be given by
g ( h ) = go6(h) + g16(h - J) + g_~6(h + J),
(3.1)
where the normalization condition requires go + gl + g-i = I.
(3.2)
From the definition (2.6), the form of the M-bond distribution at absolute zero is given by M
g~m)(h)= ~
g~,m)8(h-nJ).
(3.3)
n=-M
The coefficient gtm) is clearly given by the expansion coefficient of the
28
S. INAWASHIRO AND S. KATSURA
following polynomial expansion, M
(go + glx + g-ix-t) u = ~
g~MJx".
(3.4)
n=-M
The expression of g~M) in terms of go, gl and g_l is obtained in a similar way to KIF. The result is
I (Mn )gOM-, g'~2F, [ ( n 2 M ) , ( n - M + I
2
g(M, =
l);n+ l;4g'g-l~, ---~o /
[
M ~ n >~O,
{M'~ _M-I,I ([nI-M) (InI-M+ 1 1;4glg_,'~, \lnlJ ~o g~l 2F, 2 ' 2 ; Inl + g~ ) O>n>~-M,
(3.5)
where 2F~ denotes the hypergeometric function. Let us rewrite the integral equation (2.12) in terms of the coefficients g, and g~-J~. Using a limiting formula tim ~1 tanh- i (tanh/3hl tanh/3hz) = sgn(h,) sgn(hz) min(lh~l, Ihzl).
(3.6)
We see eq. (2.12) becomes at absolute zero
g(h) = p f 6[h - sgn(h') min(J, Ih'l)]g~Z-~)(h') dh' + (1 -
p) f 6[h + sgn(h') min(J, Ih'l)]g~Z-1)(h ') dh'.
(3.7)
Substituting the expression (3.3) into the rhs of (3.7), we obtain an equation
k
n>0
n> 0
n> 0
J
n> 0
From (3.1) and (3.8), the integral equation is reduced to a set of two algebraic equations,
go - g~Z-~ = 0,
(3.9)
gl - g-1 - (2p - 1) ~ (g~-l) _ g ~ l ) ) = 0.
(3.10)
n>O
Since g~z-l), g~-~) and gtSg~) are expressed in terms of go, gl and g_~ using (3.5), the equations (3.9) and (3.10) are used in order to determine go, g~ and g_~. Actually they form two equations for two unknowns go and g ~ - g_~. (3.9)
S P I N - G L A S S STATE O F A B I N A R Y M I X T U R E AT T = 0
29
and (3.10) agree with (7.20) and (7.20') in KIF. The solution of these equations are sufficient to the stationary conditions for the free energy. We will discuss the equivalence for the case of z = 3 and z = 4 in the next sections.
4. An expression of the free energy at T = 0
The form of g(h) given by (3.1) is used as a trial function in the variational principle. Substituting (3.1) into (2.13) and (2.14), and taking the limit of/3 --* o% we obtain flo~limfl
=
~
~
{pA(,,,+).+(1-p)A(~).,}g(.~-I)g (z-'),
(4.1)
m=-z+l n=-z+l
where A ~ ) = lim(l//3) ln{exp(_/3J)2 cosh fl(mJ + nJ) + exp(~flJ)2 cosh fl(mJ - n J)}
= ImlJ + InlJ + Amn,(+-)
(4.2)
~+-J, mn >0, A~)~ = 1 J, mn = O, ~J, mn <0,
(4.3)
and f2o - l i m / 2 t3-~
~
=2(1-1[z)J
[nlg~ ).
(4.4)
n=-z
The equation (4.1) is rewritten as
f,o = - J + =-J
n
+1
z-I
Inlg~ -')
z--I
E
E
n'~A(-)~°(z-1)°(z-I) ]
m=--z+l n=--Z4-1 I Z--I
2 N
{PA(,.+.)+(1-,',--,.-,~,m
~,
j
InlgT-')+ 2ggZ-')-(gg~-')) 2
n=--z+l
(4.5)
30
S. INAWASHIRO AND S. KATSURA
The identity Z-I
g ~ x ~ = (go + g l x + g - i x -j) n=-2
~
o~-~),~"
n=-2+l
leads to a recurrence relation, g~)= ~m - g,,s,-m. ,,(z-l) Then we obtain In[g~) = go Y~ {mlg~-l' + gl ~ Im + l[g~-l)+ g_ 1 ~] Im - l l g ~ - ' ~ n=--z
m
m
z--I
=
~,
{nigh-" + (1 - go)g~z-l) + (gl - g - 0 ~ (g~-l)_ g~Z~l)). (4.6)
n=-z+l
n>0
Then flo is rewritten as
/10 = - J [2 , ~ Inlg~' + (2g0- g~-l))g~-l, - 2(gl - g-l)
~
(g?-l)
_ g~Z-1))
n> 0
-1)/Y.
1'-
(4.7)
tn>0
3 J
Differentiating both sides of the identity relation (3.4), and multiplying by x, we obtain M
M(go + glx
+ g-IX-1)M-1(glX
--
g - i x -1) = ~ ,
ng~,M)x".
(4.8)
n=-M
The left hand side of (4.8) is expressed by gtM-l~ according to the definition (3.4), and then a recurrence relation is derived by comparison of the coefficients of the nth power of x, i.e., ng~.u) = M ( g l g t , M_~~)- g_lg~,M+~l)).
(4.9)
Carrying out the summation over n in (4.9), we have Inlgt,M) = M
(1 --go)g~M-l) + glg~_~-') + g_lgt u-l)
n=-M
+ (g, - g-l)
E (g(M-l)_g ~ - l ) ) /d"
(4.10)
n>0
Using (4.4), (4.7) and (4.10), we obtain an expression of the variational free energy, i.e. .fo = flo +.f2o = JqJ(go, gl, g-l)
(4.11)
where ~b(go, gl, g-~) = - 2 g ~ ~-1) + (g~-~))2 _ (1 - go)(gl ~-1) + g~_~-~)) + ( g t - g - O ( g t ~ - 1 ) - g ~ i -I)) - ( 2 p - 1)
( g ~ - l ) _ g~21))
.
