Free energy expansion of the spin glass with finite connectivity for ∞ RSB
aa r X i v : . [ c ond - m a t . d i s - nn ] J a n Free energy expansion of the spin glass with finiteconnectivity for ∞ RSB
Gioia Boschi and Giorgio Parisi Department of Mathematics, King’s College London, Strand, London WC2R 2LS,United Kingdom Dipartimento di Fisica, Sapienza Università di Roma, Piazzale A. Moro 2, 00185Rome, ItalyE-mail: [email protected], [email protected]
Abstract.
In this paper, we investigate the finite connectivity spin-glass problem.Our work is focused on the expansion around the point of infinite connectivity of thefree energy of a spin glass on a graph with Poissonian distributed connectivity: we areinterested to study the first-order correction to the infinite connectivity result for largevalues or the connectivity z . The same calculations for one and two replica symmetrybreakings were done in previous works; the result for the first-order correction wasdivergent in the limit of zero temperature and it was suggested that it was an artifactfor having a finite number of replica symmetry breakings. In this paper we are ableto calculate the expansion for an infinite number of replica symmetry breakings: inthe zero-temperature limit, we obtain a well defined free energy. We have shown thatcancellations of divergent terms occur in the case of an infinite number of replicasymmetry breakings and that the pathological behavior of the expansion was due onlyto the finite number of replica symmetry breakings. Keywords : Spin Glasses, Finite Connectivity, Replica Simmetry, Ground State, Free-Energy Expansion
1. Introduction
In the last 40 years, a large amount of work has been dedicated to Spin Glass models withinfinite connectivity, especially the very famous Sherrington-Kirkpatrick (SK) model[1, 2, 3]. Given also the technical difficulties, less attention has been dedicated to morerealistic finite connection mean-field models ( [4, 5, 6, 7]) with average connectivity z .The main problem common to finite connectivity systems is that the local fields are notGaussian as infinite range models, but their distribution is a more complicated function.This implies that the order parameter is a function of the overlaps of any number ofreplicas instead of the overlap over only two of them as in the SK model. Consequently,in these models, it is difficult to find the exact free energy. ree energy expansion of the spin glass with finite connectivity for ∞ RSB
2A general solution for this problem has been proposed in [8], [9] using the Bethe-Peierls cavity method. It is conjectured, but it remains unproven that this approachgives the correct result ([10, 11, 12] ). Perturbative expansions have been proposed nearthe infinite connectivity point (SK model) [13], [14], [15], [16] (i.e. the /z expansion)and near the critical temperature [17].In this paper we consider spin glasses with Poisson distributed connectivity [5] andwe study the large connectivity ( z ) expansion of the free energy for an infinite numberof replica symmetry breakings (RSBs); at the end we specalize our formulae to thezero-temperature limit. In particular, we will calculate the first term of the expansionalready investigated by Goldschmidt and De Dominicis [14] and Parisi and Tria in [16]respectively for 1RSB and 2RSB. De Dominicis and Goldschmidt found that the freeenergy expansion diverges at low temperatures for 1RSB and Parisi and Tria showedthat for 2RSB the divergence is less pronounced. This indicates that the divergentbehavior was due only to the finite number of RSBs. In this work, we show that whenan infinite number of RSBs is considered, the divergent parts of the expansion cancelout leaving only a small residual divergence due to numerical errors.This paper is organized as follows: in section (2) we describe the model and wewrite the expression for the first term of the free energy expansion as a function of /z .This term will be composed of different integrals, that we compute in section (3). Inthe same section, we study the divergent behavior of these integrals for the temperature T which goes to 0, showing analytically how we can reduce the divergences to a lineardivergence in β = 1 /T . In the last section (4) we finally combine the analytical resultswith the numerical evaluation to obtain an estimation of the first term of the free energyexpansion, showing how its different divergent components cancel out.
