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Theory of random bond model
Volume 88A, number S
PHYSICS LETTERS
15 March 1982
THEORY OF RANDOM BOND MODEL Takehiko OGUCHI a and Fumihiko TAKANA b a Department of Physics, Tokyo Institute of Technology,
Oh-okayama, Meguro-ku, Tokyo 152, Japan b Institute of Physics, The University of Tsukuba, Sakura-mura, Niihari-gun, Ibaraki 305, Japan Received 3 December 1981
The free energy of random bond model (± J model) is derived by the effective hamiltonian method. The average of the free energy over sites contains a parameter ~ concerned with the frustration effect. The second-order phase transition is shown to occur only when ~ is larger than a critical value.
Spin glasses and ±Jmodel have been investigated with a great deal of interest. In the theories by Edwards and Anderson [1], Sherrington and Kirkpatrick [2] and Katsura [3], the solutions are derived from an extremum of the free energy, hut this extremum is not a minimum but a maximum. This peculiar situation is well known [4,51, and it is stated that the free energy used is not a variational function, but rather is defined only at the physical values of order parameters. The meaning of this statement, however, is not clear, so we investigate the solutions which can be derived from the minimization of the free energy. We consider the ±Jmodel, which is described by the hamiltonian, (1)
(if)
ii
/
where the sum is over pairs of nearest neighbor spins, and takes the value J or —J with the given probabilities. In the present paper we develop a method which is an extension of the effective hamiltonian method [6] This is equivalent to the Bethe approximation. Let us assume that the effective hamiltonian for the ith spin is of the form .
= —H~o~, (2) where H, is the effective field at the site i. Following the notation by Nakanishi [7] we decompose H, as follows,
Hi=h?~+k?),
(3)
is the effective field at the site i coming from the site j, and h~’~ is the field coming from other sites. The thermal average mi of a, is calculated as
where
Tr a, exp(_I39(~) m~=
_____________=
Tr exp(—13?(1)
tanh I3h~’~ + tanh 13k~’~ tanh(J3H.) =
(4)
1
I
+
tanh i3h~’~tanh ~3k~’~
~3= 1/(kT), k the Boltzmann constant, T the temperature. We regard m, as independent thermodynamicvariables and derive the corresponding free energy f~ by a kind of Legendre transformation as follows,
where
f3f,(rn~) —ln Tr exp(—I3~1)[email protected]. = —ln 2 ++(l
+
m.)ln(l -i-rn.) +-~(1 ,n1)ln(l
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
—
—
me).
(5) 247
Volume 88A, number 5
PHYSICS LETTERS
IS March 1982
Next, let us assume that the effective hamiltonian for a pair of nearest neighbor spins i and! the form —J~1a~a1 ~
=
—
can be written in
h7)u1,
—
(6)
has been introduced in eq. (3). We can express rn~by using (6) as follows,
where
Tr cj~exp(—~3~(,1) —
tanh
13h~~~ + tanh i3h7)
—~
Tr exp(_i3~c~1) ~ +
~
.
I3h~’~ tanh
(7)
—,
where we have put KU
tanh ~
=
(8)
We can inversely express tanh The result is tanhjh~’~” A.. 1/
—
as a function of m, and rn by
using
(7) and corresponding equation for m1.
2 1 K.. 2K~. (A~. 1/1/ 4K..M.M.)~ 1 / = K-Mi +_LM~M~ —~M~M~ —
—
(9)
÷...,
where we have put M~~rn~_K~ 1rn1, M,.~rn1—K~1m~,A~1~1~—K~.
(10)
Using a similar kind of Legendre transformation, we can obtain the free energy f,1 for two spins i and j as
f3f~= —ln Tr exp(—13~~1) + j3h~’~rn. + I3h7~rn1. The free energy f of the total system per spin is given by [8] 1
~
(11)
—i~)
i3f= —
+
—~---
2N
(z
~ ln( 1
—
—
1)
~ [(1 + rn~)ln(1 + m~)÷(1
tanh2!3h7))
ln(l
—
+
—
m.) ln(1
—
1
rn~)]+-~-~ [+ln(l N(if)
—
tanh2~h~’~)
tanh f3h,c’~tanh L3h7~)+ !3(/41)m
1 + h7)m1)] (12) where z is the number of nearest neighbors. Henceforth we shall neglect the 1st and 2nd terms in (12) which are independent of the m~.Substitution of (9) into (12) gives ,
1
1
+
(zA~11)K~ /rn~+~[l_z4(3 / A~K~) / —
N(i,)(A..’JE,A3
[4K~(rn~rnf+rn~1nf)—3K~(1
+K~)rn~m~] +...).
