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arXiv:cond-mat/9808146v2 [cond-mat.stat-mech] 14 Aug 1998

Replica analysis of the p-spin interactions Ising spin-glass model

Viviane M de Oliveira and J F Fontanari Instituto de F? ?sica de S? ao Carlos Universidade de S? ao Paulo Caixa Postal 369 13560-970 S? ao Carlos SP Brazil Abstract

The thermodynamics of the in?nite-range Ising spin glass with p-spin interactions in the presence of an external magnetic ?eld h is investigated analytically using the replica method. We give emphasis to the analysis of the transition between the replica symmetric and the one-step replica symmetry breaking regimes. In particular, we derive analytical conditions for the onset of the continuous transition, as well as for the location of the tricritical point at which the transition between those two regimes becomes discontinuous. Short Title: p-spin Ising model PACS: 87.10+e, 64.60Cn

1

1

Introduction

Although the thermodynamics of the Ising spin glass with in?nite-range interactions, so-called Sherrington-Kirkpatrick (SK) model [1], has been thoroughly investigated in the last two decades [2, 3], comparatively little attention has been given to the analysis of a natural generalization of the SK model, namely, the p-spin interactions Ising spin glass. This model is described by the Hamiltonian [4, 5] Hp (S) = ? Ji1 i2 ...ip Si1 Si2 . . . Sip ? h Si (1)

1≤i1 <i2 ...<ip ≤N i

where Si = ±1, i = 1, . . . , N are Ising spins and h is the external magnetic ?eld. Here the coupling strengths are statistically independent random variables with a Gaussian distribution P Ji1 i2 ...ip = Ji i ...i N p?1 N p ?1 exp ? 1 2 p . πp! p!

2

(2)

Besides the acknowledged importance of the p-spin interactions Ising spin glass in the framework of the traditional statistical mechanics of disordered systems (it yields the celebrated random energy model in the limit p → ∞ [4] and the SK model for p = 2), it also plays a signi?cant role in the study of adaptive walks in rugged ?tness landscapes within the research program championed by Kau?man [6, 7, 8]. The thermodynamics of the SK model (p = 2) as well as that of the random energy model (p → ∞) are now well understood. In particular, for p = 2 and h = 0 the order parameter function q (x) tends to zero continuously as the temperature (2) approaches the critical value Tc = 1 at which the transition between the spinglass and the high temperature (disordered) phases takes place [2, 3]. For p → ∞ √ (∞) and h = 0, the system has a critical temperature Tc = 1/ 2 ln 2 at which it freezes completely into the ground state: q (x) is a step function with values (∞) zero and one, and with a break point at x = T /Tc [4, 5]. These results are not a?ected qualitatively by the presence of a non-zero magnetic ?eld. In particular, for the SK model the critical temperature decreases monotonically with increasing h while the transition remains continuous, in the sense that q (x) is continuous at the transition line [9, 10]. In contrast, for the random energy model the critical temperature increases with increasing h while the discontinuity in the step function q (x) decreases with increasing h and vanishes in the limit h → ∞ [4, 5]. The situation for ?nite p > 2 is considerably more complicated and so the thermodynamics of the p-spin model has been investigated for h = 0 only [11, 12].

