Appendix A Stability Analysis of the RS Solution

Transcription

1 Appendix A Stability Analysis of the RS Solution In this appendix we derive the eigenvalues of the Hessian matrix of the model in Eq for the RS ansaltz. In this appendix we use the explicit form of the Hessian and its eigenvectors in the Replica Space. This approach becomes rapidly involved as the number of RSB steps grows. In the Appendix B we will show as the problem is greatly simplified in the Replica Fourier Space. The stability of a stationary point requires that all the eigenvalues of the Hessian matrix H in the point are positive. We define the n -dimensional variations vector θ = δq ab δr ab δr aa A. where a < b so we have separeted the diagonal part of matrix R. Then the equation for the eigenvalues is H θ = λ θ. A. where here and in the following we use the notation H MN abcd β FQ M ab N cd where the free energy FQ of the model is given in Eq and where M and N could be the overlap matrices Q or R. The first derivatives of the functional are β F = ξq ab μ Qab 3 Q + Q abrab ab m σ Q + R a β F = ξr ab μ Rab 3 R + R abqab ab m σ Q + R a Q + R ab Q R m σ Q + R b m τ Q R a ab + Q + R ab + Q R m σ Q + R + m τ Q R b a m τ Q R b ab + m τ Q R b Springer International Publishing Switzerland 06 F. Antenucci Statistical Physics of Wave Interactions Springer Theses DOI 0.007/

2 30 Appendix A: Stability Analysis of the RS Solution where the first equation is valid only for a = b because Q aa by the spherical constrain. Then defining the functions x x y = ξ + 6μx + 4μy y x y = 8μxy A.3 and the auxiliary matrices cf. also Eq. 3.6 Z ab = Q + R = A ab W ab = Q R ab = B ab we obtain for the Hessian here we take the solution m τ = 0 without loss of generality cf. Sect. 3.4 H RQ abcd H QQ abcd = x Q ab R ab δ ac δ bd + Z ac Z db + W ac W db + m σ Z a m σ Z c Z bd m σ Z b m σ Z c Z ad + + c d H RR abcd = xr ab Q ab δ ac δ bd + Z ac Z db + W ac W db + A.4 m σ Z a m σ Z c Z bd m σ Z b m σ Z c Z ad + + c d = H QR abcd = yq ab R ab δ ac δ bd + Z ac Z db W ac W db + m σ Z a m σ Z c Z bd m σ Z b m σ Z c Z ad + + c d and note as the elements of the Hessian are symmetric in a b and c d. In the RS ansaltz all the matrices that appear in the Hessian have a RS structure so there are actually just a few different expressions. It s hence useful to explicitly define all the different expressions of Habcd MN for all the possible choices of M N and a b c d when non explicitly said the indices are supposed to be different: for all MN so must be a = b and c = d C MN : all different B MN : a = c b = d F MN : a = c for MN = QR RQ RR for which we can have a = b with c = d G MN : a = b = c C MN : a = b for just RR for which we can have a = b and c = d C RR : a = b c = d E RR : a = b = c = d

3 Appendix A: Stability Analysis of the RS Solution 3 so that their explicit expressions are C QQ = C RR = Z0 + W 0 4m σ Z 0 Z Z 0 C QR = C RQ = Z0 W 0 4m σ Z 0 Z Z 0 B QQ = x Q 0 R 0 + Z + W + Z 0 + W 0 m σ Z Z 0 Z + Z 0 B RR = x R 0 Q 0 + Z + W + Z 0 + W 0 m σ Z Z 0 Z + Z 0 B QR = y Q 0 R 0 + Z W + Z 0 W 0 m σ Z Z 0 Z + Z 0 F QQ = F RR = Z Z 0 + W W 0 + Z0 + W 0 m σ Z Z 0 Z + 3Z 0 F QR = F RQ = Z Z 0 W W 0 + Z0 W 0 m σ Z Z 0 Z + 3Z 0 G RR = Z Z 0 + W W 0 m σ Z Z 0 Z + Z 0 G QR = Z Z 0 W W 0 m σ Z Z 0 Z + Z 0 E RR = x R Q + Z + W 4m σ Z Z Z 0. Also in the RS ansaltz the explicit expressions of Z and W are Z = A = A A 0 A 0 A A 0 Z 0 = A 0 = A A 0 A 0 W = B = B B 0 B 0 B B 0 W 0 = B 0 = B B 0. The eigenvalue equation Eq. A. is explicitly given by where a < b c =d d c =d d c =d d H QQ abcd δq cd + H QR abcd δr cd + c H RQ abcd δq cd + H RR abcd δr cd + c H RQ aacd δq cd + H RR aacd δr cd + c B 0 H QR abcc δr cc = λδq ab H RR abcc δr cc = λδr ab H RR aacc δr cc = λδraa

