Funding/Equipment:NSF, MCC, SLOAN, UIUC, IBM, SANDIA

Crackling Noise, Glassiness, and Disorder Induced Critical Scaling In and Out of Equilibrium:

Are they the same ??? UIUCUIUC: Yang Liu, John Carpenter (now Sandia), Amit Mehta, Robert White (now UCSD), Sharon Loverde (now Northwestern), Riva Vanderveld (now Caltech).

Jim Sethna, M. Kuntz, O. Perkovic (Cornell University), Gary Friedman (Drexel U.),Alan Middleton (Syr. U.) -- theoryA. Berger, O. Hellwig (IBM/Hitachi) -- exp.Gianfranco Durin (Torino, Italy) -- exp.Andrea Mills, Mike Weissman (UIUC) -- exp.

E. Carlson (Perdue), E. Fradkin (UIUC), S. Kivelson (Stanford),D. VanHarlingen (UIUC), M. Weissman (UIUC), C.Panagopoulos (Cambridge) -- superconductorsC. Marchetti (Syr.)-- plastic CDW Y. Ben-Zion (USC), D.S. Fisher (Harvard) -- earthquakes

Magnets and Barkhausen Noise- or Martensites and acc. emissionMagnets and Barkhausen Noise- or Martensites and acc. emission

Volt

Sample

H(t)

Crackling noise

Crackling Noise / Avalanches:Crackling Noise / Avalanches:Barkhausen Noise (magnets)Barkhausen Noise (magnets)

Acoustic emission (Martensites)Acoustic emission (Martensites) (Ortin, Vives,... )

Superconductors Superconductors (P.Adams; Field,Witt,Nori,...)

Liquid He invading NucleporeLiquid He invading Nuclepore (Hallock, Lilly,Wooters...)

Rupture of fibrous MaterialsRupture of fibrous Materials

EarthquakesEarthquakes

BroadBroad Range of Range ofsizes & sizes & durations durations

s-

Big open questions:

(1) Can we go beyond universal power laws? Compare universal scaling functions in theory and experiments!!!Example: average avalanche temporal shape.

(2) What is the extent of the underlying universality classes ???

(3) What is the relation between crackling noise and glassiness? Glasses are stuck in metastable states; crackling noise comes from transitions between

metastable states. Are they the same states, or are the typical thermal metastable states different from those sampled by transitions under stress?

For the random-field model, the two types of states are governed by the same universal fixed point -- at least near the transition where the correlation lengths diverge!

(4) Experimental tests with disorder as a tuning parameter ?!!!!!!!!!!!!!!

Zero Temperature : (Equilibration time scale) >> (Experimental time scale)

The Zero Temperature Non-equilibrium Random Field Ising Model

H

Random Field Distribution

R=“strength” of the disorder

(hi)

Jumps = Avalanches = Jumps = Avalanches =

Barkhausen noiseBarkhausen noise

iii

nnjisystem ShHSSJΗ

..

..

)(force"" localnn

ji SJhtH

(t)HEach spin is always aligned with the direction of the

Clean Dirty

The Disorder Induced Critical Point

Vary disorder by 50% of critical amount: still find 2 decades of powerlaw scaling!… HUGE scaling region!!!

R

H Up

Down

disorderdisorder

+=2.030.03, 1/=4.20.3, ...

critical

LOG (AVALANCHE SIZE)

Log(#)

Dirty critical

Seen also in experiments: Berger et al, PRL 2000, J.Appl.Phys. 2004

Self-similarity at critical disorder RSelf-similarity at critical disorder Rcc=2.16J=2.16J

1003

10003

(Cross-sections of avalanches during magnetization)

CRITICAL POINT: CRITICAL POINT: system is at a fixed point system is at a fixed point under coarse graining under coarse graining transformationtransformation

(Renormalization Group)(Renormalization Group)

SimulationSimulation6-6- expansion expansion (Renormalization Group)

1/ = 2 - /3 - 0.12 + 0.13-0.34 + 5 + O(6 )

(PRL ‘93,’95, 2003 PRB ‘96, ‘99, 2002 (R)) Nature 410, 242 (2001)

Experiments and Experiments and Simulations in 3 dim. Simulations in 3 dim. (Barkhausen Noise):(Barkhausen Noise):

RESULTSRESULTS

Need Noise Exp. Tuning disorder!!!Need Noise Exp. Tuning disorder!!!

Big open questions:

(1) Can we go beyond universal power laws? Compare universal scaling functions in theory and experiments!!!Example: average avalanche temporal shape.

(2) What is the extent of the underlying universality classes ???

(3) What is the relation between crackling noise and glassiness? Glasses are stuck in metastable states; crackling noise comes from transitions between

metastable states. Are they the same states, or are the typical thermal metastable states different from those sampled by transitions under stress?