2

In this case there is a transition from the disordered phase to a partially frozen phase characterized by a step function q (x) with values zero and q1 < 1. As the temperature is lowered further, a second transition occurs, leading to a phase described by a continuous order parameter function [11, 12]. However, there are evidences that the presence of a non-zero magnetic ?eld decreases the size of the discontinuity of the order parameter q (x) leading, eventually, to a continuous phase transition. In fact, a recent analysis of the typical overlap q ? between pairs of metastable states with energy density ? indicates that q ? is a discontinuous function of ? for p > 2, and that the size of the jump in q ? increases with p and decreases with h, vanishing at ?nite values of the magnetic ?eld [13]. Moreover, a similar e?ect has already been observed in the thermodynamics analysis of the spherical p-spin interaction spin-glass model [14]. It is interesting to note that the spin-glass phase of this continuous spin model is described exactly by a step order parameter function, i.e., the one-step replica symmetry breaking is the most general solution within the Parisi scheme of replica symmetry breaking [14]. In this paper we use the replica method to study the thermodynamics of the Ising p-spin interaction spin-glass model in the presence of the magnetic ?eld h. We focus on the e?ects of h on the transition between the replica symmetric (RS) and the one-step replica symmetry breaking (1 RSB) regimes. In particular, we show that for p > 2 the discontinuous transition reported in previous analyses [11, 12] turns into a continuous one for h larger than a certain value hT . Moreover, we derive analytical conditions to determine the location of the continuous transition line, as well as that of the tricritical point at which the transition becomes discontinuous. The remainder of the paper is organized as follows. In Sec. 2 we discuss the replica formulation and present the formal equation for the average free-energy density, which is then rewritten using the RS and the 1 RSB ans¨ atze. These results are discussed very brie?y since their derivations are given in detail in Gardner’s paper [11]. We also present the solution of the 1 RSB saddle-point equations in the limit of large p, extending thus the series expansions results for non-zero h. In Sec. 3 we derive analytical conditions for locating the continuous transition and the tricritical point between the RS and 1 RSB regimes, and present the phase diagrams in the plane (T, h). Finally, some concluding remarks are presented in Sec. 4.

3

2

The replica formulation

? βf = lim 1 N ln Z (3)

We are interested in the evaluation of the average free-energy density f de?ned by

N →∞

where Z = TrS exp [?β Hp (S)] , (4) is the partition function and β is the inverse temperature. Here . . . stands for the average over the coupling strengths, and TrS denotes the summation over the 2N states of the system. As usual, the evaluation of the quenched average in Eq. (3) can be e?ectuated through the replica method: using the identity ln Z = lim

n→ 0

1 ln n

Zn

(5)

we ?rst calculate Z n for integer n, i.e. Z n = n a=1 Za , and then analytically continue to n = 0 [2, 3]. The ?nal result is simply [11] ? βf = lim extr

n→ 0

1 1 G (qab , λab ) + β 2 n 4

(6)

where G (qab , λab ) = 1 2 λab qab q p ? β2 β 2 a<b ab a<b + ln Tr{S a } exp ?β 2 1 N

N a Si i=1 T b Si T

?

λab S a S b + βh

a<b a

S a? .

?

(7)

The extremum in Eq. (6) is taken over the physical order parameter qab = a < b, (8)

which measures the overlap between two di?erent equilibrium states Sa and Sb , and over its corresponding Lagrange multiplier λab . Here, . . . T stands for a thermal average. To proceed further, next we consider two standard ans¨ atze for the structure of the saddle-point parameters.

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2.1

Replica symmetric solution

In this case we assume that the saddle-point parameters are symmetric under permutations of the replica indices, i.e., qab = q and λab = λ. With this prescription the evaluation of Eq. (6) is straightforward, resulting in the replica-symmetric freeenergy density 1 1 ? βfrs = ? β 2 λ (1 ? q ) + β 2 (1 ? q p ) + 2 4 where and √ Ξs = z λ + h dz ?z 2 /2 e Dz = √ 2π

∞ ?∞

Dz ln 2 cosh [β Ξs (z )]

(9)

(10) (11)

is the Gaussian measure. The saddle-point equations ?frs /?q = 0 and ?frs /?λ = 0 yield p (12) λ = q p ?1 2 and ∞ q= Dz tanh2 [β Ξs (z )] , (13)

?∞

respectively. The replica-symmetric solution is locally stable wherever the AlmeidaThouless condition [9], which in this case is given by [11] 1 ? β 2 (p ? 1) λ q

∞ ?∞

Dz sech4 [β Ξs (z )] > 0,

(14)

is satis?ed. In fact, since Eq. (13) has either one or three positive solutions, this stability condition is very useful to single out the physical one. In particular, for h = 0 the only stable solution is q = 0.

