Stability of the Parisi Solution for the 1 Sherrington-Kirkpatrick model near T = 0 1 0 2 A Crisanti1 and C De Dominicis2 n 1 DipartimentodiFisica,UniversitàdiRomaLa SapienzaandISC-CNR,P.le a AldoMoro2,I-00185Roma,Italy. J 2 Institut de Physique Théorique,CEA-Saclay-OrmedesMerisiers,91191Gif 7 surYvette, France 2 E-mail: andrea.crisanti@phys.uniroma1.it, cirano.de-dominicis@cea.fr ] n Abstract. TotestthestabilityoftheParisisolutionnearT =0,westudythe n spectrumoftheHessianoftheSherrington-KirkpatrickmodelnearT =0,whose - eigenvalues are the masses of the bare propagators in the expansion around the s i mean-field solution. In the limit T ≪ 1 two regions can be identified. In the d first region, for x close to 0, where x is the Parisi replica symmetry breaking . scheme parameter, the spectrum of the Hessian is not trivial and maintains the t a structureofthefullreplicasymmetrybreakingstatefoundathighertemperatures. m In the second region T ≪ x ≤ 1 as T → 0, the components of the Hessian become insensitive to changes of the overlaps and the bands typical of the full - d replicasymmetrybreakingstatecollapse. Inthisregiononlytwoeigenvalues are n found: a null one and a positive one, ensuring stability for T ≪1. In the limit T → 0 the width of the first region shrinks to zero and only the positive and o null eigenvalues survive. As byproduct we enlighten the close analogy between c [ the static Parisi replicasymmetry breaking scheme and the multiple time-scales approach of dynamics, and compute the static susceptibility showing that it 1 equals the static limit of the dynamic susceptibility computed via the modified v fluctuation dissipationtheorem. 3 3 2 PACSnumbers: 75.10.Nr,64.70.Pf 5 . 1 0 1 Submitted to: J. Phys. A: Math. Gen. 1 : v i X r a Stability of the SK model near T =0 2 1. Introduction: The physics of spin glasses is still an active field of research because the methods and techniques developed to analyze the static and dynamic properties have found application in a variety of others fields of the complex system world, such as neural networks or combinatorial optimization or glass physics. In the study of spin glasses a central role is played by the Sherrington-Kirkpatrick (SK) model [1], introduced in the middle of 70’s,as a mean-field model for spin glasses. Despite its solution,known asthe “Parisisolution”[2,3,4],wasfound30yearsago,someaspectarestillfarfrom being completely understood. In this work we discuss the spectrum of the Hessian of the fluctuations for the Parisi solution in the limit of vanishing temperature, a still not fully explored problem. The Hessian spectrum plays a centralrole non only for the stability of the Parisi solution of the mean-field SK model, but also for the study of finite dimensional systems. Its eigenvalues are indeed the masses of the “bare” propagators in the loop expansionaboutthe mean-fieldlimit. Thusthe knowledgeofthe Hessianspectrumof the SK model is a prerequisitefor any theory obtainedfroma developmentaboutthe mean-field limit. The stability of Parisisolutionfor the SK model nearits criticaltemperature T , c hasbeenestablishedlongago[5,6]byexhibitingtheeigenvaluesoftheHessianmatrix. Infewwords,onehasaRepliconbandwhoselowesteigenvaluesarezeromodes,anda Longitudinal-Anomalous(LA) band, sitting at (T T), of positive eigenvalues (both c − with) a band width of order (T T)2. The analysis was partially extended later [7] c − viathederivationofWard-Takahashiidentities,showingthatthezeroRepliconmodes would remain null in the whole low temperature phase, and hence would not ruin the stability under loop corrections to the mean-field solution. Despite these efforts a complete analysis of the stability