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Spin Glasses: Lectures 2 and 3 - Review, Notions, and Replica Symmetry Breaking

This article reviews the concepts covered in the first lecture on spin glasses, including finite-volume Gibbs distributions and thermodynamic states. It then delves into the Sherrington-Kirkpatrick model and the Parisi solution, highlighting the phenomenon of replica symmetry breaking (RSB) and its implications. The article also discusses ground states, quenched disorder, the Edwards-Anderson model, frustration, and the nature of broken symmetry in spin glasses.

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Spin Glasses: Lectures 2 and 3 - Review, Notions, and Replica Symmetry Breaking

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

  2. Ground States Quenched disorder

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

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

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

  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? One guide: the infinite-range Sherrington-Kirkpatrick (SK) model displays an exotic new type of broken symmetry, known as replica symmetry breaking (RSB). ``…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). 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? What does this mean? Look at relations between states.

  7. Another way to think of a state is as a collection of all long-time averages Thermodynamic States • A thermodynamic state is a probability measure on infinite-volume spin configurations • 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, … (These are known as correlation functions.)

  8. The Parisi solution of the SK model First feature: the Parisi solution of the SK model has many thermodynamic states! 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: The overlap qβ between pure states  and β in a volume Lis defined to be: with so that, for any, β, -qEA ≤qβ≤qEA.

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

  11. J2 J1 Replica symmetry breaking (RSB) solution of Parisi for the infinite-range (SK) model: nontrivial overlap structure and non-self-averaging. Non-self-averaging: Nontrivial overlap structure: So, when average over all coupling realizations:

  12. 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 Ultrametricity Third feature: the space of overlaps of states has an ultrametric structure. (All triangles are acute isosceles!) There are no in-between points. What kind of space has this structure? R. Rammal, G. Toulouse, and M.A. Virasoro, Rev. Mod. Phys. 58, 765 (1986)

  13. Answer: a nested (or tree-like or hierarchical) structure. Kinship relations are an obvious example. H. Simon, ``The Organization of Complex Systems’’, in Hierarchy Theory – The Challenge of Complex Systems, ed. H.H. Pattee, (George Braziller, 1973). 3 4 4

  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 for its genericity … … this is a subject of an intense and ongoing debate. F. Guerra and F.L. Toninelli, Commun. Math. Phys. 230, 71 (2002); M. Talagrand, Spin Glasses: A Challenge to Mathematicians (Springer-Verlag, 2003)

  16. In fact: the most straightforward interpretation of this statement (the ``standard RSB picture’’) --- a thermodynamic Gibbs state ρJdecomposable 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 Required because of nonexistence of thermodynamic limit for states due to chaotic size dependence (NS, Phys. Rev. B 46, 973 (1992)). 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.

  18. Metastates • 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 • Relates equilibrium (infinite-volume) thermodynamic structure to physical behavior in large finite volumes (Not trivial if many competing states because of presence of chaotic size dependence of correlations – NS, Phys. Rev. B46, 973 (1992)) Inspired by analogy with chaotic dynamical systems A probability distribution over the thermodynamic states themselves: κJ (G) 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.

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

  20. Other possible scenarios TNT (Trivial Edge-Nontrivial Spin) Overlap: Krzakala and Martin, Palassini and Young 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. 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. In some ways, this is an even stranger departure from the behavior of ordered systems than RSB. Why? Combination of disorder and physical couplings scaling to zero as N (Recall the `physical’ coupling in the SK model is Jij/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? Relationship between large finite volumes and thermodynamic limit Creation of new thermodynamic tool: the metastate

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

  24. For fixedJ, consider an infinite sequence of volumes, all with periodic boundary conditions (for example): 2 3 1 • 0 <σ4σ57>

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

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

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