The Sherrington-Kirkpatrick model for spin glasses: A new approach for the solution
TThe Sherrington-Kirkpatrick model for spin glasses: A new approach for the solution
C. D. Rodr´ıguez-Camargo,
1, 2, ∗ E. A. Mojica-Nava,
3, 2, † and N. F. Svaiter ‡ Centro de Estudios Industriales y Log´ısticos para la productividad (CEIL, MD)Programa de Ingenier´ıa IndustrialCorporaci´on Universitaria Minuto de Dios, Bogot´a AA 111021, Colombia Programa de Investigaci´on sobre Adquisici´on y An´alisis de Se˜nales (PAAS-UN)Universidad Nacional de Colombia, Bogot´a AA 055051, Colombia Departamento de Ingenier´ıa El´ectrica y Electr´onicaFacultad de Ingenier´ıaUniversidad Nacional de Colombia, Bogot´a AA 055051, Colombia Centro Brasileiro de Pesquisas F´ısicasRua Dr. Xavier Sigaud, 150, 22290-180, Rio de Janeiro, RJ, Brasil
We discuss the Sherrington-Kirkpatrick mean-field version of a spin glass within the distributionalzeta-function method (DZFM). In the DZFM, since the dominant contribution to the average freeenergy is written as a series of moments of the partition function of the model, the spin-glassmultivalley structure is obtained. Also, an exact expression for the saddle points correspondingto each valley and a global critical temperature showing the existence of many stables or at leastmetastables equilibrium states is presented. Near the critical point we obtain analytical expressionsof the order parameters that are in agreement with phenomenological results. We evaluate thelinear and nonlinear susceptibility and we find the expected singular behavior at the spin-glasscritical temperature. Furthermore, we obtain a positive definite expression for the entropy and weshow that ground-state entropy tends to zero as the temperature goes to zero. We show that oursolution is stable for each term in the expansion. Finally, we analyze the behavior of the overlapdistribution, where we find a general expression for each moment of the partition function.
I. INTRODUCTION
In statistical mechanics, we define critical points asthe points where the thermodynamic free energy or itsderivative are singular [1–4]. The simplest non-trivialmodel that presents a phase transition, from paramag-netic to ferromagnetic phase, is the Ising model [5, 6].In order to describe physical systems with antiferromag-netic and ferromagnetic interactions, the concept of spinglass was introduced [7, 8]. One of the main character-istics of such systems is that the order parameter at lowtemperatures has a random spatial structure [9–12]. An-other special characteristic is the multivalley structure inthe free-energy landscape [13]. To achieve a reasonabledescription of the rich topography of the energy surface,it has been proposed a wide set of methods such as dy-namical methods [14, 15], replica methods, and others.Furthermore, by using density functional theory [16],mode coupling theory [17], and purely thermodynami-cal method the metastable states in the equilibrium free-energy landscape in any glassy system are investigated.It is shown that near a critical point, an exponential num-ber of spin-glass-like metastables states appear in thefree-energy landscape leading to a complete dynamicalfreezing of the system [18].The Edwards and Anderson (EA) Hamiltonian of spin- ∗ [email protected] † [email protected] ‡ [email protected] glasses is defined as H = − (cid:88) (cid:104) i,j (cid:105) J ij S i S j , (1)where (cid:104) i, j (cid:105) represents nearest neighbours of a lattice and J ij satisfies some probability distribution P ( J ij ), as forexample, a Gaussian probability distribution P ( J ij ) ∼ exp[ − ( J ij − J ) / J ], being J is the mean and variance J .The infinite range version of the model, was solvedin [19], where the mean field approximation is exact. Us-ing the ansatz of replica symmetry a problem of neg-ative entropy at low temperatures appears, see for ex-ample [20]. Almeida and Thouless showed that the sta-bility condition of the replica-symmetric solution is notsatisfied in the region below a line, called the Almeida-Thouless (AT) line [21].The solution of the Sherrington-Kirkpatrick (SK)model was obtained by Parisi [22–25]. Parisi sugested thereplica symmetry breaking (RSB) as a consistent schemeto break the permutational symmetry of fictitious copiesof the system introduced by the replica method