in the zero temperature limit is still missing. Near T one can take advantage of the vanishing of the order c parameterforT =T andexpandthefreeenergy,asimplificationclearlymissingclose c tozerotemperature,wheretheorderparameterstaysfinite. MoreovertheT =0limit ishighlynon-trivial. Allthesemakethederivationof“effective”approximationsvalid for T 0 a rather difficult task [8, 9]. → In this work, anticipating the main results, we show that in the limit T 1 the ≪ spectrum of the Hessian can be divided into two regions. A first region where the spectrum maintains a structure similar to that found close to T , and a secondregion c where only two eigenvalues, one null and one positive, are found. In the limit T 0 → the width of the first region shrinks to zero, and only the second region survives. The outline of the paper is as follow. In Section 2 we describe how the Hessian of fluctuations associated with the SK model is obtained. In Section 3 we discuss the properties of the Parisi solution in the low limit T 1 and how these affect the ≪ Hessian spectrum by considering three simple cases. In Section 4 we show how spins averages, and response functions, involving any number of spins can be computed within the Parisi Replica Symmetry Breaking scheme with a finite number R of replica symmetry breaking steps. In Sections 5 and 6 using the results of Section 4 we derive the Hessian spectrum in the T 0 limit for both the Replicon and → Longitudinal-Anomalous Sectors. Finally Section 7 contains some discussions and conclusions. The two Appendices contain details on the calculation of spin averages in the continuous R limit, Appendix A, and the T 1 limit, Appendix B. → ∞ ≪ Finally in Appendix C for completeness we report the approach in terms of frozen Stability of the SK model near T =0 3 fields probability distribution functions. 2. Free energy functional, fluctuations and propagator masses The model is defined by the Hamiltonian [10] 1 H = J s s (1) ij i j −2 i,j X where s = 1 are N Ising spins located on a regular d-dimensional lattice and the i ± symmetric bonds J ,whichcouple nearest-neighborspins only,arerandomquenched ij Gaussianvariablesofzeromean. Thevarianceisproperlynormalizedtoensureawell defined thermodynamic limit N . To average over the disorder one introduces → ∞ replicas. After standard manipulations the free-energy density functional f in the thermodynamic limit is written as a function of the symmetric n n site dependent × replica overlap matrix Qab as [11]: i e−nNf/T = dQab exp Qab (2) i L{ i } Z (Yab) Yi β2 Qab = (p2+1) (Qab)2 L{ i } − 2 p p (ab) X X + lnTrsa exp β2 Qaibsasb (3) Xi (cid:16) X(ab) (cid:17) where Qab is the spatialFourier transformof Qab with respect to the site index i and p i β =1/T. The notation “(ab)” means that sum is over distinct ordered pairs a x′. To make contact with the static Parisi solution one takes → ∞ x [0,1],withx=0correspondingtothelargestpossiblerelaxationtimeandx=1− ∈ to the shortest one. With this assumption one recoversQ(0)=0 and Q(1−)=q (T), c thelargestoverlap. InbothcasesQ(1)=1,sinceitgivestheselforequal-timeoverlap. Other choices are possible, e.g., those used in [14, 15, 16, 17, 18] to tackle the T 0 → limit. We stress however that different choices just give a different parameterization ofthe functionQ(x),butdonotchangethe physics,sincethisisgivenbythepossible values q that the function Q(x) can take and by their probability distribution P(q). This propertyis calledgaugeinvariance[13,19, 14]. Inwhat follows,unless explicitly stated, we take for x the Parisi parameterization. The quadratic term (2) = β2 (p2+1) δQab 2 L − 2 p Xp X(ab)(cid:0) (cid:1) β4 + δQabδMab;cdδQcd. (10) 2 i i (abX),(cd) defines