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Replica symmetry breaking of the Ising spin glass in the Bethe and cluster approximations
Physics Letters A 356 (2006) 439–445 www.elsevier.com/locate/pla
Replica symmetry breaking of the Ising spin glass in the Bethe and cluster approximations Terufumi Yokota Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan Received 9 December 2005; accepted 4 April 2006 Available online 18 April 2006 Communicated by A.R. Bishop
Abstract The Ising spin glass model in a magnetic field is studied within the Bethe and a related cluster approximations. Using replicas, spin glass order parameter function and the de Almeida–Thouless (AT) line are obtained close to the spin glass transition temperature. The results for the Bethe approximation deviate considerably from those for the Sherrington–Kirkpatrick (SK) model. © 2006 Elsevier B.V. All rights reserved. Keywords: Ising spin glass; Bethe approximation; Cluster approximation; Replica symmetry breaking; AT line
1. Introduction The nature of the glassy phase in the Ising spin glass has been studied for a long time, but it has not been understood satisfactorily in realistic short range models (for a review, see [1]). In the mean field theory of the SK model (for a review, see [2]), Parisi’s replica symmetry breaking solution is considered to be valid. It has not been clear by now whether the concept of replica symmetry breaking is relevant to the realistic short range models. There have been several investigations toward more realistic theory of spin glass than the mean field theory of the SK model. In the Bethe approximation, some features of the replica symmetry breaking solution have been elucidated [3–5]. There exists a replica symmetry breaking instability known as the AT line in a magnetic field [3]. Close to the spin glass transition temperature, the replica symmetry breaking solution was shown to be similar to that for the SK model although details of the solution were not given [4]. Another type of cluster approximations has been proposed [6] and the approximations have been applied to obtain the replica symmetry breaking solutions close to the spin glass transition temperature in a magnetic field [7]. A cluster with a loop has been used to see effects of the frustration. The 1/d expansion has been used and the replica symmetry breaking solutions with enhanced replica symmetry breaking effects were obtained [8]. In this Letter, explicit form of the replica symmetry breaking solution for the short range Ising spin glass model in a magnetic field within the Bethe approximation close to the spin glass transition temperature is obtained. In addition, the cluster approximation [6] with a Bethe approximation type cluster is used to obtain a replica symmetry breaking solution. The solutions are compared with the previously obtained ones including the one for the SK model. The form of the solution for the Bethe approximation differs quantitatively from the one for the SK model. The slope in the spin glass order parameter function and the region of replica symmetry breaking below the AT line diminish considerably especially for smaller dimensions.
E-mail address: [email protected] (T. Yokota). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.04.008
440
T. Yokota / Physics Letters A 356 (2006) 439–445
2. Cluster approximation We study the Ising spin glass model on a d-dimensional hypercubic lattice. The Hamiltonian is given by H =− Jij Si Sj − h Si , ij
(1)
i
where the first summation is over all nearest neighbor pairs and quenched random exchange couplings Jij obey a Gaussian probability distribution with zero mean and variance J˜2 /2d. Si = ±1 are N Ising variables and h is an external field. In this section, the cluster approximation introduced by Baviera, Pasquini and Serva [6] is used to study the spin glass phase close to the transition temperature. The spins on the lattice are decomposed into a set of cluster spins. Each spin belongs to one and only one of the clusters. Using the replica method, the replicated partition function Z n is given by exp β Jij Siα Sjα + βh Siα Zn = (2) ij α
{Siα }
i
α
where Siα is the αth replica of the Ising spin on the ithe site and β is inverse temperature. To obtain the free energy per spin fd on a d-dimensional hypercubic lattice, the random average is performed in two steps. The free energy is given by 1 ln Z n N→∞ n→0 nN
−βfd = lim lim
(3)
where · · · is the random average over the couplings between different clusters and · · · is the random average over the couplings inside the clusters. Z n is given by
˜ 2 ˜)2 α β α β (β J ) (β J n J α α α Z n = exp exp β Jij Si Sj + βh Si + S i S i S j Sj 1− Nn 4 dns 2d α α α {Si }
ij
ij α<β
i
˜ 2 (β J ) nJ ≡ exp exp −βH (n) 1− Nn 4 dns α
(4)
{Si }
where nJ and ns are numbers of bonds and spins in a cluster, respectively. The pairs represented by ij and ij are the nearest neighbor pairs inside a cluster and between different clusters, respectively. In the cluster approximation, a set of variational parameters {q αβ } are used by a replacement β β β β Siα Si Sjα Sj −→ q αβ Siα Si + Sjα Sj . (5) By the replacement, the replicated Hamiltonian H (n) is changed to (n) H˜ (n) = Ωclust ,
(6)
clust
where (n)
Ωclust = −
clust ij α
Jij Siα Sjα −
clust clust β J˜2 nJ αβ α β q Si Si − h Siα . ns − nb d α (i) α<β
(7)
i
clust
clust Here nb is the number of boundary spins in a cluster and the summations clust run over nearest neighbor bonds, ij , (i) and i boundary sites and all the sites in a single cluster, respectively. Variational calculation with respect to {q αβ } gives self-consistent relation on the boundary sites q αβ = S α S β (8) where · · · is thermal average with respect to the Hamiltonian H˜ (n) . The variational parameters {q αβ } may be considered as mean fields on the boundary sites. The free energy in the cluster approximation f˜d is given by
˜)2 ˜)2 αβ 2 1 (β J 1 n n (β J J J −β f˜d = (9) q + ln exp −βΩ (n) 1− 1− + lim min − . n→0 n {q αβ } 4 dns 2 dns ns α<β
{S}
The cluster approximation method has been applied to study the Ising spin glass model using 2-spin and 4-spin clusters close to the spin glass transition temperature [7]. Here a Bethe approximation type cluster is used to compare results with those for the Bethe
T. Yokota / Physics Letters A 356 (2006) 439–445
441
Fig. 1. Bethe approximation type cluster for d = 3.
