Self-organized criticality and avalanches in spin...
Transcript of Self-organized criticality and avalanches in spin...
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Self-organized criticality and avalanches in spin glasses
Markus Müller
In collaboration with Pierre Le Doussal (LPT-ENS Paris) Kay Wiese (LPT-ENS Paris)
Weizmann Institute, 10th February, 2010
The Abdus Salam International
Center of Theoretical
Physics
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Outline • Crackling, avalanches, and “shocks” in disordered, non-linear systems; Self-organized criticality
• Avalanches in the magnetizing process (“Barkhausen noise”)
• The criticality of spin glasses at equilibrium – why to expect scale free avalanches
• Magnetization avalanches in the Sherrington- Kirkpatrick spin glass – an analytical study.
• Outlook: electron glasses, finite dimensions…
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Crackling Crackling = Response to a slow driving which occurs in a discrete set of avalanches, spanning a wide range of sizes.
Occurs often but not necessarily only out-of equilibrium.
Examples:
• Earthquakes • Crumpling paper • Vortices and vortex lattices in disordered media etc. • Disordered magnet in a changing external field magnetizes in a series of jumps
Review: Sethna, Dahmen, Myers, Nature 410, 242 (2001).
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Crackling Crackling = Response to a slow driving which occurs in a discrete set of avalanches, spanning a wide range of sizes.
Occurs often but not necessarily only out-of equilibrium.
Examples:
• Earthquakes • Crumpling paper • Vortices and vortex lattices in disordered media etc. • Disordered magnet in a changing external field magnetizes in a series of jumps
But: Not everything crackles! It is intermediate between snapping (e.g., twigs, chalk, weakly disordered ferromagnets, nucleation in clean systems) and popping (e.g., popcorn, strongly disordered ferromagnets)
Review: Sethna, Dahmen, Myers, Nature 410, 242 (2001).
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Crackling Crackling = Response to a slow driving which occurs in a discrete set of avalanches, spanning a wide range of sizes.
Occurs often but not necessarily only out-of equilibrium.
Examples:
• Earthquakes • Crumpling paper • Vortices and vortex lattices in disordered media etc. • Disordered magnet in a changing external field magnetizes in a series of jumps
But: Not everything crackles! It is intermediate between snapping (e.g., twigs, chalk, weakly disordered ferromagnets, nucleation in clean systems) and popping (e.g., popcorn, strongly disordered ferromagnets)
Review: Sethna, Dahmen, Myers, Nature 410, 242 (2001).
Crackling on all scales is generally a signature of a critical state in driven, non-linear systems. It can thus be an interesting diagnostic.
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Examples of crackling I • Gutenberg-Richter law for strength of earthquakes (jumps of driven tectonic plates)
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Examples of crackling II • Depinning of elastic interfaces
Liquid fronts, domain walls, charge density waves, vortex lattices:
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Examples of crackling II • Depinning of elastic interfaces
Liquid fronts, domain walls, charge density waves, vortex lattices:
Statistics of avalanches: - mean field theory - recent first steps and successes with FRG
find non-trivial critical power laws (without scale)
Depinning as a dynamical critical phenomenon in disordered glassy systems Sophisticated theory approach: functional RG “FRG” [D. Fisher, LeDoussal, ...]
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Examples of crackling III • Power laws due to self-organized criticality: Dynamics is attracted to a critical state, without fine-tuning of parameters
Example: sandpile model by Bak, Tang, and Wiesenfeld
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Magnetic systems
• Crackling noise in the hysteresis loop: “Barkhausen noise”
• When does crackling occur in random magnets, and why?
• What happens in frustrated spin glasses (as opposed to just dirty ferromagnets)? ?
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Magnetic systems
• Crackling noise in the hysteresis loop: “Barkhausen noise”
• When does crackling occur in random magnets, and why?
• What happens in frustrated spin glasses (as opposed to just dirty ferromagnets)?
Equilibrium avalanches in the hysteresis reflect criticality of the glass phase! Noise as a diagnostic of a critical glass state!
?
