1. Spin Glasses: Lectures 2 and 3 Parisi solution of SK model: Replica symmetry breaking (RSB) Parisi solution of SK model: Replica symmetry breaking (RSB) - Overlaps - Overlaps - Non-self-averaging - Non-self-averaging - Ultrametricity - Ultrametricity Review of first lecture Review of first lecture The Sherrington-Kirkpatrick (SK) infinite-range spin glass model The Sherrington-Kirkpatrick (SK) infinite-range spin glass model Some notions from statistical mechanics Some notions from statistical mechanics - Finite-volume Gibbs distributions - Thermodynamic states: pure, mixed, and ground states Open questions Open questions Summary of RSB solution of SK model Summary of RSB solution of SK model

2. Ground States Quenched disorder

3. The Edwards-Anderson (EA) Ising Model Site in Zd Nearest neighbor spins only The fields and couplings are i.i.d. random variables: Frustration

4. 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!

5. The Sherrington-Kirkpatrick (SK) model The fields and couplings are i.i.d. random variables: with

6. Question: If (as is widely believed) there is a phase transition with broken spin flip symmetry (in zero field), what is the nature of the broken symmetry in the low temperature phase? ``…the Gibbs equilibrium measure decomposes into a mixture of many pure states. This phenomenon was first studied in detail in the mean field theory of spin glasses, where it received the name of replica symmetry breaking. But it can be defined and easily extended to other systems, by considering an order parameter function, the overlap distribution function. This function measures the probability that two configurations of the system, picked up independently with the Gibbs measure, lie at a given distance from each other. Replica symmetry breaking is made manifest when this function is nontrivial.’’ S. Franz, M. Mézard, G. Parisi, and L. Peliti, Phys. Rev. Lett. 81, 1758 (1998). What does this mean? One guide: the infinite-range Sherrington-Kirkpatrick (SK) model displays an exotic new type of broken symmetry, known as replica symmetry breaking (RSB). To begin, RSB asserts the existence of many thermodynamic pure states unrelated by any symmetry transformation. Each of these looks ``random’’ … so how does one describe ordering in such a situation? Look at relations between states.

7. Thermodynamic States A thermodynamic state is a probability measure on infinite-volume spin configurations A thermodynamic state is a probability measure on infinite-volume spin configurations We’ll denote a state by the index α, β, γ, … We’ll denote a state by the index α, β, γ, … A given state α gives you the probability that at any moment spin 1 is up, spin 18 is down, spin 486 is down, … A given state α gives you the probability that at any moment spin 1 is up, spin 18 is down, spin 486 is down, … Another way to think of a state is as a collection of all long-time averages Another way to think of a state is as a collection of all long-time averages (These are known as correlation functions.)

8. First feature: the Parisi solution of the SK model has many thermodynamic states! The Parisi solution of the SK model G. Parisi, Phys. Rev. Lett. 43, 1754 (1979); 50, 1946 (1983)

9. Overlaps and their distribution Consider a thermodynamic state that is a mixture of pure (extremal) Gibbs states: with so that, for any , β, -q EA ≤ q  β ≤ q EA. The overlap q  β between pure states  and β in a volume  L is defined to be:

10. is a classical field defined on the interval [-L/2,L/2] It is subject to a potential like or Now add noise … classical (thermal) or quantum mechanical Their overlap density is: commonly called the Parisi overlap distribution. Example: Uniform Ising ferromagnet below T c.

11. Replica symmetry breaking (RSB) solution of Parisi for the infinite-range (SK) model: nontrivial overlap structure and non-self-averaging. Nontrivial overlap structure: Nontrivial overlap structure:Non-self-averaging: J1J1J1J1 J2J2J2J2 So, when average over all coupling realizations:

12. Ultrametricity R. Rammal, G. Toulouse, and M.A. Virasoro, Rev. Mod. Phys. 58, 765 (1986) In an ordinary metric space, any three points x, y, and z must satisfy the triangle inequality: But in an ultrametric space, all distances obey the strong triangle inequality: which is equivalent to (All triangles are acute isosceles!) There are no in-between points. What kind of space has this structure? Third feature: the space of overlaps of states has an ultrametric structure.

13. Answer: a nested (or tree-like or hierarchical) structure. Kinship relations are an obvious example. 344 H. Simon, ``The Organization of Complex Systems’’, in Hierarchy Theory – The Challenge of Complex Systems, ed. H.H. Pattee, (George Braziller, 1973).

14. ``…the Gibbs equilibrium measure decomposes into a mixture of many pure states. This phenomenon was first studied in detail in the mean field theory of spin glasses, where it received the name of replica symmetry breaking. But it can be defined and easily extended to other systems, by considering an order parameter function, the overlap distribution function. This function measures the probability that two configurations of the system, picked up independently with the Gibbs measure, lie at a given distance from each other. Replica symmetry breaking is made manifest when this function is nontrivial.’’ S. Franz, M. Mézard, G. Parisi, and L. Peliti, Phys. Rev. Lett. 81, 1758 (1998). The four main features of RSB: 1) Infinitely many thermodynamic states (unrelated by any simple symmetry transformation) 2) Infinite number of order parameters, characterizing the overlaps of the states 3) Non-self-averaging of state overlaps (sample-to-sample fluctuations) 4) Ultrametric structure of state overlaps

15. Very pretty, but is it right? And if it is, how generic is it? As a solution to the SK model, there are recent rigorous results that support the correctness of the RSB ansatz. As a solution to the SK model, there are recent rigorous results that support the correctness of the RSB ansatz. F. Guerra and F.L. Toninelli, Commun. Math. Phys. 230, 71 (2002); M. Talagrand, Spin Glasses: A Challenge to Mathematicians (Springer-Verlag, 2003) As for its genericity … As for its genericity … … this is a subject of an intense and ongoing debate.