where E [log Z ] = lim n → ( E [ Z n ] − /n . The RSB in disor-dered spin systems has a physical interpretation relatedwith the emergence of a spin glass phase characterizedby many pure states organized in an ultrametric struc-ture [26–28]. Despite the equations of the RSB-Parisiansatz were rigorously proven to be exact [29], othermethods and solutions have been explored in regimes for T = 0 or near the critical point [30–38]. Extensions of theRSB scheme on random graphs and neural networks arepresented in [39] and [40], respectively. However, within a r X i v : . [ c ond - m a t . d i s - nn ] F e b all those approaches there are open questions related tothe order parameters and spin-glass systems [41].Recently, it has been proposed an alternative methodto average the disorder-dependent free energy in statis-tical field theory called the distributional zeta functionmethod (DZFM) [42]. Within this approach, the domi-nant contribution to the average free energy is expressedas a series of the integer moments of the partition func-tion of the model [43–48].The aim of this paper is to explore the complexityof the free-energy landscape of the SK model using ofthe DZFM. Since we have an expansion of free energywhere all the integer moments of partition function arecontributing, we are able to investigate the multi-valleystructure each by each minimum. We show that theParisi results can be recovered using this formalism. Weexamine the connection between the DZFM and the phe-nomenological characterization of the spin-glass phase.We obtain an order parameter q k and m k for each mo-ment of the partition function. Furthermore, we ana-lyze the low temperature regime and the behaviour nearthe critical point where we obtain an expression of thecritical temperature for each minimum in the series rep-resentation of the averaged free energy. Afterward, weexamine the local magnetization in the regime m k > J >
0. We find the critical temperatures where themagnetization is non null and we extract the behavior ofthe linear susceptibility χ and the non linear susceptibil-ity χ . Keeping terms of O ( m k ) and O ( q k m k ) we showthat our solutions are compatible with the results thathave been obtained by phenomenological models and ex-periments. Finally, we analyze the stability due to theorder parameter, and show that our solution is stable foreach term in the expansion. We also study the overlapdistributions and show a different statistics with respectthe induced ultrametricity in the RSB.The organization of the paper is as follows: In SectionII we review the SK model, present the DZFM approachto this model, and examine our solution proposed to theorder parameters, beside a regularization procedure. Fur-thermore, we also obtain the ground-state entropy andwe show that tends to zero as T →
0. In Section IIIwe present our main result. The order parameters of theSK model derived from the DZFM are presented and weexplore its behavior for low temperatures and near thecritical point. We perform the calculus of the magnetiza-tion (main and local) and the susceptibilities. We studythe properties of the stability. Conclusions are presentedin Section IV. In the appendix A we review the replicamethod in order to compare with own. In the appendixB we review the derivation of the configurational averagefree energy from the distributional zeta function. Finally,in the appendix C we study the overlap distribution.
II. DZFM APPROACH TOSHERRINGTON-KIRKPATRICK MODEL
The Sherrington-Kirkpatrick (SK) is defined by theHamiltonian H = − (cid:88) i 14 ( k − β J q k − βJ m k + 1 k log I k + 14 β J . (28)In the Equations (27) and (28) I k = (cid:90) Dz exp (cid:20) k log 2cosh( βψ k ) − kβ J q k (cid:21) , (29)being ψ k = J √ q k z + J m k + h, (30)and Dz = dz √ π exp (cid:18) − z (cid:19) . (31)The extremization condition with respect to q k , δf /δq k = 0, gives us the following condition, − k ( k − β J q k + (cid:20) − kβ J + δδq k log (cid:90) Dz cosh k ( βψ k ) (cid:21) = 0 . The functional equation for the order parameter q k q k = 1( k − βJ √ q k (cid:82) Dz z cosh k ( βψ k )tanh( βψ k ) (cid:82) Dz cosh k ( βψ k ) − k − . Partial integration yields q k = (cid:82) Dz cosh k ( βψ k )tanh ( βψ k ) (cid:82) Dz cosh k ( βψ k ) . (32)On the same way, the condition δf /δm k = 0, gives thefollowing relation for the order parameter m k m k = (cid:82) Dz cosh k ( βψ k )tanh( βψ k ) (cid:82) Dz cosh k ( βψ k ) . (33)Notice the structural similarity of Eqs. (32) and (33)with respect to the results of the replica symmetric ansatz(A3) and (A4), and the 1RSB (A8), (A9), and (A7). InFIG. 