approximation. The cluster for d = 3 is given in Fig. 1. Contrary to the cases of 2-spin and 4-spin clusters [7], there is a site that is not boundary in the Bethe approximation type cluster. To study the spin glass phase close to the transition temperature and for a small magnetic field, the free energy is expanded up to orders q 4 and h2 q: 2 4 q αβ + A3 q αβ q αβ q βγ q γ α + A4 −β f˜d A0 + lim A2 n→0
+ B4
α =β
q
α =β =γ =α
αβ βγ γ δ δα
q
q q
+ C4
α =β
q
αβ 2 βγ 2 q
α =β =γ =α
αβγ δ all different
(βh)2 αβ + q n + D1 . 2
(10)
α =β
The order parameter function q(x) is obtained following the method in [9]: q , 0 x x , q(x) =
0
0
cx, x0 x x1 , q1 , x1 x 1,
(11)
where D1 (βh)2 , 16(A4 + B4 − C4 ) A3 , c= 4(A4 + B4 − C4 ) A2 . q1 = 3A3 q03 =
(12) (13) (14)
Spin glass transition temperature Tc without external field is obtained from A2 = 0 and A2 can be written as A2 = −A2c
T Tc
(15)
close to the transition temperature, where T = T − Tc . The slope of the order parameter function q(x) may be related to the strength of the replica symmetry breaking [7]. The AT line is given by 2 T −3 16(A4 + B4 − C4 ) A2c 3 Tc 2 h ≡ α. − Tc D1 3A3 J˜ J˜
(16)
For the Bethe approximation type cluster, ns = 2d + 1,
(17)
nb = 2d,
(18)
nJ = 2d.
(19)
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T. Yokota / Physics Letters A 356 (2006) 439–445
Table 1 Transition temperature Tc as well as c, q1 and α defined by (13), (14) and (16), respectively, near the transition temperature for d from 1 to 6 in the Bethe approximation type cluster approximation Tc /J˜ c q1 TcT−T c α
d =1
d =2
d =3
d =4
d =5
d =6
0.754 0.470 1.024 0.626
0.915 0.469 1.056 1.157
0.955 0.475 1.057 1.315
0.972 0.480 1.054 1.374
0.981 0.484 1.050 1.398
0.986 0.487 1.046 1.408
Because all the J ’s in a cluster obey the same distribution, the suffix will be omitted and the random average will be denoted by · · ·. 2 ˜2 (2d−1) , the A, B, C and D coefficients are obtained as follows: Defining μ ≡ β J 4d 2 2d μ − + μ2 1 + (2d − 1) tanh2 βJ , A2 = (20) 2d + 1 2 2 8d A3 = μ3 (21) 1 + 3(2d − 1) tanh2 βJ , 3(2d + 1) 2 4 d 4 + d(2d − 1) tanh2 βJ A4 = −μ4 2d + 1 3 3 2 2 + 4d(2d − 1)(d − 1)tanh4 βJ tanh2 βJ + d(2d − 1) tanh4 βJ (22) , 2 2 2 4d 1 + 6(2d − 1) tanh2 βJ + 2(2d − 1)(d − 1)tanh4 βJ tanh2 βJ + (2d − 1) tanh4 βJ , 2d + 1 2 2 2 8 C4 = μ4 d(2d − 1) tanh2 βJ − 4d(2d − 1)(d − 1)tanh4 βJ tanh2 βJ − d(2d − 1) tanh4 βJ , 2d + 1 2 4d 1 + tanh2 βJ + (2d − 1) tanh2 βJ . D1 = μ 2d + 1
B4 = μ4
(23) (24) (25)
Transition temperature Tc and the quantities c, q1 and α that characterize the replica symmetry breaking close to the transition temperature are given in Table 1 for the dimensions from 1 to 6. Results for the SK model are recovered when d → ∞. For the SK model, c = 1/2, q1 = −T /J˜ and α = 4/3. The results for c are smaller than that of the SK model, which may be considered as a sign of enhancement of the replica symmetry breaking [7]. The region of the replica symmetry breaking in a field is reduced for d from 1 to 3 and enhanced for higher dimensions. The results for lower dimensions are similar to those for the 2-spin cluster [7]. 