4 3 Appendix A: Stability Analysis of the RS Solution so that in the RS ansaltz it can be expressed in terms of C B F G E as N=QR { B MN F MN + C MN δn ab + F MN C MN δn ac + δn cb + c =a c =b + C MN δn cd }+ c =d d + G MR C MR δr aa + δr bb + C MR δr cc = λδm ab M = Q R c G RN C RN δn ac + C RN δn cd + N=QR c =a c =d d + E RR C RR δr aa + C RR c δr cc = λδr aa. Now we seek the eigenvectors in particular subspaces in which the eigenvalue problem is greatly simplified. The natural choices are S = δq ac = δr ac = 0 δr aa = 0 c =a c =a dims = 3 nn 3 ; S = δq cd = δr cd = 0 c =d d c =d d c dims = 5n δr cc = 0 where in S we must have for at least one a such that c δq ac = 0 c δr ac = 0 c δt ac = 0 δr aa = 0 δt aa = 0. At last the third subspace S 3 is fixed by the orthogonality with S and S and it has dims 3 = 5. In S the eigenvalue problem is reduced at the diagonalization of the following matrix D H = QQ D QR D RQ D RR

5 Appendix A: Stability Analysis of the RS Solution 33 where D MN = B MN F MN + C MN D QQ = x Q 0 R 0 + Z Z 0 + W W 0 D RR = x R 0 Q 0 + Z Z 0 + W W 0 D QR = y Q 0 R 0 + Z Z 0 W W 0 In S subspace we have as obtained summing over b in the eigenvalues equations I QQ I QR P QR H = I RQ I RR P RR P RQ P RR S RR where I MN = B MN 4F MN + 3C MN I QQ = D QQ Z Z 0 W W 0 + Z0 + W 0 + m σ Z Z 0 3 I RR = D RR Z Z 0 W W 0 + Z0 + W 0 + m σ Z Z 0 3 I QR = D QR Z Z 0 + W W 0 + Z0 W 0 + m σ Z Z 0 3 P MN = G MN C MN { P RR = Z Z 0 + W W 0 Z0 W 0 m σ Z Z 0 3 P QR = Z Z 0 W W 0 Z0 + W 0 m σ Z Z 0 3 S RR = E RR C RR S RR = x R Q + Z + W Z 0 W 0. Eventually it is easily shown that in the subspace S 3 summing over a b in the eigenvalue equations the problem reduces to the same matrix obtained in S when the limit n 0 is considered. Numerically it is obtained that the solutions with Q 0 = ±R 0 have always some unstable eigenvalues. Consider for example the case at h σ = 0: the stationarity equations also admit a solution with Q 0 = 0 and R 0 = 0; in this case the eigenvalues of H are λ Q 0 so that they are negative for any Q 0 = R 0 = 0.

6 34 Appendix A: Stability Analysis of the RS Solution RS Stability of the Solution Q 0 = R 0 In the case Q 0 R 0 the stationary equations for the RS ansaltz are given by the Eq. 3.4 so that the stability matrices reduce to D D QR H = D QR D with whose eigenvalues are given by D = ξ 5 μa q + a q + a D QR = μa q + a q a λ = D D QR = ξ μa q + a. λ = D + D QR = ξ 9 μa q + a q and H 3 = I QR P I I QR I P P P S with whose eigenvalues are I = D P I QR = D QR P P = q + ah σ a q 3 S = x a + 3q a q + 3 a λ 3 =I I QR λ λ 45 =I + I QR + S I S + II QR + I QR S 6P S. For the PM solution with a = and q = 0 so that P = 0 and the matrix H 3 is block diagonal. In this case λ 4 = λ and λ 5 = S and one easily see that the last one