1 we depict the numerical solutions of the equationssystem (32) and (33) for, recovering k B , J/k B T = 1, J /k B T = 0 . h = 0 and different k . We may evidencethat the index k would be assumed as an order param-eter to explore the metastable states in the multi-valleystructure of free energy.To obtain the ground-state entropy under the DZFM,from the free energy (27), we first derive the low-temperature form of the spin glass parameter q k for J = h = 0 and β → ∞ ( T → q k = 1 − (cid:82) Dz cosh k − ( βJ √ q k z ) (cid:82) Dz cosh k ( βJ √ q k z ) . (34)Thus, for low temperatures, (cid:90) Dz cosh k ( βJ √ q k z ) ≈ k (cid:90) Dz k e − βJ √ k qz e − kβJ √ q k z = 12 k (cid:104) e k β J q k / + k e ( k − β J q k / (cid:105) . Then, for low temperatures the spin glass parameter q yields q k = 1 − (cid:36) k ( k − (cid:104) − k e (cid:36) k ( k − (cid:105) (cid:104) k − (cid:36) k ( k − (cid:105) , (35) k . . . . . q k m k FIG. 1: Numerical solutions of (32) and (33) for J/k B T = 1, J /k B T = 0 . h = 0 and different k . Thecontinuum blue curve describes the behavior of q k andthe dashed red curve describes m k . We have atransition in these parameters for a given k .being (cid:36) k = − β J q k . After this manipulations, from(35) we have that q k → T → 0. Furthermore, byimplicit derivation ∂q k /∂T → T → 0. Within theseconsiderations, the ground-state entropy has the follow-ing form S = lim N →∞ ∞ (cid:88) k =1 ( − k +1 a k kN s k (36)where s k = e − Nk β J (cid:20)(cid:90) Dz cosh k ( βJz ) (cid:21) N g k ( β, J ) , (37)with g k ( β, J ) = 1 N k ! k + β J k ! (cid:18) − k k − ι k ( β, J ) (cid:19) . (38)being ι k ( β, J ) = (cid:82) Dz cosh k − ( βJz ) (cid:82) Dz cosh k ( βJz ) . (39)Notice that when we have a symmetric distribution, i.e. J = 0, only the values k = 2 n will contribute. Then, inthis case ( − k +1 will be always − 1. Thus, the entropy(36) will be positive if g k ( β, J ) ≤ k . By simpleinspection we have that this condition is accomplishedif (cid:82) Dz cosh k ( βJz ) ≥ (2( k − /k ) (cid:82) Dz cosh k − ( βJz )which is satisfied for all k ≥ 0. In the FIG. 2 we de-pict the behavior of (38) where we can evidence that forall k it will be always negative. With this in mind, wehave thus that each contribution in (36) will be positive. Furthermore, we can observe that with the form (36), S → β → ∞ ( T → a to regularize our quantities of interest. k − − − − g k FIG. 2: Behavior of (38) for different k , k B T /J (cid:28) N . A. Regularized quantities Since a is an arbitrary dimensionless parameter (seeAppendix B) we can use the quantity log a to regularizethe expression (27), such that we can rewrite (27) asfollows f r = 1 β ∞ (cid:88) k =1 ( − k k ! ς k . (40)Notice that (40) and the subsequent quantities arevalid on regions where ς k < 0. With this form, the totalmagnetization of the system is m r = ∞ (cid:88) k =1 ( − k k ! µ k , (41)where µ k = β J η k [( k − q k + 1] − m k (cid:37) k (42)being η k = ( k − u k + 2 m k − km k q k , (43) (cid:37) k = 1 − βJ [( k − q k − km k + 1] (44)and u k = (cid:82) Dz cosh k ( βψ )tanh ( βψ k ) (cid:82) Dz cosh k ( βψ k ) . (45)While the linear susceptibility is given, in analogy with(24) and (26), by χ r = ∞ (cid:88) k =1 ( − k β k ! µ k − q r , (46)where the spin-glass order parameter q r within the afore-mentioned regularization is defined as q r = ∞ (cid:88) k =1 ( − k β k ! (cid:8) β J η k [( k − q k + 1] − β J b k [( k − q k + 1] + c k (cid:9) , (47)being b k = 4 η k m k (cid:37) k + (cid:96) k , (48)and c k = 4 m k (cid:37) k − β J ( k − η k − m k λ k + 2 (cid:37) k βJ (1 − (cid:37) k ) . (49)In the above equation (cid:96) k = ( k − υ k + (2 − kq k ) 1 − (cid:37) k J − km k η k , (50) λ k = ( k − βJ η k − km k (1 − (cid:37) k ) , (51)and υ k = ( k − r k + 3 q k − km k u k , (52)being r k = (cid:82) Dz cosh k ( βψ k )tanh ( βψ k ) (cid:82) Dz cosh k ( βψ k ) . (53)In order to explore the advantages of the functional ex-pressions (32) and (33), and study its implications on thequantities of interest (40), (41), and (46), in the next