Daniel Stein

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Quenched Disorder, Spin Glasses, and Complexity
Daniel Stein Departments of Physics and
Mathematics New York University
Complex Systems Summer School Santa Fe Institute
June, 2008
Partially supported by US National Science
Foundation Grants DMS-01-02541, DMS-01-02587, and
DMS-06-04869
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Our guide to complexity through disorder --- the
spin glass.

• What is a spin glass?

• Why are they interesting to

-- Physics (condensed matter, statistical
mechanics)
-- Complexity

• Canonical model of disorder

• New computational techniques

• Application to other problems

• Generic aspects?

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Overview

• Lecture 1

• -- Ordered and disordered condensed matter
  systems

-- Phase transitions, ordering, and broken
symmetry
-- Magnetic systems
-- Spin glasses and their properties
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Lecture 2

• Spin glass energy and broken symmetry

• Applications

- Combinatorial optimization and traveling
salesman problem
- Simulated annealing
- Hopfield-Tank neural network computation
- Protein conformational dynamics and folding

• Geometry of interactions and the infinite-range
  model

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Lecture 3

• Parisi solution of SK model

• Replica symmetry breaking (RSB)

- Overlaps
- Non-self-averaging
- Ultrametricity

• What is the structure of short-range spin glasses?

• Are spin glasses complex systems?

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(Approximate) Timeline
Ca. 1930 Ordered Systems (crystals,
ferromagnets, superconductors, superfluids, )
Blochs theorem, broken symmetry, Goldstone
modes, single order parameter,
Ca. 1958 Disordered systems (glasses, spin
glasses, polymers, )
Localization, frustration, broken replica
symmetry, infinitely many order parameters,
metastates
Ca. 1980 Complex systems
(Condensed matter physics, computer science,
biology, economics, archaeology, )
http//sprott.physics.wisc.edu/Pickover/pc/brain-u
niverse.html
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Phases of Matter and Phase Transitions
What is a central bridge between traditional
physics and complexity studies?
Phase diagram of water
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Order parameters

• Quantifies amount and type of order in a
  system --- undergoes discontinuous (in it or its
  derivatives) change at a phase transition

(fixed pressure)
? Discontinuous jump latent heat
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Glasses
The Berkeley effect
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Magnetic Order
In magnetic materials, each atom has a tiny
magnetic moment mx arising from the quantum
mechanical spins of electrons in incompletely
filled shells.
These spins couple to magnetic fields, which
can be external (from an applied magnetic field
h), or internal (from the field arising from
other spins.
At high temperature (and in zero external field),
thermal agitation disorders the spins, leading to
a net zero field at each site
This is called the paramagnetic state.
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Magnetization is the spatial average of all of
the local (i.e., atomic) magnetic moments,
and describes the overall magnetic state of the
sample as such, it serves as a magnetic order
parameter.
So M0 in the paramagnet in the absence of an
external magnetic field.
?x
What happens when you lower the temperature?
In certain materials, there is a sharp phase
transition to a magnetically ordered state.
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What is the nature of the ordering?

• In some materials (e.g., Fe, Mn), nearby spins
  like to align these are called ferromagnets.

• In others (e.g., Cr, many metal oxides), they
  like to anti-align these are called
  antiferromagnets.

• And there are many other types as well
  (ferrimagnets, canted ferromagnets, helical
  ferromagnets, )

• Can capture both behaviors with a simple model
  energy function (Hamiltonian)

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Magnetic Phase Transitions
Phase diagram for ferromagnet
High temperature
Low temperature
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Broken symmetry
J.P. Sethna, Statistical Mechanics Entropy,
Order Parameters, and Complexity (Oxford U.
Press, 2007)
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A New State of Matter?
Prehistory The Kondo Problem (1950s 1970s)
Generated interest in dilute magnetic alloys
(CuMn, AuFe, )
Addition of ln(1/T) term to the resistivity
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Early 1970s Magnetic effects seen at greater
impurity concentrations
Cannella, Mydosh, and Budnick, J. Appl. Phys. 42,
1689 (1971)
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The Solid State Physics of Spin Glasses
Dilute magnetic alloy localized spins at
magnetic impurity sites
M.A. Ruderman and C. Kittel, Phys. Rev. 96, 99
(1954) T. Kasuya, Prog. Theor. Phys. 16, 45
(1956) K. Yosida, Phys. Rev. 106, 893 (1957).
D.L. Stein, Sci. Am. 261, 52 (1989).
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Frustration!
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Ground States
Spin Glass
Quenched disorder
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Two meta-principles
1) For these systems, disorder cannot be
treated as a perturbative effect
2) P.W. Anderson, Rev. Mod. Phys. 50, 191 (1978)
there is an important fundamental truth about
random systems we must always keep in mind no
real atom is an average atom, nor is an
experiment ever done on an ensemble of samples.
What we really need to know is the probability
distribution , not (the) average this is the
important, and deeply new, step taken here the
willingness to deal with distributions, not
averages. Most of the recent progress in
fundamental physics or amorphous materials
involves this same kind of step, which implies
that a random system is to be treated not as just
a dirty regular one, but in a fundamentally
different way.
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Rugged Energy Landscape

• Disorder and frustration

• Many metastable states

M. Goldstein, J. Chem. Phys. 51, 3728 (1969)
S.A. Kauffman, The Origins of Order (Oxford,
1993) W. Hordijk and P.F. Stadler, J. Complex
Systems 1, 39 (1998) D.L. Stein and C.M. Newman,
Phys. Rev. E 51, 5228 (1995).

• Many thermodynamic states?

C.M. Newman and D.L. Stein, Phys. Rev. E 60, 5244
(1999).

• Slow dynamics --- can get stuck in a local
  energy minimum

R.G. Palmer, Adv. Phys. 31, 669 (1982).
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Is there a phase transition to a spin glass
phase?
L.E. Wenger and P.H. Keesom, Phys. Rev. B 13,
4953 (1976).
Cannella, Mydosh, and Budnick, J. Appl. Phys. 42,
1689 (1971)
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Aging and Memory Effects
K. Binder and A.P. Young, Rev. Mod. Phys. 58, 801
(1986).
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Aging
P. Svedlinh et al., Phys. Rev. B 35, 268 (1987)
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So far lots of nice stuff

• Disorder

• Frustration

• Complicated state space --- rugged energy
  landscape

• Anomalous dynamical behavior

-- Memory effects
-- History dependence and irreversibility

• Well-defined mathematical structure

• Connections to other problems --- new insights
  and techniques

• which well start with tomorrow.