7 Appendix A: Stability Analysis of the RS Solution 35 λ 5 = S = ξ 4μ + A.5 is the relevant in this case so that the PM solution is unstable for ξ + μ >. At this line the APM solution appears with a =. For the APM solution it is numerically obtained that the relevant eigenvalue for the stability is λ. The surface where the APM phase becomes unstable is then given by the system of equations a 0 q q + ah σ = 0 a a a 0 + q a a = 0 A.6 ξ 9 μa q + a q = 0 with the usual functions defined in Eq In particular in the fieldless case q = 0 and λ reduce to λ = ξ + a. A.7

8 Appendix B Stability Analysis of the RSB Solution The stability analysis in the Replica Space [] through the direct construction of the eigenvectors cf. Appendix A becomes rapidly involved as the number of replica symmetry breaking steps increases. The analysis is greatly simplified in the Replica Fourier Space [ 3]. In this case is convenient to write the eigenvalue equation as cf. Eq. A. M abcd θ cd = λθ ab cd B. where the matrix M is M abcd H MN abcd for c = d while it is M abcc H MN abcd. Making explicit the RSB structure of the matrices we have also R+ s=0 cd c d=s M rs a b; c d θ s c d = λ θ r a b where with c d = s it is intended that the replicas c and d have overlap s i.e. c k = d k for k = 0...s while c s = d s ; also with M rs a b; c d M abcd with a b = r and c d = s; similarly for θ r a b. The Replica Fourier Transform RFT in the index r of a R-RSB matrix M r is M = R+ r=k p r M r M r then performing the RFT in the four indices the eigenvalue equation becomes Springer International Publishing Switzerland 06 F. Antenucci Statistical Physics of Wave Interactions Springer Theses DOI 0.007/

9 38 Appendix B: Stability Analysis of the RSB Solution R+ s=0 M rs â b; ĉ d θ s ĉ d = λ θ r â b. ĉ d The great benefit is that the ultrametric property of the RSB solutions implies a specific form for the RFT of a four-replica function so that the eigenvalues and an orthogonal basis of eigenvectors can be generally found [3]. In particular we can distinguish two specific forms for the matrix M entering in the eigenvalue equation: if r = s then M rs a b c d depends only on the larger cross-overlap M rs a b c d = M rs t t = max a c a d b c b d B. corresponding to the so-called Longitudinal-Anomalous sector; when instead r = s two cross-overlap are necessary so that { M rr a b c d = M rr uv u = max a c a d v = max b c b d B.3 corresponding to the so-called Replicon sector. Here we do not report all the general derivation of the eigenvalues and eigenvectors [3] but just the evaluation for the model in interest cf. Eq e.g. for the application the the SK model see Ref. [4]. In particular we explicitly report the eigenvalues for the relevant RSB solution with Q r = R r for r = 0. Moreover it is well-known that the addiction of an ordered external field does not change the stability analysis of a RSB phase see e.g. Ref. [5 6]. Indeed any addition of an external parameter coupled to a singlereplica quantity cannot alter the breaking in the replica space: the corresponding order parameters must be always replica symmetric as the magnetization m in the case of the external field h. Then for the sake of simplicity we consider here only the fieldless case so that m σ = m τ = 0 and then Q 0 = R 0 = 0 as well cf. Sect For this solution we thus have Q = Q 0 and R = R R 0 with R = Q = aq /; and so A = Q + R = + R q 0 a aq 0 and B = Q R = R 0 0 a 0 0. In particular must be 0 q R. We hence have as usual x p Q = aq R = a aq A = a aq B = B = B = a. Q = Q = aq x R = R = a aq x A = A = a aq x B.4

10 Appendix B: Stability Analysis of the RSB Solution 39 In the following it is useful to remember that the derivatives of the function are x x y = ξ + 6μx + 4μy y x y = 8μxy. B.5 Note at last as for the equivalent solution with Q = R one has just to exchange A and B in all the stability equations. Replicon Sector In the Replicon Sector the eigenvalue equation is written as with QQM rr l RQM rr l δq rr l + QRM rr δr rr = λδq rr l l l δq rr l + RRM rr δr rr = λδr rr l l l QQM rr l RRM rr l RQM rr l QRM rr l = x Q r R r + A A l = x R r Q r + A A l = y Q r R r + = RQ M rr l. A A l + B B l + B B l B B l Then for each r k l the eigenvalues in the Replicon Sector are obtained from the eigenvalue of a real symmetric matrix. The multiplicity d of each eigenvalue is dr; k l = n δ k δ l δ r k l B.6 with δ r k l = p r + δ kr+ + δ lr+ pr+ δ k = { pk p k k > r + p k k = r +. In the case of the RSB solution in the notation of Eq. B.4 we explicitly have For r = 0 we can have k = l = ork = l = ork = l =. We have