(4.12)
SPIN-GLASS STATE OF A BINARY MIXTURE AT T = 0
31
5. Stationary properties of the energy at T = 0
T w o n e w v a r i a b l e s / £ and ~ are i n t r o d u c e d b y
g0=/£,
gl-
(1 - / £ + 71) 2 ,
g-l=
(1 - / £ - 77) 2
(5.1)
/£ r e p r e s e n t s the p r o b a b i l i t y of the z e r o effective field in the single b o n d distribution and 7} r e p r e s e n t s the m a g n e t i z a t i o n per spin. /£ is a m e a s u r e of the spin glass o r d e r p a r a m e t e r q[--- (o-) 2 = I - g~Z)], w h e r e / £ = 1 f o r q = 0 w h e n ~7=0. For example q=1-[/£3+(2~)/£(1-/£)2] for z=3 and q= 1 [[£4 + 3/£2(1 --/£)2 + (~8)(1 --/£)4] f o r z = 4. -
T h e n all the coefficients g ' s are e x p r e s s e d in t e r m s o f / £ and r/. It is to be n o t e d that the p h y s i c a l regions o f / £ and W are given b y - 1 ~
(5.2)
(5.3)
T h e solutions of ~bK/£, 7 ) = 0 and ~b2(/£,7/)=0 or o f G 0 . t , r / ) = 0 and ~ ( / £ , r / ) = 0 are classified into s e v e r a l categories. (a) ~7 = 0, /£ = 0; (b) ~7 = 0, /£ # 0,/£ # 1 ; (c) ~7 = 0,/£ = 1 ; (d)/£ = 0, r / ~ 0; (e)/£ ~ 0, 7/# 0; (f) the solutions of ~b~ = 0 and 4~ = 0 which do not satisfy ~bl = 0 and ~b2 = 0. T h e solution rl = 0 c o r r e s p o n d s to a s y m m e t r i c distribution of the effective field. T h e solutions (a) and (b) c o r r e s p o n d to spin-glass states, (c) to the p a r a m a g n e t i c state, and (d) and (e) to f e r r o m a g n e t i c states. T h e f u n c t i o n s ~b(/£, r/), ~bl(/£, r/), and t~2(/£, "Q) are e x p r e s s e d by p o l y n o m i a l s o f / £ and 7} and are g i v e n f o r z = 3 and 4 in a p p e n d i x I t o g e t h e r with ~b(/£, 0) for z = 5 and 6. In this section and in a p p e n d i x I, x - p - ½is used instead of p to r e p r e s e n t the c o n c e n t r a t i o n . First we c o n c e n t r a t e on the s y m m e t r i c distribution of effective field • c o r r e s p o n d i n g to ~7 = 0. T h e n the variational e n e r g y is a f u n c t i o n of only /£, and is plotted f o r z = 3, 4, 5, 6 in fig. 1. It is to be n o t e d that the spin glass solutions of t y p e (a) h a v e a local m i n i m u m f o r z = 4 and 6 (in general, f o r e v e n z) and the spin-glass solutions of type (b) h a v e inflection points f o r z = 3 and 5 (in general f o r o d d z). W h e n w e plot V'-zE/2JN as a f u n c t i o n o f z, the energies of the spin-glass states f o r z = 3, 4, 5, 6 lie a l m o s t on a straight line which e x t r a p o l a t e s to the value of the infinitely long-ranged spin glass of
32
S. INAWASHIRO AND S. KATSURA 2E, I Z 3
-0.6 (3
-0.7
Q, -0.8
-09
-IO
0.5
1.0
/.z
Fig. I. The variational energies ~(#, 0) as functions of # for z = 3, 4, 5 and 6. For z = 3 and 5, the spin-glass solutions of type (b) have inflection points. For z = 4 and 6, the spin-glass solutions of type (a) have local minima. S h e r r i n g t o n a n d K i r k p a t r i c k 2) (fig. 2). T h e e n e r g i e s o f t h e a b o v e s o l u t i o n s a r e p l o t t e d a g a i n s t the c o n c e n t r a t i o n in fig. 3. T h e s p i n - g l a s s s o l u t i o n (a) has t h e l o w est e n e r g y f o r 0 ~< x ~< 8a (z = 3), a n d 0 ~< x ~< ~ (z = 4), a n d t h e n t h e f e r r o m a g n e t i c s o l u t i o n (d) f o r a8~< x ~< ~ (z = 3) a n d 31~< x ~< ~ (z = 4), e x c e p t t h e s o l u t i o n (c). F e a t u r e s o f s o l u t i o n s f o r e v e n z a n d for o d d z are a little d i f f e r e n t . W e a n a l y z e t h e c a s e s o f z = 3 a n d o f z = 4 in detail. F o r z = 3 w e h a v e t h e f o l l o w i n g f o u r s o l u t i o n s (cf. figs. 4 - 6 ) . (1) 77 = 0 , / z = ~ ( t y p e (b))
.~21// all ~
0'
~21// __8(3--8X) arl 2
9
t~2~/ = 0 , '
al~071
t~3~// - 1 2 < 0 . a# 3 =
T h e s o l u t i o n c o r r e s p o n d s to a n i n f l e c t i o n p o i n t w i t h r e s p e c t to /z, a n d t h e m i n i m u m f o r 0 < x < 8~ a n d t h e m a x i m u m f o r x > • w i t h r e s p e c t to r/. (2) r / = 0 , / z = 1 ( t y p e (c)) az@ = 4 > 0,
c~2--~- 16(41- x),
al~2
3,rl 2 -
02~/ 81x8 ~
= 0.
SPIN-GLASS
S T A T E O F A B I N A R Y M I X T U R E AT T = 0
33
~ E / 2N...T 0
6
0.1
5
3
4
0.2
I/Z
0.~
Fig. 2. The energy of the spin-grass state as a function of z. The energy is scaled by V~z for comparison with the value of Sherrington and Kirkpatrick.
T h e e n e r g y is t h e m i n i m u m w i t h i n t h e p h y s i c a l r e g i o n , - 1 ~ g --- ~ ~< I. (3) /x = (1 - 2x)[2x, n -- *_V'(Sx - 3)(4x - l ) / 2 x ( t y p e (e)). T h e real s o l u t i o n exists for ~
,~
~<~.
_
- ( 8 x - 3){~ - 8 ( x - b ~} > o,
0zt/, ( 8 x - 3 ) ( 4 x - 1) > 0, 0~72 = 2x 2
q.,:
Z=4 b
C
e
-0"81
Z=3 -0.9
-I.0 -0.2
-
--~0.3
i
O.4
0.5 X
Fig. 3. The energies of several states as functions of concentration (x -= PA - ~)- (a], (b), (c), (d), (e) and (f) denote the respective solutions. A f x = ]), B(.r = }) C(x = ~) D ( x = ~) denote junctions of solutions. The energies of (e) and (f) can not be distinguished in this figure.
34
S. I N A W A S H I R O
A N D S. K A T S U R A
-08
-09
c
0.5
f
f,.o
/ /
/
/ /
0.5
/ / / /
/ /
1.0
~7 Fig. 4. ~b(/x, "0), z = 3, x = 0, (b) saddle point, (c) m m .
026 _ (8x - 3)7/ O0.OT/
x
a26 00. z
az4~ O0.O'q
azO
a2q~
>0
for
~s
0~0.a ,},.j. C~'F/2 T h e s o l u t i o n c o r r e s p o n d s to a local m i n i m u m o f t h e v a r i a t i o n a l e n e r g y .
2 ~ / 1 4 x - 3 + ( 4 x - 6 ) X / 1 - 2x
I - 2x + 2X/I - 2x
(4) 0 ` =
2x+3
'
"q= -~ -
2x+3
(a ~ x ~< ~), ( t y p e (f)), a20 270`z_300`+7+Tt2(_3_4x)
forsa~
'
SPIN-GLASS STATE OF A BINARY MIXTURE AT T = 0
35
q~
b
/)
c
-I.C
/
J
J
P-
j
J
~:/
0.5
//://
1.0
77 Fig. 5. 4'(~, ~t), z = 3, 0 < x(= 0.35) < ], (b) saddle point, (c) min.
O,!~b= {_3/z2 + 6/z + I - 4x(l +/~)2 + 3r/2} = 2.02 > O, 3,r/2 a2~b = 2~{-3/z + 3 - 4x(l + ~)}, arl3gt T h e solution c o r r e s p o n d s to a saddle point o f the variational energy. W e h a v e no solutions o f the t y p e (a) and the t y p e (d) f o r z = 3. F o r z = 4, w e h a v e the following five solutions (cf. figs. 7 - I 1). (1) g = 0 , ,?--0 (type(a)), a2~b = a2 > 0,
a2--~0- 3(1 - 3x),
0tl. 2
O~l 2 -
O2~b = 0. Ol~aTi
T h e e n e r g y is a local m i n i m u m f o r x < l3 c o r r e s p o n d i n g to the spin-glass p h a s e ( G L P ) , and a saddle point f o r x > 31. (2) /~=51, 7 / = 0 (type(b)),
ou~
~ -
(1 - ~ x ) ,
-
Ol~ O'q
= o. 25
T h e e n e r g y is a m a x i m u m f o r x > ~, and a saddle point f o r x < 78.
36
S. I N A W A S H I R O
A N D S. K A T S U R A
j0 \
IO
~7
Fig. 6. ~b(/z, "0), z = 3, 8~ < x ( = 0.4) < ½, (b) max., (e) min., (f) saddle point.