2. The Large z expansion for the Random connectivity model We consider a model with N spins σ i = ± , with i ∈ { ...N } interacting with randomcouplings and each one connected with z i other spins. The coupling are defined on theedges of an Erdös-Rény graph, with links drawn with probability z/N . The distributionof the connectivities ( z i ) is Poissonian with mean z in the limit for N which goes toinfinity. The couplings J ij distributed according to the following formula: P ( J ik ) = (cid:18) − zN (cid:19) δ ( J ik ) + zN ˜ P ( J ik ) . (1)If two nodes of the graph are not connected the value of J ij is null, while if they areconnected the distribution of the values of J ij is the following: ˜ P ( J ik ) = 12 " δ J ik − √ z ! + δ J ik + 1 √ z ! ∀ i, k . (2)This distribution has been chosen to be binary for simplicity, however we can obtainslightly more complicated expressions for Gaussian distributed J ij . Given the random ree energy expansion of the spin glass with finite connectivity for ∞ RSB ln( N ) . Thereforein the infinite volume limit the graph approaches a Bethe lattice that does not havefinite size loops. For this reason the model belongs to the a mean field category, even ifit cannot be written in a simple way (as for the SK model) due to the difficulties relatedto the finite connectivity: for more explicit formulae see [12].As usual we are interested in finding the average free-energy density defined as f ( z, β ) = lim N →∞ − βN Z dJ ik P ( J ik ) ln Z J ( z, β ) , (3)with Z J ( z, β ) = X { σ } exp {− βH J [ { σ } ] } . (4)It can be shown that for z → ∞ the mean free energy density becomes the SK one lim z →∞ f ( z, β ) = f SK ( β ) . (5)The computation of the free energy can be studied with the replica trick f ( n, z, β ) = lim N →∞ − βN n ln Z n , f ( z, β ) = lim n → f ( n, z, β ) , (6)where n is the number of replicas and · denotes the average over the disorder. Thepartition function for the n replicas is Z n = Y i 24 + β n X a
24 + β n X a x s ). We recall that m ( x, z ) is the average value of themagnetization of the trajectories that taking the value z at x end up in x = 1 . We findconvenient represent the quantity q ( x ) by the tree diagram shown in Figure 1. Figure 1. Utrametric tree for the two indices overlap, by G. Parisi e F. Tria [16] Figure 2. Ultrametric trees for the 4 replicas overlap, by G. Parisi e F. Tria [16]. In the same way we can rewrite all the integrals in Eq. 28, considering the fivepossible ways in which the replicas can be organized (represented by the tree diagramsin Figure 2): q ( x , x , x ) = Z dz dz dz P ( x , z | , P ( x , z | x , z ) P ( x , z | x , z ) m ( x , z ) m ( x , z ) , (40) q ( x , x , x ) = Z dz dz dz P ( x , z | , P ( x , z | z , x ) (41) P ( x , z | z , x ) m ( x , z ) m ( x , z ) m ( x , z ) , (42) ree energy expansion of the spin glass with finite connectivity for ∞ RSB q ( x ) = Z dz P ( x , z | , m ( x , z ) , (43) q ( x , x ) = Z dz dz P ( x , z | , P ( x , z | x , z ) m ( x , z ) m ( x , z ) , (44) q ( x , x ) = Z dz dz P ( x , z | , P ( x , z | x , z ) m ( x , z ) m ( x , z ) . (45)If we now introduce the sum over the four indices we find: − n β X a
