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Exactly solvable model of a spin glass
Solid State Communications, Vol. 19, pp. 8 3 3 - 8 3 5 , 1976.
Pergamon Press.
Printed in Great Britain
EXACTLY SOLVABLE MODEL OF A SPIN GLASS T. Morita and T. Horiguchi Department of Applied Science, Faculty of Engineering, Tohoku University, Sendal 980, Japan
(Received 29 March 1976 by Y. Toyozawa) Sherrington and Kirkpatrick presented a solvable model of a spin glass. In the solution, they used a mathematically unwarranted procedure. In the present article, we show that the problem is exactly solved by starting with the virial expansion formula, and confirm the results of Sherrington and Kirkpatrick. The solution is obtained for the random lsing magnet in which the external field of each site and the exchange integral between each pair of sites are random variables. We obtain the exact thermodynamic properties for this system in the limit of nw ~ 0% assuming that the exchange integrals of a spin with O(nw) neighbours are O(n~ 1/2) and the average value of each is O(n~l). The system is found to show the spin-glass state as well as the paramagnetic and the ferromagnetic state.
WE SHALL CONSIDER the Ising system where each site is occupied by a spin and the Hamiltonian is given by
i
i>j
si for each site i takes on + 1 and - 1. In a recent paper, Sherrington and Kirkpatrick 1 gave a solution of the random system where the exchange integral Jij for all pairs i and ] are random variables whose distributions are given by the same Gaussian distribution function, assuming that the average and the average of the square are both O(1/N) where N is the total .number of the lattice sites in the system, h i was assumed to be a constant independent of the site i. They obtained the phase transitions between the paramagnetic, the ferromagnetic and the spin-glass state. The method used in the solution involved a mathematically unjustifiable procedure. In the present note, we show that the problem is solved exactly, without using such a dangerous procedure, for a more general case without assuming the Gaussian distribution for Ji~. We conclude that their results are correct. For the regular system, hi does not depend on i and Jij is a function only of the relative coordinate of the two sites i and/'. IfJij for a fixed i is a small quantity of O(1/nw) for O(n w) lattice sites/' and is zero for other sites j, an argument based on the virial expansion shows that the properties calculated by the molecular field approximation are exact in the limit of n w -+ 00.2 The argument applies equally to the systems where h i depends on i and Jij depend on i and j, if J# for each i is O(1/nw) for O(nw) lattice sitesj and is zero for other sites j. In the present paper, we apply such an argument to the random 833
systems where Jo for each i is O(1/tz 1/2) for O(nw) lattice sites/" and the average of Jij for each of those pairs i and ] is O(1/nw); Ji~ for the other pairs i and ] are assumed to be zero. Then we find that the contributions which remain in the limit n w ~ ~o are again easily evaluated. We shall start with the virial expansion formula for the free energy, applicable to non-uniform Ising systems. If we write the probability of the state si = -+ 1 of the ith site by ½(1 -+ oi), then it is written as a
13F = --(3 ~ hioi + Z ½[(1 + oi) ln (1 + oi) i
i
+ ( 1 - - o i ) ln (l -- oi)-- 21n 2] -- Sum, (2) where t3 = 1[k B T as usual and 1
Sum = ~ ~ ~13Ji, oicri 1
+ ~ 2 Z .~2&( 1 ~- o?)(1 - °7) + . . - ; (3) i
J
Here Jij = 0 when i = ]. If we take the average over the set {hi, Jij}, the two terms written explicitly on the righthand side of (3) give contributions of order N x O(nw) x O(1/nw). The dots ( . . . ) denote terms with more factors 13Jij, or others expressed by more than singly connected diagrams of three or more sites; the factor associated to a bond is 13Jm when it connects the sites k and l. The averages over the set {hi, J~i} of the contributions of those terms are estimated to be N x O(n~/2) and N x O(n~,l), respectively, which drop as n w ~ oo. In the following expressions also, dots denote the terms which give no contribution after the average over the set
Vol. 19, No. 9
EXACTLY SOLVABLE MODEL OF A SPIN GLASS
834
{hi, Jij} and the limit nw ~ ~ are taken. When Jij for each site i were O(1/nw) for O(nw) sites/', the second term of(3) would be estimated to be N x O(nw) x O(1/n~w) and drops as n w -~ ~. Then we would have the molecular field approximation. The virial expansion formula (2), which involves the external field hi only through --/3Eihio i, is stationary with respect to the variation of the distribution functions ½(1 ± oi) and hence of oi .~'4 It follows that 0 = 1 In 1+ oi 2 1--oi
/3ff1 -~ /31122Oi
d¢- . . . .