3. Bethe approximation Mottishaw studied the replica symmetry breaking in the Bethe approximation close to the spin glass transition temperature [4]. Although the replica symmetry breaking is shown to survive in the Bethe approximation, explicit form of the order parameter function was not given. To discuss the strength of the replica symmetry breaking, details of the order parameter function are needed. In this section, the order parameter function and the AT line are obtained explicitly. Because the Bethe approximation is valid on the Bethe lattice, the problem is considered on the Bethe lattice in the following. On the Bethe lattice, the order parameter qα1 α2 is insufficient to describe the spin glass phase and infinitely many order parameters qα1 α2 , . . . , qα1 ...αr , . . . are needed [4,10]. The global order parameter gn ({S α }) is used instead, which includes informations of the infinitely many order parameters. Near the spin glass transition temperature, only qα1 α2 and qα1 α2 α3 α4 are needed to obtain the order parameter function q(x) and the AT line. As the details of the derivation of the equations for the order parameters are described in Ref. [4], we will sketch it here. Because the normalization of the spin glass order parameter in Ref. [4] is different from the usual one [10], the normalization in Ref. [10] is used here to compare the results to those obtained by other approximations. The global order parameter gn ({S α }) is related to the spin glass order parameters as [10] ∞ gn S α =
coshn βJ tanhs βJ qβ1 ...βs S β1 · · · S βs .
(26)
s=0 β1 <β2 <···<βs
The equation for the global order parameter satisfies ∞
α
α α α K α α α S1 + βJ α S1 S )(gn ({S1 })) −∞ dJ ρ(J ) Tr{S1 } exp(βh
α = gn S Tr{S1α } exp(βh α S1 )(gn ({S1α }))K
(27)
T. Yokota / Physics Letters A 356 (2006) 439–445
443
where ρ(J ) is the Gaussian bond distribution and K + 1 is the number of neighbors, which is 2d in the Bethe approximation. The order parameter function qα1 ...αr satisfies
Tr{S1α } S1α1 · · · S1αr exp(βh α S1α )(gn ({S1α }))K
qα1 ...αr = (28) . Tr{S1α } exp(βh α S1α )(gn ({S1α }))K In the limit of n → 0, the equations for the order parameters are obtained as ∞ K β1 βs α1 α α s r qα1 ...αr = Tr{S1α } S1 · · · S1 tanh βJ qβ1 ...βs S1 · · · S1 exp βh S1 .
(29)
α
s=0 β1 <β2 <···<βs
)3 , only two equations for qα1 α2 and qα1 ...α4 are relevant. The equations for a small field are To order ( TcT−T c 2 qα1 β qβα2 qα1 α2 (βh)2 + Ktanh2 βJ qα1 α2 + K(K − 1) tanh2 βJ β
+ K(K − 1)tanh βJ tanh βJ 2
4
qβ1 β2 qβ1 β2 α1 α2 + 5permutations
β1 <β2
3 + K(K − 1)(K − 2) tanh2 βJ qα1 β1 qβ1 β2 qβ2 α2 β1 =β2
3 K(K − 1)(K − 2) 2 3 2 + tanh βJ (qβ1 β2 ) − 2(qα1 α2 ) , 3qα1 α2 6
(30)
β1 <β2
2 qα1 α2 α3 α4 Ktanh4 βJ qα1 α2 α3 α4 + K(K − 1) tanh2 βJ (qα1 α2 qα3 α4 + qα1 α3 qα2 α4 + qα1 α4 qα2 α3 ).
(31)
Defining tr as tr = tanh2r βJ ,
(32)
we have from Eq. (31) t12 (K − 1) (qα α qα α + qα1 α3 qα2 α4 + qα1 α4 qα2 α3 ). K(1 − t2 ) 1 2 3 4 Using this equation, Eq. (30) becomes K −1 2 (K − 1)(K − 2) 3 qα1 α2 (βh)2 + t1 qα1 α2 + qα1 β qβα2 − t1 (qα1 α2 )3 t K 1 3K 2 qα1 α2 α3 α4
(33)
β
+
(K − 1)2 t13 t2 (K − 1)(K − 2) 3 2 qα α t q (q ) + β1 β2 1 α1 α2 2K 2 K 2 (1 − t2 ) 1 2 β1 <β2
+
(qβ1 β2 )2
β1 <β2 β1 =α1 ,α2 β2 =α1 ,α2
(K − 1)(K − 2 + t2 )t13 qα1 β1 qβ1 β2 qβ2 α2 . K 2 (1 − t2 )
(34)
β1 =β2
Free energy per spin f may be obtained considering that this equation comes from the saddle point of the free energy: −nβf
1 K −1 2 (βh)2 qαβ + (t1 − 1) (qαβ )2 + t 2 4 6K 1 α =β
α =β
qαβ qβγ qγ α
α =β =γ =α
2 (K − 1)(K − 2) 3 (K − 1)(K − 2) 2 − t1 (qα1 α2 )4 + t13 qαβ 24K 2 32K 2 α =β α =β
3 2 (K − 1) t1 t2 2 2 2 2 4 + (qα1 α2 ) (qβ1 β2 ) − 4 (qαβ ) (qβγ ) − 2 (qαβ ) 16K 2 (1 − t2 ) α1 =α2
+
− 2 + t2 )t13
(K − 1)(K 8K 2 (1 − t2 )
β1 =β2
αβγ δ all different
α =β =γ =α
qαβ qβγ qγ δ qδα .