1] ++ Z β dy y [ q ( y ) − 1] + 4 Z β dy Z βy dy y [ q ( y , y ) − 1] ++6 Z β dy Z βy dy [ q ( y , y ) − , (55) ree energy expansion of the spin glass with finite connectivity for ∞ RSB A = 1 β " Z β dy Z βy dy Z βy dy + 12 Z β dy Z βy dy Z βy dy + (56) + Z β dy y + 4 Z β dy Z βy dy y + 6 Z β dy Z βy dy . If we multiply A for the proper factor from Eq. 46 we find that the contribution of thenon convergent part to the finale results is : β A = 18 β , (57)that is clearly divergent for β → ∞ .We can do the same for the two indices sum: − lim n → n X ab q ab ≡ β Z β q ( y ) dy = 1 β Z β [ q ( y ) − dy + A = I + A , (58)with I = 1 β Z β [ q ( y ) − dy (59)convergent and the term A = 1 which contributes to the β divergent term. Thedivergent part, multiplied by the factor from Eq. 46, gives: β A = β . (60)The last term to analyze is β / , which is divergent at zero temperature. If we sumtogether all the divergent terms, we can see that they cancel out, solving the problemof the divergences in β : β − β β . (61) β divergences The divergences check is not finished, we have in fact to check the superficially convergent terms. We shall see that these terms are only convergent if we do a superficialanalysis, but they are divergent as β .Let us consider to the following contribution to f ( β ) : G ( β ) ≡ − β I ( β ) + β I ( β ) . (62)Using the asymptotic behaviour of q ( y ) (i.e. − c/y ) we find that I ( β ) contains botha term proportional to β − and a term equal to − cβ − . These two terms multiplied py β give a term proportional to β and β . Similar terms are present also in I ( β ) . Weshall see now the cancellation of the β out. ree energy expansion of the spin glass with finite connectivity for ∞ RSB Let us analyse the behaviour of the firstintegral appearing in I (Eq. 54): Z β dy Z βy dy Z βy dy ( q ( y , y , y ) − . (63)Before analysing the function q ( y , y , y ) , we can manipulate it in order to simplify therest of the calculations. As we can see from the ultrametric tree in Fig. 2, the argumentof the integral is symmetric under the exchange of y and y , so we can write: Z β dy Z βy dy Z βy dy ( q ( y , y , y ) − 1) = (64) = 2 Z β dy Z βy dy Z βy dy ( q ( y , y , y ) − . As in Eq. 40, the function q ( y , y , y ) can be written as q ( y , y , y ) = Z dz dz dz P ( y , z | , P ( y , z | y , z ) (65) P ( y , z | y , z ) m ( y , z ) m ( y , z ) , with the conditions y > y and y > y . For y going to infinity: y → ∞ m ( y , z ) → , (66)the expression for q ( y , y , y ) becomes: q ( y , y , y ) = Z dz dz P ( y , z | , P ( y , z | y , z ) m ( y , z ) , (67)where: Z dz P ( y , z | , P ( y , z | y , z ) = P ( y , z | , , Z dz P ( y , z | , m ( y , z ) = q ( y ) . (68)Therefore the divergent part of the first diagram is: Z β dy Z βy dy Z βy dy [ q ( y ) − . (69)With a similar reasoning we can find the divergent part of all the other diagrams.After having multiplied them for the appropriate factors and summed them together weobtain: A ≡ β " Z β dy Z βy dy Z βy dy [ q ( y ) − (70) + 12 Z β dy Z βy dy Z βy dy [ q ( y ) − 1] ++ 4 Z β dy Z βy dy y [ q ( y ) − 1] + 6 Z β dy y Z βy dy [ q ( y ) − = − β Z β dyy [ q ( y ) − 1] + 8 1 β Z β dy [ q ( y ) − − β Z β dyy [ q ( y ) − 1] + 8 I , ree energy expansion of the spin glass with finite connectivity for ∞ RSB G ( β ) is: f = − β I + β 48 [ I − A ] + β A = (71) = − β I + β 48 [ I − A ] − β Z β dyy [ q ( y ) − 1] + β I == β 48 [ I − A ] − Z β dyy [ q ( y ) − . One could hope that everying is now convergent. Unfortunately at this point there isanother kind of divergence not yet analysed. If we take the last term of the sum in Eq.71, we can show that this diverges linearly as β goes to infinity, in fact: q ( y ) ∼ − cy , (72)so: Z β dyy [ q ( y ) − ∼ Z β dyy cy = Z β cdy = βc . (73)We can argue that the multivariable overlap has the same trend in y and that theother integrals have the same divergence problem. Therefore we can expect a reciprocalcancellation of the linear divergences in β which come out from these integrals. Howeverthis last claim can not be verified analytically, so we used numerical techniques to proveit. 