sec-tion, we shall explore the properties of these new objectsnear critical points and at the limit T → III. NEAR THE CRITICAL POINT The behavior of the solution of the equations of state(32) and (33) is determined by the parameters β , J and J . First, we shall focus on the case without external field h = 0 and symmetric distribution of J ij ( J = 0). In thiscase, we have ψ k = J √ q k z such that tanh( βψ k ) is an oddfunction. Then the magnetization vanishes ( m k = 0 forall k ) and there is no ferromagnetic phase. Afterward weexplore the solutions where J > m k > , and q k > q k issmall, we expand the right hand side of (32) and keep-ing up to O ( q k ), for q k → 0, we have cosh k ( βJ √ q k z ) ≈ kβ J q k z + ( − kβ J z + β J k z ) q k + O ( q k ),and tanh ( βJ √ q k z ) ≈ β J q k z − β J q k z + O ( q k ).Replacing the above expansions in (32) we obtain the be-havior of the numerical solutions for q k depicted in FIG.3 where we can identify the critical temperature givenwhen β J becomes one for J = 0. As we can see inFIG. 3, q k → θ , with θ (cid:28) 1, as we are approaching tothe critical point and k → ∞ . Notice that it is a paral-lel situation to the Parisi scheme where, in the limit of k → /k ) (cid:80) α (cid:54) = β q lαβ → − (cid:82) q l ( x ) dx , and thenthe order parameter is defined by (A13). . . . . . . k B T/J . . . . . . . q k k = 1 k = 2 k = 3 k = 4 k = 5 k = 6 k = 7 FIG. 3: Numerical solutions for (32) near the criticalpoint for different values of k . We can identify thecritical temperature at βJ = 1.Furthermore, using this numerical solution for a J (cid:28) ς k < T → 0. In the Fig. 4 we depict thebehavior of this ground-state entropy given by s = ∞ (cid:88) k =1 ( − k +1 k ! s k , (54)where s k = β J (cid:18) 14 ( k − q k + 12 q k − (cid:19) + 12 β J ∂q k ∂β (( k − q k + 1)+ log 2 + 1 k log (cid:90) Dz cosh k ( βJ √ q k z ) − β J (cid:18) β ∂q k ∂β + q k (cid:19) (( k − q k + 1) (55)Now, if J (cid:54) = 0 we will have four kind of solutions: i) q k = 0 and m k = 0; ii) q k > m k = 0; iii) q k > m k > 0; and iv) q k > m k < 0. To explore theregime where J > 0, and m k > 0, we expand (32) and . 00 0 . 02 0 . 04 0 . 06 0 . 08 0 . k B T/J . . . . . . s FIG. 4: Entropy for the numerical solutions for (32)and a fixed J (cid:28) q k and m k . If we neglect terms of O ( q k m k )and keep with O ( q k ) and O ( m k ), we obtain expressionsof the form q k = 2( βJ − kβJ , (56) m k = (cid:20) βJ − β J )( βJ − kβ J (cid:21) / . (57)Observe that we have, as is expected, the points oftransition determined by βJ = 1 or T C = J and βJ /J = 1. In the FIG. 5 we depict the behavior of (57)for different temperatures and values of J . Note that theinterval where it is well defined is [ J/J , J /J ]. We canevidence the appearance of a maximum for a given valueof J/k B T and k . Each curve correspond to different val-ues of k . Similar behaviors are obtained experimentally(see Refs. [13, 49–52]).With these results we are able to recover the typicalphase diagram with the regularized order parameters (41)and (47). Furthermore, keep now the terms of O ( q k m k )to observe other phenomena, the local susceptibility from(33) yields χ k = − kβm k + ( k − βq k + β, (58)while the nonlinear susceptibility reads χ k = − kβχ k + β χ k [( k − − kq k ) − k m k ]+ k ( k − βm k [( k − u k + m k (2 − kq k )]+ ( k − k − β [( k − r k + 3 q k − km k u k ] . (59)We depict the behavior of (58) in function of k B T /J fordifferent values of J /J in the FIG. 6. Notice that we ob-tain the expected discontinuity of the linear susceptibilityat the spin glass critical temperature extracted from phe-nomenological models and experimental results [13, 53–56]. Furthermore, the inset show (59). We recover the FIG. 5: Behavior of (57) for (a) J /J = 2, (b) J /J = 4,(c) J /J = 8, (16) J /J = 16. Each curve correspond to k = 1 (indigo), k = 2 (blue), k = 3 (dark cyan), k = 4(green), k = 5 (yellow), k = 6 (orange), and k = 7 (red).typical behavior of the nonlinear susceptibility. With thisresult, now we shall study the stability of our solution q α k γ k = q k . . 12 2 . 14 2 . 16 2 . 18 2 . 20 2 . 22 2 . k B T/J . . . . . . . . . χ k k B T . 15 2 . 