(3) / x = l ,
r/=0
(type(c)),
32~b= ~9/z2
12>0,
c~2---~-~-6(1-6x), &0 2 -
8-2~b = 0 . 3tzar/
O n a physical b o u n d a r y , s a y , / z + 7 / = _ 1, the f u n c t i o n qJ b e c o m e s ~b = - 1 + (1 - 2x)(1 - / z 3 ) 2, which s h o w s that the e n e r g y b e c o m e s the absolute m i n i m u m at /x = 1, and r / = 0, e x c e p t the case o f x = ½, i.e., the pure f e r r o m a g n e t , w h e r e the e n e r g y is c o n s t a n t along the b o u n d a r y # + 7 / = - 1. (4) / x = 0 , r / = _ _ _ X / 3 - 1 / x , (x>l~) ( t y p e ( d ) ) , a2O _ 24(x - ½)(x - ~) tg/.t 2 --
at/2
X2
x
,
S P I N - G L A S S STATE O F A BINARY M I X T U R E AT T = 0
37
q,
-0.9
/z // J jj
/
J
1.0
~7 Fig. 7. ~b(/.~,n), z = 4, x = 0, (a) min., (b) saddle point, (c) min.
and f 1 2 ~ = 0.
The solution represents the ferromagnetic phase. The energy is a local minimum for ~ < x < sa, and a saddle point for sa
(~ ~< x ~< ~)
(type (e)).
This solution is c o n n e c t e d with the solution (b) at q - - ~ 78, and to the solution (d) at x = ~. s It is c o n c l u d e d that the solution c o r r e s p o n d s to a saddle point. W e have no solutions o f the type (f) for z = 4.
38
S. INAWASHIRO AND S. KATSURA -0"
b
..3.~,~°, / -09
-I l '
i
kc
o.s
f'i'~o
/
. / / //
///
/
./ /
/. /
1.0 ~7
Fig. 8. ~b(/z,"0), z =4, 0
SPIN-GLASS STATE OF A BINARY MIXTURE AT T = 0
39
-0.7
(3
// I(~"
-0.9
\
r
-IO/0
. . . . z
ff
/
//./" /z
//
/ J
I1
I,O
Fig. 9. ~b(g, r/), z = 4, ~ < x(= 0.35)< ], (a) saddle point, (b) max., (c) rain., (d) min., (e) saddle point.
linear c o m b i n a t i o n s o f ~bl and ~b2, i.e., G ~ ot,bl +/3~b;,
(5.4)
G ------r4'~ + 84'2,
(5.5)
w h e r e t~, /3, y and 8 are linear p o l y n o m i a l s o f /~ and ~ f o r z = 3, and are s e c o n d - o r d e r p o l y n o m i a l s f o r z = 4, as given in a p p e n d i x II. It is s h o w n that the coefficient d e t e r m i n a n t o f the s i m u l t a n e o u s e q u a t i o n s (5.4) and (5.5) is positive definite f o r z = 4, e n s u r i n g the e q u i v a l e n c e o f ~b~ --- ~,~ = 0 and ~bl = 4,2 = 0. O n the o t h e r hand, the coefficient d e t e r m i n a n t f o r z = 3 is not a l w a y s n o n - z e r o , w h i c h allows the solution o f the t y p e (f) to c o m e f r o m ~ , = ~b~ = 0 w h i c h are o f higher o r d e r in /~ and ~/ than 4,~ = 4,2 = 0. T h e e n e r g y o f the solution (f), h o w e v e r , is quite close to that o f (e). T h e coefficient d e t e r m i n a n t s f o r z = 3 and 4 are also s h o w n in a p p e n d i x II.
40
S. I N A W A S H I R O A N D S. K A T S U R A
-07 I C]
//
/
A~//
\ 0.5
o p.
/o5
1.0 Fig. 10. Ik(#, n), z = 4, l < x ( = 0.4) < ~, (a) saddle point, (b) max., (c) min., (d) saddle point.
6. Conclusions The integral equations for the single-bond distribution was derived intuitively making clear the physical meaning of the pair approximation. Further, we have introduced the concept of multi-bond distribution of the effective fields, which will be useful for further development of a clutter approximation. For the binary bond mixture, the exact f o r m of the single-bond distribution at T = 0 is represented by a linear combination of 8-functions with two variational p a r a m e t e r s / x and n. The variational energy as a function o f / z with a fixed value of 9 = 0 has a local minimum at /x = 0 for an even z, and an inflection point a t / x # 0 for an odd z, corresponding to the spin-glass states. It has a minimum a t / x = 0 which represents the paramagnetic state, of which the energy is equal to the ferromagnetic ground state. F r o m the curves of the energy for z = 3, 4, 5 and 6 shown in fig. 1, it is conjectured that the stationary point of the type (b) a p p r o a c h e s to /z = 0 for both even and odd z as z increases, and that the
SPIN-GLASS STATE OF A BINARY MIXTURE AT T = 0
41
d~c 1.0
~7 Fig. 11. ~b(/.t, 71), z = 4, x = ~, (a) saddle point, (b) max., (c) min., (d) min.
energy has only one m a x i m u m at 9. = 0 and only one minimum at/~ = 1 in the limit z ~ ~. The characteristic feature of this limiting c u r v e agrees with those of the free energy as a function of the p a r a m e t e r {(tr)2}~ obtained by Thouless, A n d e r s o n and PalmerS). This suggests strongly that the m a x i m u m of the free energy of the spin-glass state 7) of SK is attributed not to the replica method, but to the m e a n field approximation. By taking a closed loop instead of the pair in our cluster approximation, we expect that the energy of the paramagnetic state will be raised c o m p a r e d with the value of the ferromagnetic ground state and the situation will be improved. We do not k n o w at present which of the improved paramagnetic state or the spin-glass state will have the lower energy. If still the f o r m e r is the case, the spin-glass state is to be regarded as a metastable state. The variational energy as a function o f / x and T/ allows the spin-glass state or the ferromagnetic state to o c c u r at least as a metastable state according to the concentration p a r a m e t e r x. The spin-glass state c o r r e s p o n d s to the local minimum of the energy with respect t o / z and ~7 for 0 ~< x ~<13for z = 4, and to the inflection point with respect to # and the local minimum with respect to ~7
42
s. INAWASHIRO AND S. KATSURA
f o r 0 ~< x ~ ~s f o r z = 3. T h e f e r r o m a g n e t i c s t a t e h a s a l o w e r e n e r g y t h a n t h a t o f t h e s p i n - g l a s s s t a t e f o r as < x - < ~ f o r z = 3, a n d f o r i3 < X ~~< l2 f o r z = 4, a n d is c o n n e c t e d t o t h e s p i n - g l a s s s t a t e a n d t h e p u r e f e r r o m a g n e t i c s t a t e at t h e b o t h e n d s o f t h e r a n g e o f x, r e s p e c t i v e l y .
Acknowledgements The authors acknowledge valuable discussions with Professor P r o f e s s o r F. T a k a n o , S. F u j i k i , a n d Dr. F. M a t s u b a r a .
Appendix I z=3:
¢,(~,,
n) = ~,'-
5 ~ ' + ~ , 2 _ ~ _ 4~
+ r/2{ - 2a/z2 + 3/z + 21- 2x(1 + / z ) 2} + ,/2 4' 4~(/~, rl) -_ ~/z2 _ 2/z T± 2 l- - 2 1 1_2 / ,
~5~(/z,r/) = T/{I - 2x(l + 7/)}. Spin-glass solution is type (b),/~ = 31.
¢,(~,0)
-
23
-- 27,
g~Z-~)(h) =
~ 8 ( h ) + 9z[8(h - J ) + 8(h + J ) ] + ~9[tS(h - 2 J ) + 8 ( h + 2J)].
z=4: I//(~U,,, n ) ----" '~'4t "L6 -- 15/'L5 -'}- ~4t Z4 -- 5 t `£3 "~ 4~jU"2 --
-- T~2(2]~-t.L4 -- 9/z 3 - ~) - ~4,}~4(1 - 3 / z 2) -q- X,0 2(-- 2~/.£4 -- 9/z 2 - ~ + 3~72 + 3/z 2r/2 - 211~4),
~b2(/z, r/) = r/{1 - x(3/x 2 + 3 - r/2)}. S p i n - g l a s s s o l u t i o n is t y p e (a).
¢,(0, 0) =
g~-')(h)
3 --4~
= 8a[~(h - J ) + 8(h + J ) ] + 8~[8(h - 3 J ) + ~(h + 3J)].