For the random-field model, the two types of states are governed by the same universal fixed point -- at least near the transition where the correlation lengths diverge!

(4) Experimental tests with disorder as a tuning parameter ?!!!!!!!!!!!!!!

Universal Scaling Functions:

T

V(t)

AVERAGE :

AVALANCHE SIZE

DISTRIBUTION:UNIVERSAL SCALING FUNCTION

EXPERIMENTAL SCALING FUNCTION

Mills, Weissman

Asymmetric!?!

Eddy currents ?

Zapperi, Durin et al.Nature Phys. 1, 46-49 (2005); K.D. News and ViewsNature Phys. 1, 13-14(2005)

Big open questions:

(1) Can we go beyond universal power laws? Compare universal scaling functions in theory and experiments!!!Example: average avalanche temporal shape.

(2) What is the extent of the underlying universality classes ???

(3) What is the relation between crackling noise and glassiness? Glasses are stuck in metastable states; crackling noise comes from transitions between

metastable states. Are they the same states, or are the typical thermal metastable states different from those sampled by transitions under stress?

For the random-field model, the two types of states are governed by the same universal fixed point -- at least near the transition where the correlation lengths diverge!

(4) Experimental tests with disorder as a tuning parameter ?!!!!!!!!!!!!!!

Huge Universality Class!!! (Details don’t matter!)

i

iinn

jisystem ShHSSJΗ..

(t)H

Replace J by random couplings Use different distribution of hi or

replace hi by random anisotropies

Other Universality classes ?

i

iinn

jisystem ShHSSJΗ..

(t)H + long range forces+

2 DynamicsSingle domain wall; Nattermann, Robbins, Ji, Zapperi, Ciseau, Durin, Stanley, Urbach et al., Narayan, Sethna, ...

With nucleation of new domains

Magnets (Sethna,KD,Myers, Nature 2001), plastic charge density wave depinning (Marchetti, KD PRB 2002), earthquakes (Fisher, KD, Ramanathan, Ben-Zion, PRL 1997, Mehta, BenZion, KD 2005), maybe superconductors… (Carlson, KD, Fradkin, Kivelson, PRL 2005), …plasticity (Zaiser 2006, KD,Ben-Zion,Uhl 2008)… others ?

History Induced Critical Behavior at R=Rc:

John Carpenter, K.D., A. Mills, M. Weissman, A. Berger, O. Hellwig (2004), und J. Carpenter, K.D., Gary Friedman, et al. (2001).

Universal Exponents (Predictions):

Also: Avalanche Size Distribution, Finite Size Scaling, Exponent relations etc.

EXPERIMENTS ???

Hmax

Comparison to first Experiments:ExperimentSimulation

Andreas Berger (Hitachi)

CoPtCrB

Need: associated Barkhausen Noise Experiments!!!

Different Behavior with long range interactions: (critical domain wall motion) Barkhausen noise experiments:

Olav Hellwig, Hitachi Co/Pt multilayer

Mills, Weissman, UIUC Fe21Co64B15 amorph

SameCutoff

Simulation Experiment

Demagnetizing curves in the T=0 Random Field Ising Model:Demagnetizing curves in the T=0 Random Field Ising Model:John Carpenter, KD, PRB 2003 (R)

Zapperi et al., PRB 2004

Need: Experiments on Barkhausen Noise for various disorders ???

Same disorder induced critical point as saturation loop:

Avalanche Size Distribution M vs R

R=3.3 R=2.5 203803

Big open questions:

(1) Can we go beyond universal power laws? Compare universal scaling functions in theory and experiments!!!Example: average avalanche temporal shape.

(2) What is the extent of the underlying universality class ???

(3) What is the relation between crackling noise and glassiness? Glasses are stuck in metastable states; crackling noise comes from transitions between

metastable states. Are they the same states, or are the typical thermal metastable states different from those sampled by transitions under stress?

For the random-field model, the two types of states are governed by the same universal fixed point -- at least near the transition where the correlation lengths diverge!

(4) Experimental tests with disorder as a tuning parameter ?!!!!!!!!!!!!!!

