September 9, 2002 ( ) [1] K. Hukushima and Y. Iba, cond-mat/ [2] H. Takayama and K. Hukushima, cond-mat/020
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1 September 9, 2002 ( ) [1] and Y. Iba, cond-mat/ [2] H. Takayama and, cond-mat/ Typeset by FoilTEX
2 Today s Contents Against Temperature Chaos in Spin Glasses Against the against /09/13 1
3 ... 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
4 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
5 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, rejuvenation T 1 memory T /09/13 4
6 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, rejuvenation T 1 memory T /09/13 5
7 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
8 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
9 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
10 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, (2002), (cond-mat/ ) 9
11 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, (2001), Against temperature chaos in naive Thouless-Anderson-Palmer equations A. Billoire and E. Marinari, cond-mat/ , Overlap Among States at Different Temperatures in the SK Model. 10
12 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
13 The first eigenvector of a sample with N = 64 2 T/J = 1.8 T/J =
14 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= Tc/J=2.0 ) r( T,L T 0 /J=1.0=0.5T c ζ=1.3(1) L T 1/ζ L= (L ) r( T ) 0 = ( ) 2002/09/13 13
15 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
16 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
17 rejuvenation rejuvenation χ J. Hammann, et al : J.Phys.Soc.Jpn. 69 (2000)Suppl. A,
18 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, MC J. Phys. Soc. Jpn. 66 Suppl. A (2000)355) SG R[T2,T1 ](t;t w1) (a) t,t ~ T 1 =0.8 T 2 =0.6 17
19 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
20 cumulative chaos crossover Overlap Length L ovlp = (δt ) 1/ζ Twin experimets: Jönsson-Yoshino-Nordblad, PRL89,(2002) 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
21 Effective time : t eff T 1 t w = T w (MC Simulation) ( ) R 1 = L(t w, T 1 ) R 2 = L(t eff w, T 2) 0.04 L(t, T ) T t Komori-Yoshino-Takayama(1999) χ"(t;ω) Takayama-Hukushima;cond-mat/ T 1 =0.7 T 2 =0.5 ω 1 =64,t w1 = t t sh1 =0 t sh1 = t sh1 = t sh1 = R 1 = R 2 ( )cumulative memory chaos regime t w R 2 = L ovlp ( T 2 T 1 ) 2002/09/13 20
22 our MC Results Takayama-Hukushima;cond-mat/ ,R2 eff R ovlp RT2/L /ζ~ 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
23 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
( ) URL: December 2, 2003
( ) URL: http://dbs.c.u-tokyo.ac.jp/~fukushima mailto:hukusima@phys.c.u-tokyo.ac.jp December 2, 2003 Today s Contents Summary 2003/12/02 1 Cannella Mydosh(1972) Edwards Anderson(1975): Model Hamiltonian:
References: 3 June 21, 2002 K. Hukushima and H. Kawamura, Phys.Rev.E, 61, R1008 (2000). M. Matsumoto, K. Hukushima,
References: 3 mailto:hukusima@issp.u-tokyo.ac.jp June 21, 2002 K. Hukushima and H. Kawamura, Phys.Rev.E, 61, R1008 (2000). M. Matsumoto, K. Hukushima, and H. Takayama, cond-mat/0204225. Typeset by FoilTEX
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