mailto:hukusima@issp.u-tokyo.ac.jp September 9, 2002 ( ) [1] and Y. Iba, cond-mat/0207123. [2] H. Takayama and, cond-mat/0205276. Typeset by FoilTEX
Today s Contents Against Temperature Chaos in Spin Glasses Against the against... 2002/09/13 1
... ferromagnetic model below T c Overlap function between two valleys Temperature T c State 1 State 2 T+δT T q 12 = q 21 = m 2 (T ) q 11 = q 22 = +m 2 (T ) Overlap between an equilibrium state at T and T + δt q = ±m(t )m(t + δt ), varying smoothly with T Temperature Chaos The equilibrium states at different temperatures are TOTALLY DIFFERENT. = The overlap q is ZERO 2002/09/13 2
Disordered Systems P (q) = αβ δ(q qαβ ) δ droplet picture, Mean-field picture and more... = No Chaos Free energy NO LEVEL CROSSING A state dominatioin the partition function at T ALSO dominates it at T + δt. Chaotic case Free energy Temperature Temperature chaos as level crossings Temperature 2002/09/13 3
Experiment: Memory and Chaos Effects in Spin Glasses K. Jonason, et al : Phys. Rev. Lett. 81, 3243 (1998). J. Hammann, et al : J.Phys.Soc.Jpn. 69 (2000)Suppl. A, 206 211. rejuvenation T 1 memory T 1 2002/09/13 4
Experiment: Memory and Chaos Effects in Spin Glasses K. Jonason, et al : Phys. Rev. Lett. 81, 3243 (1998). J. Hammann, et al : J.Phys.Soc.Jpn. 69 (2000)Suppl. A, 206 211. rejuvenation T 1 memory T 1 2002/09/13 5
Chaotic Nature of the Spin-Glass Phase A. J. Bray and M. A. Moore: Phys. Rev. Lett. 58, 57 (1987). D. S. Fisher and D. A. Huse: Phys. Rev. B 38, 386 (1988). F Free-energy difference at T F (T ) = E T S ΥL θ θ: stiffness exponent Υ: T dependent stiffness constant Change the temperature to T + δt F (T + δt ) E (T + δt ) S ΥL θ δt S. L Entropy difference of the droplet surface S ±L d s/2 : d s : fractal dimension If d s /2 > θ, F (T + δt ) ΥL θ + δt L d s/2 can CHANGE the sign. = The equilibrium state should change on a length scale L(δT ) δt 1 ds/2 θ. 6
Chaos exponent ζ and Stiffness Exponent θ Chaos exponent: ζ = d s /2 θ > 0 = CHAOS............. Lyapunov exponent Stiffness exponent θ: mean-field picture (mean-field model): θ = 0. short-ranged SG model in three dimensions: θ 0.2. d s /2 (d 1)/2 = Temperature Chaos likely occurs in SG systems. Temperature T c T+δT T 2002/09/13 7
Sensitivity of the Spin Glass order parameter Original Hamiltonial : H 1 [σ] Perturbed Hamiltonial H 2 [τ] = H 1 [τ] + P [τ] total Hamiltonial H[σ, τ] = β 1 (H 1 [σ] + H 2 [τ]) P [τ] = (β 2 /β 1 1)H 1 [τ] : P [τ] = p i τ i P [τ] = p ij J ij τ iτ j r(p ) = σ i τ i 2 H σi aσb i 2 H 1 τi aτ i b 2 H 2 P = 0, H 1 = H 2 r(p = 0) = 1. The spin-glass phase is CHAOTIC, if lim p 0 lim r(p ) < 1. N 2002/09/13 8
Numerical Examinations {m i } q = σ i τ i r. TAP eq. NaiveTAP eq. : F. Ritort, PRB50,6844(1994) J ij : M. NeyNifle, PRB57,492(1998) chaotic : Sales-Yoshino, RPE65, 066131 (2002), (cond-mat/0203371) 9
