Aging of the Ising EA spin-glass mod

el under a magnetic field

--- Numerical vs. Real Experiments ---

Hajime Takayama

J-F-Seminar_Paris, Sep. 2005

Institute for Solid State Physics, University of Tokyo

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. (CdCr0.17In0.30S4) Vincent et al (1995)

numerical exp. (3D Gaussian Ising EA model)

h ~ 10Gauss t ~ 300min

h ~ 0.2Tc

t ~ 4000MCs

Are the two really common phenomena?

hsim ~ 103 hexp

in micro. time units100 106 1013 1017

( with 1 MCs ~ 10-12 s )

sim. exp.

Do further numerical experiments!

Could the comparison be made quantitative?

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 fieldP. 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.

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 Jij) ：　 Tc ≃ 0.95J

　･ time by 1 MCssystem:N=L3 with L=24, and with periodic boundary condition

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’)

Zero-Field-Cooled Magnetization

As h becomes larger, the smaller becomes tcr.

Characteristic Time Regimes

1) tw > t > 0: (isothermal) isobaric aging in h=0 h=0

T=0.8

T=0.4

Komori, Yoshino, HT (’99)

RT

Mean size of SG domains, RT,h(t), grows.

thermal activation processJ. 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

t’=tcr

1) 2) 3)

3) t’ > tcr : isobaric aging in h>0

“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=tw.

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 withthem, clusters, .. ; subdomains growth

Time-Length Scale Conversion Before h-shift :

After h-shift:

Kovacs-like (or transient) effect will be a priori taken into account by –ah2 in the above exponent.

At t’=tcr, 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 tcr < tw for large h?

Actually, for a small h, , and so 　　　　　　 are observed.

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:

Scaling Analysis of Rcr/Lh vs Rw/Lh

Before h-shift :

After h-shift:

Rw/Lh

Rcr/Lh

No SG state in equilibrium in h > 0

Crossover from SG to Paramagnetic States

at T=0.4 – 0.8 andh=0.1 - 0.75 are allwell scaled

aT scales data at each TlT(=bl) those at different T

　 Paramagnetic state is realized 　 at t’~ 105 MCs for h=0.75.

h ~ a few tens Oe

Semi-Quantitative Comparison with Experiments

Dynamical crossover condition

semi-quantitative comparison

in m.t.u100 106 1013 1017

Let’s extend simulational results to 1017 MCs and compare with real experimental results

or

dynamical crossover scenario

open: exp.solid: simu. with cT=1.6

in micro. time units100 106 1013 1017

common behavior even semi-quantitatively !!

Deviation of ZFCM from FCM: Aruga-Ito ('94)

Irreversibility in FCM and ZFCM (in large h)

h ~ (1-T/Tc)3/2

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)

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

Fisobaric

isobaric

h=0

shift-up

shift-down (h-cut)

h>0

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: Fe0.55Mn0.45TiO3

Can the FCM cusp experimentally observed be interpreted as the occurrence of a phase transition, or as thermal blocking (dynamical crossover)?

Characteristic features of FCM and ZFCM observed in real experiments

Fe0.55Mn0.45TiO3

Tirr

1) Tirr depends on a cooling rate.

Tirr: onset of irreversibilityT* : peak of FCMTc : transition temp.

T*

estimated from high temps. ac data in h=0

2) FCM exhibits a peak at T* (~Tc).

3) FCM’s with different cooling rates cross with each other at T < T*

CuMn canonical SG

Lundgren et al, (1985)

Corresponding numerical experiments

Tirr(a) Tirr

Tirr

T*Tc

1) Tirr depends on a cooling rate. T*

Tc

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 !

3D I EA

rate###: cooling by ΔT=0.01 with ### MCs at each T

FCM behavior at a stop of cooling

3D I EA Fe0.55Mn0.45TiO3

4) FCM increases at a stop at T* < T < Tirr .

At T < Tirr , not only ZFCM but also FCM states are out-of equilibrium.

5) FCM decreases at a stop at T<T* .

6) FCM upturn at a stop close to T*.

3D I EA

Fe0.55Mn0.45TiO3 AuFe canonical SG

Lundgren et al, (‘85)

FCM upturn is considered a SG common property.

Our Interpretation of FCM Behavior 1) ~ 6)

in equilibrium

Tirr(cooling rate; h)thermal blocking of spin clusters with SG SRO ofwhich are separated from each other and

are polarized under Zeeman energy alone .

ξ*

high T

out-of equilibrium

When cooling is stopped:

blocked clusters become larger and are further polarized,

and so FCM increases. : 4)

ξ*

1) slower cooling rate: lower Tirr, and larger ξ* and FCM

By further cooling: further blocking of spin clusters of sizes smaller than

T* (~Tc)

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 Lh (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 Lh 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) )

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 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 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.

Comment. II. Power-Law-Growth of RT(t)

Fisher-Huse theory

RT(t) ~ (ln t)1/ψ

free-energy barrier against droplet overturn ΔBR ~ Rψ

growth law

numerical simulation

RT(t) ~ t1/z(T)

ΔBR ~ ln R

f-energy change by overturnΔFR ~ Rθ ΔFR ~ Rθ

asymptotic regime near equilibrium

pre-asymptotic regime far from equilibrium

(θ<ψ) (θ>ψ=0)