Replica mean-field theory for Levy spin-glasses

0.12 MB·English

by  A. Engel

Replica mean-field theory for Levy spin-glasses
A. Engel∗
Institut fu¨r Physik, Carl-von-Ossietzky-Universtit¨at, 26111 Oldenburg, Germany
Infinite-rangespin-glassmodelswithLevy-distributedinteractionsshowafreezingtransitionsim-
ilar to disordered spin systems on finite connectivity random graphs. It is shown that despite
7 diverging moments of the local field distribution this transition can be analyzed within the replica
0 approachbyworkingatimaginarytemperatureandusingavariantofthereplicamethoddeveloped
0 for diluted systems and optimization problems. The replica-symmetric self-consistent equation for
2 the distribution of local fields illustrates how the long tail in the distribution of coupling strengths

n gives rise to a significant fraction of strong bonds per spin which form a percolating backbone at
a thetransition temperature.
J
0 PACSnumbers: 02.50.-r,05.20.-y,89.75.-k
1

]
n
n I. INTRODUCTION
-
s
i Spin-glasses are model systems of statistical mechanics in which simple degrees of freedom interact via random
d
. couplings drawn from a given probability distribution [1]. The ensuing interplay between disorder and frustration
t
a gives rise to peculiar static and dynamic properties which made spin-glasses paradigms for complex systems with
m competing interactions. In this way the concepts and techniques developed for their theoretical understanding [2]
- became invaluable tools in the quantitative analysis of problems originating from such diverse fields as algorithmic
d complexity [3, 4, 5], game theory [6, 7], artificial neural networks [8], and cryptography [9].
n
In the present note we study a spin-glass for which the couplings strengths are drawn from a Levy-distribution.
o
The main characteristic of these distributions are power-law tails resulting in diverging moments. Compared to
c
[ the extensively studied spin-glass models with Gaussian [10] or other finite moment distributions [11, 12, 13] Levy-
distributed couplings are interesting for several reasons. On the one hand the comparatively large fraction of strong
1
bonds givesrise to a mechanismforthe glasstransitionwhich is differentfromthe usualscenario. Onthe otherhand
v
these systems pose new challenges to the theoretical analysis because the diverging second moments invalidate the
7
central limit theorem which is at the bottom of many mean-field techniques. Related issues of interest include the
9
1 spectral theory of random matrices with Levy-distributed entries [14, 15] and relaxation and transport on scale-free
1 networks[16]. It is also interesting to note that the characteristicproperties of the Cauchy-distributionhave recently
0 enabled progress in the mathematically rigorous analysis of matrix games with random pay-off matrices [17].
7 The model consideredbelow with the help ofthe replicamethod wasanalyzedpreviouslyby CizeauandBouchaud
0
(CB) using the cavity approach [18]. In their paper CB remark that they resorted to the cavity method because
/
t they were not able to make progress within the replica framework. This might have been caused by the fact that
a
m at that time the central quantity in the replica method was the second moment of the local field distribution, the
so-calledEdwards-Andersonparameter [19], which for Levy-distributedcouplings is likely to diverge. Later a variant
-
d of the replica method wasdeveloped to deal with non-Gaussianlocalfield distributions characteristicfor diluted spin
n glasses and complex optimization problems [20]. Until now this approachwas used only in situations where the local
o field distribution is inadequately characterized by its second moment alone and higher moments of the distribution
c are needed for a complete description. Here we show that the method may also be adapted to situations where the
:
v moments not even exist.
i
X
r II. THE MODEL
a
We consider a system of N Ising spins S = 1, i=1,...,N with Hamiltonian
i
±
1
H( S )= J S S , (1)
{ i} −2N1/α ij i j
X
(i,j)
∗Electronicaddress: [email protected]
2
where the sum is over all pairs of spins. The couplings J = J are i.i.d. random variables drawn from a Levy
ij ji
distribution P (J) defined by its characteristic function [22]
α
P˜ (k):= dJ e−ikJ P (J)=e−|k|α (2)
α α
Z
with the real parameter α (0,2]. The thermodynamic properties of the system are described by the ensemble
∈
averagedfree energy
1
f(β):= lim lnZ(β), (3)
−N→∞βN
with the partition function
Z(β):= exp( βH( S )). (4)
i
− { }
X
{Si}
Here β denotes the inverse temperature and the overbar stands for the average over the random couplings J .