α =β
(35)
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T. Yokota / Physics Letters A 356 (2006) 439–445
Table 2 Transition temperature Tc as well as c, q1 and α defined by (13), (14) and (16), respectively, near the transition temperature for d from 2 to 6 in the Bethe approximation Tc /J˜ c q1 TcT−T c α
d =2
d =3
d =4
d =5
d =6
0.593 0.162 0.788 0.473
0.738 0.219 0.855 0.828
0.806 0.259 0.889 1.005
0.846 0.289 0.910 1.104
0.872 0.312 0.924 1.165
In the limit of d → ∞ and n → 0, this becomes −nβf
1 1 (βh)2 qαβ + (β J˜)2 − 1 (qαβ )2 + 2 4 6 α =β
−
1 1 (qα1 α2 )4 + 24 8 α =β
α =β
qαβ qβγ qγ α
α =β =γ =α
qαβ qβγ qγ δ qδα .
(36)
αβγ δ all different
This is just the free energy of the SK model near the spin glass transition temperature. Using the notation defined in Eq. (10), the A, B, C and D coefficients are given by 1 A2 (t1 − 1), 4 K −1 2 t , A3 6K 1 (K − 1)(K − 2 + 2t2 K − t2 )t13 , A4 − 24K 2 (1 − t2 ) (K − 1)(K − 2 + t2 )t13 B4 , 8K 2 (1 − t2 ) (K − 1)2 t13 t2 C4 − , 4K 2 (1 − t2 ) D1 1.
(37) (38) (39) (40) (41) (42)
Transition temperature Tc and the quantities c, q1 and α defined in Eqs. (11), (13), (14) and (16) are given in Table 2 for the dimensions from 2 to 6. The explicit form of α is given by 3 Tc 2 4K 4 (K − 2 + 2Kt2 − t2 ) 2 β J tanh β J 1 − tanh β J . α (43) c c c 3(K − 1)2 (1 − t2 ) J˜ This is the same instability line as that obtained by Thouless [3]. The slope of the order parameter function c reduces considerably compared to the value 1/2 for the SK model. Also the region of replica symmetry breaking under the AT line is much smaller than that for the SK model. 4. Conclusions Bethe approximation and a cluster approximation with a Bethe approximation type cluster are applied to the Ising spin glass model close to the transition temperature in a field. The spin glass order parameter function q(x) and the AT line are obtained. The transition temperature for the Bethe approximation is much lower than that for the corresponding cluster approximation. Although it is still higher than the realistic short range model [1], fluctuation effects seem to be included more properly. The spin glass properties characterized by the order parameter function q(x) and the locus of the AT line change considerably from those for the SK model. The reduction of the slope for q(x) may be considered as increase of the replica symmetry breaking [7]. On the contrary, the region of the replica symmetry breaking in a field reduces in the Bethe approximation that does not possess a loop. A loop that include frustration effects explicitly might change the conclusion for the Bethe approximation [7,8]. References [1] N. Kawashima, H. Rieger, in: H.T. Diep (Ed.), Frustrated Spin Systems, World Scientific, Singapore, 2004, Chapter 9. [2] M. Mézard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore, 1987. [3] D.J. Thouless, Phys. Rev. Lett. 56 (1986) 1082.
T. Yokota / Physics Letters A 356 (2006) 439–445
[4] [5] [6] [7] [8] [9] [10]
P. Mottishaw, Europhys. Lett. 4 (1987) 333. M. Mézard, G. Parisi, Eur. Phys. J. B 20 (2001) 217. R. Baviera, M. Pasquini, M. Serva, J. Phys. A 31 (1998) 4127. T. Yokota, Physica A 363 (2006) 161. A. Georges, M. Mézard, S.S. Yedidia, Phys. Rev. Lett. 64 (1990) 2937. D.J. Thouless, J.R.L. de Almeida, J.M. Kosterlitz, J. Phys. C 13 (1980) 3271. Y.Y. Goldschmidt, C. De Dominicis, Phys. Rev. B 41 (1990) 2184.
