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Aging of the Ising EA spin-glass model under a magnetic field

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

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Aging of the Ising EA spin-glass model under a magnetic field

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

  2. There have been so many qualitativelysimilar phenomena observed both in real and numerical experiments on spin-glass slow dynamics (in a magnetic field). ac susceptibility after field shifts real exp. (CdCr0.17In0.30S4) Vincent et al (1995) numerical exp. (3D Gaussian Ising EA model) h~ 10Gauss t ~ 300min h~ 0.2Tc t ~ 4000MCs hsim~ 103hexp ( with 1 MCs ~ 10-12 s ) 100 106 1013 1017 in micro. time units sim. exp. Are the two really common phenomena? Could the comparison be made quantitative? Do further numerical experiments!

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

  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 units: ･ T, h(Zeeman energy) by J (width of Jij)：Tc≃ 0.95J ･time by 1 MCs system: N=L3with L=24, and with periodic boundary condition HT and K. Hukushima: J. Phys. Soc. Jpn. 73 (2004) 2077

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

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

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

  8. “Subdomains-within-Domain” Picture for Transient Regime We suppose: After the h-shift,SG subdomains in local equilibrium in (T,h) of a mean size grow within each domainwhich has grown under (T,0) up to t=tw. Its growth law is expected to be similar to but with a certain modification reflecting the difference in initial spin configurations. 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 : J. Kisker et al (’96); E. Marinari et al (’98); Komori, Yoshino, HT (’99) After h-shift: At t’=tcr, i.e., at crossover, we expect that holds and that the system crossovers to isobaric aging under (T,h). Actually, for a small h, , and soare observed. How we can interpret the results tcr < tw for large h? Kovacs-like (or transient) effect will be a prioritaken into account by –ah2 in the above exponent.

  10. Field Crossover Length in Droplet picture In equilibrium Droplet excitation under field h Zeeman energy : free-energy gap : SG state is unstable! Field crossover length Lh:

  11. Scaling Analysis of Rcr/Lh vs Rw/Lh Before h-shift : After h-shift: Rcr/Lh Rw/Lh

  12. Crossover from SG to Paramagnetic States at T=0.4 – 0.8 and h=0.1 - 0.75 are all well scaled aT scales data at each T lT(=bl) those at different T Paramagnetic state is realized at t’~ 105 MCs for h=0.75. h~ a few tens Oe No SG state in equilibrium in h > 0

  13. Semi-Quantitative Comparison with Experiments Dynamical crossover condition or semi-quantitative comparison 100 106 1013 1017 in m.t.u Let’s extend simulational results to 1017 MCs and compare with real experimental results

  14. Irreversibility in FCM and ZFCM (in large h) Deviation of ZFCM from FCM: Aruga-Ito ('94) dynamical crossover scenario open: exp. solid: simu. with cT=1.6 h ~ (1-T/Tc)3/2 100 106 1013 1017 in micro. time units common behavior even semi-quantitatively !!

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

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

  17. 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 Can the FCM cusp experimentally observed be interpreted as the occurrence of a phase transition, or as thermal blocking (dynamical crossover)? • simulation: 3D Ising EA model • experiment: Fe0.55Mn0.45TiO3 P. E. JönssonandHT : J. Phys. Soc. Jpn. 74 (2005) 1131.

  18. Characteristic features of FCM and ZFCM observed in real experiments Fe0.55Mn0.45TiO3 T* Tirr: onset of irreversibility T* : peak of FCM Tc : transition temp. Tirr estimated from high temps. ac data in h=0 CuMn canonical SG 1) Tirr depends on a cooling rate. 2) FCM exhibits a peak at T* (~Tc). 3) FCM’s with different cooling rates cross with each other at T < T* Lundgren et al, (1985)

  19. Corresponding numerical experiments Tc 3D I EA T* Tirr Tirr (a) Tirr rate###: cooling by ΔT=0.01 with ### MCs at each T 1) Tirr depends on a cooling rate. Tc T* 2) FCM exhibits a peak at T* (>Tc). (checked for rate104 and 33333) 3) FCM’s with different c-rates don’t cross yet, but at T < T* m/hr-slower <m/hr-faster !

  20. FCM behavior at a stop of cooling 3D I EA Fe0.55Mn0.45TiO3 4)FCM increases at a stop at T* < T < Tirr . 5)FCM decreases at a stop at T<T* . At T < Tirr , not only ZFCM but also FCM states are out-of equilibrium.

  21. 6) FCM upturn at a stop close to T*. Fe0.55Mn0.45TiO3 AuFe canonical SG Lundgren et al, (‘85) 3D I EA FCM upturn is considered a SG common property.

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

  23. out-of equilibrium T* (~Tc) SRO clusters thermally blocked become in touch with each other. 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) For the slower cooling rate with the larger ξ*, the larger is, maybe, the reconstruction (crossing of FCM’s 3) ) transient! When cooling is stopped: SG SLO in local equilibrium of (T,h), , increases until it reaches field crossover lengthLh(so FCM decreases), and then the paramagneticbehavior 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 Lh at lower T: 5)

  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 variousout-of-equilibrium behavior of FCM in Ising spin glass FexMn1-xTiO3 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 crossoverscenario, 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 RT(t) numerical simulation Fisher-Huse theory RT(t) ~ (ln t)1/ψ RT(t) ~ t1/z(T) growth law free-energy barrier against droplet overturn ΔBR ~ Rψ ΔBR ~ ln R ΔFR ~ Rθ f-energy change by overturn ΔFR ~ Rθ asymptotic regime near equilibrium pre-asymptotic regime far from equilibrium (θ<ψ) (θ>ψ=0)

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