Spin Glasses and Complexity: Lecture 2

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Spin Glasses and Complexity Lecture 2

• Brief review of yesterdays lecture

• Spin glass energy and broken symmetry

• Applications

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

• Geometry of interactions and the infinite-range
  model

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Homogeneous systems possess symmetries that
greatly simplify mathematical analysis and
physical understanding--- Blochs theorem, broken
symmetry, order parameter, Goldstone modes,
Examples crystals, ferromagnets,
superconductors and superfluids, liquid crystals,
ferroelectrics,
But for glasses, spin glasses, and other systems
with quenched disorder many new ideas and
concepts have been proposed, but so far no
universal ones
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Spin Glasses a prototype disordered system?
Dilute magnetic alloy, e.g., CuMn
Frustration
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Ground States
Spin Glass
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The Edwards-Anderson (EA) Ising Model
Site in Zd
Coupling realization
The couplings are i.i.d. random variables
Nearest neighbor spins only
Site in Zd
S.F. Edwards and P.W. Anderson, J. Phys. F 5, 965
(1975).
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Broken symmetry in the spin glass
EA 75 A low-temperature spin glass phase should
be described by presence of temporal order
(freezing) along with absence of spatial disorder.
But there are some surprises in store
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• The most fundamental questions remain unanswered
• Is there a phase transition?
• What is the nature of low-temperature phase
  (broken symmetry, order parameter)?
• How does one account for the anomalous dynamical
  behavior (slow relaxation, memory, aging)?

Important not only for physics, but may lend
important concepts to other areas
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• Quenched disorder
• Frustration
• Combinatorially huge possible number of
  configurations, or states, or outcomes
• Many statistically equivalent ground states
  (more or less equally good optimal solutions)?
• Slow equilibration
• Memory, aging

(NP-complete)
Applications to combinatorial optimization (graph
theory) problems, neural networks, biological
evolution, protein dynamics and folding,
Example the traveling salesman problem

• N5 ? 12 tours
• N10 ? 181,440 tours
• N50 ? Forget it.

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Simulated annealing

• Cost function (plays role of energy function)

• Quenched disorder
• Frustration
• Combinatorially huge possible number of
  configurations, or states, or outcomes
• Many statistically equivalent ground states
  (more or less equally good optimal solutions)

- TSP length of a tour
- Placement in computer design
- k-SAT
Many of these resemble spin glass Hamiltonian!

• Add a temperature, and treat problem like a
  statistical mechanical problem

Metropolis algorithm
S. Kirkpatrick, C.D. Gelatt, Jr., and M.P.
Vecchi, Science 220, 671 (1983)
M. Mézard, G. Parisi, and R. Zecchina, Science
297, 812 (2002)
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• Construct a cooling schedule

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Neural circuit computation

• Circuit element (neuron) can be in one of two
  states (on/off 0/1, spin up/spin down)

• Dynamics of neurons given by

J.J. Hopfield and D.W. Tank, Science 233, 625
(1986)
W.S. McCullough and W.H. Pitts, Bull. Math.
Biophys. 5, 115 (1943)
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• Dynamics corresponds to an energy function

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Protein Conformational Dynamics
Myoglobin
D.L. Stein, ed., Spin Glasses and Biology (World
Scientific, Singapore, 1992)
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• Fluctuations important for biological processes
  (e.g., ligand diffusion)

• Recombination experiments imply many
  conformational substates

A. Osterman et al., Nature 404, 205 (2000)
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Spin Glass Model of Protein Conformational
Substates
D.L. Stein, Proc. Natl. Acad. Sci. USA 82, 3670
(1985)
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Protein Folding

• Levinthal paradox

• Principle of minimal frustration

J.D. Bryngelson and P.G. Wolynes, Proc. Natl.
Acad. Sci. USA 84, 7524 (1987)
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Folding landscapes as a rough funnel
Used to develop algorithms for structure
prediction (J. Pillardy et al., PNAS 98, 2329
2001) designing knowledge-based potentials for
fold recognition etc.
C.L. Brooks III, J.N. Onuchic, and D.J. Wales,
Science 293, 612 (2001)
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Back to spin glasses proper
By now, its (hopefully) clear that understanding
the behavior of these systems is important not
only for condensed matter physics and statistical
mechanics, but for many other fields as well
so we will now turn to examine what we know
about them.
Unfortunately, understanding their nature has
been very difficult --- theoretically,
experimentally, and numerically!
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The Geometry of Information Propagation
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The Sherrington-Kirkpatrick (SK) Model
Infinite-range model no geometry left!
Mean-field model infinite-dimensional model.
Phase transition with Tc1. What is the
thermodynamic structure of the low-temperature
phase?
Broken replica symmetry --- one of the biggest
surprises of all.
Stay tuned
D. Sherrington and S. Kirkpatrick, Phys. Rev.
Lett. 35, 1792 (1975).
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