1. Lectures on Spin Glasses Daniel Stein Department of Physics and Courant Institute New York University Workshop on Disordered Systems, Random Spatial Processes, and Some Applications Institut Henri Poincaré Paris, January 5 – March 30, 2015 Overview: Lectures 1 and 2: Introduction and background: experiments and theory Lectures 3 and 4: Infinite-range and short-range spin glasses Lectures 5 and 6: Applications to computer science, neural networks, etc.

2. Main Goals What is a spin glass? What is a spin glass? Why are spin glasses of interest to: Why are spin glasses of interest to: -- Physicists (condensed matter), Mathematicians (probability theory), and Mathematical Physicists (statistical mechanics) -- “Complexity scientists’’ -- “Complexity scientists’’ 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? Provide a general sense of mathematical and physical research into spin glasses, with an emphasis on the following questions:

3. 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

4. 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.

5. 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)

6. 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

7. 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):

8. 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 )

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

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

11. Early 1970 ’ s: Magnetic effects seen at greater impurity concentrations Cannella, Mydosh, and Budnick, J. Appl. Phys. 42, 1689 (1971) Magnetic susceptibility χ = lim Δh->0 (ΔM/Δh)

12. Ground States Quenched disorder Crystal Glass Ferromagnet Spin Glass

13. 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).

14. 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)

15. The Edwards-Anderson (EA) Ising Model Site in Zd Nearest neighbor spins only The fields and couplings are i.i.d. random variables: 1975: Theory Rears Its Head

16. Frustration!

17. ``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).

18. EA conjecture: Spin glasses (and glasses, …) are characterized by broken symmetry in time but not in space. Broken symmetry in the spin glass But remember: this was a conjecture!

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

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

21. 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.

22. Spin glasses represent a gap in our understanding of condensed matter. Bishop George Berkeley 1685-1753 Dr. Samuel Johnson Dr. Samuel Johnson1709-1784 Question: Why are solids rigid? Primary question: Why should we be interested? “I refute it thus!”

23. “Ordinary” glasses: no change in symmetry, no phase transition. And for spin glasses, the situation is even more problematic.

24. 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.’’