20 2 . k B T/J − − − χ k ( k B T ) FIG. 6: Behavior of (58) for J /J = 3 (indigo), J /J = 4 (dark cyan), J /J = 5 (yellow), J /J = 6(red), and k = 3 in function of k B T /J . Note theappearance of the discontinuity at the spin glass criticaltemperature. Inset show the typical behavior of thenonlinear susceptibility (59).With the expressions (57) and (56) we can explore theregions where (28) is negative. Near the critical point inFIG. 7 we depict the regions of validity of our regularizedquantities. We can evidence that the region is increasingas the disorder becomes stronger (increasing value of J ). . . . . . J/k B T . . . . . J / J a ) − − − − − − . . . . . J/k B T . . . . . J / J b ) − − − − − − − . . . . . J/k B T . . . . . J / J c ) − − − − − − − . . . . . J/k B T . . . . . J / J d ) − − − − − − FIG. 7: Regions of validity of regularized free energy (40) and subsequent regularized quantities for (a) J = 1, (b) J = 2, (c) J = 3, (d) J = 4. A. Stability For the analysis of the stability, let us explore the Hes-sian matrix for each term in the series expansion of freeenergy, for h = 0. Thus, for each k the Hessian evaluatedin q k = (cid:104)(cid:104) S α k S β k (cid:105)(cid:105) is δ ( βf ) δq α k β k δq µ k ν k = ( − k +1 a k k ! k e w k δ w k δq α k β k δq µ k ν k , (60)where w k = β J (cid:88) α k <γ k q α k γ k S α k S γ k + βJ (cid:88) α k m α k S α k , (61)and δ w k δq α k β k δq µ k ν k = β J δ ( α k β k ) , ( µ k ν k ) − β J [ (cid:104)(cid:104) S α k S β k S µ k S ν k (cid:105)(cid:105)− (cid:104)(cid:104) S α k S β k (cid:105)(cid:105)(cid:104)(cid:104) S µ k S ν k (cid:105)(cid:105) ] . (62)Following the standard procedure [35, 57], we havethree kind of contributions in (60): δ ( βf ) δq α k β k δq µ k ν k = H ( k )1 if ( α k β k ) = ( µ k ν k ) H ( k )2 if α k = µ k , β k (cid:54) = ν k H ( k )2 if α k (cid:54) = µ k , β k = ν k H ( k )3 if α k (cid:54) = µ k , β k (cid:54) = ν k , (63) where each contribution is given by H ( k )1 = ( − k +1 a k k ! k e w k [ β J − β J (1 − q k )] , H ( k )2 = ( − k +1 a k k ! k e w k [ − β J ( q k − q k )] , (64) H ( k )3 = ( − k +1 a k k ! k e w k [ − β J ( r k − q k )] , with r k defined by (53).We proceed to study the eigenvalues of (60) the sub-space spanned by the vector with all equal componentsis a good eigenspace, called longitudinal (scalar) space,which at the same time, can be one can be divided inother two orthogonal eigenspaces called, respectively, the anomalous (vectorial) and replicon (tensorial).Let us study the condition of stability for eacheigenspace. First, for the longitudinal space, the eigen-value is given by λ ( k ) L = H ( k )1 + 2( k − H ( k )2 + ( k − k − H ( k )3 . (65)We must divide the study in two cases. The first one,when k + 1 = 2 n , with n ∈ N . The second one, k +1 = 2 n + 1, with n ∈ N . For the first case, in the spinglass phase, the stability condition is given by q n − > (2 n − / ( n + ( n/ − n > k + 1 = 2 n + 1, we have q n > ( n − /n ( n − / n > T → k + 1 = 2 n . Nearthe critical point 1 > β J .For the anomalous space, the eigenvalue is λ ( k ) A = H ( k )1 + ( k − H ( k )2 − ( k − H ( k )3 . (66)To go further, we discuss two situations. For k + 1 =2 n , with n ∈ N , in the spin glass phase, the stabilitycondition is given for all n > . 5. On the other hand,when k + 1 = 2 n + 1, we have that the solution is stablefor all n > 2. In the limit T → k + 1 = 2 n . Near the critical point 1 > β J . In thelimit T → k + 1 = 2 n . Nearthe critical point 1 > β J .Finally, the eigenvalue of the replicon space yields λ ( k ) R = H ( k )1 − H ( k )2 + H ( k )3 . (67)Which gives us our version for the AT line of stability,that is, for k + 1 = 2 n , in the spin glass phase (cid:18) TJ (cid:19) > (cid:82) Dz cosh n − ( βψ ) (cid:82) Dz cosh n − ( βψ ) , (68)and for k + 1 = 2 n + 1 (cid:18) TJ (cid:19) < (cid:82) Dz cosh n − ( βψ ) (cid:82) Dz cosh n ( βψ ) . (69)Here, near the critical point the solution is stable if1 > β J . In the limit T → k .In FIG. (8) we depict numerical solutions of (32) when wecan observe the behavior in the regions of stability, nearthe critical point, and for low fields and temperatures for k = 0 , , , , , 5. Note that the discontinuity near thecritical point is getting sharp as k increases. IV. CONCLUSIONS Afterward the Parisi RSB scheme was consolidated asa solution to avoid unphysical scenarios obtained by thereplica ansatz in the replica method, a wide set of re-sults exploring each steep of the scheme, were