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Avalanches in ferromagnetic films Direct Observation of Barkhausen Avalanche in (ferro) Co Thin Films
Kim, Choe, and Shin (PRL 2003)
P s( ) = Asτ
τ =43
Distribution of magnetization jumps
Cizeau et al.: Theoretical model with dipolar long range interactions (believed to be crucial to get criticality)
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Model ferromagnets Random field Ising model (short range):
• Generically non-critical • Scale free avalanches require fine tuning of disorder and field
H = −J sis j −<ij>∑ hisi −
i∑ hext si
i∑
Δ = hi2
hext ,crit
Dahmen, Sethna Vives, Planes
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Model ferromagnets Random field Ising model (short range):
• Generically non-critical • Scale free avalanches require fine tuning of disorder and field
H = −J sis j −<ij>∑ hisi −
i∑ hext si
i∑
Δ = hi2
hext ,crit
Experiment (disordered ferro) Berger et al. (2000)
Dahmen, Sethna Vives, Planes
(T tunes effective disorder)
Experiment:
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Why is the random Ising model generally non-critical?
Pazmandi, Zarand, Zimanyi (PRL 1999):
Elastic manifolds and ferros with dipolar interactions:
They have strong frustration: - long range interactions with varying signs and/or - strong configurational constraints
Glassy systems with arbitrarily high barriers, metastable states
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Why is the random Ising model generally non-critical?
Pazmandi, Zarand, Zimanyi (PRL 1999):
Elastic manifolds and ferros with dipolar interactions:
They have strong frustration: - long range interactions with varying signs and/or - strong configurational constraints
Glassy systems with arbitrarily high barriers, metastable states
In contrast: RFIM is known not to have a spin glass phase (Krzakala, Ricci-Tersenghi, Zdeborova)
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Why is the random Ising model generally non-critical?
Pazmandi, Zarand, Zimanyi (PRL 1999):
Elastic manifolds and ferros with dipolar interactions:
They have strong frustration: - long range interactions with varying signs and/or - strong configurational constraints
Glassy systems with arbitrarily high barriers, metastable states
In contrast: RFIM is known not to have a spin glass phase (Krzakala, Ricci-Tersenghi, Zdeborova)
Look at spin glasses! (Frustration + disorder = glass and criticality? yes!)
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SK criticality
Canonical spin glass model: Sherrington-Kirkpatrick (SK) model - fully connected
H =12
Jijsis j − hext sii∑
i, j∑ , Jij : random Gaussian Jij
2 = J 2 N
• Extremely intricate mean field version of the Edwards-Anderson model in finite dimensions (but duc = 8)
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SK criticality
Canonical spin glass model: Sherrington-Kirkpatrick (SK) model - fully connected
• Extremely intricate mean field version of the Edwards-Anderson model in finite dimensions (but duc = 8) • Known facts: - There is a thermodynamic transition at Tc to a glass phase: - no global magnetization, but broken Ising symmetry: <si> ≠ 0, - measured by Edwards Anderson order parameter QEA =
1N
si2
i∑
H =12
Jijsis j − hext sii∑
i, j∑ , Jij : random Gaussian Jij
2 = J 2 N
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SK criticality
Canonical spin glass model: Sherrington-Kirkpatrick (SK) model - fully connected
• Extremely intricate mean field version of the Edwards-Anderson model in finite dimensions (but duc = 8) • Known facts: - There is a thermodynamic transition at Tc to a glass phase: - no global magnetization, but broken Ising symmetry: <si> ≠ 0, - measured by Edwards Anderson order parameter - Multitude of metastable states, separated by barriers - Correct equilibrium solution by G. Parisi : Replica symmetry breaking
QEA =1N
si2
i∑
H =12
Jijsis j − hext sii∑
i, j∑ , Jij : random Gaussian Jij
2 = J 2 N
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SK criticality
Canonical spin glass model: Sherrington-Kirkpatrick (SK) model - fully connected
• Extremely intricate mean field version of the Edwards-Anderson model in finite dimensions (but duc = 8) • Known facts: - There is a thermodynamic transition at Tc to a glass phase: - no global magnetization, but broken Ising symmetry: <si> ≠ 0, - measured by Edwards Anderson order parameter - Multitude of metastable states, separated by barriers - Correct equilibrium solution by G. Parisi : Replica symmetry breaking - Glass phase is always critical! (Kondor, DeDominicis)
QEA =1N
si2
i∑
H =12
Jijsis j − hext sii∑
i, j∑ , Jij : random Gaussian Jij
2 = J 2 N
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SK criticality – local fields H =
12
Jijsis j − hext sii∑
i, j∑
Thouless, Anderson and Palmer, (1977); Palmer and Pond (1979) Parisi (1979), Bray, Moore (1980) Sommers and Dupont (1984) Dobrosavljevic, Pastor (1999) Pazmandi, Zarand, Zimanyi (1999) MM, Pankov (2007)
λi ≡ −
∂H∂si
= − Jijs j + hextj≠ i∑
Linear “Coulomb” gap in the distribution of local fields
A first indication of criticality!