ij
III. REPLICA THEORY
To calculate the average in (3) we employ the replica trick [19]
Zn 1
lnZ = lim − . (5)
n→0 n
As usual we aim at calculating Zn for integer n by replicating the system n times, S Sa , a = 1,...,n, and
{ i} 7→ { i}
then try to continue the results to real n in order to perform the limit n 0.
DuetothealgebraicdecayP (J) J −α−1 ofthedistributionP (J)fo→rlarge J theaverageZn(β)doesnotexist
α α
for real β. On the other hand, for a∼pu|re|ly imaginary temperature, β = ik, k |R|, we find from the very definition
− ∈
of P (J), cf. (2)
α
k α α
Zn( ik)= exp | | SaSa + (1) . (6)
− {XSia} (cid:16)− 2N Xi,j (cid:12)(cid:12)Xa i j(cid:12)(cid:12) O (cid:17)
(cid:12) (cid:12)
Note that the replica Hamiltonian is extensive which justifies a-posteriori the scaling of the interaction strengths
with N used in (1). The determination of Zn can be reduced to an effective single site problem by introducing the
distributions
1
c(S~)= δ(S~ ,S~), (7)
i
N
Xi
where S~ = Sa stands for a spin vector with n components. We find after standard manipulations [20]
{ }
k α
Zn( ik)= dc(S~)δ( c(S~) 1)exp N c(S~)lnc(S~)+ | | c(S~)c(S~′)S~ S~′ α . (8)
− Z YS~ XS~ − (cid:16)− hXS~ 2 SX~,S~′ | · | i(cid:17)
In the thermodynamic limit, N , the integral in (8) can be calculated by the saddle-point method. The
→ ∞
corresponding self-consistent equation for c(~σ) has the form
c(~σ)=Λ(n)exp k α c(S~)S~ ~σ α , (9)
(cid:16)−| | XS~ | · | (cid:17)
where the Lagrange parameter Λ(n) enforces the constraint c(S~)=1 resulting from (7).
S~
P
3
IV. REPLICA SYMMETRY
Within the replicasymmetricapproximationone assumesthatthe solutionof(9)is symmetricunder permutations
of the replica indices implying that the saddle-point value of c(S~) depends only on the sum, s := Sa, of the
a
components of the vectorS~. It is then convenientto determine the distribution oflocal magnetic fieldsPP(h) from its
relation to c(s) as given by [20]
ds s
c(s)= dhP(h)e−ikhs P(h)= eish c( ). (10)
Z Z 2π k
Note that the P(h) defined in this way is normalized only after the limit n 0 is taken. The distribution of local
→
magnetic fields is equivalentto the free energyf(β) since allthermodynamicproperties maybe derivedfromsuitable
averageswith P(h) [21].
To get an equation for P(h) from (9) we need to calculate
drdrˆ
e−ikhs S~ ~σ α = rαeirrˆ exp ikhs irˆS~ ~σ (11)
XS~ | · | Z 2π | | XS~ (cid:16)− − · (cid:17)
drdrˆ
= rαeirrˆ exp iSa(kh+rˆσa) (12)
Z 2π | | XS~ Ya (cid:16)− (cid:17)
= drdrˆ rαeirrˆ[2cos(kh+rˆ)]n+2σ [2cos(kh rˆ)]n−2σ (13)
Z 2π | | −
σ
drdrˆ cos(kh+rˆ) 2
rαeirrˆ , (14)
→Z 2π | | (cid:20)cos(kh rˆ)(cid:21)
−
where the limit n 0 was performed in the last line and σ := σa. Using Λ(n) 1 for n 0 [20] we therefore
→ a → →
find from (9) in the replica symmetric approximation P
drdrˆ σ cos(kh+rˆ)
c(σ)=exp k α dhP(h) rαexp irrˆ+ ln . (15)
(cid:18)−| | Z Z 2π | | (cid:16) 2 cos(kh rˆ)(cid:17)(cid:19)
−
Using this result in (10) and performing the transformations r r/k,rˆ rˆk we get
7→ 7→
ds drdrˆ s cos(kh′+krˆ)
P(h)= exp ish dh′P(h′) rαexp irrˆ+ ln . (16)
Z 2π (cid:18) −Z Z 2π | | (cid:16) 2k cos(kh′ krˆ)(cid:17)(cid:19)
−
We are now in the position to continue this result back to real values of the temperature by simply setting k = iβ.
