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

Avalanches in Equilibrium and NonequilibriumIn the same Universality Class ???

UIUCUIUC: John Carpenter (now Sandia), Amit Mehta, Robert White (now San Diego), Matthew Delgado, Yang Liu, Andrew Missel, Geoffrey Poore, Alex Travesset (now Assis. Prof. Iowa/Ames Lab)REUREU’’ss: Ma’ayan Bresler (Princeton), Sharon Loverde(now Northwestern), Riva Vanderveld (now Cornell)

Jim Sethna, M. Kuntz, O. Perkovic (Cornell University), A. Middleton -- theoryYehuda Ben Zion (USC, Earth and PlanetarySciences), (D.S. Fisher (Harvard)) - earthquakesA. Berger, O. Hellwig (IBM/Hitachi) -- exp.Gianfranco Durin (Turino, 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 D. Nelson (Harvard) and N. Shnerb (Israel) - spreading Bacteria Colonies

Magnets and Magnets and BarkhausenBarkhausen NoiseNoise-- or or MartensitesMartensites and acc. emissionand acc. emission

Volt

Sample

H(t)

Crackling noise

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

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

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

Liquid He invading Liquid He invading NucleporeNuclepore (Hallock, Lilly,Wooters...)

Rupture of fibrous MaterialsRupture of fibrous Materials

EarthquakesEarthquakesBroadBroad Range ofRange of

sizes & sizes & durations durations

s-τ

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

ρ(hi)

Jumps = Avalanches = Jumps = Avalanches =

BarkhausenBarkhausen noisenoise

( )∑∑ +−−=i

iinn

jisystem 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

τ+σβδ=2.03±0.03, 1/σ=4.2 ±0.3, ...

critical

LOG (AVALANCHE SIZE)

Log(#)

Dirty critical

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

SelfSelf--similarity at critical disorder similarity at critical disorder RRcc=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)

Why powerlaw at (Hc, Rc ) ? Renormalization Group (Wilson, Fisher, Kadanoff) history dependent, deterministic time dependent description: H=H(t)= Ωt with Ω→0

Average and expand around mean field theory

Random fields→ probability distribution for paths

Martin Siggia Rose / Bausch Janssen Wagner Formalismus

soft spins: Renormalization group: (6-ε) expansion

Langevin dynamics:

Generating Functional:

Path integral:

Product over all times, average over the random fields:

Transformation to local fields:

Random field averaged probability distribution for path η : P( η )

Narayan, Middleton PRB 1993, Narayan, Fisher, PRB 1992, Zippelius PRB 1984, Sompolinsky, Zippelius PRB 1992, De Dominicis, PRB 1978, Bausch, Janssen, Wagner, ZPB 1976, Martin Siggia Rose, PRA 1973

a

Effective Action:

Momentum Shell Renormalization:

1. Integrate over momentum shell2.rescale

New Fixed Point (to O(ε))

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 ?( )∑∑ +−−=

iii

nnjisystem ShHSSJΗ

..

(t)H+

2 Dynamics

+ long range forces

Single 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 (Mehta, BenZion, KD 2005), maybe superconductors… (Carlson, KD, Fradkin, Kivelson, PRL 2005), …others?

SimulationSimulation66--εε expansion expansion (Renormalization Group)

1/ν = 2 - ε/3 - 0.1ε2 + 0.1ε3-0.3ε4 + ε5 + O(ε6 )

(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. ((BarkhausenBarkhausen Noise):Noise):

RESULTSRESULTS

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

SURPRIZINGLY SIMILAR: EQUILIBRIUM and NONEQUILIBRIUM RFIM: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 wrong 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

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

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

Same disorder induced critical point as saturation loop:

Avalanche Size Distribution ΔM vs R

R=3.3 R=2.5 203803

•Conclusions

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

suggest that the equilibrium and non-equilibriumtransition 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)