445
Replica symmetry breaking of the Ising spin glass in the Bethe and cluster approximations Terufumi Yokota Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan Received 9 December 2005; accepted 4 April 2006 Available online 18 April 2006 Communicated by A.R. Bishop
Abstract The Ising spin glass model in a magnetic field is studied within the Bethe and a related cluster approximations. Using replicas, spin glass order parameter function and the de Almeida–Thouless (AT) line are obtained close to the spin glass transition temperature. The results for the Bethe approximation deviate considerably from those for the Sherrington–Kirkpatrick (SK) model. © 2006 Elsevier B.V. All rights reserved. Keywords: Ising spin glass; Bethe approximation; Cluster approximation; Replica symmetry breaking; AT line
1. Introduction The nature of the glassy phase in the Ising spin glass has been studied for a long time, but it has not been understood satisfactorily in realistic short range models (for a review, see [1]). In the mean field theory of the SK model (for a review, see [2]), Parisi’s replica symmetry breaking solution is considered to be valid. It has not been clear by now whether the concept of replica symmetry breaking is relevant to the realistic short range models. There have been several investigations toward more realistic theory of spin glass than the mean field theory of the SK model. In the Bethe approximation, some features of the replica symmetry breaking solution have been elucidated [3–5]. There exists a replica symmetry breaking instability known as the AT line in a magnetic field [3]. Close to the spin glass transition temperature, the replica symmetry breaking solution was shown to be similar to that for the SK model although details of the solution were not given [4]. Another type of cluster approximations has been proposed [6] and the approximations have been applied to obtain the replica symmetry breaking solutions close to the spin glass transition temperature in a magnetic field [7]. A cluster with a loop has been used to see effects of the frustration. The 1/d expansion has been used and the replica symmetry breaking solutions with enhanced replica symmetry breaking effects were obtained [8]. In this Letter, explicit form of the replica symmetry breaking solution for the short range Ising spin glass model in a magnetic field within the Bethe approximation close to the spin glass transition temperature is obtained. In addition, the cluster approximation [6] with a Bethe approximation type cluster is used to obtain a replica symmetry breaking solution. The solutions are compared with the previously obtained ones including the one for the SK model. The form of the solution for the Bethe approximation differs quantitatively from the one for the SK model. The slope in the spin glass order parameter function and the region of replica symmetry breaking below the AT line diminish considerably especially for smaller dimensions.
E-mail address: [email protected] (T. Yokota). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.04.008
440
T. Yokota / Physics Letters A 356 (2006) 439–445
2. Cluster approximation We study the Ising spin glass model on a d-dimensional hypercubic lattice. The Hamiltonian is given by H =− Jij Si Sj − h Si , ij
(1)
i
where the first summation is over all nearest neighbor pairs and quenched random exchange couplings Jij obey a Gaussian probability distribution with zero mean and variance J˜2 /2d. Si = ±1 are N Ising variables and h is an external field. In this section, the cluster approximation introduced by Baviera, Pasquini and Serva [6] is used to study the spin glass phase close to the transition temperature. The spins on the lattice are decomposed into a set of cluster spins. Each spin belongs to one and only one of the clusters. Using the replica method, the replicated partition function Z n is given by exp β Jij Siα Sjα + βh Siα Zn = (2) ij α
{Siα }
i
α
where Siα is the αth replica of the Ising spin on the ithe site and β is inverse temperature. To obtain the free energy per spin fd on a d-dimensional hypercubic lattice, the random average is performed in two steps. The free energy is given by 1 ln Z n N→∞ n→0 nN
−βfd = lim lim
(3)
where · · · is the random average over the couplings between different clusters and · · · is the random average over the couplings inside the clusters. Z n is given by
˜ 2 ˜)2 α β α β (β J ) (β J n J α α α Z n = exp exp β Jij Si Sj + βh Si + S i S i S j Sj 1− Nn 4 dns 2d α α α {Si }
ij
ij α<β
i
˜ 2 (β J ) nJ ≡ exp exp −βH (n) 1− Nn 4 dns α
(4)
{Si }
where nJ and ns are numbers of bonds and spins in a cluster, respectively. The pairs represented by ij and ij are the nearest neighbor pairs inside a cluster and between different clusters, respectively. In the cluster approximation, a set of variational parameters {q αβ } are used by a replacement β β β β Siα Si Sjα Sj −→ q αβ Siα Si + Sjα Sj . (5) By the replacement, the replicated Hamiltonian H (n) is changed to (n) H˜ (n) = Ωclust ,
(6)
clust
where (n)
Ωclust = −
clust ij α
Jij Siα Sjα −
clust clust β J˜2 nJ αβ α β q Si Si − h Siα . ns − nb d α (i) α<β
(7)
i
clust
clust Here nb is the number of boundary spins in a cluster and the summations clust run over nearest neighbor bonds, ij , (i) and i boundary sites and all the sites in a single cluster, respectively. Variational calculation with respect to {q αβ } gives self-consistent relation on the boundary sites q αβ = S α S β (8) where · · · is thermal average with respect to the Hamiltonian H˜ (n) . The variational parameters {q αβ } may be considered as mean fields on the boundary sites. The free energy in the cluster approximation f˜d is given by
˜)2 ˜)2 αβ 2 1 (β J 1 n n (β J J J −β f˜d = (9) q + ln exp −βΩ (n) 1− 1− + lim min − . n→0 n {q αβ } 4 dns 2 dns ns α<β
{S}
The cluster approximation method has been applied to study the Ising spin glass model using 2-spin and 4-spin clusters close to the spin glass transition temperature [7]. Here a Bethe approximation type cluster is used to compare results with those for the Bethe
T. Yokota / Physics Letters A 356 (2006) 439–445
441
Fig. 1. Bethe approximation type cluster for d = 3.