discover-ing new physics of the Sherrington-Kirkpatrick model forspin glasses. Within this framework a great theoreticaland numerical effort were carry out to study the physicsof the complex free-energy landscape and order parame-ters under different domains. In this paper we adopt thedistributional zeta function method (DZFM). There is aseries representation of the average free energy where allof the moments of the partition function contribute.Within the DZFM we obtain the multi-valley struc-ture of the average free energy. We obtained the self-consistent integral equations for each order parameter q k and m k , which have a similar structure presented in thereplica symmetric ansatz and for the first steeps of the Parisi RSB. We perform a study of this parameter nearthe critical point and in the low temperatures regime.For the critical point we obtained a critical temperaturefor each k . Since we are dealing with a series represen-tation, an asymptotic analysis was made to recover theknown result T C = J . In the evaluation of the localmagnetizations and linear susceptibilities we have foundsimilar behaviors described in phenomenological modelsand experimental results. In particular we obtain thebehavior of reentrant phase in the local magnetizationgiven by two critical temperatures determined by J and J . On the other hand, with an upper order expansion,keeping terms of order O ( m k ) and O ( q k m k ), we haveobtained the typical discontinuity of the linear suscep-tibility at the spin-glass critical temperature. Further-more, with numerical solutions of the integral equationswe were able to recover the behavior of the continuumlimit of the Parisi RSB.For the low temperature regime, we have shown thatfor each k the limit result for q k is 1 as has been in-vestigated in the literature. Furthermore, before takethe limit β → ∞ , we have obtained a composed expres-sion for each k . The main result in this regime is theconstruction of a positive definite series representation ofthe ground state entropy. We show that the ground-stateentropy goes to zero as temperature tends to zero, thatis, in the low temperature regime, β → ∞ , we obtain S → longitu-dinal (scalar) space, which at the same time, can be onecan be divided in other two orthogonal eigenspaces called,respectively, the anomalous (vectorial) and replicon (ten-sorial), for each k . From this structure we obtained ourgeneralized version for the AT conditions for stability.Finally, we study the distribution of the overlaps. Weobtained for each k a structure similar (but not equal) tothe ultrametricity extracted from the Parisi solution inthe continuum limit. However, we have that as k → ∞ the overlaps are becoming statistically independent. Thisdistribution is valid for k > T / J h / J q k (a) T / J h / J q k (b) T / J h / J q k (c) T / J h / J q k (d) T / J h / J q k (e) T / J h / J q k (f) FIG. 8: Numerical solutions of (32) for (a) k = 0 (replica symmetric ansatz), (b) k = 1, (c) k = 2, (d) k = 3, (e) k = 4, and (f) k = 5. Here, the curves projected on each plane are level curves used to delimit the stability regions. ACKNOWLEDGMENTS This paper was partially supported by the VIII Convo-catoria para el Desarrollo y Fortalecimiento de los Gru-pos de Investigaci´on en Uniminuto with code C119-173and Industrial Engineering Program from the Corpo-raci´on Universitaria Minuto de Dios (Uniminuto, Colom-bia) (C.D.R.C.), and Conselho Nacional de Desenvolvi-mento Cient´ıfico e Tecnol´ogico - CNPq, 303436/2015-8(N.F.S.). We would like to thanks G. Krein and B. F.Svaiter for useful discussions about the mathematical andphysical implications in whole the construction of thispaper. APPENDIX In this appendix we include further developments forthe free energy from the DZFM and the overlap distri-bution under our formalism. Appendix A: Replica method To study the properties and phase structure of the SKmodel it is necessary to compute the configurational aver-aged free energy defined by (8). To bypass the average ofthe logarithm it is usually employed the replica method.After the construction of Z k , the expected value of thepartition function’s k -th power E [ Z k ( J ij )] is evaluated by integrating over the disorder field on the new model (col-lection of replicas). Notice that in Z k , integration overquenched random couplings