We introduce the order parameter 72 as the site average of rn~and a parameter tween signs of and mi, m1 as follows, 2
m~~ where ~
Kijmimj
EK7
K~rn?rnJ
tanh i3J. The parameter
~K374,
(13) ~ expressing the correlation be-
(14)
E is the ratio of the difference of numbers of right (satisfied) bonds and wrong
(unsatisfied) bonds to the total number of bonds. It becomes 1 in the case of no frustration (all bonds are right), 248
Volume 88A, number 5
PHYSICS LETTERS
15 March 1982
and becomes smaller with increasing frustrations. Using (13) and (14) we obtain 2
1—Z~K+(Z— 1)K 2)
2(3_8~K+6K2__K4)) 72+l(izK
(1—K2)3
(15)
74+
2(1 —K phase transition occurs when the coefficient of 72 changes the sign. In (15), it occurs at The second-order =
{z~ [(z~)2 —
—
4(z
-
1)] l/2} /2(z
In the case of no frustration K~(Z_
-
(16)
1).
(~ 1), (16) is reduced to =
1)_i,
(17)
which is the well-known result for the pure ferromagnetic case in the pair approximation. When ~ becomes smaller, Kc increases until ~ reaches the value ~ defined by ~~2(z—
(18)
l)~2z~.
For ~ <~, K~becomes complex, and no phase transition occurs. In the ferromagnetic phase, where all spins align parallel to each other in the ground state, ~ is given by ~=CA_CB=l_2CB,
(19)
where CA and CB are the concentrations of ferromagnetic and antiferromagnetic bonds, respectively. From (18)
and (19) we obtain the critical concentration (cB)C of the antiferromagnetic bonds, beyond which the ferromagnetic phase cannot exist. The numerical values of (cB)C are shown in table 1. The Curie point at (cB)c is given by =
(z — l)1~’2.
(20)
This happens to be the same expression as that of the glass like phase obtained by Matsubara et al. [9] and by
Katsura [10] . The fact that T~is finite at the critical concentration for the ferromagnetic phase is obtained by the method of high-temperature series expansion by Okamoto et al. [11). As mentioned above, ~ is the quantity introduced to take into account the correlations between nearest neighbors. Consequently in a spin glass theory for the ±Jmodel of refs. [3,9,10] , ~is zero because these correlations are not considered. In the theory of random ordered phase [12,13] , the rough estimation of~gives the value of 0.5 for the honeycomb lattice. These values are smaller than ~ and neither spin glass phase nor random ordered phase can exist stably. Kirkpatrick [14] calculated the energy of the ground state which is not ferromagnetic. His results are —4J/3 and —3J/2 per bond for square lattice and simple cubic lattice, respectively. We can estimate the
values of ~ as (—4J/3)/(—2J) = 0.667 and (—3J/2)/(—3J) = 0.5 for each lattice. If we use these values of ~ in the present approximation, we cannot obtain the phase transition to this phase because these values are smaller than ~c.
In conclusion, we have developed a new method which is based on the principle of minimization of the free enTable 1 Critical value ~ and critical concentration (CB)C for each lattice. z is the number of nearest neighbors.
z
(CB)c 3
0.943
0.029
4 6
0.866 0.745
0.067
8 12
0.661 0.553
0.127 0.169 0.223 249
Volume 88A, numberS
PHYSICS LETTERS
15 March 1982
ergy. However, it turns out that neither the spin glass phase nor the random ordered phase are stable in the present approximation. This result might be due to the roughness of the present approximation. In order to check this point, it is necessary to improve the present pair approximation towards the square approximation so as to include the effect of frustrations directly. The authors thank Dr. K. Nakanishi for stimulating discussions. References Ill S.F. Edwards and P.W. Anderson, J. Phys. F5 (1975) 965. [21 D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1792. [3] S. Katsura, S. Inawashiro and S. Fujiki, Physica 99A (1979) 193. [4] J.R.L. de Almeida and D.J. Thouless, J. Phys. All (1978) 983. [5] G. Parisi, J. Phys. A13 (1980) 1101. [61 T. Oguchi and!. Ono, J. Phys. Soc. Japan 21(1966) 2178. [7] K. Nakanishi, Phys. Rev. 23 (1981) 3514. 18] T. Morita, Physica 98A (1979) 566. [9] F. Matsubara and M. Sakata, Prog. Theor. Phys. 55 (1976) 672. [10] S. Katsura, Prog. Theor. Phys. 55 (1976) 1049. [11] S. Okamoto, T. Oguchi and Y. Ueno, submitted to Prog. Theor. Phys. [12] Y. Ueno and T. Oguchi, J. Phys. Soc. Japan 40 (1976) 1513. [13] T. Oguchi and Y. Ueno, J. Phys. Soc. Japan 43 (1977) 764. [14] S. Kirkpatrick, Phys. Rev. B16 (1977) 4630.