2.2

Replica symmetry broken solution

Following Parisi’s scheme [3], we carry out the ?rst step of replica symmetry breaking by dividing the n replicas into n/m groups of m replicas and setting qab = q1 , λab = λ1 if a and b belong to the same group and qab = q0 , λab = λ0 otherwise. The physical meaning of the saddle-point parameters is the following q0 = 1 N

N a Si i=1 T b Si T

a < b,

(15)

5

q1 =

1 N

N a Si i=1 2 T

,

(16)

2 . Hence q is the overlap between a pair of di?erent equilibrium and m = 1 ? a Pa 0 states, q1 is the overlap of an equilibrium state with itself (q1 ≥ q0 ), and m is the probability of ?nding two copies of the system in two di?erent states (Pa is just the Gibbs probability measure for the state Sa ). We note that in the limit n → 0, the parameter m is constrained to the range 0 ≤ m ≤ 1. Using this prescription, Eq. (6) becomes

1 p p ? βfrsb = ? β 2 [2λ1 (1 ? q1 + mq1 ) ? 2mq0 λ0 ? 1 + (1 ? m) q1 + mq0 ] 4 ∞ 1 ∞ Dz1 coshm β Ξ (17) Dz0 ln + ln 2 + m ?∞ ?∞ where Ξ = z1 λ1 ? λ0 + z0 λ0 + h. The saddle-point equations ?frsb /?qk = 0 yield p p ?1 λk = qk 2 (19) (18)

for k = 0, 1. The saddle-point parameters q0 , q1 and m are given by the equations

∞

q0 = q1 = and

?∞ ∞ ?∞

Dz0 tanh β Ξ 2 z, Dz0 tanh2 β Ξ

(20) (21)

z

1 2 1 p p β (p ? 1) (q1 ? q0 ) = ? 2 4 m 1 + m where we have introduced the notation ...

z

∞ ?∞ ∞ ?∞

∞

Dz0 ln

?∞

Dz1 coshm β Ξ (22)

Dz0 ln cosh β Ξ z ,

=

∞ m ?∞ Dz1 (. . .) cosh β Ξ . ∞ m ?∞ Dz1 cosh β Ξ

(23)

It is clear from these equations that the replica-symmetric saddle-point q0 = q1 = q is a solution for any value of m. In general, however, the 1 RSB equations will

6

admit a di?erent solution. In particular, in the limit p → ∞ the solution is (∞) q0 = tanh2 (βmh), q1 = 1 and m = βc /β where βc = 1/Tc is the solution of the equation [5] 1 2 (24) 4 βc = ln 2 cosh βc h ? βc h tanh βc h. Below Tc , the entropy vanishes and m sticks to its maximum value, namely m = 1, signaling the existence of a frozen phase in accordance with the physical meaning of m mentioned before. It is instructive to consider the ?nite p corrections to the in?nite-p solution by expanding the 1 RSB equations around that solution. Thus, extending the results of Gardner [11] for non-zero h, we ?nd q0 = tanh2 (βmh) ?1 + 2ξm sech (βmh)

2

(∞)

?

e?β

2 m2 p/4

1 2 2 pβ

2

?

?,

(25)

mξm e?β m p/4 q1 = 1 ? sech (βmh) 1 1?m pβ 2

2

(26)

and

1 2 4β

=

1 [ln 2 cosh (βmh) ? βmh tanh (βmh)] + Λm m2

1 2 2 pβ

(27) (28)

where Λm = ? ξm sech (βmh) e?β if β 2 m2 < 8 | ln tanh (βmh) | and Λm = β 2 p tanh2p (βmh) otherwise. Here,

2 m2 p/4

βmh , sinh (2βmh)

(29)

1 ∞ 1 m ξm = ? √ i 2i ? m 2π i=0 ∞ 1 dz [2 cosh (mz ) ? 2m coshm (z )] . = √ 2π ?∞

(30)

where we have used the extended de?nition of the binomial coe?cient to real m [15]. At this stage we already can realize the existence of two solutions of a quite di?erent nature, signaling then the non-trivial role played by the magnetic ?eld in the thermodynamics of the p-spin model.