yields a system defined by k replicas which are no more statistically independent.The average value in the presence of the quenched dis-order is then obtained in the limit of a zero-componentfield theory taking the limit k → E [log Z ( J ij )] = lim k → E [ Z k ( J ij )] − k , (A1)or E [log Z ( J ij )] = lim k → ∂∂k E [ Z k ( J ij )] . (A2)The standard ansatz restricts the search for a minimumto a q α k γ k invariant under a subgroup of the permutationgroup of k elements. In the Parisi’s RSB scheme, if P k isthe group of permutations of k elements, we can considerthe chain P k ⊃ ( P m ) k/m ⊗ P k/m , P m ⊃ ( P m ) m /m ⊗ P m /m , and so on. For instance, in the first step of replica sym-metry breaking (1RSB), an invariant 6 × q α k γ k is q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q . × q α k γ k matrix invariant under ( P ) ⊗ ( P / ) ⊗ P / is q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q . The SK solutions correspond to using the replica-symmetric solution where q α k γ k = q and m α k = m ,within the well known relation from the replica method(A1), the order parameters q and m read, respectively, q = 1 − (cid:90) Dz sech ( βψ ) = (cid:90) Dz tanh ( βψ ) (A3)and m = (cid:90) Dz tanh( βψ ) (A4)being Dz = dz √ π exp (cid:18) − z (cid:19) (A5)and ψ = J √ qz + J m + h. (A6)In this scenario, we have the spin glass transition at T = J when J = m = h = 0. According to (A3) q tendsto one as T → 0. However, the ground-state entropy is − / π . In order to avoid the unphysical results of thereplica-symmetric solution, the replica symmetry break-ing (RSB) was introduced. The variational parametersin the 1RSB are m = (cid:90) Du (cid:82) Dv cosh m Ξ tanhΞ (cid:82) Dv cosh m Ξ , (A7) q = (cid:90) Du (cid:32) (cid:82) Dv cosh m Ξ tanhΞ (cid:82) Dv cosh m Ξ (cid:33) , (A8)and q = (cid:90) Du (cid:82) Dv cosh m Ξ tanh Ξ (cid:82) Dv cosh m Ξ , (A9)beingΞ = β ( J √ q u + J √ q − q v + J m + h ) . Under this formalism the entropy per spin at J = 0, T = 0 reduces from − . 16 (= − / π ) for the RS solutionto − . 01 for the 1RSB. In the full RSB solution, since one obtains a K → ∞ number of order parameters defined by (cid:88) α k (cid:54) = γ k q lα k γ k = q l k + ( q l − q l ) m km +( q l − q l ) m m m km + · · · − q lK k. (A10)Rewriting the last expression we get (cid:88) α k (cid:54) = γ k q lα k γ k = k K (cid:88) j =0 ( m j − m j +1 ) q lj , (A11)where l is an arbitrary integer and m = k , m K +1 k → 0, we may use the replacement m j − m j +1 → − dx to find1 k (cid:88) α k (cid:54) = β k q lα k β k → − (cid:90) q l ( x ) dx. (A12)Near the critical point this parameter has the followingbehavior q ( x ) = (cid:40) x if 0 ≤ x ≤ x = 2 q (1) q (1) if x ≤ x ≤ , (A13)being q (1) = | θ | + O ( θ ) , θ (cid:28) . (A14) Appendix B: Free energy from DZFM In this appendix we review the alternative approach tocompute the configurational average of the free energy ofdisordered systems presented in [42–44]. We begin withthe definition of the generalized ζ -function given by ζ µ,f ( s ) = (cid:90) X f ( x ) − s dµ ( x ) , (B1)where the triplet ( X, A , µ ) is a measure space, f : X → (0 , ∞ ) is measurable, and s ∈ C such that f − s ∈ L ( µ ),where in the above integral f − s = exp( − s log( f )) is ob-tained using the principal branch of the logarithm. In thesituation where f ( J ij ) = Z ( J ij ) and dµ ( J ij ) we obtainthe definition of the distributional zeta function Φ( s ) asΦ( s ) = (cid:90) d [ J ij ] P ( J ij ) 1 Z ( J ij ) s . (B2)Following the usual steps of the spectral zeta function,the configurational average free energy can be written as f = lim N →∞ N β dds Φ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 + , Re( s ) ≥ , (B3)2where Φ( s ) is well defined. The procedure use the Euler’sintegral representation of the gamma function1 Z ( J ij ) s = 1Γ( s ) (cid:90) ∞ dt t s − e − Z ( J ij ) t , (B4)for Re( s ) ≥ 0. Substituting (B4) into (B2), we haveΦ( s ) = 1Γ( s ) (cid:90) d [ J ij ] P ( J ij ) (cid:90) ∞ dt t s − e − Z ( J ij ) t . (B5)To proceed, it is assumed the commutativity of theconfigurational average, differentiation and integration.Now, we take a > + Φ whereΦ ( s ) = 1Γ( s ) (cid:90) d [ J ij ] P ( J ij ) (cid:90) a dt t s − e − Z ( J ij ) t (B6)andΦ ( s ) = 1Γ( s ) (cid:90) d [ J ij ] P ( J ij ) (cid:90) ∞ a dt t s − e − Z ( J ij ) t , (B7)being a a dimensionless parameter. The configurationalaverage free energy can be written as f = lim N →∞ N β (cid:26) dds Φ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 + + dds Φ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 + (cid:27) . (B8)In the innermost integral in Φ , the series represen-tation for the exponential converges uniformly (for each J ij ), so that we can reverse the order of integration andsummation to obtainΦ ( s ) = (cid:90) d [ J ij ] P ( J ij ) 1Γ( s ) ∞ (cid:88) k =0 ( − k a k + s k !