4. Evaluation of f at T = 0 and conclusions Given that is not possible to further simplify the problem to check analytically if areciprocal cancellation of the terms leads to a convergent f , we will evaluate numerically(details can be found in Appendix B) the different integrals appearing in Eq. (71) whichcan be written in the following way: f = 148 " Z β dy Z βy dy Z βy dy [ q ( y , y , y ) − q ( y )]+ (74) + 12 Z β dy Z βy dy Z βy dy [ q ( y , y , y ) − q ( y )] ++ Z β dy y [ q ( y ) − 1] + 4 Z β dy Z βy dy y [ q ( y , y ) − q ( y )] ++ 6 Z β dy y Z βy dy [ q ( y , y ) − q ( y )] − Z β dyy [ q ( y ) − C C C C C C . In order to evaluate the magnetizations appearing in the integrals (e.g. the ones in Eq.67) we generated trajectories of the following form z i +1 = z i + √ ǫη ( q i ) − y ( q i ) m ( q i , z i ) ǫ , (75)with ǫ = q i +1 − q i and q i which goes from 0 to q MAX , following the branching point ofthe trees in Fig. 2 (see Appendix B). In the generation of the trajectories the values of ree energy expansion of the spin glass with finite connectivity for ∞ RSB . 150 0 . 175 0 . 200 0 . 225 0 . 250 0 . 275 0 . 300 0 . /y − . − . − . − . − . C ǫ = 1 / ǫ = 1 / . 150 0 . 175 0 . 200 0 . 225 0 . 250 0 . 275 0 . 300 0 . /y − . − . − . − . − . − . − . − . C ǫ = 1 / ǫ = 1 / . 150 0 . 175 0 . 200 0 . 225 0 . 250 0 . 275 0 . 300 0 . /y − − − − − C ǫ = 1 / ǫ = 1 / . 150 0 . 175 0 . 200 0 . 225 0 . 250 0 . 275 0 . 300 0 . /y − . − . − . − . − . − . − . C ǫ = 1 / ǫ = 1 / . 150 0 . 175 0 . 200 0 . 225 0 . 250 0 . 275 0 . 300 0 . /y − . − . − . − . − . − . − . C ǫ = 1 / ǫ = 1 / . 150 0 . 175 0 . 200 0 . 225 0 . 250 0 . 275 0 . 300 0 . /y − . − . − . − . − . − . − . − . C ǫ = 1 / ǫ = 1 / Figure 3. The figure shows the numerical solution of the integrals in Eq. 74 infunction of /y for different values of the discretization interval ǫ . y ( q i ) and m ( q i , z i ) are obtained through interpolation from a table of results for q ( x ) , y ( q ) and m ( q, z ) at T = 0 calculated with the same method of [26], with 40 RSB. Inthis way we are able to calculate the integrals in Eq. 74 at T = 0 (see Appendix B).We will now focus on the behaviour of the divergent terms in function of y .In Fig. 3 we plot the solution of the different integrals of Eq. 74 in function of /y ∼ T for ǫ = 1 / and ǫ = 1 / . The figure shows that all the terms divergefor T → . However, when we plot f (Fig. 4) we can notice that there are greatcancellations between the different terms. Looking at Fig. 4 we can see that f appearsto have a "residual" divergence for T → . This divergence can be addressed to theapproximations done to compute numerically the integrals, which are the discretizationof the differential equation in Eq. 75 and the finite, even if large, number of replicasymmetry breakings used to estimate q ( x ) . The effect of the finite size of ǫ is evidentlooking at Fig. 4, where we compare the values of f for ǫ = 1 / and ǫ = 1 / . For ǫ = 1 / we can see that the divergence of f tends to decrease as a result of a bettercancellation between the different integrals. This improvement with the decreasing of ǫ ree energy expansion of the spin glass with finite connectivity for ∞ RSB . 150 0 . 175 0 . 200 0 . 225 0 . 250 0 . 275 0 . 