(5)
i
if2 = ~ ~2 J}(1 -- 4 ) + . . . .
(6)
J The dots on the right-hand side of (4) denote /3~so~ + . . . . which is O(n~I/2). We can regard that (4) determines oi as a function of C/t, qJ2, t)s . . . . . The function will be denoted by f(qJ~, ~2 . . . . ): Oi
Oi :
= f(l~l, ~J2,"" ")"
(7)
When ~2 = . . . = 0, it is given by f ( ~ l , 0 , . . . ) = tanh (/3q~0.
(14)
By definition of the magnetization of the/th site oy = Tr sj e-13/-//Tre -~u, Ooj/~hj =/3(1 -- o~). Similarly we have ~ / O h i = 3(1 - 8~), so that the leading terms of ~2 written explicitly in (13) cancel out. ~Jt, ~ : , . . . depend on the set {hi, Jij}. We shall introduce the probability distribution function P($) of ~ = ~t. It is defined by
where the subscripts h, J of ( . . . ) h , a mean the average with respect to the set {hi, Jij}. When the quantity to be averaged depends only on the set {hi} or {Jij}, the average will be denoted as ( . . . ) h or ( . . . ) j , accordingly. Then the average of an arbitrary function g(oi) of oi is calculated by oo
@(oi))h,d = f g[tanh (/3~)]P(~) d~ + O(nwl/2), -
(16)
oo
by noting (8). We shall now calculate the distribution function P(ff). By the aid of the integral representation of the delta function, we write (15) as follows:
(8) 1
If we apply this formula (7) with (5) and (6) to % we have oi = f ( h j + ~
f(t~l, ~2,'-.)-
(4)
where
~J~ = hi + ~ Jijoj + . . . .
Now we have
Jyk%+...,~]fl~(l--o~)+~
x H (exp ( - iqJij8j + . . . ) ) n , a dq, J
.... ...)
= ~ + O°~J~io i + . . . . Ohj
e iqo ( e - i q h i ) h -¢o
(93
Here
(17)
where the average of the product is replaced by the product of the averages; the terms omitted being estimated to be O(1/nw). The average in the last product of (17) is estimated as follows:
1 -- iq(Jij)aml ~ ½q2 (j~ij)dm z 4- O(nw3/2) exp [--iq(Jij)arnl --½q2(J})am2 + O(nwa/2)], (18) /3~k(1 -- o~) + . . . . . . .
),
(10)
that is the magnetization of the jth site when the ith site is deleted from the system. Substituting (9) in the place of %. on the right-hand side of (4) with (5) and (6), we have 1 1 + o____-_t/ 3 ~ , + ~ o , + . . . . 01) 0 = -~lnl_oi where
~ ~2
= h~ + Z s o ~ s + . . . .
~ J~-~/3(1 f _~)
ml = (oi)h,a,
m2 = (o~)h,a.
(19)
By substituting (18) into (17), we have
P(~O) = ~
F(q) exp [iq(~ - - J l m l ) -- ~q2J~rn2] dq, - 0~
(2o)
in the limit nw -->0% where
F(q) = (e-iqhi)h,
(21)
(12)
i =
where
J1 = ~ (Jo)a, j ~F.]+ . . . . 3hjJ
(13)
J2 = Z (J'~}J. i
(22)
If the distribution function of the field h i is given, we
EXACTLY SOLVABLE MODEL OF A SPIN GLASS
Vol. 19, No. 9
calculate the right-hand sides of (19) by using the formula (16) and (20) with (21). Then (19) constitute the set of equations determining ml and m2. The average of (2) with respect to the set {hi, Ju} is taken after eliminating h i in terms of 1o/} with the aid of (11). We obtain the following result for the average free energy per site f =
x j t a n h ~ [ ( h = +J~m2)l/2x +h, + J , m l ] } e -~=/2 dx, - =
m2
d
(23)
Either if hi is equal to the same constant h for all i, or if it is a random variable with the Gaussian distribution, then we have
F(q) = exp ( iqh, -- ½q2h2),
(24)
where
h2 = (h~)h--h'~ 2.