the “bare” propagators of the theory. This quadratic form in δQab contains i the Hessian matrix Mab;cd =δKr β2δMab;cd (ab);(cd)− =δKr β2 sasbscsd sasb scsd (11) (ab);(cd)− h i−h ih i of the SK model whose eigenvalues rulehthe stability of the meani-field solution, and give the masses of the “bare” propagators. Terms with higher powers of δQab in the i expansion (5) defines the interaction vertices of the theory. In the reminder of this paper we shall consider the eigenvalue spectrum of the Hessian matrix Mab;cd of the SK model for the Parisi solution in the very low temperature limit T 1. ≪ ‡ TheequalityQab=hsasbithatfollowsfromthestationarityconditionisvalidonlyfora6=b. For consistency onedefines Qaa=QR+1=1. Stability of the SK model near T =0 5 2.1. The Hessian Mab;cd: the Replicon and the Longitudinal-Anomalous Sectors With 4 replicas the Hessian is characterized by 3 overlaps. We can distinguish two different geometries: (i) The Longitudinal-Anomalous (LA) Sector. This is characterized by the two overlaps r = a b and s = c d and, if r = s, the single cross-overlap t = ∩ ∩ 6 max[a c,a d,b c,b d]. Then we denote the matrix element in the LA Sector as ∩ ∩ ∩ ∩ Mab;cd =Mr;s, r,s=0,1,...R; t=0,1,...R. (12) t Note that t=R+1 if a=c or a=d or b=c or b=d. (ii)TheRepliconSector. Inthiscasea b=c d=r,andthegeometryischaracterized ∩ ∩ by the two cross-overlaps u = max[a c,a d] ∩ ∩ u,v r+1 (13) v = max[b c,b d] ≥ ∩ ∩ For the Replicon Sector the matrix elements are denoted as Mab;cd =Mr;r, u,v r+1. (14) u;v ≥ The element Mr;r, however, contains contribution from both the Replicon and LA u;v Sectors, and one has [20] Mr;r = Mr;r+Mr;r+Mr;r Mr;r (15) u;v R u;v u v − r wherethefirstistheRepliconcontributionwhiletheotherscomefromtheLASector. ThelattercanbeprojectedoutbytakingthedoubleReplicaFourierTransform(RFT) on the cross-overlapsu,v: R+1R+1 Mr;r = p p Mr;r Mr;r Mr;r +Mr;r . (16) kˆ,ˆl u v u;v− u−1;v− u;v−1 u−1;v−1 u=k v=l X X (cid:2) (cid:3) TheLAtermsindeedcancelinthisexpressionandonecanreplaceMr;r inthedouble u;v RFT by Mr;r. This in turns implies that the inverse double RFT of Mr;r yields the R u;v kˆ,ˆl Replicon contribution Mr;r and not Mr;r. R u;v u;v 3. How things work near T =0: simplest cases The equation for Q(x) is rather difficult to solve by analytical and/or numerical methods for T 0. The origin of this difficulty can be traced back to the fact → that, as the temperature decreases towards T =0, the probability of finding overlaps Qab sensibly smallerthanq (T)=1 αT2+ (T3), withα=1.575..., vanisheswith c − O T [21, 22]. There is however a finite probability x 0.524... that Qab q (T). c c ≃ ≤ Consequenceofthis theorderparameterfunctionQ(x)intheParisiparameterization develops for T 1 a boundary layer of thickness δ T close to x = 0, as shown ≪ ∼ in Fig.1. From the Figure we see that for very small T the function Q(x) is slowly varying for δ x x . However, in the boundary layer 0 < x δ, it undergoes an c ≪ ≤ ≤ abrupt and rapid change. In the limit T 0 the thickness δ T 0 and the order → ∼ → parameter function becomes discontinuous at x=0. Uniform approximate solutions valid for T 1 can be constructed by using the ≪ boundarylayertheory,thatisbystudyingtheproblemseparatelyinside(innerregion) and outside (outer region) the boundary layer [23]. One then introduces the notion Stability of the SK model near T =0 6 Q(x) Inner region Outer region 1 0.8 0.6 0.4 δ 0.2 x c 0 0 0.2 0.4 0.6 0.8 x Figure 1. Shape of the order parameter function Q(x) for T ≪1 in the Parisi parameterization. The horizontal arrow shows the extent of the boundary layer ofthickness δ∼T asT →0. of the inner and outer limit of the solution. The outer limit is obtained by