Against Temperature Chaos... I. Kondor, J. Phys. A 22, L163 (1989) On chaos in spin glass zero-th loop order one loop : I. Kondor and A. Végsõ, J. Phys. A 26 L641 (1993). A. Billoire and E. Marinari, J. Phys. A 33, L265 (2000), Evidences Against Temperature Chaos in Mean Field and Realistic Spin Glasses T. Rizzo, J. Phys. A 34, 5531 (2001), Against Choas in Temperature in Mean-Field Spin Glass Models. R. Mulet, A. Pagnani, and G. Parisi, Phys. Rev. B 63, 184438 (2001), Against temperature chaos in naive Thouless-Anderson-Palmer equations A. Billoire and E. Marinari, cond-mat/0202473, Overlap Among States at Different Temperatures in the SK Model. 10
Our strategy: Eigenmode Analysis of susceptibility matrix χ ij = 2 F ({h i }) h i h j = h=0 S j h i = β S i S j h=0 O(N) 4 SG O(N) 11
The first eigenvector of a sample with N = 64 2 T/J = 1.8 T/J = 0.4 12
4d ± J Ising EA Model using the dual trick method r T0 ( T, L) = F (L/L ovl ) with r T0 ( T ) i e(t 0) L ovl = T 1/ζ T0)ei (1) (T0+ T) r( T)= 1 N iei (1) ( 1 N T 0 /J=1.0=0.5T c i e (T 0+ T ) i T=T T 0 L=4 6 8 10 Tc/J=2.0 ) r( T,L T 0 /J=1.0=0.5T c ζ=1.3(1) L T 1/ζ L=4 6 8 10 (L ) r( T ) 0 = ( ) 2002/09/13 13
4d ± J Ising EA Model using the dual trick method Extensive r T0 ( T, L) = F (L/L ovl ) with L ovl = T 1/ζ... ζ 1.3 (M. Ney-Nifle,PRB57, 492(1998)) r1 and r2 factor L T 1/ζ 14
r( T ) T = 0 1 Bray-Moore L Entropy gain p T Ld s/2 ΥL θ with L ovl = (Υ/ T ) 1/ζ = (L/L ovl ) ζ r O(1) r 1 TL ζ (L/L ovl ) ζ L ζ T linear OK r 1 r p TL ζ 15
rejuvenation rejuvenation χ J. Hammann, et al : J.Phys.Soc.Jpn. 69 (2000)Suppl. A, 206 211. 16
cumulative memory effect MC simulations rejuvenation cumulative memory effect: (a) (Komori-Yoshino-Takayama, ω;t;tw1) χ ~ [T2,T1](τ t w1 =256 (eff) t w1 =824 t2) χ ~ i( t,t ~ Kmori-Yoshino-Takayama : t 2 J.Phys.Soc.Jpn. 69 (2000)Suppl. A, 228 237.... MC J. Phys. Soc. Jpn. 66 Suppl. A (2000)355) SG R[T2,T1 ](t;t w1) 4 3 2 1 (a) 10 100 1000 10000 100000 t,t ~ T 1 =0.8 T 2 =0.6 17
L SG t w SG t L t W L L 1 > L 2 T 1 > T 2 t 1 < ( ) t 2 T 1 > T 2 Effective time T 1 t 1 T 2 t 2 length scale L[t] L[t] time scale > T 2 T 1 T 2 18
cumulative chaos crossover Overlap Length L ovlp = (δt ) 1/ζ Twin experimets: Jönsson-Yoshino-Nordblad, PRL89,(2002)097201. L ovlp T 2 large Length chaotic Length L ovlp small cumulative Τ Length T 1 L[t] Overlap length AgMn 1/ζ 2.6 L ovlp Leff/L T L Ti (t w )/L T 2002/09/13 19
Effective time : t eff T 1 t w = T 2 0.046 w (MC Simulation) ( ) 0.044 0.042 R 1 = L(t w, T 1 ) R 2 = L(t eff w, T 2) 0.04 L(t, T ) T t 0.034 Komori-Yoshino-Takayama(1999) χ"(t;ω) Takayama-Hukushima;cond-mat/0205276 0.048 0.038 0.036 T 1 =0.7 T 2 =0.5 ω 1 =64,t w1 =512 0 5000 10000 15000 20000 t t sh1 =0 t sh1 =2600 512 t sh1 =3300 512 t sh1 =4000 512 R 1 = R 2 ( )cumulative memory chaos regime t w R 2 = L ovlp ( T 2 T 1 ) 2002/09/13 20
our MC Results Takayama-Hukushima;cond-mat/0205276,R2 eff R1 2.2 2 1.8 1.6 0.7 0.6 0.6 0.7 0.6 0.5 0.5 0.6 0.5 0.4 0.4 0.5 0.7 0.5 0.5 0.7 0.6 0.4 0.7 0.4 0.4 0.7 ovlp RT2/L 0.4 0.7 0.5 0.7 0.6 0.7 0.5 0.4 0.6 0.4 0.7 0.4 0.7 0.5 0.7 0.6 1/ζ~1.2 1.4 1.2 1.2 1.4 1.6 1.8 2 2.2 R 1,R 2 eff T = 0.3 t w R 1 = R 2 1/ζ 1.0 R T1 /L ovlp + cumulative + cumulative 21
Summary ζ L T1 (t w ) L ovlp : cumulative memory L T1 (t w ) L ovlp : cumulative L T1 (t w ) L ovlp : chaotic behavior rejuvenation? 22