(25)
in the former case, hi = h and/Y2 = 0. Substituting (24) into (20), we have
1
[ 2(h2 + J=m2)
]
(26) The right-hand sides of (19) are now calculated by using (16) and (26). They give, in the limit nw ~ oo, 1 ml
--
(21r)1/2
× ~ t a n h 2 ~[(h2 +J2m~)l/2x +hi + J l m l ] } e -~'/2 dx
_oo
PGb) = [2rr(h2 + J=m=)] x/zexp
-
-~
--/3 -I ; In [2 cosh (/3q0]P(ff) d~O.
= ~vli)h,
(27) 1
1 ~J1 mx2 -- ¼/3J2(1 -- m2) z
hi
835
(2701/2
(28)
This set of equations (27) and (28) determines the values of mx and m2. Then the free energy of the system is calculated by (23) and (26). Under zero external field h i = 0, hi = h2 = 0. If m2 = 0, ml must then be zero by (28); representing the paramagnetic state (P). If hi = h2 = 0 and m2 = 0, ml may be zero, the spin-glass state (SG), or non-zero, the ferromagnetic state (F). In the spin-glass state, (28) determines m2, where m~ is put equal to zero. If we put hi = h2 = 0 in our results, e.g. in (23) and (26)-(28), they reduce to the corresponding results given in reference 1, which were obtained under the assumption of the Gaussian distribution for di.i. Thus we confirm the results given in reference 1, which were derived by invoking a procedure which is not warranted to give correct results; note that no reason was given not to allow us to add a non-zero function f(n) which vanishes for integral values n = 1, 2, 3 . . . . and has nonzero derivative f'(n) at n --* 0, before taking the limit n ~ 0 in (8) of reference 1 ; e.g. f(z) can be sin Orz)/z x f~(z) or 1/(zP(z)) x f ~ ( z ) i f f l ( z ) i s finite at z = 1,2 . . . . .
REFERENCES 1.
SHERRINGTON D. & KIRKPATRICK S., Phys. Rev. Lett. 35, 1792 (1975).
2.
KATSURA S., Adv. Phys. 12,391 (1963).
3.
F A R R E L L R.A., MORITA T. & MEIJER P.H.E., J. Chem. Phys. 4 5 , 3 4 9 (1966).
4.
MORITA T. & HIROIKI K.,Prog. Theor. Phys. 2 5 , 5 3 7 ( 1 9 6 1 ) .
Pergamon Press.
Printed in Great Britain
EXACTLY SOLVABLE MODEL OF A SPIN GLASS T. Morita and T. Horiguchi Department of Applied Science, Faculty of Engineering, Tohoku University, Sendal 980, Japan
(Received 29 March 1976 by Y. Toyozawa) Sherrington and Kirkpatrick presented a solvable model of a spin glass. In the solution, they used a mathematically unwarranted procedure. In the present article, we show that the problem is exactly solved by starting with the virial expansion formula, and confirm the results of Sherrington and Kirkpatrick. The solution is obtained for the random lsing magnet in which the external field of each site and the exchange integral between each pair of sites are random variables. We obtain the exact thermodynamic properties for this system in the limit of nw ~ 0% assuming that the exchange integrals of a spin with O(nw) neighbours are O(n~ 1/2) and the average value of each is O(n~l). The system is found to show the spin-glass state as well as the paramagnetic and the ferromagnetic state.