In this way we find the following self-consistent equation for the replica symmetric field distribution P(h) of a Levy
spin-glass at inverse temperature β
ds drdrˆ s coshβ(h′+rˆ)
P(h)= exp ish dh′P(h′) rαexp irrˆ i ln . (17)
Z 2π (cid:18) −Z Z 2π | | (cid:16) − 2β coshβ(h′ rˆ)(cid:17)(cid:19)
−
V. SPIN GLASS TRANSITION
From (17) we infer that the paramagnetic field distribution, P(h)=δ(h), is always a solution. To test its stability
weplugintother.h.s. of(17)adistributionP (h)withasmallsecondmoment,ǫ := dhP (h)h2 1,calculatethe
0 0 0
≪
l.h.s. (to be denoted by P1(h)) by linearizing in ǫ0 and compare the new second moRment, ǫ1 := dhP1(h)h2, with
ǫ0. We find ǫ1 > ǫ0, i.e. instability of the paramagnetic state, if the temperature T is smaller thRan a critical value
T determined by
f,α
drdrˆ Γ(α+1) α+1 drˆ
(T )α = rαeirrˆtanh2rˆ= cos( π) tanh2rˆ. (18)
f,α −Z 2π | | − π 2 Z rˆα+1
| |
This result for the freezing temperature is essentially the same as the one obtained by CB using the cavity method
[18]. Our somewhat more detailed prefactor ensures that the limit α 2 correctly reproduces the value TSK = √2
→ f
of the SK-model [10]. The dependence of T on α is shown in fig. 1.
f,α
4
1.5
1.4
1.3
1.2
α
Tf, 1.1
1
0.9
0.8
0.7
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
α
FIG.1: Freezing temperatureTc,α of an infinite-rangespin-glass with Levy-distributedcouplings as function of theparameter
α of the Levy-distribution defined in (2). For the scaling of the coupling strength with N as chosen in (1) there is a finite
transition temperature for all values of α. In thelimit α→2 theresult for the SK-modelis recovered.
The peculiarities of the spin-glass transition in the present system are apparent from the similarity between (17)
andanalogousresults for stronglydiluted spinglassesanddisorderedspinsystems onrandomgraphs[11, 20, 21]. To
make this analogy more explicit we rewrite (17) in a form that allows to perform the s-integration to obtain
ds ∞ ( 1)d d dr drˆ s d coshβ(h +rˆ)
P(h)= eish − dh P(h ) i i r αeirirˆi exp i ln i i
i i i
Z 2π Xd=0 d! Z iY=1(cid:16) 2π | | (cid:17) (cid:16)− 2β Xi=0 coshβ(hi−rˆi)(cid:17)
∞ ( 1)d d dr drˆ 1 d
= − dh P(h ) i i r αeirirˆi δ h tanh−1(tanhβh tanhβrˆ) . (19)
i i i i i
Xd=0 d! Z Yi=1(cid:16) 2π | | (cid:17) (cid:16) − β Xi=0 (cid:17)
Thisformoftheself-consistentequationissimilartothosederivedwithinthecavityapproachforsystemswithlocally
tree-like topology [5, 11, 20] and may also form a suitable starting point for a numerical determination of P(h) using
a population-dynamical algorithm [21].
VI. DISCUSSION
Infinite-range spin-glasseswith Levy-distributed couplings are interesting examples of classicaldisorderedsystems.
