Funding/Equipment:

1
Crackling Noise, Glassiness, and Disorder Induced
Critical Scaling In and Out of Equilibrium
Are they the same ???
UIUC Yang Liu, John Carpenter (now Sandia),
Amit Mehta, Robert White (now UCSD), Sharon
Loverde (now Northwestern), Riva Vanderveld (now
Caltech).
E. Carlson (Perdue), E. Fradkin (UIUC), S.
Kivelson (Stanford), D. VanHarlingen (UIUC), M.
Weissman (UIUC), C.Panagopoulos (Cambridge) --
superconductors C. Marchetti (Syr.)-- plastic CDW
Y. Ben-Zion (USC), D.S. Fisher (Harvard) --
earthquakes
Jim Sethna, M. Kuntz, O. Perkovic (Cornell
University), Gary Friedman (Drexel U.), Alan
Middleton (Syr. U.) --
theory A. Berger, O. Hellwig (IBM/Hitachi) --
exp. Gianfranco Durin (Torino, Italy) --
exp. Andrea Mills, Mike Weissman (UIUC) -- exp.
Funding/Equipment NSF,
MCC, SLOAN, UIUC, IBM, SANDIA
2
Magnets and Barkhausen Noise- or Martensites and
acc. emission
Crackling noise
3
Crackling Noise / Avalanches
Barkhausen Noise (magnets) Acoustic
emission (Martensites) (Ortin, Vives,...
) Superconductors (P.Adams Field,Witt,Nori,...) L
iquid He invading Nuclepore (Hallock,
Lilly,Wooters...) Rupture of fibrous
Materials Earthquakes
s-?
4
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
?!!!!!!!!!!!!!! 
5




H
Each spin is always aligned with the direction of
the
(t)


Zero Temperature (Equilibration time
scale) gtgt (Experimental time scale)
Jumps Avalanches Barkhausen noise
6
Seen also in experiments Berger et al, PRL 2000,
J.Appl.Phys. 2004
The Disorder Induced Critical Point
disorder
critical
????2.03?0.03, 1/?4.2??0.3, ...
Vary disorder by 50 of critical amount still
find 2 decades of powerlaw scaling! HUGE
scaling region!!!
7
Self-similarity at critical disorder Rc2.16J
(Cross-sections of avalanches during
magnetization)
1003
CRITICAL POINT system is at a fixed point under
coarse graining transformation (Renormalization
Group)
10003
8
RESULTS
Simulation
6-? expansion (Renormalization Group)
1/? 2 - ?/3 - 0.1?2 0.1?3-0.3?4 ?5 O(?6 )
(PRL 93,95, 2003 PRB 96, 99, 2002 (R))
Nature 410, 242 (2001)
Experiments and Simulations in 3 dim. (Barkhausen
Noise)
Need Noise Exp. Tuning disorder!!!
9
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
?!!!!!!!!!!!!!! 
10
Universal Scaling Functions
AVALANCHE SIZE DISTRIBUTION
V(t)
UNIVERSAL SCALING FUNCTION
T
AVERAGE
11
EXPERIMENTAL SCALING FUNCTION
Asymmetric!?!
Eddy currents ? Zapperi, Durin et al. Nature
Phys. 1, 46-49 (2005) K.D. News and
Views Nature Phys. 1, 13-14 (2005)
Mills, Weissman
12
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
?!!!!!!!!!!!!!! 
13
Huge Universality Class!!! (Details dont matter!)
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 ?
Other Universality classes ?
long range forces
2 Dynamics
Single domain wall Nattermann, Robbins, Ji,
Zapperi, Ciseau, Durin, Stanley, Urbach et al.,
Narayan, Sethna, ...
With nucleation of new domains
14
History Induced Critical Behavior at RRc
John Carpenter, K.D., A. Mills, M. Weissman, A.
Berger, O. Hellwig (2004), und J. Carpenter,
K.D., Gary Friedman, et al. (2001).
Hmax
Also Avalanche Size Distribution, Finite Size
Scaling, Exponent relations etc.
EXPERIMENTS ???
Universal Exponents (Predictions)
15
Comparison to first Experiments
Experiment
Simulation
Need associated Barkhausen Noise Experiments!!!
Andreas Berger (Hitachi) CoPtCrB
Different Behavior with long range interactions
(critical domain wall motion) Barkhausen noise
experiments
Simulation
Experiment
Olav Hellwig, Hitachi Co/Pt multilayer
Mills, Weissman, UIUC Fe21Co64B15 amorph
SameCutoff
16
Demagnetizing curves in the T0 Random Field
Ising Model
John Carpenter, KD, PRB 2003 (R) Zapperi et al.,
PRB 2004
Same disorder induced critical point as
saturation loop
Avalanche Size Distribution
DM vs R
Need Experiments on Barkhausen Noise for various
disorders ???
R3.3
R2.5
203
803
17
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
?!!!!!!!!!!!!!! 
18

