1. Aging of the Ising EA spin-glass model under a magnetic field --- Numerical vs. Real Experiments --- Hajime Takayama J-F-Seminar_Paris, Sep. 2005 Institute for Solid State Physics, University of Tokyo

2. There have been so many qualitatively similar phenomena observed both in real and numerical experiments on spin-glass slow dynamics (in a magnetic field). ac susceptibility after field shifts real exp. (CdCr 0.17 In 0.30 S 4 ) Vincent et al (1995) numerical exp. (3D Gaussian Ising EA model) h ~ 10Gauss t ~ 300min h ~ 0.2T c t ~ 4000MCs Are the two really common phenomena? h sim ~ 10 3 h exp in micro. time units 10 0 10 6 10 13 10 17 ( with 1 MCs ~ 10 -12 s ) sim.exp. Do further numerical experiments! Could the comparison be made quantitative?

3. K. Hukushima (U. Tokyo) Outline 1. Introduction (previous slide) 2. Field-shift aging protocol in 3D Ising EA model 3. Field-cooled magnetization in a small field P. E. Jönsson --- Instability of the SG phase in a static magnetic field --- 4. Conclusion (now in RIKEN) HT and KH: J. Phys. Soc. Jpn. 73 (2004) 2077. PEJ and HT: J. Phys. Soc. Jpn. 74 (2005) 1131.

4. 2) Field-shift protocol in 3D Ising EA Model --- Instability of the SG phase under a static field --- Simulation: Standard (Heat-Bath) Monte Carlo method on 3D Gaussian Ising EA model HT and K. Hukushima: J. Phys. Soc. Jpn. 73 (2004) 2077 units: ･ T, h (Zeeman energy) by J (width of J ij ) ： T c ≃ 0.95J ･ time by 1 MCs system: N=L 3 with L=24, and with periodic boundary condition

5. Lundgren et al ('83) CuMn: Granberg et al (’88) peak position of S(t) : waiting time Field-Shift Aging Protocol for small h Simulation S(t’)

6. Zero-Field-Cooled Magnetization As h becomes larger, the smaller becomes t cr.

7. Characteristic Time Regimes 1)t w > t > 0: (isothermal) isobaric aging in h=0 h=0 T=0.8 T=0.4 Komori, Yoshino, HT (’99) RTRT Mean size of SG domains, R T,h (t), grows. thermal activation process J. Kisker et al (’96), E. Marinari et al (’98) 2) t cr > t’=t-t w > 0: transient t’ ≃ t cr ： Crossover from h=0 to h>0 t’=t cr 1) 2) 3) 3) t’ > t cr : isobaric aging in h>0

8. “Subdomains-within-Domain” Picture We suppose: After the h-shift, SG subdomains in local equilibrium in (T,h) of a mean size grow within each domain which has grown under (T,0) up to t=t w. Its growth law is expected to be similar to but with a certain modification reflecting the difference in initial spin configurations. for Transient Regime In the mean-field language, they are at different locations in phase space, separated by a free-energy barrier. Energy change in a T-shift-down process (Kovacs effect) The system adjusts itself to a h/T-shift by first individual spins, then spins pairing with them, clusters,.. ; subdomains growth

9. Time-Length Scale Conversion Before h-shift : After h-shift: Kovacs-like (or transient) effect will be a priori taken into account by –ah 2 in the above exponent. At t’=t cr, i.e., at crossover, we expect that holds and that the system crossovers to isobaric aging under (T,h). J. Kisker et al (’96); E. Marinari et al (’98);Komori, Yoshino, HT (’99) How we can interpret the results t cr < t w for large h? Actually, for a small h,, and so are observed.

10. Field Crossover Length in Droplet picture In equilibriumDroplet excitationunder field h Zeeman energy : free-energy gap : SG state is unstable! Field crossover length L h :

11. Scaling Analysis of R cr /L h vs R w /L h Before h-shift : After h-shift: R w /L h R cr /L h

12. No SG state in equilibrium in h > 0 Crossover from SG to Paramagnetic States at T=0.4 – 0.8 and h=0.1 - 0.75 are all well scaled a T scales data at each T l T (=b l ) those at different T Paramagnetic state is realized at t’~ 10 5 MCs for h=0.75. h ~ a few tens Oe

13. Semi-Quantitative Comparison with Experiments Dynamical crossover condition semi-quantitative comparison in m.t.u 10 0 10 6 10 13 10 17 Let’s extend simulational results to 10 17 MCs and compare with real experimental results or

14. dynamical crossover scenario open: exp. solid: simu. with c T =1.6 in micro. time units 10 0 10 6 10 13 10 17 common behavior even semi-quantitatively !! Deviation of ZFCM from FCM: Aruga-Ito ('94) Irreversibility in FCM and ZFCM (in large h) h ~ (1-T/T c ) 3/2