Local field on spin i:
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The linear pseudogap in SK Thouless (1977)
The distribution of local fields must vanish at λ=0 at T = 0!
Stability of ground state with respect to flipping of a pair:
• Suppose pseudogap P λ( )∝ λγ
γ ≥ 1 → At least linear pseudogap!
→ Smallest local fields λmin ∝ N −1 1+γ
λ1 λ2
• 2-spin flip cost Ecos t ∝ λ1 + λ2 − N −1 2 ~ N −1 1+γ − N −1 2 >!
0
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The linear pseudogap in SK Thouless (1977)
The distribution of local fields must vanish at λ=0 at T = 0!
Stability of ground state with respect to flipping of a pair:
• Suppose pseudogap P λ( )∝ λγ
γ ≥ 1 → At least linear pseudogap!
→ Smallest local fields λmin ∝ N −1 1+γ
λ1 λ2
• 2-spin flip cost Ecos t ∝ λ1 + λ2 − N −1 2 ~ N −1 1+γ − N −1 2 >!
0
• But: γ = 1! Largest possible density of soft spins!
Distribution is so critical that flipping the first spin by an increase of can trigger a large avalanche!
Δhext = λmin
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“Living on the edge”
Size distribution of avalanches:
• Avalanches are large
• Only cutoff: system size (N1/2)
• Power law: Sign of Self-Organized Criticality
Pazmandi, Zarand, Zimanyi (1999)
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Review: Criticality and RSB
Edwards, Anderson (1974)
SK-model
Replica trick −βF = lnZ = limn→0
Zn −1n
H = Jijsis ji< j∑
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Review: Criticality and RSB
−βF = lnZ = limn→0
Zn −1n
Zn = exp −β H sa⎡⎣ ⎤⎦a=1
n∑( ) = exp β 2Nn
4+β 2N2
siasi
b / Ni∑⎛⎝⎜
⎞⎠⎟
2
1≤a<b≤n∑
⎡
⎣⎢⎢
⎤
⎦⎥⎥
=dQab
2π / β 2Na<b∏∫ exp −NA Q[ ]( )
H = Jijsis ji< j∑
Edwards, Anderson (1974)
A Q[ ] = −nβ 2 4 + β 2 2 Qab2
1≤a<b≤n∑ − log exp β 2QabS
aSb( )Sa{ }∑⎡
⎣⎢⎢
⎤
⎦⎥⎥
Free energy functional
SK-model
Replica trick
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Review: Criticality and RSB
SK-model
Replica trick
Parisi ansatz for the saddle point: Hierarchical replica symmetry breaking
Parisi (1979)
−βF = lnZ = limn→0
Zn −1n
Zn = exp −β H sa⎡⎣ ⎤⎦a=1
n∑( ) = exp β 2Nn
4+β 2N2
siasi
b / Ni∑⎛⎝⎜
⎞⎠⎟
2
1≤a<b≤n∑
⎡
⎣⎢⎢
⎤
⎦⎥⎥
=dQab
2π / β 2Na<b∏∫ exp −NA Q[ ]( )
A Q[ ] = −nβ 2 4 + β 2 2 Qab2
1≤a<b≤n∑ − log exp β 2QabS
aSb( )Sa{ }∑⎡
⎣⎢⎢
⎤
⎦⎥⎥
Edwards, Anderson (1974)
Free energy functional
H = Jijsis ji< j∑
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SK criticality Important features of the solution in the glass phase:
• The free energy functional is only marginally stable! Collection of zero modes of the Hessian
• Critical spin-spin correlations in the whole glass phase! Found numerically also in finite dimensions!
∂2A∂Qab∂Qcd
sis j2~ 1rijα
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SK criticality Important features of the solution in the glass phase:
• The free energy functional is only marginally stable! Collection of zero modes of the Hessian
• Critical spin-spin correlations in the whole glass phase! Found numerically also in finite dimensions!
• Hierarchical structure of phase space and time scales
Free energy landscape ∂2A
∂Qab∂Qcd
sis j2~ 1rijα
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SK criticality Important features of the solution in the glass phase:
• The free energy functional is only marginally stable! Collection of zero modes of the Hessian
• Critical spin-spin correlations in the whole glass phase! Found numerically also in finite dimensions!