WE SHALL CONSIDER the Ising system where each site is occupied by a spin and the Hamiltonian is given by
i
i>j
si for each site i takes on + 1 and - 1. In a recent paper, Sherrington and Kirkpatrick 1 gave a solution of the random system where the exchange integral Jij for all pairs i and ] are random variables whose distributions are given by the same Gaussian distribution function, assuming that the average and the average of the square are both O(1/N) where N is the total .number of the lattice sites in the system, h i was assumed to be a constant independent of the site i. They obtained the phase transitions between the paramagnetic, the ferromagnetic and the spin-glass state. The method used in the solution involved a mathematically unjustifiable procedure. In the present note, we show that the problem is solved exactly, without using such a dangerous procedure, for a more general case without assuming the Gaussian distribution for Ji~. We conclude that their results are correct. For the regular system, hi does not depend on i and Jij is a function only of the relative coordinate of the two sites i and/'. IfJij for a fixed i is a small quantity of O(1/nw) for O(n w) lattice sites/' and is zero for other sites j, an argument based on the virial expansion shows that the properties calculated by the molecular field approximation are exact in the limit of n w -+ 00.2 The argument applies equally to the systems where h i depends on i and Jij depend on i and j, if J# for each i is O(1/nw) for O(nw) lattice sitesj and is zero for other sites j. In the present paper, we apply such an argument to the random 833
systems where Jo for each i is O(1/tz 1/2) for O(nw) lattice sites/" and the average of Jij for each of those pairs i and ] is O(1/nw); Ji~ for the other pairs i and ] are assumed to be zero. Then we find that the contributions which remain in the limit n w ~ ~o are again easily evaluated. We shall start with the virial expansion formula for the free energy, applicable to non-uniform Ising systems. If we write the probability of the state si = -+ 1 of the ith site by ½(1 -+ oi), then it is written as a
13F = --(3 ~ hioi + Z ½[(1 + oi) ln (1 + oi) i
i
+ ( 1 - - o i ) ln (l -- oi)-- 21n 2] -- Sum, (2) where t3 = 1[k B T as usual and 1
Sum = ~ ~ ~13Ji, oicri 1
+ ~ 2 Z .~2&( 1 ~- o?)(1 - °7) + . . - ; (3) i
J
Here Jij = 0 when i = ]. If we take the average over the set {hi, Jij}, the two terms written explicitly on the righthand side of (3) give contributions of order N x O(nw) x O(1/nw). The dots ( . . . ) denote terms with more factors 13Jij, or others expressed by more than singly connected diagrams of three or more sites; the factor associated to a bond is 13Jm when it connects the sites k and l. The averages over the set {hi, J~i} of the contributions of those terms are estimated to be N x O(n~/2) and N x O(n~,l), respectively, which drop as n w ~ oo. In the following expressions also, dots denote the terms which give no contribution after the average over the set
Vol. 19, No. 9
EXACTLY SOLVABLE MODEL OF A SPIN GLASS
834
{hi, Jij} and the limit nw ~ ~ are taken. When Jij for each site i were O(1/nw) for O(nw) sites/', the second term of(3) would be estimated to be N x O(nw) x O(1/n~w) and drops as n w -~ ~. Then we would have the molecular field approximation. The virial expansion formula (2), which involves the external field hi only through --/3Eihio i, is stationary with respect to the variation of the distribution functions ½(1 ± oi) and hence of oi .~'4 It follows that 0 = 1 In 1+ oi 2 1--oi
/3ff1 -~ /31122Oi
d¢- . . . .
(5)
i
if2 = ~ ~2 J}(1 -- 4 ) + . . . .
(6)
J The dots on the right-hand side of (4) denote /3~so~ + . . . . which is O(n~I/2). We can regard that (4) determines oi as a function of C/t, qJ2, t)s . . . . . The function will be denoted by f(qJ~, ~2 . . . . ): Oi
Oi :
= f(l~l, ~J2,"" ")"
(7)
When ~2 = . . . = 0, it is given by f ( ~ l , 0 , . . . ) = tanh (/3q~0.
(14)
By definition of the magnetization of the/th site oy = Tr sj e-13/-//Tre -~u, Ooj/~hj =/3(1 -- o~). Similarly we have ~ / O h i = 3(1 - 8~), so that the leading terms of ~2 written explicitly in (13) cancel out. ~Jt, ~ : , . . . depend on the set {hi, Jij}. We shall introduce the probability distribution function P($) of ~ = ~t. It is defined by
where the subscripts h, J of ( . . . ) h , a mean the average with respect to the set {hi, Jij}. When the quantity to be averaged depends only on the set {hi} or {Jij}, the average will be denoted as ( . . . ) h or ( . . . ) j , accordingly. Then the average of an arbitrary function g(oi) of oi is calculated by oo
@(oi))h,d = f g[tanh (/3~)]P(~) d~ + O(nwl/2), -
(16)
oo
by noting (8). We shall now calculate the distribution function P(ff). By the aid of the integral representation of the delta function, we write (15) as follows:
(8) 1
If we apply this formula (7) with (5) and (6) to % we have oi = f ( h j + ~
f(t~l, ~2,'-.)-
(4)
where
~J~ = hi + ~ Jijoj + . . . .
Now we have
Jyk%+...,~]fl~(l--o~)+~
x H (exp ( - iqJij8j + . . . ) ) n , a dq, J
.... ...)