11 40 Appendix B: Stability Analysis of the RSB Solution QQM 00 l QRM 00 l = RR M 00 l = RQ M 00 l so the two eigenvalues for every choice of k l are 00 l = ξ + A A l = ξ + + A A l B B l = A A l B B l 00 l = ξ + B B l. B.7 The dominant eigenvalue is obtained for = l = and it is explicitly given by 00 = ξ + a [ q x]. B.8 The dimension of the subspace r = 0is d0 = d0; + d0; + d0; + d0; = = x + x n. x For r = we can have only k = l = and then QQM = RRM = x QRM = RQM = y and we obtain the two eigenvalues aq aq aq aq + A + B + A B = ξ 9 μa q + a q = ξ μa q + a B.9 and because R > q we have that between the two the leading eigenvalue for the stability is. The dimension of the subspace r = isd = x 3n. Longitudinal-Anomalous Sector In the Longitudinal-Anomalous sector the eigenvalues are labelled by k = 0...R + and they are the solution of the eigenvalue problem

12 Appendix B: Stability Analysis of the RSB Solution 4 R+ s=0 R+ s=0 [ δ rs QQ M rr r+ + 4 δk s QQM rs δq s + + δ rs QR M rr r+ + ] 4 δk s QRM rs δr s = λδq r [ δ rs RQ M rr r+ + 4 δk s RQM rs δq s + + δ rs RR M rr r+ + ] 4 δk s RRM rs δr s = λδr r for r = 0...R and R+ s=0 4 δk s RQM R+s δq s + RRM R+s δr s where δq R+ = 0. Note that in the first set of equations it is intended that and we have defined δ k s p k s M rs M rs l max r+ max l r+ p k s+ pk s { p s The coefficients in the previous eigenvalue equations are with C = A B QQM rs RRM rs QRM rs RQM rs = A M rs = A M rs = A M rs = QR M rs + B M rs + B M rs B M rs = λδr R+ s k p s s > k. x a δ rs δ rr+ C M rs C M rs = 4 C t C = 4 C t C k > t = minr s + t u=k [ C p u u C ] u k t = minr s

13 4 Appendix B: Stability Analysis of the RSB Solution where as usual the component of the inverse matrix can be expressed as C t = t k=0 p k [ C ]. C+ For any k = 0...R + the eigenvalues of the Longitudinal-Anomalous are then obtained from the eigenvalues of a R + 3 R + 3. Each eigenvalue has multiplicity dk = nδ k δ k = { pk p k k > 0 p 0 k = 0. In the case of the RSB solution in the notation of Eq. B.4 we explicitly have For k = 0 the coefficients C M rs from t = minr s A M 00 = A M 0 A M are we have r s = 0 but they depend only = A M 0 = A M A M B M rs = 0 = A xa A A A = A A A + xa A A = 0 for r s = B M = B and also T V = Q R TVM 00 = TV M 0 TVM = TV M 0 = TV M = 0 = A xa A A A RRM = A A A + xa A A + B x a The eigenvalue equations are then written in this case as QQM 00 δq0 + QRM 00 = δr0 λδq0 RQM 00 δq0 + RRM 00 = δr0 λδr0 [ QQM + ] [ 4 δ QQM δq + QRM + ] 4 δ QRM δr +

14 Appendix B: Stability Analysis of the RSB Solution δ QRM = δr λδq [ RQM + ] [ 4 δ RQM δq + RRM + ] 4 δ RRM δr δ RRM = δr λδr [ 4 δ RQM δq + 4 δ RRM δr + ] 4 δ RRM δr = λδr where δ = x and δ = ; then the matrix is factorized in two block matrices of matrix for the directions δq and a 3 3 matrix for the remaining δq δr δr. 0 δr0 In particular the matrix is the same of the Replicon Sector with r = 0cf. the eigenvalues in Eq. B.8. The remaining 3 3matrixis where T V = Q R M TV M M D xm M O xm M M O xm M D xm M xm xm = TV M M D QQ M = RRM = x M O QR M = RQM = y = A xa A A A aq aq aq aq M R + A + B + A B M R RR M = A A A + xa A A + B x a the three eigenvalues of this matrix are LA 0 =M D M O LA 0 3 =Ɣ ± Ɣ 8 B.0 where Ɣ = 4M D + 4M O + M R 4 xm = 6 [ M D M R + M O M R + xm xm ] M R. B. The eigenvalue LA 0 is equal to of the Replicon Sector and it is larger than. Then the relevant eigenvalues for k = 0are LA 0 3.