15. Comment: h-Shift-down Process Before h-cut : After h-cut: All the parameters are common to the shift-up process! : h-independent : h-dependent h-shift-uph-shift-down (h-cut)

16. h-Shift-down vs. h-Shift-up Before h-cut : After h-cut: All the parameters are common to the shift-up process! Similar free-energy landscapes!? : h-independent : h-dependent phase space F isobaric h=0 shift-up shift-down (h-cut) h>0

17. P. E. Jönsson and HT : J. Phys. Soc. Jpn. 74 (2005) 1131. 3) Field-Cooled-Magnetization in a Small Field --- Cusp in FCM and irreversibility of ZFCM--- CuMn: Nagata et al (’79) one of the most typical SG phenomena (a) simulation: 3D Ising EA model (b) experiment: Fe 0.55 Mn 0.45 TiO 3 Can the FCM cusp experimentally observed be interpreted as the occurrence of a phase transition, or as thermal blocking (dynamical crossover)?

18. Characteristic features of FCM and ZFCM observed in real experiments Fe 0.55 Mn 0.45 TiO 3 T irr 1) T irr depends on a cooling rate. T irr : onset of irreversibility T* : peak of FCM T c : transition temp. T* estimated from high temps. ac data in h=0 2) FCM exhibits a peak at T* (~T c ). 3) FCM’s with different cooling rates cross with each other at T < T* CuMn canonical SG Lundgren et al, (1985)

19. Corresponding numerical experiments T irr (a) T irr T* TcTc 1) T irr depends on a cooling rate. T*T* TcTc 2) FCM exhibits a peak at T* (>T c ). (checked for rate10 4 and 33333) 3) FCM’s with different c-rates don’t cross yet, but at T < T* m/h r-slower < m/h r-faster ! 3D I EA rate###: cooling by ΔT=0.01 with ### MCs at each T

20. FCM behavior at a stop of cooling 3D I EAFe 0.55 Mn 0.45 TiO 3 4) FCM increases at a stop at T* < T < T irr. At T < T irr, not only ZFCM but also FCM states are out-of equilibrium. 5) FCM decreases at a stop at T<T*.

21. 6) FCM upturn at a stop close to T*. 3D I EA Fe 0.55 Mn 0.45 TiO 3 AuFe canonical SG Lundgren et al, (‘85) FCM upturn is considered a SG common property.

22. Our Interpretation of FCM Behavior 1) ~ 6) in equilibrium T irr (cooling rate; h) thermal blocking of spin clusters with SG SRO of which are separated from each other and are polarized under Zeeman energy alone. ξ*ξ* high T out-of equilibrium When cooling is stopped: blocked clusters become larger and are further polarized, and so FCM increases. : 4) ξ*ξ* 1) slower cooling rate: lower T irr, and larger ξ* and FCM By further cooling: further blocking of spin clusters of sizes smaller than

23. T* (~T c ) Reconstruction of the clusters takes place under SG stiffness energy which now becomes effective, and FCM exhibits a peak (more than a cusp) at T* ! : 2) out-of equilibrium When cooling is stopped: SG SLO in local equilibrium of (T,h),, increases until it reaches field crossover length L h (so FCM decreases), and then the paramagnetic behavior is resumed (so FCM increases)  FCM upturn behavior 6) Spins don’t know longer-ranged equilibrium configurations a priori, but find them only through shorter- range order (Kovacs effect) FCM upturn can be observed only at T close to T* since it takes more than an astronomical time for to reach L h at lower T : 5) transient! SRO clusters thermally blocked become in touch with each other. For the slower cooling rate with the larger ξ*, the larger is, maybe, the reconstruction (crossing of FCM’s 3) )

24. 4) Conclusion From simulation on h-shift aging processes, we reach to the dynamical crossover (from SG to paramagnetic) scenario, or the absence of the equilibrium SG phase, for 3D Ising spin glasses under a static field. The result is consistent even semi-quantitatively with real experiments. Not only the onset of irreversibility in FCM and ZFCM, but also various out-of-equilibrium behavior of FCM in Ising spin glass Fe x Mn 1-x TiO 3 under small fields are examined. The results are at least qualitatively consistent with the numerical experiment. The FCM cusp-like behavior is argued to be consistent with our dynamical crossover scenario, or it is essentially due to thermal blocking. Numerical Experiments (numerical simulation based on a model as microscopic as possible) are indispensable to properly understand “glassy dynamics” (slow dynamics of a cooperative origin + thermal blocking) observed in complex systems.

25. Comment. II. Power-Law-Growth of R T (t) Fisher-Huse theory R T (t) ~ (ln t) 1/ψ free-energy barrier against droplet overturn ΔB R ~ R ψ growth law numerical simulation R T (t) ~ t 1/z(T) ΔB R ~ ln R f-energy change by overturn ΔF R ~ R θ asymptotic regime near equilibrium pre-asymptotic regime far from equilibrium (θ<ψ)(θ<ψ) (θ>ψ=0)