( k + s ) Z k ( J ij ) . (B9)The term k = 0 in (B9) contains a removable singu-larity at s = 0 since s Γ( s ) = Γ( s + 1), so that we canwriteΦ ( s ) = a s Γ( s + 1) + 1Γ( s ) ∞ (cid:88) k =0 ( − k a k + s k !( k + s ) E [ Z k ] . (B10)The function Γ( s ) has a pole at s = 0 with residue 1,therefore − dds Φ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 + = ∞ (cid:88) k =0 ( − k +1 a k k ! k E [ Z k ] + f ( a ) , (B11)where f ( a ) = − dds (cid:18) a s Γ( s + 1) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) s =0 + = − log( a ) + γ e (B12)being γ e the Euler’s constant. The derivative of Φ isgiven by dds Φ ( s ) (cid:12)(cid:12)(cid:12)(cid:12) s =0 + = (cid:90) d [ J ij ] P ( J ij ) (cid:90) ∞ a dtt e − Z ( J ij ) t = R ( a ) . (B13) The asymptotic behavior of R ( a ) is related to the in-complete gamma function defined asΓ( α, x ) = (cid:90) ∞ x e − t t α − dt. (B14)The asymptotic representation for | x | → ∞ and − π/ < arg x < π/ α, x ) ∼ x α − e − x (cid:20) α − x + ( α − α − x + ... (cid:21) . (B15)Defining Z c as the partition function of a system where P ( J ij ) = c , where c ∈ R is a constant such that c < J ij for all stochastic variable J ij defined by (3), we have abound for R ( a ) given by | R ( a ) | ≤ ( Z c a ) − exp( − Z c a ). Appendix C: Overlap distribution For completeness in our analysis we present the overlapdistribution under our formalism. In the Parisi RSB thisdistribution is associated with the continuum parameter x of the order parameter q ( x ). Furthermore, the jointdistribution shows that the symmetry breaking gener-ates a ultrametric structure. The explicit calculation of P ( Q , Q , Q ) gives the result P ( Q , Q , Q ) = P ( Q ) x ( Q ) δ ( Q − Q ) δ ( Q − Q )+ 12 P ( Q ) P ( Q )Θ( Q − Q ) δ ( Q − Q )+ 12 P ( Q ) P ( Q )Θ( Q − Q ) δ ( Q − Q )+ 12 P ( Q ) P ( Q )Θ( Q − Q ) δ ( Q − Q ) . (C1)For any Q c < Q max the integrated contributions inthe volume 0 ≤ Q i ≤ Q C of all four terms are equal.It follows that, among all the triangles, 1 / / (cid:104) σ i (cid:105) = m i = (cid:88) α k w α k m α k i , (C2)where the α k ’s label the pure states and w α k are theirstatistical weights which could be written as w α k = e − F αk , (C3)being F α k the free energy of the pure state α .The overlap between two states α k and γ k is definedby Q α k γ k = 1 N N (cid:88) i =1 m α k i m γ k i . (C4)3We can observe that 0 ≤ | Q α k γ k | ≤ 1. To describethe statistics of the overlaps between all the pairs of purestates it is natural to introduce the probability distribu-tion function P J ( Q ) = (cid:88) α k γ k w α k w γ k δ ( Q α k γ k − Q ) . (C5)The function P J ( Q ) could depend on the concrete re-alization of the quenched interactions J ij . The averageover the disorder is P ( Q ) = E [ P J ( Q )]. The function P ( Q ) gives the probability to find two pure states having mutual overlap equal to Q . In terms of the entries of thematrix q α k γ k we have P ( Q ) = 2 k ( k − (cid:88) α k <γ k δ ( q α k γ k − Q ) . (C6)To explore the metric of the space of pure states weconsider the distribution function P ( Q , Q , Q ) whichdescribes the joint statistics of the overlaps of arbitrarythree pure states. We have then P ( Q , Q , Q ) = 1 k ( k − k − (cid:88) α k (cid:54) = γ k (cid:54) = σ k δ ( q α k γ k − Q ) δ ( q α k σ k − Q ) δ ( q γ k σ k − Q ) . (C7)To perform the calculus, we use the Fourier transform of the function P ( Q , Q , Q ) g ( y , y , y ) = (cid:90) dQ dQ dQ P ( Q , Q , Q ) e iQ y + iQ y + iQ y . (C8)Thus we have g ( y , y , y ) = 1 k ( k − k − (cid:88) α k (cid:54) = γ k (cid:54) = σ k e iq αkσk y + iq αkσk y + iq γkσk y = 1 k ( k − k − 2) Tr[ A ( y ) A ( y ) A ( y )] (C9)where A α k γ k ( y ) = (cid:40) e iq αkγk y if α k (cid:54) = γ k α k = γ k . 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