• Hierarchical structure of phase space and time scales
• Replica symmetry is broken continuously (at all scales) A continuous function Q(x), n<x<1, parametrizes Qab
Free energy landscape ∂2A
∂Qab∂Qcd
sis j2~ 1rijα
Qab =
x
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SK criticality Important features of the solution in the glass phase:
• The free energy functional is only marginally stable! Collection of zero modes of the Hessian
• Critical spin-spin correlations in the whole glass phase! Found numerically also in finite dimensions!
• Hierarchical structure of phase space and time scales
• Replica symmetry is broken continuously (at all scales) A continuous function Q(x), n<x<1, parametrizes Qab
• Marginality is directly related to the linear pseudogap The pseudogap can be calculated analytically at low T (Pankov)
Free energy landscape ∂2A
∂Qab∂Qcd
sis j2~ 1rijα
Qab =
x
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After so much critical preparation:�
• Understand shocks in spin glasses
• Calculate avalanche distribution analytically!
• Confirm the direct connection of scale free avalanches and thermodynamic criticality!
→ Barkhausen noise as a diagnostic tool for a glass phase?
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Stepwise response and shocks in spin glass models
p-spin models [akin to supercooled liquids] - no continuous, but 1-step RSB - Glassy, but much simpler and non-critical
Yoshino, Rizzo (2008)
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Stepwise response and shocks in spin glass models
p-spin models [akin to supercooled liquids] - no continuous, but 1-step RSB - Glassy, but much simpler and non-critical
m
Fα h( ) = Fα h = 0( ) − hMα
Yoshino, Rizzo (2008)
Free energy of metastable state α: Equilibrium jump/shock when two states cross: Fα hshock( ) = Fβ hshock( )
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Stepwise response and shocks in spin glass models
m
Fα h( ) = Fα h = 0( ) − hMαFree energy of metastable state α: Equilibrium jump/shock when two states cross: Fα hshock( ) = Fβ hshock( )
Mesoscopic effect: Susceptibility has spikes and does not self-average!
p-spin models [akin to supercooled liquids] - no continuous, but 1-step RSB - Glassy, but much simpler and non-critical
Yoshino, Rizzo (2008)
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A simple picture of shocks Balents, Bouchaud, Mezard
u↔ hV u( )↔ F h( )f u( ) = − ′V u( )↔ m h( ) = − ′F h( )
Displacement Effective potential Force
Magnetic field Free energy magnetization
Elastic system Magnetic system
T = 0
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Detecting shocks
2nd cumulant of the magnetization (T = 0)
Non-analytic cusp! • Reflects the probability of shocks. • The cusp is rounded at finite T.
Yoshino, Rizzo (2008)
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Detecting shocks
2nd cumulant of the magnetization (T = 0)
Closely related effects:
• Functional renormalization group for collectively pinned elastic manifolds (e.g. vortex lattices): Cusp in cumulants of effective potential Veff(ucm) at T = 0 and beyond collective pinning scale L> Llarkin
Non-analytic cusp! • Reflects the probability of shocks. • The cusp is rounded at finite T.
Yoshino, Rizzo (2008)
D. Fisher (1986) LeDoussal, Wiese Balents, Bouchaud,
Mézard LeDoussal, MM, Wiese
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Detecting shocks
2nd cumulant of the magnetization (T = 0)
Closely related effects:
• Functional renormalization group for collectively pinned elastic manifolds (e.g. vortex lattices): Cusp in cumulants of effective potential Veff(ucm) at T = 0 and beyond collective pinning scale L> Llarkin • Turbulence: Shocks in the velocity field v(x), rounded only by finite viscosity η (akin to T above)
Non-analytic cusp! • Reflects the probability of shocks. • The cusp is rounded at finite T.
Yoshino, Rizzo (2008)
D. Fisher (1986) LeDoussal, Wiese Balents, Bouchaud,
Mézard LeDoussal, MM, Wiese
Bouchaud, Mézard, Parisi
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Strategy of calculation kth cumulant of magnetization difference
Shock density
Avalanche size cumulants
Calculate
Natural scales:
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Strategy of calculation Calculate
Calculate effective potential of n replicas:
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Strategy of calculation Calculate
Calculate effective potential of n replicas: Easy to extract in the replica limit n→ 0
Qab =
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Strategy of calculation Calculate
Calculate effective potential of n replicas: Easy to extract in the replica limit n→ 0
limit:
Saddle point Qab, sum over replica permutations!