15 44 Appendix B: Stability Analysis of the RSB Solution For k = the coefficients C M rs are all the same as for k = 0. This is because A 0 and B 0 are both zero so the additional elements of k = 0 are indeed zero. We also have that δ 0 = δ = x and δ 0 = δ = so all the matrix elements for k = arethesameofk = 0 and so the stability analysis. For k = the coefficients C M rs are A M 00 = A M 0 A M = A M 0 = A M A M B M rs = 0 = 4 A A A = A + 4 A = A A A A A = 0 for r s = B M = B and also T V = Q R TVM 00 = TV M 0 TVM = TV M 0 = TV M = 0 = 4 A A A RRM = A A A A + B x a. In this case as well there are two diagonal blocks. The first is the matrix for δq and that is indeed the same matrix obtained in the Replicon Sector for 0 δr0 r = 0 with l = and k = cf. Eq. B.8. The remaining 3 3 matrix is given by [ QQM + ] [ 4 δ QQM δq + QRM + ] 4 δ QRM δr δ δr = λδq [ RQM + ] [ 4 δ RQM δq + RRM + ] 4 δ RRM δr + [ 4 δ RQM δq + 4 δ RRM + 4 δ δr + 4 δ QRM RRM RRM δr δr = λδr ] = λδr

16 Appendix B: Stability Analysis of the RSB Solution 45 with δ = x and δ = ; then we have the matrix where T V = Q R N TV M 4N D xn 4N O xn N 4N O xn 4N D xn N 4 xn xn N R = TV M = 4 A N D QQ M = RRM = x N O QR M = RQM = y N R RR M whose eigenvalues are with = A A A A A A + B aq aq aq aq x a + A + B + A B LA = N D N O LA 3 =Ɣ ± Ɣ B. 8 Ɣ = 4N D + 4N O + N R xn = 6 [ N D N R + N O N R + xn xn ] N R. B.3 The eigenvalue LA is equal to of the Replicon Sector and it is indeed larger than. Then the relevant eigenvalues for k = are LA 3. In conclusion it is found numerically that the eigenvalues LA 0 3 and LA 3 are in general complex but always with a positive real part when the RSB solution Q 0 = R 0 exists. The relevant eigenvalues in the Replicon Sector are: 00 cf. Eq. B.8 which controls the fluctuations with respect to Q 0 = 0 and then marks the appearance of a Full RSB phase in particular a FRSB phase is expected as found in the case with real spins [5]; note also that for x = wehave that 00 reduces to the relevant eigenvalue λ of the RS solution cf. Eq. A.7 as a consequence the RS and RSB phase critical lines meet at the tricritical point; cf. Eq. B.9 that controls the fluctuations with respect to Q and leads to the marginal condition see Sect. 3.7.

17 46 Appendix B: Stability Analysis of the RSB Solution The analysis for the alternative chance of a RSB solution with R 0 = R = 0 and then A r = B r for r = 0 proceeds similarly to the previous analysis. In this case one founds LA 0 3 < LA 3 < 0 so that this kind of solution is always unstable in the Longitudinal-Anomalous Sector. References. T. Temesvari C. De Dominicis I. Kondor Block diagonalizing ultrametric matrices. J. Phys. A: Math. Gen C. De Dominicis D.M. Carlucci T. Temesvári Replica Fourier transforms on ultrametric trees and block-diagonalizing multi-replica matrices. J. Phys. I Fr A. Crisanti C. De Dominicis Replica Fourier transform: properties and applications. Nucl. Phys. B A. Crisanti C. De Dominicis Stability of the Parisi solution for the Sherrington-Kirkpatrick model near T = 0. J. Phys. A: Math. Theor A. Crisanti L. Leuzzi Exactly solvable spin-glass models with ferromagnetic couplings: the spherical multi-p-spin model in a self-induced field. Nucl. Phys. B A. Crisanti H.-J. Sommers The spherical p-spin interaction spin glass model: the statics. Zeitschrift für Physik B Condensed Matter