SURPRIZINGLY SIMILAR: EQUILIBRIUM and NONEQUILIBRIUM RFIM (Vives,Ortin,Perez Reche et al.):

1. SAME MEAN FIELD EXPONENTS (β=1/2, ν=1/2, δ=3) τ=3/2 and σ=1/2 (Liu, KD)

2. ABOUT SAME SIMULATION EXPONENTS in 3D and even in 4D (WITHIN ERRORBARS)

3. 6-ε expansion of noneq. mapped to all orders in ε to eq. expansion (KD, Sethna)

4. SAME SIMULATION EXPONENTS for GROUND STATE & DEMAGN. STATE (Zapperi et al.)

5. SAME EXP. AND SCALING FUNCTS FOR DEMAGN. CURVE & SAT. LOOP (Carpenter, KD)

6. Middleton's no-passing rule: flipped spin cannot flip back with increasing field. (Liu, KD)

Liu, KD 2006

Ground state “Avalanches”:

Surprizing Result: same Avalanche Exponents And Scaling Function

In Equilibrium and Nonequilibrium

•Result: ~Same scaling of Avalanche Surface Area Distribution

Integrated avalanche surface distribution for avalanche size S

Liu, KD, 2006

•Conclusions on comparison to equilibrium:

Avalanche exponents and spatial structures (fractal dimensions and anisotropy measures), etc. all strongly suggest that the equilibrium

and non-equilibrium transitions of the T=0 RFIM belong to the same universality

class… !!!???

•Thanks to A. A. Middleton and J.P. Sethna(Yang Liu, KD, condmat 2006)

Summary on magnets and further results:

clean disordered

Renormalization group (huge universality class)

finite sweep rate effects (model indep. theory)

history induced critical behavior like equilibrium critical behavior ?!?!?!

Second Spectra (mean field theory), universal scaling functions, return point memory, temperature effects...

Earthquakes,...

Plasticity…

S-

sizesize Ssize

Num-ber

EXPERIMENTS ???!!!

Universal Gutenberg Richter Scaling behavior

OR

Characteristic Earthquake distribution ?

Magnitude ~ 2/3 Log(total displacement)

GR exponent

RG: MEAN FIELD THEORY EXACT IN THE PHYSICAL DIMENSION

BRITTLE

PATCHES

3D FAULT ZONE

MODEL(BEN-ZION & RICE ’93)

Fisher, KD, Ben-Zion, et al. PRL 78 (1997)

Mean field exponent: b=0.75

for Frequency~ 10-bM

Phase Diagram

small events scale as moment~area

cracklike large events: moment~area3/2

Mehta, KD, Ben-Zion, PRE 2005

Universal Scaling Functions: Data from Susan Bilek, see

Mehta, KD, Ben-Zion, 2005

Big open questions:

(1) What is the relation between crackling noise and glassiness?

Glasses are stuck in metastable states; crackling noise comes from transitions between metastable states.

Are they the same states, or are the typical thermal metastable states different from those sampled by transitions under stress?

For the random-field model, the two types of states are governed by the same universal fixed point -- at least near the transition where the correlation lengths diverge!

(2) Can we go beyond power laws? If universality is correct, we should be able to predict all kinds

of things about crackling systems -- up to the distribution of shapes of avalanches.

Example: average avalanche temporal shape.

(3) What is the extent of the underlying universality class???

(4) Experimental tests ?!!!!!!!!!!!!!!

Conclusions and Outlook:

-“Disorder” effects similar in Magnets and Earthquakes!

- Renormalization Group for universal predictions (for eq: mean field theory exact in the physical dimension)

- Exponents the same, universal scaling functions seem similar

- Similar phase diagrams (disorder in magnets same role as dynamic weakening/strengthening in Ben-Zion-Rice earthquake model)

NEED:

- improved models (geometry, correlations in disorder, fault network, friction laws, seismic emission,...)

- more time resolved data for small earthquakes, (moment rates), time vs. diameter, slip vs. M, regularity, parameters for Mrunaway,...

Ω>0

=0 t s

IIIIII

slow

Fast

Sweep Rate Regimes of Barkhausen Noise

inc

reas

ing

Region I (“slow”)•power spectra unchanged •Pulse statistics are affected in general

No pulse statistics for >t

Region II (intermediate)•Power spectra unchanged.•Only temporal overlap

Region III (“fast”)•Low Freq. PS changed due to spatial overlap

White,KD PRL2003

How are finite sweeprate effects to be understood?(White, KD PRL 2003)

(MODEL “INDEPENDENT” THEORY): Superposition of power law distributed avalanches

In slow regime dependence of pulse exponents on sweep rate explained by temporal overlap of pulses

=0 >0

o < 2 unchanged

o = 2

o >2

() = o - c unchanged for large durations

T-

D(T

)

T

ABBM,Durin

Durin, Zapperi

???

Results for

Duration T

D(T

)( 0) 2

Linear change in exponent

Agrees with Experiment:Plot from Durin and Zapperi

Cond-mat/0404512v1

() = o - c

() = 1.5 - c

Data on amophous FeCoB ribbons (courtesy of G. Durin)

Plots from Durin and Zapperi reviewCond-mat/0404512v1

shapes

Shift in frequency of the maximum in the low frequency regime of the power spectra

Overlap in space -> parallel dynamicsLow freq. cutoff for this:

2max ~

z

z

2

z

z

Ranges from 0.5 in mf (ABBM) to 0.43 in systems w\o dipolar fields.