choosing a fixed x outside the boundary layer, that is in δ x 1, and allowing T 0. ≪ ≤ → Similarly the inner limit is obtained by taking T 0 with x δ. This limit is → ≤ conveniently expressed introducing an inner variable a, such as a = x/δ, in terms of which the solution is slowly varying inside the boundary layer as T 0. The inner → andoutersolutionsarethencombinedtogetherbymatchingthemintheintermediate limit x 0, x/δ and T 0. The inner solution Q(a) is a smooth function of → → ∞ → a for T 0 varying between 0 and q 1 [14, 24, 16, 17], similar to Q(x) at finite c → ≃ temperature. In the rest of this paper we shall concentrate on outer solution since as T 0 it covers the overwhelming part of the interval [0,1]. → The behavior of Q(x) for T 1 has strong consequences on other relevant ≪ quantities, such as, e.g, the four-spin correlation entering into the Hessian matrix. We shall make this more quantitative in the next Sections. Here the only feature we whish to retain is that in the outer region for T x < x and T 0, the function c ≪ → Q(x) is driven closer and closer to Q(x ) = q (T) as T approaches zero. It can be c c shown [25], see also Appendix B, that for T x x and T 0 c ≪ ≤ → c Q(x)=1 c(βx)−2+ α T2+ (βx)−3,T3 , (17) − x2 − O (cid:18) c (cid:19) (cid:0) (cid:1) where c = 0.4108... and α = lim (1 q(x ))/T2. We note that the breakpoint T→0 c − x depends on T. The dependence is however very weak for low temperatures [22] c and the approximation x (T) x = 0.524... is rather good for T 0. From this c c ≃ ∼ expression we see that the variation of Q(x) in the outer region is Q(x ) Q(x) T 2 x 2 c − c 1 (18) Q(x) ≃ (cid:18)x(cid:19) " −(cid:18)xc(cid:19) # so that one can safely take the approximation Q(x) Q(x ) = q (T) as T 0, the c c ∼ → errorbeing (T2)atleast. GoingbacktoRstepsofReplica SymmetryBreakingthis O Stability of the SK model near T =0 7 r s s r a c b a b c (a) (b) r s a c b (c) Figure 2. Treeconfigurationforreplicasa,b,cwitha∩b=r. approximation translates into Q Q =q (T)=1 αT2+ (T3), T 0 (19) r R c ∼ − O → for all r in the outer region, that is such that T x(Q ) = p and T 0, or, r r ≪ → equivalently, for fixed r =0 and T 0. We shall make this insensitivity with respect 6 → to the overlaps r in the T 1 limit more precise in the next Sections. Here we ≪ just discuss the consequence of the insensitivity on the elements of the Hessian by considering some simple cases. Suppose the two pairs of replicas are equal: (a,b) = (c,d). In this case from eq. (11) one constructs the simplest Hessian component: Mab;ab =1 β2 (sasb)2 sasb sasb − h i−h ih i =1 β2h1 Qab 2 i (20) − − that for the overlap a b=r givhes (cid:0) (cid:1) i ∩ Mr;r =1 β2(1 Q2). (21) R+1;R+1 − − r Insensitivity implies that for fixed r and T 0 we have → Mr;r MR;R =1 2α+ (T2), T 0. (22) R+1;R+1 ∼ R+1;R+1 − O → The next simple case is when only three replicas are different, in which case we have Mab;ac = β2 sbsc sasb sasc , b=c. (23) − h i−h ih i 6 Ultrametricityimposesthatthehthreereplicasa,b,cwiitha b=rcanbeonlydisposed ∩ as shown in Fig. 2. The LA geometries (a) and (b) lead for T 0 and fixed r and s → to Q Q Q Q (1 Q ) Mr;s = r− r s R − R = α+ (T2), (24) R+1 − T2 ≃− T2 − O Stability of the SK model near T =0 8 while the Replicon geometry (c) yields Q Q2 Q (1 Q ) Mr;r = s− r R − R = α+ (T2). (25) R+1;s − T2 ≃− T2 − O We shall see below that insensitivity implies that Mr;r Mr;r 0, and that all s − r ∼ Replicon components vanish. Then from eq. (15) and (25) it follows MR;R = α+ (T2), T 0. (26) R+1 − O → Similarly from (15) and (22) one obtains 2MR;R MR;R =1 2α+ (T2) (27) R+1− R − O which combined with (26) gives MR;R = 1+ (T2), T 0. (28) R − O → The general case with four different replicas cannot be reduced to simple forms and the expressionofthe four-spinaveragesis required. This will be derivedthe next Section. 