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A quite interesting property of the 1 RSB solution, which can easily be veri?ed numerically, is that q0 = 0 for h = 0 and p > 2, indicating thus that the equilibrium states are completely uncorrelated. Moreover, this result has greatly facilitated both the numerical and analytical analyses of the model, since the integrals over z0 in Eqs. (20)-(22) can be can carried out trivially in that case [11, 12]. However, as explicitly shown by Eq. (25) the non-zero magnetic ?eld induces correlations between di?erent equilibrium states so that q0 is no longer zero in this case. For p = 3, we present in Figs. 1, 2 and 3 the temperature dependence of the RS and 1 RSB saddle-point parameters for h = 0, 0.5 and 1, respectively. As mentioned before, for h = 0 we ?nd q = q0 = 0. The size of the jump in q1 (3) decreases with increasing h and disappears altogether for h ≥ hT ≈ 0.57. Of particular interest is the temperature dependence of the saddle-point parameter m: at the discontinuous transition it reaches its maximal value, namely, m = 1, while at the continuous transition it assumes a certain value m = mc ≤ 1, which depends on T and p. As expected, the behavior pattern depicted in Fig. 3 is very similar to that found in the analysis of the magnetic properties of the SK model [10], as the transition is continuous in that model. We note that since m plays no role in the RS solution, the curve for m must end at the transition lines. The location of the transition lines as well as the characterization of the critical values of the saddle-point parameters are discussed in detail in the next section.

3

Transition lines

As indicated in the ?gures presented before, there are two qualitatively di?erent types of transition between the RS and the 1 RSB regimes which we will discuss separately in the sequel.

3.1

Continuous transition line

The location of the continuous transition between the RS and the 1 RSB solution is determined by solving the 1 RSB equations in the limit of small q1 ? q0 . More pointedly, subtracting Eq. (20) from Eq. (21) and keeping terms up to the order (q1 ? q0 )2 yields 2 B0 ( q 0 ) 2q0 (31) q1 ? q0 = 2 β (p ? 1) λ0 B2 (q0 , m) where B0 (q0 ) = 1 ? β 2 (p ? 1) λ0 q0

∞ ?∞

Dz sech4 [β Ξ0 (z )] ,

(32)

8

and B2 (q0 , m) = p ? 2 + 4β 2 (p ? 1) λ0 (3 ? 2m) ?2β (p ? 1) λ0 (8 ? 5m) Here Ξ0 (z ) = z λ0 + h (34) with λ0 given by Eq. (19). We note that both B0 and B2 are negative quantities in the 1 RSB regime. Since at the continuous transition q1 → q0 → q , where q is the replica symmetric saddle-point (13), the transition line is given by the condition B0 (q ) = 0 (35)

2 ∞ ?∞ ∞ ?∞

Dz sech4 [β Ξ0 (z )] (33)

Dz sech6 [β Ξ0 (z )] .

which, as expected, coincides with the replica-symmetric stability line given by Eq. (14). To specify the value of m at the critical line, denoted by mc , we expand Eq. (22) for small q1 ? q0 (in this case we must keep terms up to the order (q1 ? q0 )3 ) and then subtract it from Eq. (20). Using the condition (35) together with q0 → q yields B2 (q, 1) (36) 1 ? mc = ? B4 (q ) where

∞

B4 (q ) = 12 (p ? 2)

?∞

Dz sech2 [β Ξs (z )] tanh2 [β Ξs (z )]

∞ ?∞

+2β 2 (p ? 1) λ

Dz sech6 [β Ξs (z )]

(37)

with Ξs and λ given by Eqs. (10) and (12), respectively. Eq. (36) holds provided that mc ≤ 1 and so the continuous transition line must end at a tricritical point, whose location is obtained by solving B2 (q, 1) = 0 (38)

and Eq. (35) simultaneously. As usual, the denominator in Eq. (31) vanishes at the tricritical point.