N →∞
Qab =
ha = ha N
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Calculation
• k’th cumulant: k groups of n → 0 replicas with the same ha • integral representation of W[h] and the magnetization cumulants • limit T → 0: extract non-analytic contribution from shocks
Sum over replica permutations π in S(n) [quite a challenge!]
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Calculation Sum over replica permutations π in S(n) [quite a challenge!]
• k’th cumulant: k groups of n → 0 replicas with the same ha • integral representation of W[h] and the magnetization cumulants • limit T → 0: extract non-analytic contribution from shocks
Final result:
• picture of mesoscopic avalanches ~ N1/2 fully confirmed • obtain critical probability distribution of avalanche sizes
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Calculation
• k’th cumulant: k groups of n → 0 replicas with the same ha • integral representation of W[h] and the magnetization cumulants • limit T → 0: extract non-analytic contribution from shocks
Final result:
• picture of mesoscopic avalanches ~ N1/2 fully confirmed • obtain critical probability distribution of avalanche sizes
Avalanche exponent
τ = 1
Sum over replica permutations π in S(n) [quite a challenge!]
![Page 48: Self-organized criticality and avalanches in spin …users.ictp.it/~markusm/Talks/Avalanches_Weizmann.pdfOutline • Crackling, avalanches, and “shocks” in disordered, non-linear](https://reader033.fdocuments.us/reader033/viewer/2022050218/5f639c576838c91eb76fc2bd/html5/thumbnails/48.jpg)
Calculation
• k’th cumulant: k groups of n → 0 replicas with the same ha • integral representation of W[h] and the magnetization cumulants • limit T → 0: extract non-analytic contribution from shocks
Final result:
• picture of mesoscopic avalanches ~ N1/2 fully confirmed • obtain critical probability distribution of avalanche sizes
Avalanche exponent
τ = 1
Obtained from low T solution of the SK model
Sum over replica permutations π in S(n) [quite a challenge!]
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Comparison with numerics Analytical result (shocks in equilibrium)
Avalanches in the hysteresis loop (slowly driven, out-of-equilibrium)
Pazmandi, Zarand, Zimanyi (1999)
Log(δm)
Log(δm)
Log(δm
P(δ
m))
Log(
P(δm
)) Many qualitative features agree between
analytics (equilibrium) and numerics (out-of equilibrium)
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Remarks Analytical result (shocks in equilibrium)
Important remarks
• The power law arises because of the criticality of the glass
• It receives contributions from all scales and distances within the hierarchical organization of states
• Nearly no dependence on the external field, except in the cutoff scale:
The spin glass is critical even in finite field.
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Conclusion
Spin glass criticality (in the SK model) is prominently reflected in scale free response to a slow magnetic
field change.
There is a deep connection between various manifestations of this criticality:
Soft gap – avalanches – spin-spin correlations – abundant collective low energy excitations
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Outlook • Finite d spin glasses:
- Is criticality revealed in avalanches? - Experimental probe for criticality via Barkhausen noise? - Beyond mean field (SK): Functional RG for spin glasses?
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Outlook • Finite d spin glasses:
- Is criticality revealed in avalanches? - Experimental probe for criticality via Barkhausen noise? - Beyond mean field (SK): Functional RG for spin glasses?
• Coulomb glasses: (Localized electrons with Coulomb interactions and disorder)
Close analogies with SK model:
- Critical soft gap (Efros-Shklovskii) - Infinite avalanches (~ L) at T = 0 - Mean field: full replica symmetry breaking and critical correlations predicted - Distribution of avalanches?
![Page 54: Self-organized criticality and avalanches in spin …users.ictp.it/~markusm/Talks/Avalanches_Weizmann.pdfOutline • Crackling, avalanches, and “shocks” in disordered, non-linear](https://reader033.fdocuments.us/reader033/viewer/2022050218/5f639c576838c91eb76fc2bd/html5/thumbnails/54.jpg)
Outlook • Finite d spin glasses:
- Is criticality revealed in avalanches? - Experimental probe for criticality via Barkhausen noise? - Beyond mean field (SK): Functional RG for spin glasses?
• Coulomb glasses: (Localized electrons with Coulomb interactions and disorder)
Close analogies with SK model:
- Critical soft gap (Efros-Shklovskii) - Infinite avalanches (~ L) at T = 0 - Mean field: full replica symmetry breaking and critical correlations predicted - Distribution of avalanches?
• Avalanches in other complex systems (computer science, optimization, economy, etc)