• SURPRIZINGLY SIMILAR EQUILIBRIUM and
  NONEQUILIBRIUM RFIM (Vives,Ortin,Perez Reche et
  al.)
• SAME MEAN FIELD EXPONENTS (ß1/2, ?1/2, d3)
  t3/2 and s1/2 (Liu, KD)
• ABOUT SAME SIMULATION EXPONENTS in 3D and even in
  4D (WITHIN ERRORBARS)
• 6-e expansion of noneq. mapped to all orders in
  e to eq. expansion (KD, Sethna)
• SAME SIMULATION EXPONENTS for GROUND STATE
  DEMAGN. STATE (Zapperi et al.)
• SAME EXP. AND SCALING FUNCTS FOR DEMAGN. CURVE
  SAT. LOOP (Carpenter, KD)
• Middleton's no-passing rule flipped spin cannot
  flip back with increasing field. (Liu, KD)

19
Surprizing Result same Avalanche Exponents And
Scaling Function In Equilibrium and
Nonequilibrium
Ground state Avalanches
Liu, KD 2006
20

• Result Same scaling of Avalanche Surface Area
  Distribution

Liu, KD, 2006
21

• 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 T0 RFIM
belong to the same universality
class !!!???

• Thanks to A. A. Middleton and J.P. Sethna

(Yang Liu, KD, condmat 2006)
22
Summary on magnets and further results
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
23
Universal Gutenberg Richter Scaling behavior
OR Characteristic Earthquake
distribution ?





GR exponent


Magnitude 2/3 Log(total displacement)

24
MODEL
(BEN-ZION RICE 93)
3D FAULT ZONE
Fisher, KD, Ben-Zion, et al. PRL 78 (1997)
RG MEAN FIELD THEORY EXACT IN THE PHYSICAL
DIMENSION
BRITTLE PATCHES
25
Phase Diagram
Mean field exponent b0.75 for Frequency 10-bM
cracklike large events momentarea3/2
small events scale as momentarea
Mehta, KD, Ben-Zion, PRE 2005
26
Data from Susan Bilek, see Mehta, KD, Ben-Zion,
2005
Universal Scaling Functions
27

• Big open questions
• 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 ?!!!!!!!!!!!!!! 

28
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,...
29
?gt0
30
Sweep Rate Regimes of Barkhausen Noise
White,KD PRL2003
Region III (fast)

• Low Freq. PS changed due to spatial overlap

? increasing
Region II (intermediate)

• Power spectra unchanged.
• Only temporal overlap

Fast
Region I (slow)
III
II
I

• power spectra unchanged
• Pulse statistics are affected in general

slow
?s
?
?0
?t
No pulse statistics for ?gt?t
31
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
T- a
D(T)
T
Durin, Zapperi
ABBM,Durin
???
32
Results for
Linear change in exponent
a(W) ao - cW
Agrees with Experiment Plot from Durin and
Zapperi Cond-mat/0404512v1
D(T)
?(W) 1.5 - cW/2
Duration T
33
Plots from Durin and Zapperi review Cond-mat/04045
12v1
Data on amophous FeCoB ribbons (courtesy of G.
Durin)
shapes
34
Shift in frequency of the maximum in the low
frequency regime of the power spectra
Overlap in space -gt parallel dynamics Low freq.
cutoff for this
Ranges from 0.5 in mf (ABBM) to 0.43 in systems
w\o dipolar fields.
35
OUTLOOK
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)
36
Plots from Durin and Zapperi review Cond-mat/04045
12v1
Data on amophous FeSi ribbons (courtesy of G.
Durin)
shapes
37
Temperature gt 0
Scaling Theory and Simulations using the random
field Ising model near the zero temperature far
from eq. critical regime (in progress)
38
Robert White, Alex Travesset, Matthew Delgado,
KD, 2004
Increasing Temp
39
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).
40
Experiments-
Dirty
Phase transition? Disorder is important! Noise
experiments?
Clean
A.Berger et.al. PRL 85 4176(2000)
41
Gd(0001)/W(110)-films
annealing study
data analysis fit of experimental data to
42
Gd(0001)/W(110)-films
Non-equilibrium Phase diagram
43
The Algorithm- (adiabatic. Finite sweep rate)
Raise field until one spin flips
Flip spins due to field increase
If no spins have flipped in the previous time
step. Avalanche has ended
44
Simulation Overview-
Run Time Scaling O(NlogN)
Memory Scaling O(N)
M. Kuntz, et al.. Computing in Science and
Engineering 1, 73 (1999). pdf
45
So?...
46
Avalanche end
time
Quiet space
Resulting Duration distribution at finite driving
rate...
47
Results
For our Model (ztneRFIM)-
Criteria for expected changes is met
experimentally for soft ferromagnets 2
2Berttoti,Durin,Magni, J. Chem. Phys. 75, 5490
(1994)