T. M o r i t a ,
SPIN-GLASS STATE OF A BINARY MIXTURE AT T =0
43
Solution o f t y p e (b). g~,~-l)(h) = }8(h) + ~ [ 8 ( h - J ) + 8(h + J)] + ~ [ 8 ( h - 2J) + B(h + 2J)] + ~ [ 8 ( h - 3J) + B(h + 3J)].
z=5:
~(~, 0) = ~ , ' - ~ 7 + ~ , _ +~,,
~ _~.
~,3+~,2_~
Spin glass solution is t y p e (b),/~ = 0.22879296. 4J(0.2287929, O) = 0.676531626, g~-~)(h) = 0.22879298(h) + 0.175887918(h - J ) + 8(h + J)]
+ 0.13513514218(h - 2J) + 8(h + 2J)] + 0.05247175715(h - 3J) + 6(h + 3J)] + 0.02210874518(h - 4 J ) + 8(h + 4J)].
z=6: ~ ( ~ , O) ----~ "
~0 _ Z,Z~..s 0,9 ---'-~,6~" s -- s ' ~
- ~ + ~ , - ~ +
7 + 3"~ ~"
~g2-~.
Spin-glass solution is t y p e (a). 4,(0, O) = - ~, g~'-~(h) = ~[B(h - J ) + 8 ( h + J)] + ~ [ 8 ( h - 3J) + 8(h + 3J)]
4- ~ [ 8 ( h -- 5J) q- t$(h + 5J)].
A p p e n d i x II T h e coefficients of (5.4) and (5.5) are given as follows: z=3: a =6/z -2, /3 = - 3 ' = 2"0, 8 = 2(tz + 1). I °t 3"
~1 = 4(3/z2 + 2/x + r / 2 - 1),
Z=4: a = 3(5/z 2 -- 4/X + 1 -- .q2),
44
S. INAWASHIRO AND S. KATSURA
/3 = - 3' = 6~n, a = 3(tz 2 + 1 - n2). ~l = 914/~'+ r/2 + { ( / z - 1)2 + rl 2}2].
Appendix III T h e c o r r e s p o n d e n c e of the notations of the p r e s e n t p a p e r and the p r e v i o u s p a p e r ( K I F ) is as follows. T h e e n e r g y for the s y m m e t r i c case (gn = g-n in (4.12)) is a n n o u n c e d in K I F . [By a p p l y i n g (4.10) to (A.8) in K I F we get (A.9) in KIF.] Present paper
J'
KIF
.1/2 JA/2 H~
J
hik
h!Z 1) ~hik
flh~ z-I) gtZ-l'(hi)
H~'
Li
Lik
G*(H*)
Present paper KIF Present paper KIF
g(hik) G*(H/~)
go g~-'~ tz
a.
X~ g~ -')
~, g~-')
n> 0
n<0
a
v
rl 2q/z'
2f2o/J 2flo/J ~b x 4E3/7.JAN 4(E1 + E2)/ZJAN 4EG/:ZJAN q (A.7)
(8.2') + (A.4)
(A.9)
References 1) S.F. Edwards and P.W. Anderson, J. Phys. F 5 (1975) 965. 2) D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1792. 3) F. Matsubara and M. Sakata, Prog. Theor. Phys. 55 (1976) 672. See also S. Katsura and S. Fujiki, J. Phys. C 12 (1979) 1087. 4) T. Morita, Physica 98A (1979) 566. 5) S. Katsura, S. Inawashiro and S. Fujiki, Physica 99A (1979) 193. 6) F. Matsubara, Prog. Theor. Phys. 51 (1974) 1694. 7) K.H. Fischer, Solid. St. Commun. 18 (1976) 1515. See also J. Chalupa, Phys. Rev. 17 (1978) 4335. 8) D.J. Thouless, P.W. Anderson and R.G. Palmer, Phil. Mag. 35 (1977) 593. 9) R. Fisch and A.B. Harris, Phys. Rev. Lett. 38 (1977) 785. 10) A.J. Bray, M.A. Moore and P. Reed, J. Phys. C 11 (1978) 1187. A.J. Bray and M.A. Moore, J. Phys. C 12 (1979) 79. 11) B.W. Southern, A.P. Young and P. Pfeuty, J. Phys. C 12 (1979) 683 and references therein.
THE ENERGY OF THE SPIN-GLASS STATE OF A BINARY MIXTURE AT T = 0 AND ITS VARIATIONAL PROPERTIES Sakari INAWASHIRO and Shigetoshi KATSURA
Department of Applied Physics, Tohoku University, Sendai, Japan Received 17 Jury 1979
In the random-bond model of Ising spins, the concept of a multiple-bond distribution of effective field was introduced in the pair approximation. The integral equation for a single-bond distribution was derived intuitively. The variational energy at T = 0 is expressed in terms of two parameters /~ and ~ where /~ is the probability of zero effective field in the single-bond distribution and v/ is the magnetization per spin. For 7/= 0, the energy of the spin-glass state corresponds to a local minimum as a function of /z, for an even z (number of the nearest neighbours) and to an inflection point for an odd z. It was shown that the spin-glass state corresponds to a local minimum with respect to /~ and y/ for z = 4, to an inflection point with respect to/~ and a local minimum with respect to 7/for z = 3. It is conjectured that the maximum of the energy of the spin-glass state of Sherrington and Kirkpatrick is attributed not to the replica method, but to the mean field approximation. Stationary properties of the energy as a function of both ~t and ~ were examined in detail.
1. Introduction E d w a r d s a n d A n d e r s o n ~) p r o p o s e d a s p i n - g l a s s m o d e l in w h i c h s p i n s a r e r a n d o m b u t f r o z e n . S h e r r i n g t o n a n d K i r k p a t r i c k 2) ( r e f e r r e d as S K h e r e a f t e r ) s o l v e d a r a n d o m Ising m o d e l w i t h infinitely l o n g - r a n g e d i n t e r a c t i o n b y t h e r e p l i c a m e t h o d s h o w i n g t h e s p i n - g l a s s state. A b i n a r y m i x t u r e o f a r a n d o m I s i n g m o d e l w a s a l s o s h o w n to h a v e t h e s p i n - g l a s s s t a t e ( g l a s s - l i k e p h a s e , G L P ) b y M a t s u b a r a a n d S a k a t a 3) i n d e p e n d e n t l y o f EA~). M o r i t a 4) p r o p o s e d a v a r i a t i o n a l f r e e e n e r g y o f a r a n d o m - b o n d m o d e l in a p a i r a p p r o x i m a t i o n a n d d e r i v e d a n i n t e g r a l e q u a t i o n f o r a d i s t r i b u t i o n f u n c t i o n o f t h e e f f e c t i v e fields. K a t s u r a , I n a w a s h i r o a n d F u j i k i 5) ( r e f e r r e d as K I F h e r e a f t e r ) s h o w e d t h a t t h e i n t e g r a l e q u a t i o n b y M o r i t a 4) is e q u i v a l e n t to t h a t i n t r o d u c e d b y M a t s u b a r a 6) p r e v i o u s l y , a n d t h e y d e r i v e d t h e r e s u l t o f S K 2) f o r a l o n g - r a n g e d i n t e r a c t i o n . F u r t h e r t h e y 5) o b t a i n e d t h e e n e r g y o f the s p i n - g l a s s s t a t e at T = 0 f o r a s h o r t r a n g e d b i n a r y m i x t u r e o f f e r r o - a n d a n t i f e r r o m a g n e t i c b o n d s , b y s o l v i n g the integral e q u a t i o n o f t h e p a i r a p p r o x i m a t i o n e x a c t l y . As for the stability of the spin glass state, much controversies are debated a m o n g a u t h o r s . F i s c h e r 7) s h o w e d t h a t t h e v a r i a t i o n a l f r e e e n e r g y is n o t m i n i m u m b u t m a x i m u m in the s p i n - g l a s s state. T h o u l e s s , A n d e r s o n a n d 24
SPIN-GLASS STATE OF A BINARY MIXTURE AT T = 0
25
Palmer s) modified SK theory so as to have zero entropy at T = 0. Their free energys) of the spin-glass state also has a maximum below the glass transition temperature, and a minimum in the paramagnetic state. The lower critical dimensionality is supposed to be 4 by Fisch and Harris 9) and by Bray, Moore and Reed~°). Southern, Young and Pfeuty H) did not obtain the spin glass transition in two dimensions but did in three dimensions. The purpose of the present paper is to obtain the energy at T = 0 of the spin-glass phase as well as the ferromagnetic phase of a binary mixture and to investigate the stability of these phases. In section 2, we give a simplified derivation of the integral equation making clear the meaning of the pair approximation. In section 3, a variational form of the solution is introduced, and a set of algebraic equations are derived from the integral equation. In section 4, the free energy is expressed in terms of two variational parameters. In section 5 the solutions are classified into several categories, and the stationary properties of these solutions are examined. Section 6 is devoted to conclusions and discussions.