approximation. The cluster for d = 3 is given in Fig. 1. Contrary to the cases of 2-spin and 4-spin clusters [7], there is a site that is not boundary in the Bethe approximation type cluster. To study the spin glass phase close to the transition temperature and for a small magnetic field, the free energy is expanded up to orders q 4 and h2 q: 2 4 q αβ + A3 q αβ q αβ q βγ q γ α + A4 −β f˜d A0 + lim A2 n→0
+ B4
α =β
q
α =β =γ =α
αβ βγ γ δ δα
q
q q
+ C4
α =β
q
αβ 2 βγ 2 q
α =β =γ =α
αβγ δ all different
(βh)2 αβ + q n + D1 . 2
(10)
α =β
The order parameter function q(x) is obtained following the method in [9]: q , 0 x x , q(x) =
0
0
cx, x0 x x1 , q1 , x1 x 1,
(11)
where D1 (βh)2 , 16(A4 + B4 − C4 ) A3 , c= 4(A4 + B4 − C4 ) A2 . q1 = 3A3 q03 =
(12) (13) (14)
Spin glass transition temperature Tc without external field is obtained from A2 = 0 and A2 can be written as A2 = −A2c
T Tc
(15)
close to the transition temperature, where T = T − Tc . The slope of the order parameter function q(x) may be related to the strength of the replica symmetry breaking [7]. The AT line is given by 2 T −3 16(A4 + B4 − C4 ) A2c 3 Tc 2 h ≡ α. − Tc D1 3A3 J˜ J˜
(16)
For the Bethe approximation type cluster, ns = 2d + 1,
(17)
nb = 2d,
(18)
nJ = 2d.
(19)
442
T. Yokota / Physics Letters A 356 (2006) 439–445
Table 1 Transition temperature Tc as well as c, q1 and α defined by (13), (14) and (16), respectively, near the transition temperature for d from 1 to 6 in the Bethe approximation type cluster approximation Tc /J˜ c q1 TcT−T c α
d =1
d =2
d =3
d =4
d =5
d =6
0.754 0.470 1.024 0.626
0.915 0.469 1.056 1.157
0.955 0.475 1.057 1.315
0.972 0.480 1.054 1.374
0.981 0.484 1.050 1.398
0.986 0.487 1.046 1.408
Because all the J ’s in a cluster obey the same distribution, the suffix will be omitted and the random average will be denoted by · · ·. 2 ˜2 (2d−1) , the A, B, C and D coefficients are obtained as follows: Defining μ ≡ β J 4d 2 2d μ − + μ2 1 + (2d − 1) tanh2 βJ , A2 = (20) 2d + 1 2 2 8d A3 = μ3 (21) 1 + 3(2d − 1) tanh2 βJ , 3(2d + 1) 2 4 d 4 + d(2d − 1) tanh2 βJ A4 = −μ4 2d + 1 3 3 2 2 + 4d(2d − 1)(d − 1)tanh4 βJ tanh2 βJ + d(2d − 1) tanh4 βJ (22) , 2 2 2 4d 1 + 6(2d − 1) tanh2 βJ + 2(2d − 1)(d − 1)tanh4 βJ tanh2 βJ + (2d − 1) tanh4 βJ , 2d + 1 2 2 2 8 C4 = μ4 d(2d − 1) tanh2 βJ − 4d(2d − 1)(d − 1)tanh4 βJ tanh2 βJ − d(2d − 1) tanh4 βJ , 2d + 1 2 4d 1 + tanh2 βJ + (2d − 1) tanh2 βJ . D1 = μ 2d + 1
B4 = μ4
(23) (24) (25)
Transition temperature Tc and the quantities c, q1 and α that characterize the replica symmetry breaking close to the transition temperature are given in Table 1 for the dimensions from 1 to 6. Results for the SK model are recovered when d → ∞. For the SK model, c = 1/2, q1 = −T /J˜ and α = 4/3. The results for c are smaller than that of the SK model, which may be considered as a sign of enhancement of the replica symmetry breaking [7]. The region of the replica symmetry breaking in a field is reduced for d from 1 to 3 and enhanced for higher dimensions. The results for lower dimensions are similar to those for the 2-spin cluster [7]. 