300 0 . /y . . . . . . . . . f ǫ = 1 / ǫ = 1 / Figure 4. Numerical solution of f in function of /y for different values of thediscretization interval ǫ . suggests that for ǫ → the divergence disappears leaving a finite value for f . Giventhe divergences due to these errors we couldn’t calculate the values of f at small T,we used a linear extrapolation to find its value at T = 0 . Using the last two estimatedpoints (1 /y ) = 0 . e (1 /y ) = 0 . and their corresponding ordinates, we obtained: f ( T = 0) = 0 . ± . . (76)The error being purely statistical. Looking at the curve obtained by G.Parisi e F. Tria in[16], we can notice that in that case an extrapolation at T = 0 would lead to a greatervalue respect to the one evaluated with our data. The reasons for this discrepancycan be attributed to the underestimation due to the finite step ǫ used to evaluate thetrajectories in Eq. 75, that could bring to a systematic error of the same order of thestatistical one. The value we get for f ( T = 0) , i.e. . ± . is not far from thenumerical value of ≈ . found in [24].We can conclude that there are analytical and numerical evidence that the ex-pansion of the free energy around the point of infinite connectivity can be successfullycomputed at low temperature. Similar, but albeit more difficult, computations can bedone in mean field model of structural glasses in high, but finite dimensions. Appendix A. The definition of the q ’s functions The functions that appears in 28, corresponding to the trees in 2, are defined as q ( x , x , x ) = q abcd | a ∧ b = x ,c ∧ d = x ,a ∧ c = b ∧ c = a ∧ d = b ∧ d = x (A.1) q ( x , x , x ) = q abcd | a ∧ b = x ,a ∧ c = b ∧ c = x ,a ∧ d = b ∧ d = c ∧ d = x (A.2) q ( x ) = q abcd | a ∧ b = a ∧ c = a ∧ d = b ∧ c = b ∧ d = c ∧ d = x (A.3) q ( x , x ) = q abcd | a ∧ b = a ∧ c = x ,a ∧ d = b ∧ d = c ∧ d = x (A.4) q ( x , x ) = q abcd | a ∧ b == x ,a ∧ c = a ∧ d = b ∧ c = b ∧ d = c ∧ d = x (A.5) ree energy expansion of the spin glass with finite connectivity for ∞ RSB Appendix B. Details of the numerical evaluation of f We want to evaluate integrals of the form of Eq. 28. Let us take the following one(corresponding to the second tree in Fig. 2) Z β dy Z βy dy Z βy dy q ( y , y , y ) . (B.1)We can rewrite it in the following way Z λ dq dy dq Z λq dq dy dq Z λq dq dy dq q ( y , y , y ) , (B.2)imposing a cut off λ to control the divergence of the different terms for λ → . Theargument of the integral is described in Eq. 41. In order to evaluate it we generaterandomly q , q and q in the interval [0 , a large number M of times such that q > q e q > q . For each random generation of the three variables we calculate the argumentof the integral in these points and than take the average over all the M generations. Foreach generation j we will have q ( y , y , y ) j = m η ( q ) j m η ( q ) j m η ( q ) j m ω ( q ) j m ω ( q ) j m ω ( q ) j , (B.3)such that Z β dy Z βy dy Z βy dy q ( y , y , y ) = (B.4) = 1 M M X j m η ( q ) j m η ( q ) j m η ( q ) j m ω ( q ) j m ω ( q ) j m ω ( q ) j . In order to evaluate m η ( q ) and m ω ( q ) we need to take two random walks (with noises η and ω ) using Eq. (75) which starts from q = 0 , pass trough q and then branches intwo leaves ending at q and q , with q < λ e q < λ (see figure 2). We should noticethat it is sufficient to average the generation of the random overlaps, used to performthe integration, in order to perform also the average over the different trajectories. Thesame procedure has been applied to calculate the other integrals appearing in Eq. 74. Aknowledgments This project has received funding from the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation programme (grant agreementNo [694925]). References [1] D. Sherrington and S. Kirkpatrick, “Solvable model of a spin-glass,” Physical review letters , vol. 35,no. 26, p. 1792, 1975.[2] M. Mézard, G. Parisi, and M. Virasoro, Spin glass theory and beyond: An Introduction to theReplica Method and Its Applications , vol. 9. World Scientific Publishing Company, 1987.[3] F. Morone, F. Caltagirone, E. Harrison, and G. 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