2. Distribution of effective fields
We consider a spin in an Ising model with random exchange-interactions between nearest neighbours. The number of the nearest neighbours of a lattice site is denoted by z. In an effective field approximation, the spin is ~onsidered to be placed in an effective field, i.e., a z-bond field which is ~onsisted of z contributions from the nearest neighbours. The contribution through a single bond is called a single-bond field. Then the z-bond field is equal to the sum of z single-bond fields. Then an effective Hamiltonian of the spin is given by HI = -trjh ~ --trl
~ . hlk,
(2.1)
k=l
where h
where
(2.2)
26
S. INAWASHIRO AND S. KATSURA Z--I
h~-'~ = ~--I hik, (i = 1, 2),
(2.3)
which is a sum of ( z - 1) single-bond fields from the outside of the pair upon the ith spin of the pair. N o w we consider a spin of the pair. The effect of the other spin through the exchange bond Y' is represented by a reduced field, i.e.,
h = (1//3) tanh-| [tanh /3J' tanh(/3 ~=th2k) ].
(2.4)
Due to a random distribution of the exchange integral, the single-bond field has a distribution denoted by g(h~k). A pair approximation requires a selfconsistent condition that the distribution of the reduced field is equal to that of the single-bond field, i.e., g(h)=
f 3[h-(l//3)tanh-'Itanh /3J' tanh(/3 ~
h2k)}]
Z--I
× P(J') dJ' l-I g(hEk)dhzk,
(2.5)
k=l
where we have assumed that the distribution of the single-bond field is independent of the location of spins and bonds. A distribution of M-bond field is defined by
g(M'(h) = f 6 ( h - k~,= l h,k ) kfi= l g(h,D dh,k,
(2.6)
Then the integral equation (2.5) is rewritten as
g(h) = f 8[h -
(1//3) tanh-~{tanh/3J' tanh/3h'}]P(J')
dJ'gtZ-~)(h') dh'.
(2.7)
It was shown by Morita 4) that the integral equation (2.5) is derived through the variational principle from a free energy of the random system, which is given by f = f, + f2,
(2.8)
where
Is = (- 11/3)f
ln[Tr{exp(-
z-1
z-I
k=l
m=l
/3HI2)}]P(J') dJ'
× 1-I g(h,k) dhnk FI g(h2m)dh2m,
(2.9)
and .f2 = (2//3)(1 -
l/z) f
ln[Tr{exp(-/3H1)}1 l~I g(h~k) dhlk. k=l
(2.10)
SPIN-GLASS STATE OF A BINARY MIXTURE AT T--0
27
In this paper, we consider a binary mixture, where the distribution of the exchange interaction is given by
P ( J ' ) = p6(J' - J) + (1 - p ) 8 ( J ' × J),
(J > 0),
(2.1 I)
where p and ( l - p ) represent the concentrations of the ferromagnetic and antiferromagnetic bonds, respectively. The integral equation (2.7) is written as
g ( h ) = p f 6[h - (fiB) tanh-I{tanh OJ tanh flh ,}]g(Z-I)( h ,) dh' + (1 - p ) f 6 [h + ( 1/B) tanh-I{tanh flJ tanh flh'}]gtZ-~)(h') dh'. (2.12) Using the distributions of ( z - 1)-bond and z-bond fields and the distribution of the exchange interaction, fl and [2 are expressed as /~ = -(1/fl) f [p ln{exp(flJ)2 cosh fl(hl + h2) + exp(-/3J)2 cosh fl(hl - h2)} + (1 - p) ln{exp(-/3J)2 cosh fl(hl + hz) + exp(flJ)2 cosh fl(hl - hz)}]
× g~-l)(hOg~Z-l)(h2) dhl dh2,
(2.13)
and f2 = (2/fl)(1 - I/z) f In(2 cosh flh)gtZ)(h) dh.
(2.14)
(2.13) and (2.14) are the free energy used in KIF.
3. The integral equation at T = 0
At absolute zero temperature, the distribution of the single-bond field is assumed to be given by
g ( h ) = go6(h) + g16(h - J) + g_~6(h + J),
(3.1)
where the normalization condition requires go + gl + g-i = I.
(3.2)
From the definition (2.6), the form of the M-bond distribution at absolute zero is given by M
g~m)(h)= ~
g~,m)8(h-nJ).
(3.3)
n=-M
The coefficient gtm) is clearly given by the expansion coefficient of the
28
S. INAWASHIRO AND S. KATSURA
following polynomial expansion, M
(go + glx + g-ix-t) u = ~
g~MJx".
(3.4)
n=-M
The expression of g~M) in terms of go, gl and g_l is obtained in a similar way to KIF. The result is
I (Mn )gOM-, g'~2F, [ ( n 2 M ) , ( n - M + I
2
g(M, =
l);n+ l;4g'g-l~, ---~o /
[
M ~ n >~O,
{M'~ _M-I,I ([nI-M) (InI-M+ 1 1;4glg_,'~, \lnlJ ~o g~l 2F, 2 ' 2 ; Inl + g~ ) O>n>~-M,
(3.5)
where 2F~ denotes the hypergeometric function. Let us rewrite the integral equation (2.12) in terms of the coefficients g, and g~-J~. Using a limiting formula tim ~1 tanh- i (tanh/3hl tanh/3hz) = sgn(h,) sgn(hz) min(lh~l, Ihzl).
(3.6)
We see eq. (2.12) becomes at absolute zero
g(h) = p f 6[h - sgn(h') min(J, Ih'l)]g~Z-~)(h') dh' + (1 -
p) f 6[h + sgn(h') min(J, Ih'l)]g~Z-1)(h ') dh'.
(3.7)
Substituting the expression (3.3) into the rhs of (3.7), we obtain an equation
k
n>0
n> 0
n> 0
J
n> 0
From (3.1) and (3.8), the integral equation is reduced to a set of two algebraic equations,
go - g~Z-~ = 0,
(3.9)
gl - g-1 - (2p - 1) ~ (g~-l) _ g ~ l ) ) = 0.
(3.10)
n>O
Since g~z-l), g~-~) and gtSg~) are expressed in terms of go, gl and g_~ using (3.5), the equations (3.9) and (3.10) are used in order to determine go, g~ and g_~. Actually they form two equations for two unknowns go and g ~ - g_~. (3.9)
S P I N - G L A S S STATE O F A B I N A R Y M I X T U R E AT T = 0
29
and (3.10) agree with (7.20) and (7.20') in KIF. The solution of these equations are sufficient to the stationary conditions for the free energy. We will discuss the equivalence for the case of z = 3 and z = 4 in the next sections.
4. An expression of the free energy at T = 0
The form of g(h) given by (3.1) is used as a trial function in the variational principle. Substituting (3.1) into (2.13) and (2.14), and taking the limit of/3 --* o% we obtain flo~limfl
=
~
~
{pA(,,,+).+(1-p)A(~).,}g(.~-I)g (z-'),
(4.1)
m=-z+l n=-z+l
where A ~ ) = lim(l//3) ln{exp(_/3J)2 cosh fl(mJ + nJ) + exp(~flJ)2 cosh fl(mJ - n J)}
= ImlJ + InlJ + Amn,(+-)
(4.2)
~+-J, mn >0, A~)~ = 1 J, mn = O, ~J, mn <0,
(4.3)
and f2o - l i m / 2 t3-~
~
=2(1-1[z)J
[nlg~ ).