16. In fact: the most straightforward interpretation of this statement (the ``standard RSB picture’’) --- a thermodynamic Gibbs state ρ J decomposable into pure states whose overlaps are non-self-averaging --- cannot happen in any finite dimension. Reason essentially the same as why (e.g.) one can’t have a phase transition for some coupling realizations and infinitely many for others. Follows from the ergodic theorem for translation-invariant functions on certain probability distributions. C.M. Newman and D.L. Stein, Phys. Rev. Lett. 76, 515 (1996); J. Phys.: Condensed Matter 15, R1319 (2003).

17. So what sort of “mean field picture” is allowed in short-range spin glasses? Maximal mean-field picture: “nonstandard RSB scenario” (NS, Phys. Rev. Lett. 76, 4821 (1996) and subsequent publications). To properly deal with statistical mechanics of spin glasses, need new tool: the metastate M. Aizenman and J. Wehr, Commun. Math. Phys. 130, 489 (1990); C.M. Newman and D.L. Stein, Phys. Rev. Lett. 76, 4821 (1996) and subsequent papers. Required because of nonexistence of thermodynamic limit for states due to chaotic size dependence (NS, Phys. Rev. B 46, 973 (1992)).

18. Metastates A useful tool for analyzing competition of many thermodynamic states in a single system A useful tool for analyzing competition of many thermodynamic states in a single system Provides a natural framework for understanding how this (or other) thermodynamic structures could arise in short-range systems Provides a natural framework for understanding how this (or other) thermodynamic structures could arise in short-range systems Relates equilibrium (infinite-volume) thermodynamic structure to physical behavior in large finite volumes Relates equilibrium (infinite-volume) thermodynamic structure to physical behavior in large finite volumes A probability distribution over the thermodynamic states themselves: κ J (  Metastate: Gibbs state : Gibbs state: Spin configuration M. Aizenman and J. Wehr, Commun. Math. Phys. 130, 489 (1990); C.M. Newman and D.L. Stein, Phys. Rev. Lett. 76, 4821 (1996) and subsequent papers. (Not trivial if many competing states because of presence of chaotic size dependence of correlations – NS, Phys. Rev. B 46, 973 (1992)) Inspired by analogy with chaotic dynamical systems

19. For fixed J, consider an infinite sequence of volumes, all with periodic boundary conditions (for example): 2222 1111 3333 0 123 And, when averaged over all volumes: Note: This is all for a single coupling realization.

20. Other possible scenarios Droplet/scaling (McMillan, Bray and Moore, Fisher and Huse): The PBC metastate is supported on a single , which consists solely of a pair of global spin-reversed pure states: Droplet/scaling (McMillan, Bray and Moore, Fisher and Huse): The PBC metastate is supported on a single , which consists solely of a pair of global spin-reversed pure states: Chaotic pairs (Newman and Stein): the metastate is supported on uncountably many  ’ s, but each  consists of a single pair of pure states. Chaotic pairs (Newman and Stein): the metastate is supported on uncountably many  ’ s, but each  consists of a single pair of pure states. TNT (Trivial Edge-Nontrivial Spin) Overlap: Krzakala and Martin, Palassini and Young Extensive numerical work over several decades by Binder, Bray, Domany, Franz, Hartmann, Hed, Katzgraber, Krzakala, Machta, Marinari, Martin, Mezard, Middleton, M. Moore, Palassini, Parisi, Young, and many others

21. Evidence (though no proof yet) that RSB does not describe low-temperature ordering of any realistic spin glass model, at any temperature and in any finite dimension. Why? Combination of disorder and physical couplings scaling to zero as N  In some ways, this is an even stranger departure from the behavior of ordered systems than RSB. (Recall the `physical’ coupling in the SK model is J ij /  N)

22. So … where do we stand? On the one hand, many of the most basic questions remain unanswered: existence of a phase transition, number of ground states/pure states, stability of spin glass phase to magnetic field, … On the other … We now understand a great deal about how spin glass states can (and cannot) be organized Differences from ordered systems: d→∞ limit singular (?): universality? Creation of new thermodynamic tool: the metastate Relationship between large finite volumes and thermodynamic limit

23. Thank you! Questions? If you’re interested in learning more, check out (or better, buy) “Spin Glasses and Complexity”, DLS and CMN, Princeton University Press

24. For fixed J, consider an infinite sequence of volumes, all with periodic boundary conditions (for example): 2222 1111 3333 0

25. Is there a phase transition (AT line) in a magnetic field? Is there a phase transition (AT line) in a magnetic field? Scaling/droplet: no Chaotic pairs: yes (Presumably) T=0 behavior of interfaces T=0 behavior of interfaces

26. Open Questions Is there a thermodynamic phase transition to a spin glass phase? Is there a thermodynamic phase transition to a spin glass phase? Most workers in field think so. If yes: And if so, does the low-temperature phase display broken spin-flip symmetry? How many thermodynamic phases are there? How many thermodynamic phases are there? If many, what is their structure and organization? If many, what is their structure and organization? What happens when a small magnetic field is turned on? What happens when a small magnetic field is turned on? And in particular – is it mean-field-like?