Local field distributions in spin glasses

1
Local field distributions in spin glasses

• David Sherrington
• Rudolf Peierls Centre for Theoretical Physics
• University of Oxford
• with
• Stefan Boettcher (Emory U)
• Helmut Katzgraber (ETHZ)

2
Relationship between local field distributions
and spin glass character?

• Issue
• Mean-field vs. finite-range/finite-dimension?
• Replica symmetry vs. RSB?
• Equilibrium vs. metastable?
• Conclusion
• All P(h) remarkably similar
• But characteristic variations near h0, T0
• Scaling laws

3
Spin glasses

• Random bond Gaussian
• Ising
• Infinite-range/SK
• n.n. EA
• 1-d 1/r?
• Effectively interpolating long to short-range as
  function of ?
• T0 equilibrium random quench metastable

4
Local field distribution

• Concentrate mainly on T 0

5
Motivation

• SK (T0)
• But Parisi theory requires Full RSB

Simulations KS 78
Finite K
What about systems without RSB or even s.g.?
e.g. finite-d EA?
Coulomb glass pseudo-gap ? finite-T phase
transition?
Oppermann S 05
Gap
(Oppermann, Schmidt S)
6
SK
T0 with normalization Tc1
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EA(T0, TMF1)
Similar for other d
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Kotliar-Anderson-Stein
1-d Ising chain, with long-range quenched-random
interactions
Interpolates between SK (? 0) and short-range
SG(RSB) to SG(no RSB?) to no Tgt0 order
TMF1
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KAS
Note Most plots T0.05, But difference believed
small
c.f. SK Thomsen et al 86
Note P(h0) zero only for d1, ? but
d1special unfrustrated
Similar in T too
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P(h,T) KAS
c.f. SK
Again, rather similar
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PKAS(h0)
Finite-T shift
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Non-equilibrated quench
SK
EA
Note Parisi observed N-1/2 for SK (95). Also
Eastham et al.(06), Horner (07)
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So

• P(h,T) very similar for all Gaussian-random
  frustrated Ising systems
• Independently of phase transition, RSB
• Except near h0, T0
• P(h0,T0) scales as z -1/2

14
Questions

• What about Coulomb Glass?
• Experimental evidence for pseudo-gap
• Predicted theoretically by Efros-Shklovskii
• TAP-like theory
• d-dependent form ?(?) (?- ?)-(d-1)
• Finite-temperature phase transition?
• Analogue of spin glass in a field
• Other manifestations of glassiness

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Or an amorphous antiferromagnet?
Khanna S 1980
Dense random-packed hard spheres Spins at
centres First minimum of r.d.f. at a
Similar shape again But no phase transition?
c.f. random antiferromagnet No finite-T order
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Aside SK K(N)

• Aspelmeier et.al.07
• Finite-N SK ? Can truncate RSB at KN1/6
• Ground state energy deviations different for
  finite N, finite K
• Finite-N simn ? e0-0.7632 0.70N-2/3
  (Boettcher 05)
• Finite-K RSB ? e0-0.7632 - 0.0467K-4 (Oppermann
  et al. 07)
• i.e. deviations of opposite sign
• ? need terms beyond mean field
• self-consistent self-energy corrections to
  propagators
• T0 p(h) deviations also different
• Finite-N simn ? p(h) raised at low h
• p(h0) N-1/2 (Boettcher et al. 08)
• Finite-K RSB ?p(h) hole at low h
• width K-5/3 (Oppermann et al. 07)
• Self-consistent theory needs to correlate these
  too

Note
Now
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Conclusion

• Local field distributions are not the sharpest
  tool in the box
• But not the bluntest either.
• Their study has provided some surprises and some
  cautions
• Still challenges for solution of finite-N SK
  field theory, as well as finite-d EA
• Topics which have interested both Giorgio and
  Edouard
• And questions in extensions to other random
  frustrated systems

18
Happy birthday

Giorgio
Édouard
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Brézin-Parisi matrix











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