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 , DMS , and DMS Quenched Disorder, Spin Glasses, and Complexity

Our guide to complexity through disorder --- the spin glass. What is a spin glass? What is a spin glass? Why are they interesting to: Why are they interesting to: -- Physics (condensed matter, statistical mechanics) -- Complexity Canonical model of disorder Canonical model of disorder New computational techniques New computational techniques Application to other problems Application to other problems Generic aspects? Generic aspects?

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

Spin glass energy and broken symmetry Spin glass energy and broken symmetry Applications 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 Geometry of interactions and the infinite-range model Lecture 2

Lecture 3 Parisi solution of SK model Parisi solution of SK model Replica symmetry breaking (RSB) Replica symmetry breaking (RSB) - Overlaps - Non-self-averaging - Ultrametricity What is the structure of short-range spin glasses? What is the structure of short-range spin glasses? Are spin glasses complex systems? Are spin glasses complex systems?

(Approximate) Timeline Ca Ordered Systems (crystals, ferromagnets, superconductors, superfluids, …) Bloch’s theorem, broken symmetry, Goldstone modes, single order parameter, … Ca Disordered systems (glasses, spin glasses, polymers, …) Localization, frustration, broken replica symmetry, infinitely many order parameters, metastates … Ca Complex systems (Condensed matter physics, computer science, biology, economics, archaeology, …)

Phases of Matter and Phase Transitions Phase diagram of water 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?

Quantifies ``amount’’ and ``type’’ of order in a system --- undergoes discontinuous (in it or its derivatives) change at a phase transition Order parameters  Discontinuous jump – latent heat (fixed pressure)

Glasses The ``Berkeley effect’’

Magnetic Order In magnetic materials, each atom has a tiny magnetic moment m x 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: (at high temperature) This is called the paramagnetic state.

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. What happens when you lower the temperature? In certain materials, there is a sharp phase transition to a magnetically ordered state. Single spin orientation at different times – averages to zero in short time: xxxx

What is the nature of the ordering? In some materials (e.g., Fe, Mn), nearby spins ``like’’ to align; these are called ferromagnets. 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. 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, …) And there are many other types as well (ferrimagnets, canted ferromagnets, helical ferromagnets, …) Can capture both behaviors with a simple model energy function (Hamiltonian): Can capture both behaviors with a simple model energy function (Hamiltonian):

Magnetic Phase Transitions High temperature Low temperature Phase diagram for ferromagnet

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

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

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

The Solid State Physics of Spin Glasses Dilute magnetic alloy: localized spins at magnetic impurity sites D.L. Stein, Sci. Am. 261, 52 (1989). 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).

Frustration!

Ground States Quenched disorder Crystal Glass Ferromagnet Spin Glass

1) For these systems, disorder cannot be treated as a perturbative effect Two ``meta-principles’’ 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.’’

``Rugged’’ Energy Landscape Many metastable states Many metastable states Many thermodynamic states? Many thermodynamic states? Slow dynamics --- can get ``stuck’’ in a local energy minimum Slow dynamics --- can get ``stuck’’ in a local energy minimum Disorder and frustration … Disorder and frustration … R.G. Palmer, Adv. Phys. 31, 669 (1982). 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). C.M. Newman and D.L. Stein, Phys. Rev. E 60, 5244 (1999).

Is there a phase transition to a ``spin glass phase’’? tyesno L.E. Wenger and P.H. Keesom, Phys. Rev. B 13, 4953 (1976). Cannella, Mydosh, and Budnick, J. Appl. Phys. 42, 1689 (1971)

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

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

So far … lots of nice stuff Disorder Disorder Frustration Frustration Complicated state space --- rugged energy landscape Complicated state space --- rugged energy landscape Anomalous dynamical behavior Anomalous dynamical behavior -- Memory effects -- History dependence and irreversibility Connections to other problems --- new insights and techniques Connections to other problems --- new insights and techniques Well-defined mathematical structure Well-defined mathematical structure … which we’ll start with tomorrow. … which we’ll start with tomorrow.