(4.4)
n=-z
The equation (4.1) is rewritten as
f,o = - J + =-J
n
+1
z-I
Inlg~ -')
z--I
E
E
n'~A(-)~°(z-1)°(z-I) ]
m=--z+l n=--Z4-1 I Z--I
2 N
{PA(,.+.)+(1-,',--,.-,~,m
~,
j
InlgT-')+ 2ggZ-')-(gg~-')) 2
n=--z+l
(4.5)
30
S. INAWASHIRO AND S. KATSURA
The identity Z-I
g ~ x ~ = (go + g l x + g - i x -j) n=-2
~
o~-~),~"
n=-2+l
leads to a recurrence relation, g~)= ~m - g,,s,-m. ,,(z-l) Then we obtain In[g~) = go Y~ {mlg~-l' + gl ~ Im + l[g~-l)+ g_ 1 ~] Im - l l g ~ - ' ~ n=--z
m
m
z--I
=
~,
{nigh-" + (1 - go)g~z-l) + (gl - g - 0 ~ (g~-l)_ g~Z~l)). (4.6)
n=-z+l
n>0
Then flo is rewritten as
/10 = - J [2 , ~ Inlg~' + (2g0- g~-l))g~-l, - 2(gl - g-l)
~
(g?-l)
_ g~Z-1))
n> 0
-1)/Y.
1'-
(4.7)
tn>0
3 J
Differentiating both sides of the identity relation (3.4), and multiplying by x, we obtain M
M(go + glx
+ g-IX-1)M-1(glX
--
g - i x -1) = ~ ,
ng~,M)x".
(4.8)
n=-M
The left hand side of (4.8) is expressed by gtM-l~ according to the definition (3.4), and then a recurrence relation is derived by comparison of the coefficients of the nth power of x, i.e., ng~.u) = M ( g l g t , M_~~)- g_lg~,M+~l)).
(4.9)
Carrying out the summation over n in (4.9), we have Inlgt,M) = M
(1 --go)g~M-l) + glg~_~-') + g_lgt u-l)
n=-M
+ (g, - g-l)
E (g(M-l)_g ~ - l ) ) /d"
(4.10)
n>0
Using (4.4), (4.7) and (4.10), we obtain an expression of the variational free energy, i.e. .fo = flo +.f2o = JqJ(go, gl, g-l)
(4.11)
where ~b(go, gl, g-~) = - 2 g ~ ~-1) + (g~-~))2 _ (1 - go)(gl ~-1) + g~_~-~)) + ( g t - g - O ( g t ~ - 1 ) - g ~ i -I)) - ( 2 p - 1)
( g ~ - l ) _ g~21))
.
(4.12)
SPIN-GLASS STATE OF A BINARY MIXTURE AT T = 0
31
5. Stationary properties of the energy at T = 0
T w o n e w v a r i a b l e s / £ and ~ are i n t r o d u c e d b y
g0=/£,
gl-
(1 - / £ + 71) 2 ,
g-l=
(1 - / £ - 77) 2
(5.1)
/£ r e p r e s e n t s the p r o b a b i l i t y of the z e r o effective field in the single b o n d distribution and 7} r e p r e s e n t s the m a g n e t i z a t i o n per spin. /£ is a m e a s u r e of the spin glass o r d e r p a r a m e t e r q[--- (o-) 2 = I - g~Z)], w h e r e / £ = 1 f o r q = 0 w h e n ~7=0. For example q=1-[/£3+(2~)/£(1-/£)2] for z=3 and q= 1 [[£4 + 3/£2(1 --/£)2 + (~8)(1 --/£)4] f o r z = 4. -
T h e n all the coefficients g ' s are e x p r e s s e d in t e r m s o f / £ and r/. It is to be n o t e d that the p h y s i c a l regions o f / £ and W are given b y - 1 ~
(5.2)
(5.3)
T h e solutions of ~bK/£, 7 ) = 0 and ~b2(/£,7/)=0 or o f G 0 . t , r / ) = 0 and ~ ( / £ , r / ) = 0 are classified into s e v e r a l categories. (a) ~7 = 0, /£ = 0; (b) ~7 = 0, /£ # 0,/£ # 1 ; (c) ~7 = 0,/£ = 1 ; (d)/£ = 0, r / ~ 0; (e)/£ ~ 0, 7/# 0; (f) the solutions of ~b~ = 0 and 4~ = 0 which do not satisfy ~bl = 0 and ~b2 = 0. T h e solution rl = 0 c o r r e s p o n d s to a s y m m e t r i c distribution of the effective field. T h e solutions (a) and (b) c o r r e s p o n d to spin-glass states, (c) to the p a r a m a g n e t i c state, and (d) and (e) to f e r r o m a g n e t i c states. T h e f u n c t i o n s ~b(/£, r/), ~bl(/£, r/), and t~2(/£, "Q) are e x p r e s s e d by p o l y n o m i a l s o f / £ and 7} and are g i v e n f o r z = 3 and 4 in a p p e n d i x I t o g e t h e r with ~b(/£, 0) for z = 5 and 6. In this section and in a p p e n d i x I, x - p - ½is used instead of p to r e p r e s e n t the c o n c e n t r a t i o n . First we c o n c e n t r a t e on the s y m m e t r i c distribution of effective field • c o r r e s p o n d i n g to ~7 = 0. T h e n the variational e n e r g y is a f u n c t i o n of only /£, and is plotted f o r z = 3, 4, 5, 6 in fig. 1. It is to be n o t e d that the spin glass solutions of t y p e (a) h a v e a local m i n i m u m f o r z = 4 and 6 (in general, f o r e v e n z) and the spin-glass solutions of type (b) h a v e inflection points f o r z = 3 and 5 (in general f o r o d d z). W h e n w e plot V'-zE/2JN as a f u n c t i o n o f z, the energies of the spin-glass states f o r z = 3, 4, 5, 6 lie a l m o s t on a straight line which e x t r a p o l a t e s to the value of the infinitely long-ranged spin glass of
32
S. INAWASHIRO AND S. KATSURA 2E, I Z 3
-0.6 (3
-0.7
Q, -0.8
-09
-IO
0.5
1.0
/.z
Fig. I. The variational energies ~(#, 0) as functions of # for z = 3, 4, 5 and 6. For z = 3 and 5, the spin-glass solutions of type (b) have inflection points. For z = 4 and 6, the spin-glass solutions of type (a) have local minima. S h e r r i n g t o n a n d K i r k p a t r i c k 2) (fig. 2). T h e e n e r g i e s o f t h e a b o v e s o l u t i o n s a r e p l o t t e d a g a i n s t the c o n c e n t r a t i o n in fig. 3. T h e s p i n - g l a s s s o l u t i o n (a) has t h e l o w est e n e r g y f o r 0 ~< x ~< 8a (z = 3), a n d 0 ~< x ~< ~ (z = 4), a n d t h e n t h e f e r r o m a g n e t i c s o l u t i o n (d) f o r a8~< x ~< ~ (z = 3) a n d 31~< x ~< ~ (z = 4), e x c e p t t h e s o l u t i o n (c). F e a t u r e s o f s o l u t i o n s f o r e v e n z a n d for o d d z are a little d i f f e r e n t . W e a n a l y z e t h e c a s e s o f z = 3 a n d o f z = 4 in detail. F o r z = 3 w e h a v e t h e f o l l o w i n g f o u r s o l u t i o n s (cf. figs. 4 - 6 ) . (1) 77 = 0 , / z = ~ ( t y p e (b))
.~21// all ~
0'
~21// __8(3--8X) arl 2
9
t~2~/ = 0 , '
al~071
t~3~// - 1 2 < 0 . a# 3 =
T h e s o l u t i o n c o r r e s p o n d s to a n i n f l e c t i o n p o i n t w i t h r e s p e c t to /z, a n d t h e m i n i m u m f o r 0 < x < 8~ a n d t h e m a x i m u m f o r x > • w i t h r e s p e c t to r/. (2) r / = 0 , / z = 1 ( t y p e (c)) az@ = 4 > 0,
c~2--~- 16(41- x),
al~2
3,rl 2 -
02~/ 81x8 ~
= 0.