3. Bethe approximation Mottishaw studied the replica symmetry breaking in the Bethe approximation close to the spin glass transition temperature [4]. Although the replica symmetry breaking is shown to survive in the Bethe approximation, explicit form of the order parameter function was not given. To discuss the strength of the replica symmetry breaking, details of the order parameter function are needed. In this section, the order parameter function and the AT line are obtained explicitly. Because the Bethe approximation is valid on the Bethe lattice, the problem is considered on the Bethe lattice in the following. On the Bethe lattice, the order parameter qα1 α2 is insufficient to describe the spin glass phase and infinitely many order parameters qα1 α2 , . . . , qα1 ...αr , . . . are needed [4,10]. The global order parameter gn ({S α }) is used instead, which includes informations of the infinitely many order parameters. Near the spin glass transition temperature, only qα1 α2 and qα1 α2 α3 α4 are needed to obtain the order parameter function q(x) and the AT line. As the details of the derivation of the equations for the order parameters are described in Ref. [4], we will sketch it here. Because the normalization of the spin glass order parameter in Ref. [4] is different from the usual one [10], the normalization in Ref. [10] is used here to compare the results to those obtained by other approximations. The global order parameter gn ({S α }) is related to the spin glass order parameters as [10] ∞ gn S α =
coshn βJ tanhs βJ qβ1 ...βs S β1 · · · S βs .
(26)
s=0 β1 <β2 <···<βs
The equation for the global order parameter satisfies ∞
α
α α α K α α α S1 + βJ α S1 S )(gn ({S1 })) −∞ dJ ρ(J ) Tr{S1 } exp(βh
α = gn S Tr{S1α } exp(βh α S1 )(gn ({S1α }))K
(27)
T. Yokota / Physics Letters A 356 (2006) 439–445
443
where ρ(J ) is the Gaussian bond distribution and K + 1 is the number of neighbors, which is 2d in the Bethe approximation. The order parameter function qα1 ...αr satisfies
Tr{S1α } S1α1 · · · S1αr exp(βh α S1α )(gn ({S1α }))K
qα1 ...αr = (28) . Tr{S1α } exp(βh α S1α )(gn ({S1α }))K In the limit of n → 0, the equations for the order parameters are obtained as ∞ K β1 βs α1 α α s r qα1 ...αr = Tr{S1α } S1 · · · S1 tanh βJ qβ1 ...βs S1 · · · S1 exp βh S1 .
(29)
α
s=0 β1 <β2 <···<βs
)3 , only two equations for qα1 α2 and qα1 ...α4 are relevant. The equations for a small field are To order ( TcT−T c 2 qα1 β qβα2 qα1 α2 (βh)2 + Ktanh2 βJ qα1 α2 + K(K − 1) tanh2 βJ β
+ K(K − 1)tanh βJ tanh βJ 2
4
qβ1 β2 qβ1 β2 α1 α2 + 5permutations
β1 <β2
3 + K(K − 1)(K − 2) tanh2 βJ qα1 β1 qβ1 β2 qβ2 α2 β1 =β2
3 K(K − 1)(K − 2) 2 3 2 + tanh βJ (qβ1 β2 ) − 2(qα1 α2 ) , 3qα1 α2 6
(30)
β1 <β2
2 qα1 α2 α3 α4 Ktanh4 βJ qα1 α2 α3 α4 + K(K − 1) tanh2 βJ (qα1 α2 qα3 α4 + qα1 α3 qα2 α4 + qα1 α4 qα2 α3 ).
(31)
Defining tr as tr = tanh2r βJ ,
(32)
we have from Eq. (31) t12 (K − 1) (qα α qα α + qα1 α3 qα2 α4 + qα1 α4 qα2 α3 ). K(1 − t2 ) 1 2 3 4 Using this equation, Eq. (30) becomes K −1 2 (K − 1)(K − 2) 3 qα1 α2 (βh)2 + t1 qα1 α2 + qα1 β qβα2 − t1 (qα1 α2 )3 t K 1 3K 2 qα1 α2 α3 α4
(33)
β
+
(K − 1)2 t13 t2 (K − 1)(K − 2) 3 2 qα α t q (q ) + β1 β2 1 α1 α2 2K 2 K 2 (1 − t2 ) 1 2 β1 <β2
+
(qβ1 β2 )2
β1 <β2 β1 =α1 ,α2 β2 =α1 ,α2
(K − 1)(K − 2 + t2 )t13 qα1 β1 qβ1 β2 qβ2 α2 . K 2 (1 − t2 )
(34)
β1 =β2
Free energy per spin f may be obtained considering that this equation comes from the saddle point of the free energy: −nβf
1 K −1 2 (βh)2 qαβ + (t1 − 1) (qαβ )2 + t 2 4 6K 1 α =β
α =β
qαβ qβγ qγ α
α =β =γ =α
2 (K − 1)(K − 2) 3 (K − 1)(K − 2) 2 − t1 (qα1 α2 )4 + t13 qαβ 24K 2 32K 2 α =β α =β
3 2 (K − 1) t1 t2 2 2 2 2 4 + (qα1 α2 ) (qβ1 β2 ) − 4 (qαβ ) (qβγ ) − 2 (qαβ ) 16K 2 (1 − t2 ) α1 =α2
+
− 2 + t2 )t13
(K − 1)(K 8K 2 (1 − t2 )
β1 =β2
αβγ δ all different
α =β =γ =α
qαβ qβγ qγ δ qδα .