250
PHYSICS LETTERS
15 March 1982
THEORY OF RANDOM BOND MODEL Takehiko OGUCHI a and Fumihiko TAKANA b a Department of Physics, Tokyo Institute of Technology,
Oh-okayama, Meguro-ku, Tokyo 152, Japan b Institute of Physics, The University of Tsukuba, Sakura-mura, Niihari-gun, Ibaraki 305, Japan Received 3 December 1981
The free energy of random bond model (± J model) is derived by the effective hamiltonian method. The average of the free energy over sites contains a parameter ~ concerned with the frustration effect. The second-order phase transition is shown to occur only when ~ is larger than a critical value.
Spin glasses and ±Jmodel have been investigated with a great deal of interest. In the theories by Edwards and Anderson [1], Sherrington and Kirkpatrick [2] and Katsura [3], the solutions are derived from an extremum of the free energy, hut this extremum is not a minimum but a maximum. This peculiar situation is well known [4,51, and it is stated that the free energy used is not a variational function, but rather is defined only at the physical values of order parameters. The meaning of this statement, however, is not clear, so we investigate the solutions which can be derived from the minimization of the free energy. We consider the ±Jmodel, which is described by the hamiltonian, (1)
(if)
ii
/
where the sum is over pairs of nearest neighbor spins, and takes the value J or —J with the given probabilities. In the present paper we develop a method which is an extension of the effective hamiltonian method [6] This is equivalent to the Bethe approximation. Let us assume that the effective hamiltonian for the ith spin is of the form .
= —H~o~, (2) where H, is the effective field at the site i. Following the notation by Nakanishi [7] we decompose H, as follows,
Hi=h?~+k?),
(3)
is the effective field at the site i coming from the site j, and h~’~ is the field coming from other sites. The thermal average mi of a, is calculated as
where
Tr a, exp(_I39(~) m~=
_____________=
Tr exp(—13?(1)
tanh I3h~’~ + tanh 13k~’~ tanh(J3H.) =
(4)
1
I
+
tanh i3h~’~tanh ~3k~’~
~3= 1/(kT), k the Boltzmann constant, T the temperature. We regard m, as independent thermodynamicvariables and derive the corresponding free energy f~ by a kind of Legendre transformation as follows,
where
f3f,(rn~) —ln Tr exp(—I3~1)[email protected]. = —ln 2 ++(l
+
m.)ln(l -i-rn.) +-~(1 ,n1)ln(l
0 031-9163/82/0000—0000/$02.75 © 1982 North-Holland
—
—
me).
(5) 247
Volume 88A, number 5
PHYSICS LETTERS
IS March 1982
Next, let us assume that the effective hamiltonian for a pair of nearest neighbor spins i and! the form —J~1a~a1 ~
=
—
can be written in
h7)u1,
—
(6)
has been introduced in eq. (3). We can express rn~by using (6) as follows,
where
Tr cj~exp(—~3~(,1) —
tanh
13h~~~ + tanh i3h7)
—~
Tr exp(_i3~c~1) ~ +
~
.
I3h~’~ tanh
(7)
—,
where we have put KU
tanh ~
=
(8)
We can inversely express tanh The result is tanhjh~’~” A.. 1/
—
as a function of m, and rn by
using
(7) and corresponding equation for m1.
2 1 K.. 2K~. (A~. 1/1/ 4K..M.M.)~ 1 / = K-Mi +_LM~M~ —~M~M~ —
—
(9)
÷...,
where we have put M~~rn~_K~ 1rn1, M,.~rn1—K~1m~,A~1~1~—K~.
(10)
Using a similar kind of Legendre transformation, we can obtain the free energy f,1 for two spins i and j as
f3f~= —ln Tr exp(—13~~1) + j3h~’~rn. + I3h7~rn1. The free energy f of the total system per spin is given by [8] 1
~
(11)
—i~)
i3f= —
+
—~---
2N
(z
~ ln( 1
—
—
1)
~ [(1 + rn~)ln(1 + m~)÷(1
tanh2!3h7))
ln(l
—
+
—
m.) ln(1
—
1
rn~)]+-~-~ [+ln(l N(if)
—
tanh2~h~’~)
tanh f3h,c’~tanh L3h7~)+ !3(/41)m
1 + h7)m1)] (12) where z is the number of nearest neighbors. Henceforth we shall neglect the 1st and 2nd terms in (12) which are independent of the m~.Substitution of (9) into (12) gives ,
1
1
+
(zA~11)K~ /rn~+~[l_z4(3 / A~K~) / —
N(i,)(A..’JE,A3
[4K~(rn~rnf+rn~1nf)—3K~(1
+K~)rn~m~] +...).