= ~ + O°~J~io i + . . . . Ohj
e iqo ( e - i q h i ) h -¢o
(93
Here
(17)
where the average of the product is replaced by the product of the averages; the terms omitted being estimated to be O(1/nw). The average in the last product of (17) is estimated as follows:
1 -- iq(Jij)aml ~ ½q2 (j~ij)dm z 4- O(nw3/2) exp [--iq(Jij)arnl --½q2(J})am2 + O(nwa/2)], (18) /3~k(1 -- o~) + . . . . . . .
),
(10)
that is the magnetization of the jth site when the ith site is deleted from the system. Substituting (9) in the place of %. on the right-hand side of (4) with (5) and (6), we have 1 1 + o____-_t/ 3 ~ , + ~ o , + . . . . 01) 0 = -~lnl_oi where
~ ~2
= h~ + Z s o ~ s + . . . .
~ J~-~/3(1 f _~)
ml = (oi)h,a,
m2 = (o~)h,a.
(19)
By substituting (18) into (17), we have
P(~O) = ~
F(q) exp [iq(~ - - J l m l ) -- ~q2J~rn2] dq, - 0~
(2o)
in the limit nw -->0% where
F(q) = (e-iqhi)h,
(21)
(12)
i =
where
J1 = ~ (Jo)a, j ~F.]+ . . . . 3hjJ
(13)
J2 = Z (J'~}J. i
(22)
If the distribution function of the field h i is given, we
EXACTLY SOLVABLE MODEL OF A SPIN GLASS
Vol. 19, No. 9
calculate the right-hand sides of (19) by using the formula (16) and (20) with (21). Then (19) constitute the set of equations determining ml and m2. The average of (2) with respect to the set {hi, Ju} is taken after eliminating h i in terms of 1o/} with the aid of (11). We obtain the following result for the average free energy per site f =
x j t a n h ~ [ ( h = +J~m2)l/2x +h, + J , m l ] } e -~=/2 dx, - =
m2
d
(23)
Either if hi is equal to the same constant h for all i, or if it is a random variable with the Gaussian distribution, then we have
F(q) = exp ( iqh, -- ½q2h2),
(24)
where
h2 = (h~)h--h'~ 2.
(25)
in the former case, hi = h and/Y2 = 0. Substituting (24) into (20), we have
1
[ 2(h2 + J=m2)
]
(26) The right-hand sides of (19) are now calculated by using (16) and (26). They give, in the limit nw ~ oo, 1 ml
--
(21r)1/2
× ~ t a n h 2 ~[(h2 +J2m~)l/2x +hi + J l m l ] } e -~'/2 dx
_oo
PGb) = [2rr(h2 + J=m=)] x/zexp
-
-~
--/3 -I ; In [2 cosh (/3q0]P(ff) d~O.
= ~vli)h,
(27) 1
1 ~J1 mx2 -- ¼/3J2(1 -- m2) z
hi
835
(2701/2
(28)
This set of equations (27) and (28) determines the values of mx and m2. Then the free energy of the system is calculated by (23) and (26). Under zero external field h i = 0, hi = h2 = 0. If m2 = 0, ml must then be zero by (28); representing the paramagnetic state (P). If hi = h2 = 0 and m2 = 0, ml may be zero, the spin-glass state (SG), or non-zero, the ferromagnetic state (F). In the spin-glass state, (28) determines m2, where m~ is put equal to zero. If we put hi = h2 = 0 in our results, e.g. in (23) and (26)-(28), they reduce to the corresponding results given in reference 1, which were obtained under the assumption of the Gaussian distribution for di.i. Thus we confirm the results given in reference 1, which were derived by invoking a procedure which is not warranted to give correct results; note that no reason was given not to allow us to add a non-zero function f(n) which vanishes for integral values n = 1, 2, 3 . . . . and has nonzero derivative f'(n) at n --* 0, before taking the limit n ~ 0 in (8) of reference 1 ; e.g. f(z) can be sin Orz)/z x f~(z) or 1/(zP(z)) x f ~ ( z ) i f f l ( z ) i s finite at z = 1,2 . . . . .
REFERENCES 1.
SHERRINGTON D. & KIRKPATRICK S., Phys. Rev. Lett. 35, 1792 (1975).
2.
KATSURA S., Adv. Phys. 12,391 (1963).
3.
F A R R E L L R.A., MORITA T. & MEIJER P.H.E., J. Chem. Phys. 4 5 , 3 4 9 (1966).
4.
MORITA T. & HIROIKI K.,Prog. Theor. Phys. 2 5 , 5 3 7 ( 1 9 6 1 ) .