SPIN-GLASS
S T A T E O F A B I N A R Y M I X T U R E AT T = 0
33
~ E / 2N...T 0
6
0.1
5
3
4
0.2
I/Z
0.~
Fig. 2. The energy of the spin-grass state as a function of z. The energy is scaled by V~z for comparison with the value of Sherrington and Kirkpatrick.
T h e e n e r g y is t h e m i n i m u m w i t h i n t h e p h y s i c a l r e g i o n , - 1 ~ g --- ~ ~< I. (3) /x = (1 - 2x)[2x, n -- *_V'(Sx - 3)(4x - l ) / 2 x ( t y p e (e)). T h e real s o l u t i o n exists for ~
,~
~<~.
_
- ( 8 x - 3){~ - 8 ( x - b ~} > o,
0zt/, ( 8 x - 3 ) ( 4 x - 1) > 0, 0~72 = 2x 2
q.,:
Z=4 b
C
e
-0"81
Z=3 -0.9
-I.0 -0.2
-
--~0.3
i
O.4
0.5 X
Fig. 3. The energies of several states as functions of concentration (x -= PA - ~)- (a], (b), (c), (d), (e) and (f) denote the respective solutions. A f x = ]), B(.r = }) C(x = ~) D ( x = ~) denote junctions of solutions. The energies of (e) and (f) can not be distinguished in this figure.
34
S. I N A W A S H I R O
A N D S. K A T S U R A
-08
-09
c
0.5
f
f,.o
/ /
/
/ /
0.5
/ / / /
/ /
1.0
~7 Fig. 4. ~b(/x, "0), z = 3, x = 0, (b) saddle point, (c) m m .
026 _ (8x - 3)7/ O0.OT/
x
a26 00. z
az4~ O0.O'q
azO
a2q~
>0
for
~s
0~0.a ,},.j. C~'F/2 T h e s o l u t i o n c o r r e s p o n d s to a local m i n i m u m o f t h e v a r i a t i o n a l e n e r g y .
2 ~ / 1 4 x - 3 + ( 4 x - 6 ) X / 1 - 2x
I - 2x + 2X/I - 2x
(4) 0 ` =
2x+3
'
"q= -~ -
2x+3
(a ~ x ~< ~), ( t y p e (f)), a20 270`z_300`+7+Tt2(_3_4x)
forsa~
'
SPIN-GLASS STATE OF A BINARY MIXTURE AT T = 0
35
q~
b
/)
c
-I.C
/
J
J
P-
j
J
~:/
0.5
//://
1.0
77 Fig. 5. 4'(~, ~t), z = 3, 0 < x(= 0.35) < ], (b) saddle point, (c) min.
O,!~b= {_3/z2 + 6/z + I - 4x(l +/~)2 + 3r/2} = 2.02 > O, 3,r/2 a2~b = 2~{-3/z + 3 - 4x(l + ~)}, arl3gt T h e solution c o r r e s p o n d s to a saddle point o f the variational energy. W e h a v e no solutions o f the t y p e (a) and the t y p e (d) f o r z = 3. F o r z = 4, w e h a v e the following five solutions (cf. figs. 7 - I 1). (1) g = 0 , ,?--0 (type(a)), a2~b = a2 > 0,
a2--~0- 3(1 - 3x),
0tl. 2
O~l 2 -
O2~b = 0. Ol~aTi
T h e e n e r g y is a local m i n i m u m f o r x < l3 c o r r e s p o n d i n g to the spin-glass p h a s e ( G L P ) , and a saddle point f o r x > 31. (2) /~=51, 7 / = 0 (type(b)),
ou~
~ -
(1 - ~ x ) ,
-
Ol~ O'q
= o. 25
T h e e n e r g y is a m a x i m u m f o r x > ~, and a saddle point f o r x < 78.
36
S. I N A W A S H I R O
A N D S. K A T S U R A
j0 \
IO
~7
Fig. 6. ~b(/z, "0), z = 3, 8~ < x ( = 0.4) < ½, (b) max., (e) min., (f) saddle point.
(3) / x = l ,
r/=0
(type(c)),
32~b= ~9/z2
12>0,
c~2---~-~-6(1-6x), &0 2 -
8-2~b = 0 . 3tzar/
O n a physical b o u n d a r y , s a y , / z + 7 / = _ 1, the f u n c t i o n qJ b e c o m e s ~b = - 1 + (1 - 2x)(1 - / z 3 ) 2, which s h o w s that the e n e r g y b e c o m e s the absolute m i n i m u m at /x = 1, and r / = 0, e x c e p t the case o f x = ½, i.e., the pure f e r r o m a g n e t , w h e r e the e n e r g y is c o n s t a n t along the b o u n d a r y # + 7 / = - 1. (4) / x = 0 , r / = _ _ _ X / 3 - 1 / x , (x>l~) ( t y p e ( d ) ) , a2O _ 24(x - ½)(x - ~) tg/.t 2 --
at/2
X2
x
,
S P I N - G L A S S STATE O F A BINARY M I X T U R E AT T = 0
37
q,
-0.9
/z // J jj
/
J
1.0
~7 Fig. 7. ~b(/.~,n), z = 4, x = 0, (a) min., (b) saddle point, (c) min.
and f 1 2 ~ = 0.
The solution represents the ferromagnetic phase. The energy is a local minimum for ~ < x < sa, and a saddle point for sa
(~ ~< x ~< ~)
(type (e)).
This solution is c o n n e c t e d with the solution (b) at q - - ~ 78, and to the solution (d) at x = ~. s It is c o n c l u d e d that the solution c o r r e s p o n d s to a saddle point. W e have no solutions o f the type (f) for z = 4.
38
S. INAWASHIRO AND S. KATSURA -0"
b
..3.~,~°, / -09
-I l '
i
kc
o.s
f'i'~o
/
. / / //
///
/
./ /
/. /
1.0 ~7
Fig. 8. ~b(/z,"0), z =4, 0
SPIN-GLASS STATE OF A BINARY MIXTURE AT T = 0
39
-0.7
(3
// I(~"
-0.9
\
r
-IO/0
. . . . z
ff
/
//./" /z
//
/ J
I1
I,O
Fig. 9. ~b(g, r/), z = 4, ~ < x(= 0.35)< ], (a) saddle point, (b) max., (c) rain., (d) min., (e) saddle point.
linear c o m b i n a t i o n s o f ~bl and ~b2, i.e., G ~ ot,bl +/3~b;,
(5.4)
G ------r4'~ + 84'2,
(5.5)
w h e r e t~, /3, y and 8 are linear p o l y n o m i a l s o f /~ and ~ f o r z = 3, and are s e c o n d - o r d e r p o l y n o m i a l s f o r z = 4, as given in a p p e n d i x II. It is s h o w n that the coefficient d e t e r m i n a n t o f the s i m u l t a n e o u s e q u a t i o n s (5.4) and (5.5) is positive definite f o r z = 4, e n s u r i n g the e q u i v a l e n c e o f ~b~ --- ~,~ = 0 and ~bl = 4,2 = 0. O n the o t h e r hand, the coefficient d e t e r m i n a n t f o r z = 3 is not a l w a y s n o n - z e r o , w h i c h allows the solution o f the t y p e (f) to c o m e f r o m ~ , = ~b~ = 0 w h i c h are o f higher o r d e r in /~ and ~/ than 4,~ = 4,2 = 0. T h e e n e r g y o f the solution (f), h o w e v e r , is quite close to that o f (e). T h e coefficient d e t e r m i n a n t s f o r z = 3 and 4 are also s h o w n in a p p e n d i x II.