The broad variations in coupling strengths brought about by the power-law tails in the Levy-distribution violate
the Lindeberg condition for the application of the central limit theorem and give rise to non-Gaussian cavity field
distributions with divergingmoments. We haveshownthat it is nevertheless possible to derive the replica symmetric
properties of the system in a compact way by using the replica method as developed for the treatment of strongly
dilutedspinglassesandoptimizationproblems[20]whichfocusesfromthe startonthe complete distributionoffields
rather than on its moments.
Due to the long tails in the distribution of coupling strengths Levy spin-glasses interpolate between systems with
many, i.e. (N), weak couplings per spin as the Sherrington-Kirkpatrick model and systems with few, i.e. (1),
O O
strong couplings per spin as the Viana-Bray model. The majority of the N 1 random interactions coupled to each
−
spin are very weak (of order N−1/α). These weak couplings will influence only the very low temperature behaviour
which may be expected to be similar to that of the SK-model. On the other hand the largest of N random numbers
drawn independently from the distribution (2) is of order N1/α [23] and hence every spin also shares a fraction of
5
strong bonds, J = (1), which are for J > 1/β practically frozen. With decreasing temperature a growing
ij ij
O | |
backbone of frozen bonds builds up that percolates at the transition temperature T [18]. The mechanism for the
f,α
freezing transition is hence rather different from that operating in the Sherrington-Kirkpatrick model and resembles
the one taking place in disordered spin systems on random graphs with local tree-structure.
Acknowledgments
I would like to thank Daniel Grieser, R´emi Monasson and Martin Weigt for clarifying discussions.
[1] BinderK. and Young. A.P., Rev. Mod. Phys. 58, 801 (1986)
[2] Mezard M., Parisi G., and Virasoro M. A., Spin-glass Theory and Beyond (World Scientific, Singapore, 1987)
[3] Monasson R.,Zecchina R., Kirkpatrick S., Selman B., and TroyanskiL., Nature 400, 133 (1999)
[4] Mezard M., Parisi G., and ZecchinaR., Science 297, 812 (2002)
[5] HartmannA.K.andWeigtM.,PhaseTransitionsinCombinatorialOptimizationProblems(WileyVCH,Weinheim,2005)
[6] Diederich S. and OpperM., Phys. Rev. A39, 4333 (1989)
[7] Berg J. and Engel A., Phys. Rev. Lett. 81, 4999 (1998)
[8] Engel A. and Van den Broeck C., Statistical Mechanics of Learning, (Cambrigde University Press, Cambrigde, 2001)
[9] Mislovaty R., Klein E., KanterI., and Kinzel W., Phys. Rev. Lett. 91, 118701 (2003)
[10] Sherrington D. and Kirkpatrick S., Phys. Rev. Lett. 35, 1972 (1975)
[11] Viana L. and Bray A.J., J. Phys. C18, 3037 (1985)
[12] KanterI. and Sompolinski H., Phys. Rev. Lett. 58, 164 (1987)
[13] Mezard M. and Parisi G., Europhys. Lett. 3, 1067 (1987)
[14] BurdaZ., Jurkiewicz J., Nowak M. A., Papp G., and Zahed I., cond-mat/0602087
[15] Biroli G., Bouchaud J.-P., and Potters M., cond-mat/0609070
[16] Albert R.and Barabasi A.-L., Rev. Mod. Phys. 74, 47 (2002)
[17] RobertsD. P. Int. J. Game Theory 34, 167 (2006)
[18] Cizeau P. and Bouchaud J.-P., J. Phys. A26, L187 (1993)
[19] Edwards S.F. and Anderson P. W., J. Phys. F5, 965 (1975)
[20] Monasson R.,J. Phys. A31, 513 (1998)
[21] Mezard M. and Parisi G., Eur. Phys. J. B20, 217 (2001)
[22] GnedenkoB.V.andKolmogorov A.N.,LimitDistributions for Sums of Independent Random Variables(Addison-Wesley,
ReadingMA, 1954)
[23] Bouchaud J.-P. and Georges A., Phys. Rep. 195, 127 (1990), App.B