We introduce the order parameter 72 as the site average of rn~and a parameter tween signs of and mi, m1 as follows, 2
m~~ where ~
Kijmimj
EK7
K~rn?rnJ
tanh i3J. The parameter
~K374,
(13) ~ expressing the correlation be-
(14)
E is the ratio of the difference of numbers of right (satisfied) bonds and wrong
(unsatisfied) bonds to the total number of bonds. It becomes 1 in the case of no frustration (all bonds are right), 248
Volume 88A, number 5
PHYSICS LETTERS
15 March 1982
and becomes smaller with increasing frustrations. Using (13) and (14) we obtain 2
1—Z~K+(Z— 1)K 2)
2(3_8~K+6K2__K4)) 72+l(izK
(1—K2)3
(15)
74+
2(1 —K phase transition occurs when the coefficient of 72 changes the sign. In (15), it occurs at The second-order =
{z~ [(z~)2 —
—
4(z
-
1)] l/2} /2(z
In the case of no frustration K~(Z_
-
(16)
1).
(~ 1), (16) is reduced to =
1)_i,
(17)
which is the well-known result for the pure ferromagnetic case in the pair approximation. When ~ becomes smaller, Kc increases until ~ reaches the value ~ defined by ~~2(z—
(18)
l)~2z~.
For ~ <~, K~becomes complex, and no phase transition occurs. In the ferromagnetic phase, where all spins align parallel to each other in the ground state, ~ is given by ~=CA_CB=l_2CB,
(19)
where CA and CB are the concentrations of ferromagnetic and antiferromagnetic bonds, respectively. From (18)
and (19) we obtain the critical concentration (cB)C of the antiferromagnetic bonds, beyond which the ferromagnetic phase cannot exist. The numerical values of (cB)C are shown in table 1. The Curie point at (cB)c is given by =
(z — l)1~’2.
(20)
This happens to be the same expression as that of the glass like phase obtained by Matsubara et al. [9] and by
Katsura [10] . The fact that T~is finite at the critical concentration for the ferromagnetic phase is obtained by the method of high-temperature series expansion by Okamoto et al. [11). As mentioned above, ~ is the quantity introduced to take into account the correlations between nearest neighbors. Consequently in a spin glass theory for the ±Jmodel of refs. [3,9,10] , ~is zero because these correlations are not considered. In the theory of random ordered phase [12,13] , the rough estimation of~gives the value of 0.5 for the honeycomb lattice. These values are smaller than ~ and neither spin glass phase nor random ordered phase can exist stably. Kirkpatrick [14] calculated the energy of the ground state which is not ferromagnetic. His results are —4J/3 and —3J/2 per bond for square lattice and simple cubic lattice, respectively. We can estimate the
values of ~ as (—4J/3)/(—2J) = 0.667 and (—3J/2)/(—3J) = 0.5 for each lattice. If we use these values of ~ in the present approximation, we cannot obtain the phase transition to this phase because these values are smaller than ~c.
In conclusion, we have developed a new method which is based on the principle of minimization of the free enTable 1 Critical value ~ and critical concentration (CB)C for each lattice. z is the number of nearest neighbors.
z
(CB)c 3
0.943
0.029
4 6
0.866 0.745
0.067
8 12
0.661 0.553
0.127 0.169 0.223 249
Volume 88A, numberS
PHYSICS LETTERS
15 March 1982
ergy. However, it turns out that neither the spin glass phase nor the random ordered phase are stable in the present approximation. This result might be due to the roughness of the present approximation. In order to check this point, it is necessary to improve the present pair approximation towards the square approximation so as to include the effect of frustrations directly. The authors thank Dr. K. Nakanishi for stimulating discussions. References Ill S.F. Edwards and P.W. Anderson, J. Phys. F5 (1975) 965. [21 D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1792. [3] S. Katsura, S. Inawashiro and S. Fujiki, Physica 99A (1979) 193. [4] J.R.L. de Almeida and D.J. Thouless, J. Phys. All (1978) 983. [5] G. Parisi, J. Phys. A13 (1980) 1101. [61 T. Oguchi and!. Ono, J. Phys. Soc. Japan 21(1966) 2178. [7] K. Nakanishi, Phys. Rev. 23 (1981) 3514. 18] T. Morita, Physica 98A (1979) 566. [9] F. Matsubara and M. Sakata, Prog. Theor. Phys. 55 (1976) 672. [10] S. Katsura, Prog. Theor. Phys. 55 (1976) 1049. [11] S. Okamoto, T. Oguchi and Y. Ueno, submitted to Prog. Theor. Phys. [12] Y. Ueno and T. Oguchi, J. Phys. Soc. Japan 40 (1976) 1513. [13] T. Oguchi and Y. Ueno, J. Phys. Soc. Japan 43 (1977) 764. [14] S. Kirkpatrick, Phys. Rev. B16 (1977) 4630.
250