Daniel Stein Departments of Physics and Mathematics New York University

1. 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

2. 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?

3. Overview Lecture 1 -- Ordered and disordered condensed matter systems -- Phase transitions, ordering, and broken symmetry -- Magnetic systems -- Spin glasses and their properties

4. 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

5. 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?

6. (Approximate) Timeline Ca. 1930+ Ordered Systems (crystals, ferromagnets, superconductors, superfluids, …) Bloch’s 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-universe.html

7. Specific heat C = (amount of heat needed to add or subtract to change the temperature by an amount) What is a central bridge between traditional physics and complexity studies? Phases of Matter and Phase Transitions Phase diagram of water

8. 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

9. Glasses The ``Berkeley effect’’

10. (at high temperature) 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.

11. Single spin orientation at different times – averages to zero in short time: 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 M=0 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.

12. 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):

13. Magnetic Phase Transitions Phase diagram for ferromagnet High temperature Low temperature

14. Broken symmetry J.P. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity (Oxford U. Press, 2007)

15. A New State of Matter? Prehistory: The Kondo Problem (1950’s – 1970’s) Generated interest in dilute magnetic alloys (CuMn, AuFe, …) Addition of ln(1/T) term to the resistivity

16. Early 1970’s: Magnetic effects seen at greater impurity concentrations Cannella, Mydosh, and Budnick, J. Appl. Phys.42, 1689 (1971)

17. 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).

18. Frustration!

19. Crystal Glass Ferromagnet Ground States Spin Glass Quenched disorder

20. 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.’’

21. ``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).

22. yes t no 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)

23. Aging and Memory Effects K. Binder and A.P. Young, Rev. Mod. Phys. 58, 801 (1986).

24. Aging P. Svedlinh et al., Phys. Rev. B 35, 268 (1987)

25. 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 … which we’ll start with tomorrow. Connections to other problems --- new insights and techniques