1. Spin Glasses and Complexity: Lecture 3 Work done in collaboration with Charles Newman, Courant Institute, New York University Partially supported by US National Science Foundation Grants DMS-01-02541, DMS-01-02587, and DMS-06-04869 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?

2. Where we left off: spin glasses (and glasses, …) are characterized by broken symmetry in time but not in space. Broken symmetry in the spin glass But remember: this remains a conjecture!

3. 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 (that is, q EA >0)? 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? (In other words, how many order parameters are needed to describe the symmetry of the low-temperature phase?)

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

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

6. J. Bürki, R.E. Goldstein and C.A. Stafford, Phys. Rev. Lett. 91, 254501 (2003). 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 low temperature phase? And how is it affected by the addition of a small external field? ``…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).

7. 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)

8. 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.)

9. Overlaps and their distribution with so that, for any , β, -q EA ≤ q  β ≤ q EA. The overlap q  β between states  and β in a volume  L is defined to be: Second feature: relationships between states are characterized by their overlaps. M. Mézard et al., Phys. Rev. Lett. 52, 1156 (1984); J. Phys. (Paris) 45, 843 (1984)

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 … … there are numerous indications that the SK model is pathological and that the RSB symmetry-breaking structure does not apply to realistic spin glasses.

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. Other possible scenarios Droplet/scaling (Macmillan, Bray and Moore, Fisher and Huse): there is only a single pair of global spin-reversed pure states. Droplet/scaling (Macmillan, Bray and Moore, Fisher and Huse): there is only a single pair of global spin-reversed pure states. Chaotic pairs (Newman and Stein): like RSB, there are uncountably many states, but each consists of a single pair of pure states. Chaotic pairs (Newman and Stein): like RSB, there are uncountably many states, but each consists of a single pair of pure states.

18. So RSB is unlikely to hold for any realistic spin glass model, at any temperature 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)

19. D.L. Stein, ``Spin Glasses: Still Complex After All These Years?’’, in Quantum Decoherence and Entropy in Complex Systems, ed. T.-H. Elze (Springer, 2004). Are Spin Glasses Complex Systems? Most of the ``classic’’ features that earned spin glasses the title ``complex system’’ are still intact: Most of the ``classic’’ features that earned spin glasses the title ``complex system’’ are still intact: -- many metastable states -- anomalous dynamics (irreversibility, history dependence, memory effects, aging … memory effects, aging … -- ``rugged energy landscape’’ -- disorder and frustration Connections to problems in computer science, biology, economics, … Connections to problems in computer science, biology, economics, … RSB structure can hold in a variety of nonphysical problems (random TSP, k-SAT, …) RSB structure can hold in a variety of nonphysical problems (random TSP, k-SAT, …) Hierarchies as emergent property! Hierarchies as emergent property!

20. But there are also some important and interesting newly discovered properties … Singular d→ ∞ limit Singular d→ ∞ limit Absence of straightforward thermodynamic limit for states and ``chaotic size dependence’’ Absence of straightforward thermodynamic limit for states and ``chaotic size dependence’’ C.M. Newman and D.L. Stein, Phys. Rev. B 46, 973 (1992). -- Analogy between behavior of correlation functions  x ,  x  y , … as volume increases and phase space trajectory of chaotic dynamical system -- Led to concept of metastate Connection between finite and infinite volumes far more subtle than in homogeneous systems Connection between finite and infinite volumes far more subtle than in homogeneous systems