3.2

Discontinuous transition line

The location of the discontinuous transition line is determined by equating the free energies of the RS and 1 RSB solutions, given by Eqs. (9) and (17), respectively.

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This task is greatly facilitated in this case by noting that setting m = 1 in Eq. (20) yields q0 = q . Moreover, since for m = 1 Eq. (17) becomes independent of q1 (and λ1 ) one has frsb (m = 1) = frs . Thus, for ?xed h the temperature at which the discontinuous transition takes place is obtained by solving the 1 RSB saddle-point equations with m = 1 for q1 , q0 = q , and T = Tc .

3.3

Analysis of the results

The phase diagrams in the plane (T, h) are presented in Figs. 4, 5, and 6 for p = 2, 3 and 10, respectively. The solid curves are the RS stability condition, Eq. (14), whose upper branch coincides with the continuous transition line, Eq. (35). The discontinuous transition lines (dot-dashed curves) join the continuous ones at the tricritical points (full circles). We also present the lines at which the entropy of the RS solution vanishes (short-dashed curves), which for h = 0 intersect the √ temperature axis at T = 1/ 2 ln 2 ≈ 0.60, whatever the value of p > 2. We note that for p → ∞ the condition for the vanishing of the RS entropy yields exactly the discontinuous transition line for the random energy model, Eq. (24). The agreement between these lines is already very good for p = 10 and h not too near ht , as illustrated in Fig. 6. Since our results are valid for real, though physically meaningless, values of p ≥ 2 as well, we present in Fig. 7 the value of the saddle-point parameter m at the continuous transition line, given by Eq. (36), for several values of p. As expected, for p > 2 we ?nd mc = 1 at the tricritical points. We mention that, despite the numerous studies of the SK model, we are not aware of any calculation of the 1 RSB saddle-point parameter m over the Almeida-Thouless stability line. In Figs. 8 and 9 we present the values of T and h at the tricritical point, respectively, as functions of the real variable p. For √ p → 2 we ?nd ht → 0 and Tt → 1, while for large p we √ ?nd that ht increases like p ln p, Tt like p/ ln p and 1 ? qt goes to zero like 1/ p ln p . These results indicate that the phase diagrams in the plane (T, h) do not display the two types of transitions only in the extreme cases p = 2 and p → ∞.

4

Conclusion

Some comments regarding the validity of the 1 RSB solution are in order. The stability analysis of that solution carried out for h = 0 indicates that it becomes unstable for low temperatures [11]. A seemly simpler approach to check the physical soundness of the 1 RSB solution is to evaluate numerically its entropy. This

10

procedure, however, has proved very elusive: since the entropy becomes negative when it is of order e?βp , the numerical precision required to determine the temperature at which it vanishes T ′′ is exceedingly large. For instance, for h = 0 we ?nd T ′′ = 0.10, 0.087 (0.19) and 0.034 (0.18) for p = 2, 3 and 5, respectively. The numbers between parentheses are the numerical estimates of ref. [12]. As our numerical results are in good agreement with that of ref. [16] for p = 2, and also are consistent with the trend of decreasing T ′′ with increasing p, we think they are the correct ones. Already for p > 5, however, we have failed to obtain reliable estimates for T ′′ . Since the precision problem becomes much worse for non-zero h, due to the numerical evaluation of the double integrals, we refrain from presenting the estimates for T ′′ in that case. Although for ?nite p the 1 RSB solution certainly does not describe correctly the low temperature phase of the p-spin Ising spin glass, it probably yields the correct solution near the transition line delimiting the replica symmetric and the replica symmetry breaking regimes. In fact, according to Gardner [11], considering further steps of replica symmetry breaking within Parisi’s scheme will result in a new continuous transition between the 1 RSB regime and a more complex regime, described by a continuous order parameter function. In this sense, we think that our results regarding the transition lines between the RS and the 1 RSB regimes are not mere artifacts of the replica method but indeed describe genuine features of the thermodynamics of the in?nite range p-spin Ising spin glass in a magnetic ?eld. Acknowledgments This work was supported in part by Conselho Nacional de Desenvolvimento Cient? ??co e Tecnol? ogico (CNPq). VMO holds a FAPESP fellowship.