40
S. I N A W A S H I R O A N D S. K A T S U R A
-07 I C]
//
/
A~//
\ 0.5
o p.
/o5
1.0 Fig. 10. Ik(#, n), z = 4, l < x ( = 0.4) < ~, (a) saddle point, (b) max., (c) min., (d) saddle point.
6. Conclusions The integral equations for the single-bond distribution was derived intuitively making clear the physical meaning of the pair approximation. Further, we have introduced the concept of multi-bond distribution of the effective fields, which will be useful for further development of a clutter approximation. For the binary bond mixture, the exact f o r m of the single-bond distribution at T = 0 is represented by a linear combination of 8-functions with two variational p a r a m e t e r s / x and n. The variational energy as a function o f / z with a fixed value of 9 = 0 has a local minimum at /x = 0 for an even z, and an inflection point a t / x # 0 for an odd z, corresponding to the spin-glass states. It has a minimum a t / x = 0 which represents the paramagnetic state, of which the energy is equal to the ferromagnetic ground state. F r o m the curves of the energy for z = 3, 4, 5 and 6 shown in fig. 1, it is conjectured that the stationary point of the type (b) a p p r o a c h e s to /z = 0 for both even and odd z as z increases, and that the
SPIN-GLASS STATE OF A BINARY MIXTURE AT T = 0
41
d~c 1.0
~7 Fig. 11. ~b(/.t, 71), z = 4, x = ~, (a) saddle point, (b) max., (c) min., (d) min.
energy has only one m a x i m u m at 9. = 0 and only one minimum at/~ = 1 in the limit z ~ ~. The characteristic feature of this limiting c u r v e agrees with those of the free energy as a function of the p a r a m e t e r {(tr)2}~ obtained by Thouless, A n d e r s o n and PalmerS). This suggests strongly that the m a x i m u m of the free energy of the spin-glass state 7) of SK is attributed not to the replica method, but to the m e a n field approximation. By taking a closed loop instead of the pair in our cluster approximation, we expect that the energy of the paramagnetic state will be raised c o m p a r e d with the value of the ferromagnetic ground state and the situation will be improved. We do not k n o w at present which of the improved paramagnetic state or the spin-glass state will have the lower energy. If still the f o r m e r is the case, the spin-glass state is to be regarded as a metastable state. The variational energy as a function o f / x and T/ allows the spin-glass state or the ferromagnetic state to o c c u r at least as a metastable state according to the concentration p a r a m e t e r x. The spin-glass state c o r r e s p o n d s to the local minimum of the energy with respect t o / z and ~7 for 0 ~< x ~<13for z = 4, and to the inflection point with respect to # and the local minimum with respect to ~7
42
s. INAWASHIRO AND S. KATSURA
f o r 0 ~< x ~ ~s f o r z = 3. T h e f e r r o m a g n e t i c s t a t e h a s a l o w e r e n e r g y t h a n t h a t o f t h e s p i n - g l a s s s t a t e f o r as < x - < ~ f o r z = 3, a n d f o r i3 < X ~~< l2 f o r z = 4, a n d is c o n n e c t e d t o t h e s p i n - g l a s s s t a t e a n d t h e p u r e f e r r o m a g n e t i c s t a t e at t h e b o t h e n d s o f t h e r a n g e o f x, r e s p e c t i v e l y .
Acknowledgements The authors acknowledge valuable discussions with Professor P r o f e s s o r F. T a k a n o , S. F u j i k i , a n d Dr. F. M a t s u b a r a .
Appendix I z=3:
¢,(~,,
n) = ~,'-
5 ~ ' + ~ , 2 _ ~ _ 4~
+ r/2{ - 2a/z2 + 3/z + 21- 2x(1 + / z ) 2} + ,/2 4' 4~(/~, rl) -_ ~/z2 _ 2/z T± 2 l- - 2 1 1_2 / ,
~5~(/z,r/) = T/{I - 2x(l + 7/)}. Spin-glass solution is type (b),/~ = 31.
¢,(~,0)
-
23
-- 27,
g~Z-~)(h) =
~ 8 ( h ) + 9z[8(h - J ) + 8(h + J ) ] + ~9[tS(h - 2 J ) + 8 ( h + 2J)].
z=4: I//(~U,,, n ) ----" '~'4t "L6 -- 15/'L5 -'}- ~4t Z4 -- 5 t `£3 "~ 4~jU"2 --
-- T~2(2]~-t.L4 -- 9/z 3 - ~) - ~4,}~4(1 - 3 / z 2) -q- X,0 2(-- 2~/.£4 -- 9/z 2 - ~ + 3~72 + 3/z 2r/2 - 211~4),
~b2(/z, r/) = r/{1 - x(3/x 2 + 3 - r/2)}. S p i n - g l a s s s o l u t i o n is t y p e (a).
¢,(0, 0) =
g~-')(h)
3 --4~
= 8a[~(h - J ) + 8(h + J ) ] + 8~[8(h - 3 J ) + ~(h + 3J)].
T. M o r i t a ,
SPIN-GLASS STATE OF A BINARY MIXTURE AT T =0
43
Solution o f t y p e (b). g~,~-l)(h) = }8(h) + ~ [ 8 ( h - J ) + 8(h + J)] + ~ [ 8 ( h - 2J) + B(h + 2J)] + ~ [ 8 ( h - 3J) + B(h + 3J)].
z=5:
~(~, 0) = ~ , ' - ~ 7 + ~ , _ +~,,
~ _~.
~,3+~,2_~
Spin glass solution is t y p e (b),/~ = 0.22879296. 4J(0.2287929, O) = 0.676531626, g~-~)(h) = 0.22879298(h) + 0.175887918(h - J ) + 8(h + J)]
+ 0.13513514218(h - 2J) + 8(h + 2J)] + 0.05247175715(h - 3J) + 6(h + 3J)] + 0.02210874518(h - 4 J ) + 8(h + 4J)].
z=6: ~ ( ~ , O) ----~ "
~0 _ Z,Z~..s 0,9 ---'-~,6~" s -- s ' ~
- ~ + ~ , - ~ +
7 + 3"~ ~"
~g2-~.
Spin-glass solution is t y p e (a). 4,(0, O) = - ~, g~'-~(h) = ~[B(h - J ) + 8 ( h + J)] + ~ [ 8 ( h - 3J) + 8(h + 3J)]
4- ~ [ 8 ( h -- 5J) q- t$(h + 5J)].
A p p e n d i x II T h e coefficients of (5.4) and (5.5) are given as follows: z=3: a =6/z -2, /3 = - 3 ' = 2"0, 8 = 2(tz + 1). I °t 3"
~1 = 4(3/z2 + 2/x + r / 2 - 1),
Z=4: a = 3(5/z 2 -- 4/X + 1 -- .q2),
44
S. INAWASHIRO AND S. KATSURA
/3 = - 3' = 6~n, a = 3(tz 2 + 1 - n2). ~l = 914/~'+ r/2 + { ( / z - 1)2 + rl 2}2].
Appendix III T h e c o r r e s p o n d e n c e of the notations of the p r e s e n t p a p e r and the p r e v i o u s p a p e r ( K I F ) is as follows. T h e e n e r g y for the s y m m e t r i c case (gn = g-n in (4.12)) is a n n o u n c e d in K I F . [By a p p l y i n g (4.10) to (A.8) in K I F we get (A.9) in KIF.] Present paper
J'
KIF
.1/2 JA/2 H~
J
hik
h!Z 1) ~hik
flh~ z-I) gtZ-l'(hi)
H~'
Li
Lik
G*(H*)
Present paper KIF Present paper KIF
g(hik) G*(H/~)
go g~-'~ tz
a.
X~ g~ -')
~, g~-')
n> 0
n<0
a
v
rl 2q/z'
2f2o/J 2flo/J ~b x 4E3/7.JAN 4(E1 + E2)/ZJAN 4EG/:ZJAN q (A.7)
(8.2') + (A.4)
(A.9)
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