4. Spin Averages The evaluation of the Hessian components requires the computation of the four-spin averages sasbscsd for a generic geometry of the four replicas. This can be done by h i introducing the generating function 1 (b)=exp nG−1(b) =Trsa exp Λabsasb+ basa (29) Z 2 ! ab a (cid:8) (cid:9) X X whereΛ ,equaltoβ2Q withβ =1/T fortheSKmodel,isagenericn nsymmetric ab ab × matrix with Parisi’s block structure, Λ =λ , r =0,...,R+1. (30) ab|a∩b=r r Spin averages follow from differentiation 1 ∂ ∂ sasb = lim (b) . (31) h ···i n→0Z(b) h∂ba ∂bb ···i Z (cid:12)(cid:12)b1=···=bn=0 Introducing the “block indexes” ak, (cid:12) (cid:12) p k a=(a ,a ,...,a ), a =0,... 1 (32) 0 1 R k p − k+1 wherep ,withn=p >p > >p >p =1,aretheblocksizes,thegenerating k 0 1 R R+1 ··· function can be written as multiple integrals over independent Gaussian variables: § R (b)= (α) expG (bR+b ) (33) Z DR R α α Z 0 Y where (α) is the short-hand notation for: R D R t−1 (α) Dzt DR ≡ α Z t=0Z 0 Y Y R = Dz (34) α0,...,αt−1 tY=0Z α0,.Y..,αt−1 § WeuseGreeklettersforsummedreplicaindexes Stability of the SK model near T =0 9 and zt = z are independent Gaussian random variables of zero mean and α α0,...,αt−1 variance one: Dz dz e−z2/2. (35) ≡ √2π The function G (b) is the “free energy” of a single spin in a field b: R expG (b)=Tr exp(bs)=2coshb. (36) R s and the frozen (random) field bR, given by α R bR = ∆λ zt (37) α t α t=0 Xp where ∆λ =λ λ , keeps track of the contributions from the various blocks. t t t−1 Inserting th−e form (33) of (b) into eq. (31), and noticing that differentiation Z with respect to b can be replaced by differentiation with respect to bR, we obtain a a R 1 ∂ ∂ sasb = lim (α) expG (bR) h ···i n→0 (0) ∂bR∂bR ··· DR R α Z (cid:20) a b (cid:21)Z 0 Y R 1 = lim (α) expG (bR) n→0 (0) DR R α Z Z 0 Y ∂ ∂ G (bR) G (bR) (38) × ∂bR R a ∂bR R b ··· (cid:20) a (cid:21)(cid:20) b (cid:21) Foranygivengeometryofthereplicasa,b,c...theintegralscannowbeperformed recursively from scale R up to scale 0. To illustrate the procedure let us consider R (0)= (α) expG (bR). (39) Z DR R α Z 0 Y The field bR can be written as α R−1 bR = ∆λ zR+ ∆λ zt α R α t α t=0 p Xp = ∆λ zR+bR−1. (40) R α α Then splitting out thepzR-integrals, and recalling that zR depends only upon indexes α α α ,...,α , one has: 0 R−1 R−1 (0)= (α) DzRexp p G ( ∆λ zR+bR−1) . (41) Z DR−1 α R R R α α Z Y0 Z n p o This structure suggests introducing quantities G (b) as r exp p G (br−1) = Dzr exp p G ( ∆λ zr +br−1) (42) r r−1 α α r r r α α (cid:8) (cid:9) Z n p o so that eq. (39) can be written as R−1 (0)= (α) exp p G (bR−1) , (43) Z DR−1 R R−1 α Z 0 Y (cid:8) (cid:9) Stability of the SK model near T =0 10 thathasthesameformof(39)providedR R 1. Theentireprocesscanbeiterated → − up to level 0 and leads to (0)=exp p G (0) = Dzexp p G ( λ z) . (44) 0 −1 0 0 0 Z { } Z n p o In the limit p =n 0 one recovers the usual expression [12] 0 → G (0)= DzG ( λ z). (45) −1 0 0 Z p Equation(42) hasaninteresting“physical”interpretation. The quantityG (bR) R isthefreeenergyofasystemofonespin,i.e. ofsize1,inthereplicaspaceinpresence of the frozen field bR, that is with all random (Gaussian) zr held fixed. To move one levelup,R R 1,wehavetounfreezeandintegrateoverzR,whilekeepingallother → − fields zt with t < R frozen. The fields zt with t < R give the effective action, under the form of a (random) field, of the spins sb on the spin sa with a b=t

Stability of the Parisi Solution for the Sherrington-Kirkpatrick model near T=0

Similar Stability of the Parisi Solution for the Sherrington-Kirkpatrick model near T=0