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References

[1] Sherrington D and Kirkpatrick S 1975 Phys. Rev. Lett. 35 1972 [2] Binder K and Young A P 1986 Rev. Mod. Phys. 58 801 [3] M? ezard M, Parisi G and Virasoro M A 1987 Spin Glass Theory and Beyond (Singapore: World Scienti?c) [4] Derrida B 1981 Phys. Rev. B 24 2613 [5] Gross D J and M? ezard M 1984 Nuc. Phys. B 240 431 [6] Kau?man S A 1993 The Origins of Order (Oxford: Oxford University Press) [7] Amitrano C, Peliti L and Saber M 1989 J. Mol. Evol. 29 513 [8] Weinberger E D and Stadler P F 1993 J. Theor. Biol. 163 255 [9] Almeida J R L and Thouless D J 1978 J. Phys. A: Math. Gen. 11 983 [10] Parisi G 1980 J. Phys. A: Math. Gen. 13 1887 [11] Gardner E 1985 Nuc. Phys. B 257 747 [12] Stariolo D A 1990 Physica A 166 6229 [13] Oliveira V M and Fontanari J F 1997 J. Phys. A: Math. Gen. 30 8445 [14] Crisanti A and Sommers H J 1992 Z. Phys. B 87 341 [15] Feller W 1957 An Introduction to Probability Theory and its Applications, vol. I (New York: Wiley) [16] Parisi G 1980 J. Phys. A: Math. Gen. 13 1101

12

Figure captions

Fig. 1 One-step replica symmetry breaking saddle-point parameters m (solid curve) and q1 (short-dashed curve) as a function of the temperature T for p = 3 and h = 0. In this case q0 = q = 0. The discontinuous transition occurs at T ≈ 0.65. Fig. 2 One-step replica symmetry breaking saddle-point parameters m (solid curve), q0 (long-dashed curve), q1 (short-dashed curve) as a function of the temperature T for p = 3 and h = 0.5. The dash-dotted curve is the replica symmetric saddle-point parameter q . The discontinuous transition occurs at T ≈ 0.74. Fig. 3 Same as Fig. 2 but for h = 1. The continuous transition occurs at T ≈ 0.66, at which m ≈ 0.87. Fig. 4 Phase diagram in the plane (T, h) for p = 2. The RS saddle-point is locally unstable inside the region delimited by the solid curve, which coincides with the continuous transition line between the RS and the 1 RSB regimes. The short-dashed curve delimits the region inside which the RS entropy is negative. Fig. 5 Same as Fig. 4 but for p = 3. The RS stability line coincides with the continuous transition line in the branch above the tricritical point (full circle), located at Tt ≈ 0.74 and ht ≈ 0.57. The dot-dashed curve is the discontinuous transition line. The convention is the same as for Fig. 4. Fig. 6 Same as Fig. 5 but for p = 10. The tricritical point (full circle) is located at Tt ≈ 1.01 and ht ≈ 3.07. Fig. 7 Saddle-point parameter m at the continuous transition line for (from bottom to top at T = 0.5) p = 2, 2.01, 2.1, 10 and 3. Fig. 8 Temperature at the tricritical point Tt as a function of p > 2. Only integer values of p have physical meaning. Fig. 9 Magnetic ?eld at the tricritical point ht as a function of p > 2. Only integer values of p have physical meaning.

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