1. Spin Glasses and Complexity: Lecture 2 Brief review of yesterday’s lecture Brief review of yesterday’s lecture Spin glass energy and broken symmetry Spin glass energy and broken symmetry Applications 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 Geometry of interactions and the infinite-range model

2. Homogeneous systems possess symmetries that greatly simplify mathematical analysis and physical understanding--- Bloch’s theorem, broken symmetry, order parameter, Goldstone modes, … 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 Examples: crystals, ferromagnets, superconductors and superfluids, liquid crystals, ferroelectrics, …

3. Spin Glasses – a prototype disordered system? Dilute magnetic alloy, e.g., CuMn Frustration

4. Ground States Crystal Glass Ferromagnet Spin Glass

5. The Edwards-Anderson (EA) Ising Model Site in Zd Nearest neighbor spins only Coupling realization The couplings are i.i.d. random variables: S.F. Edwards and P.W. Anderson, J. Phys. F 5, 965 (1975).

6. EA ’75: A low-temperature spin glass phase should be described by presence of temporal order (freezing) along with absence of spatial disorder. Broken symmetry in the spin glass But there are some surprises in store …

7. The most fundamental questions remain unanswered: Is there a phase transition? Is there a phase transition? What is the nature of low-temperature phase (broken symmetry, order parameter)? 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…)? 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 … V. Cannella and J.A. Mydosh, Phys. Rev. B 6, 4220 (1972). L.E. Wenger and P.H. Keesom, Phys. Rev. B 13, 4053 (1976)

8. Quenched disorder Quenched disorder Frustration Frustration Combinatorially huge possible number of configurations, or states, or outcomes Combinatorially huge possible number of configurations, or states, or outcomes Many statistically equivalent `ground’ states (more or less equally good optimal solutions)? Many statistically equivalent `ground’ states (more or less equally good optimal solutions)? Slow equilibration Slow equilibration Memory, aging … Memory, aging … Applications to combinatorial optimization (graph theory) problems, neural networks, biological evolution, protein dynamics and folding, … Example – the traveling salesman problem N=5  12 tours N=5  12 tours N=10  181,440 tours N=10  181,440 tours N=50  Forget it. N=50  Forget it. (NP-complete)

9. Simulated annealing S. Kirkpatrick, C.D. Gelatt, Jr., and M.P. Vecchi, Science 220, 671 (1983) Cost function (plays role of energy function) Cost function (plays role of energy function) Quenched disorder Quenched disorder Frustration Frustration Combinatorially huge possible number of configurations, or states, or outcomes Combinatorially huge possible number of configurations, or states, or outcomes Many statistically equivalent `ground’ states (more or less equally good optimal solutions) 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 Add a ``temperature’’, and treat problem like a statistical mechanical problem Metropolis algorithm M. Mézard, G. Parisi, and R. Zecchina, Science 297, 812 (2002)

10. Construct a ``cooling schedule’’ Construct a ``cooling schedule’’

12. Neural circuit computation J.J. Hopfield and D.W. Tank, Science 233, 625 (1986) Circuit element (``neuron’’) can be in one of two states (on/off: 0/1, spin up/spin down) Circuit element (``neuron’’) can be in one of two states (on/off: 0/1, spin up/spin down) W.S. McCullough and W.H. Pitts, Bull. Math. Biophys. 5, 115 (1943) Dynamics of ``neurons’’ given by Dynamics of ``neurons’’ given by where is the potential of neuron i.

13. Dynamics corresponds to an energy function Dynamics corresponds to an energy function

14. D.L. Stein, ed., Spin Glasses and Biology (World Scientific, Singapore, 1992) Protein Conformational Dynamics Myoglobin

15. Fluctuations important for biological processes (e.g., ligand diffusion) Fluctuations important for biological processes (e.g., ligand diffusion) Recombination experiments imply many conformational substates Recombination experiments imply many conformational substates A. Osterman et al., Nature 404, 205 (2000)

16. D.L. Stein, Proc. Natl. Acad. Sci. USA 82, 3670 (1985) Spin Glass Model of Protein Conformational Substates

17. Protein Folding Levinthal paradox Levinthal paradox ``Principle of minimal frustration’’ ``Principle of minimal frustration’’ J.D. Bryngelson and P.G. Wolynes, Proc. Natl. Acad. Sci. USA 84, 7524 (1987)

18. Folding landscapes as a ``rough funnel’’ C.L. Brooks III, J.N. Onuchic, and D.J. Wales, Science 293, 612 (2001) Used to develop algorithms for structure prediction (J. Pillardy et al., PNAS 98, 2329 2001); designing ``knowledge-based potentials for fold recognition; etc.

19. Back to spin glasses proper … By now, it’s (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!

20. The Geometry of ``Information Propagation’’

21. The Sherrington-Kirkpatrick (SK) Model Phase transition with T c =1. What is the thermodynamic structure of the low-temperature phase? ``Infinite-range’’ model – no geometry left! D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1975). ``Mean-field’’ model; infinite-dimensional model. Broken replica symmetry --- one of the biggest surprises of all. Stay tuned …