α =β
(35)
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T. Yokota / Physics Letters A 356 (2006) 439–445
Table 2 Transition temperature Tc as well as c, q1 and α defined by (13), (14) and (16), respectively, near the transition temperature for d from 2 to 6 in the Bethe approximation Tc /J˜ c q1 TcT−T c α
d =2
d =3
d =4
d =5
d =6
0.593 0.162 0.788 0.473
0.738 0.219 0.855 0.828
0.806 0.259 0.889 1.005
0.846 0.289 0.910 1.104
0.872 0.312 0.924 1.165
In the limit of d → ∞ and n → 0, this becomes −nβf
1 1 (βh)2 qαβ + (β J˜)2 − 1 (qαβ )2 + 2 4 6 α =β
−
1 1 (qα1 α2 )4 + 24 8 α =β
α =β
qαβ qβγ qγ α
α =β =γ =α
qαβ qβγ qγ δ qδα .
(36)
αβγ δ all different
This is just the free energy of the SK model near the spin glass transition temperature. Using the notation defined in Eq. (10), the A, B, C and D coefficients are given by 1 A2 (t1 − 1), 4 K −1 2 t , A3 6K 1 (K − 1)(K − 2 + 2t2 K − t2 )t13 , A4 − 24K 2 (1 − t2 ) (K − 1)(K − 2 + t2 )t13 B4 , 8K 2 (1 − t2 ) (K − 1)2 t13 t2 C4 − , 4K 2 (1 − t2 ) D1 1.
(37) (38) (39) (40) (41) (42)
Transition temperature Tc and the quantities c, q1 and α defined in Eqs. (11), (13), (14) and (16) are given in Table 2 for the dimensions from 2 to 6. The explicit form of α is given by 3 Tc 2 4K 4 (K − 2 + 2Kt2 − t2 ) 2 β J tanh β J 1 − tanh β J . α (43) c c c 3(K − 1)2 (1 − t2 ) J˜ This is the same instability line as that obtained by Thouless [3]. The slope of the order parameter function c reduces considerably compared to the value 1/2 for the SK model. Also the region of replica symmetry breaking under the AT line is much smaller than that for the SK model. 4. Conclusions Bethe approximation and a cluster approximation with a Bethe approximation type cluster are applied to the Ising spin glass model close to the transition temperature in a field. The spin glass order parameter function q(x) and the AT line are obtained. The transition temperature for the Bethe approximation is much lower than that for the corresponding cluster approximation. Although it is still higher than the realistic short range model [1], fluctuation effects seem to be included more properly. The spin glass properties characterized by the order parameter function q(x) and the locus of the AT line change considerably from those for the SK model. The reduction of the slope for q(x) may be considered as increase of the replica symmetry breaking [7]. On the contrary, the region of the replica symmetry breaking in a field reduces in the Bethe approximation that does not possess a loop. A loop that include frustration effects explicitly might change the conclusion for the Bethe approximation [7,8]. References [1] N. Kawashima, H. Rieger, in: H.T. Diep (Ed.), Frustrated Spin Systems, World Scientific, Singapore, 2004, Chapter 9. [2] M. Mézard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond, World Scientific, Singapore, 1987. [3] D.J. Thouless, Phys. Rev. Lett. 56 (1986) 1082.
T. Yokota / Physics Letters A 356 (2006) 439–445
[4] [5] [6] [7] [8] [9] [10]
P. Mottishaw, Europhys. Lett. 4 (1987) 333. M. Mézard, G. Parisi, Eur. Phys. J. B 20 (2001) 217. R. Baviera, M. Pasquini, M. Serva, J. Phys. A 31 (1998) 4127. T. Yokota, Physica A 363 (2006) 161. A. Georges, M. Mézard, S.S. Yedidia, Phys. Rev. Lett. 64 (1990) 2937. D.J. Thouless, J.R.L. de Almeida, J.M. Kosterlitz, J. Phys. C 13 (1980) 3271. Y.Y. Goldschmidt, C. De Dominicis, Phys. Rev. B 41 (1990) 2184.
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