Ferroelectrics: relaxor ferroelectrics, collab. with Mike Weissman (UIUC)

Superconductors: collaboration with E. Fradkin (UIUC), S. Kivelson (Stanford), E. Carlson (Purdue), D. van Harlingen (UIUC), M. Weissman (UIUC), and C. Panagopoulos (Cambridge)

Earthquakes: expecting more data (with Ben-Zion), models for fault networks, aftershocks (with A. Mehta, Ben-Zion).

Charge Density Wave plastic depinning (with C. Marchetti)

Spreading population fronts in disordered environments (population Biology) (with J. Carpenter, A. Missel, N. Shnerb, D. Nelson, IGB group lead by Nigel Goldenfeld)

OUTLOOK:

Data on amophous FeSi ribbons (courtesy of G. Durin)

Plots from Durin and Zapperi reviewCond-mat/0404512v1

shapes

Temperature > 0Scaling Theory and

Simulations using the random field Ising model near the zero temperature far from eq. critical regime (in progress)

Increasing Temp

Robert White, Alex Travesset, Matthew Delgado, KD, 2004

OUTLOOK

Magnets: collaboration with A. Berger (IBM/Hitachi), M. Weissman (UIUC), G. Durin (Torino, Italy), comparing experiments to model predictions (tuning disorder, sweeprate, temperature, history).

Testing scaling theory at finite temperature through simulations, etc. (with M. Delgado)

Interesting Memory effects !! (Return Point Memory, with Amit Mehta).

Renormalization Group treatment for history induced critical scaling (?)

Connection ground state scaling/demagnetization state scaling (with G. Poore, Yang Liu, A. Middleton, and J.P. Sethna)

(Exchange Bias Systems ? With A. Berger).

Experiments-

“Clean”

“Dirty”

A.Berger et.al. PRL 85 4176(2000)

Disorder

H Up

Down

Phase transition?Phase transition?

Disorder is important!Disorder is important!

Noise experiments?Noise experiments?

-200 -100 0 100 200

mag. field (Oe)

Tanneal

= 540 K

550 K

560 K

570 K

580 K

590 K

mag

neti

zati

on

(arb

. u

nit

s)

Gd(0001)/W(110)-films

annealing study data analysisfit of experimental data to:

-200 -100 0 100 200

mag. field (Oe)

Tanneal

= 540 K

550 K

560 K

570 K

580 K

590 K

mag

neti

zati

on

(arb

. u

nit

s)

M(H ) Mtan 1H HC

1

A2 tan

1 H HC

2

0

1

500 550 600 650

mag

neti

zati

on

ju

mp

M

Tanneal

(K)

M ~ ((T-Tc)/T

c)

0.668

Gd(0001)/W(110)-films

Non-equilibrium Phase diagram

Flip spins due to field increase

Time increases by one time step

Increment field by the sweep

rate times the time step i.e. dh

Flip nearest neighbors of recently flipped spins

The Algorithm- (adiabatic…. Finite sweep rate)

Raise field until one spin flips

Flip nearest neighbors of recently flipped spins

(if called for)

If no spins have flipped in the previous time step. Avalanche has ended

If spins flipped avalanche continues

Begin (Hext= -)

System size ~ 10003

Simulation Overview-

Run Time Scaling O(NlogN)

Memory Scaling O(N)

Equilibrium RFIMMC methods size ~ 253 M. Kuntz, et al.. Computing in Science and Engineering 1, 73 (1999). pdf

Sweep rate at which spatial overlap dominates.

z

d

s S

L

Ha 1*)()(

1~

So?...

ts Separated pulses in time No Spatial overlap

Superposition of Adiabatic Avalanches!

Sweep rate at which temporal overlap dominatesand we get no separated avalanches.

CTHat 2*)(

1

)(

1~ 1-d Percolation

timeQuiet space

Avalanche begin Avalanche end

Resulting Duration distribution at finite driving rate...

oo

oTTc

eTTP

22*

~),(2o

2o cTTP 2~),(

?2o ?2o

Results

[2]Berttoti,Durin,Magni, J. Chem. Phys. 75, 5490 (1994)

For our Model (ztneRFIM)-

05.05.2 o Really large systems - (expect) no change in exponentsWe find-

co ~)(

We can’t tune o ...But argument for expected changesare general and only depend on a fewassumptions.

Criteria for expected changes is met experimentally for soft ferromagnets [2]