Spin Glasses and Complexity: Spin Glasses and Complexity: Lecture 3Lecture 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 modelParisi solution of SK model

• Replica symmetry breaking (RSB)Replica symmetry breaking (RSB)

- Overlaps- Overlaps

- Non-self-averaging- Non-self-averaging

- Ultrametricity- Ultrametricity

• What What isis the structure of short-range spin glasses? the structure of short-range spin glasses?

• Are spin glasses complex systems?Are spin glasses complex systems?

Where we left off: spin glasses (and glasses, …) are characterized by Where we left off: spin glasses (and glasses, …) are characterized by broken symmetry in broken symmetry in timetime but not in but not in spacespace..

Broken symmetry in the spin glassBroken symmetry in the spin glass

01

lim

01

lim

1

2

1

N

ii

N

N

ii

N

SN

q

SN

M

EA

But remember: this remains a conjecture!But remember: this remains a conjecture!

Open QuestionsOpen 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:Most workers in field think so. If yes:

And if so, does the low-temperature phase display And if so, does the low-temperature phase display broken spin-flip symmetry (that is, qbroken spin-flip symmetry (that is, qEAEA>0)?>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?And in particular – is it mean-field-like?

(In other words, how many order parameters are needed to (In other words, how many order parameters are needed to describe the symmetry of the low-temperature phase?)describe the symmetry of the low-temperature phase?)

The Edwards-Anderson (EA) Ising Model

Site in Zd

x

xxxy

yxxy hJ hJ,

H

Site in Site in ZZdd

Nearest neighbor spins onlyNearest neighbor spins only

1Coupling and field realizationCoupling and field realization

The fields and couplings are i.i.d. random variables:The fields and couplings are i.i.d. random variables:

]2/exp[2

1)( 2

xyxy JJP

]2/exp[2

1)( 2

2

2 xx hhP

i

iiji

ji hJ ijNN

1

h,J,

H

The Sherrington-Kirkpatrick (SK) modelThe Sherrington-Kirkpatrick (SK) model

The fields and couplings are i.i.d. random variables:The fields and couplings are i.i.d. random variables:

]2/exp[2

1)( 2

ijij JJP

]2/exp[2

1)( 2

2

2

ii hhP

Nji ,,1, withwith

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 ``…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 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 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 can be defined and easily extended to other systems, by considering an order parameter function, the overlap distribution function. This function measures parameter function, the overlap distribution function. This function measures the probability that two configurations of the system, picked up independently the probability that two configurations of the system, picked up independently

with the Gibbs measure, lie at a given distance from each other. Replica with the Gibbs measure, lie at a given distance from each other. Replica

symmetry breaking is made manifest when this function is nontrivial.’’ symmetry breaking is made manifest when this function is nontrivial.’’

S. Franz, M. MS. Franz, M. Méézard, G. Parisi, and L. Peliti, zard, G. Parisi, and L. Peliti, Phys. Rev. Lett.Phys. Rev. Lett. 8181, 1758 (1998)., 1758 (1998).

What does this mean?What does this mean?

One guide: the infinite-range Sherrington-Kirkpatrick (SK) model displays an exotic One guide: the infinite-range Sherrington-Kirkpatrick (SK) model displays an exotic new type of broken symmetry, known as new type of broken symmetry, known as replica symmetry breakingreplica symmetry breaking (RSB). (RSB).

First feature: the Parisi solution of the SK model has First feature: the Parisi solution of the SK model has manymany thermodynamic states! thermodynamic states!

The Parisi solution of the SK modelThe Parisi solution of the SK model

G. Parisi, G. Parisi, Phys. Rev. Lett.Phys. Rev. Lett. 4343, 1754 (1979); , 1754 (1979); 5050, 1946 (1983), 1946 (1983)

iii many for

Thermodynamic StatesThermodynamic States

• A thermodynamic state is a A thermodynamic state is a probability measureprobability measure on infinite-volume spin on infinite-volume spin configurationsconfigurations

• We’ll denote a state by the index We’ll denote a state by the index αα, , ββ, , γγ, …, …

• A given state A given state αα gives you the probability that at any moment spin 1 is gives you the probability that at any moment spin 1 is up, spin 18 is down, spin 486 is down, …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

,,, zyxyxx

(These are known as (These are known as correlation functionscorrelation functions.).)

Overlaps and their distributionOverlaps and their distribution

EAx

x

L

xxx

L

qq

q

L

L

21

1

withwith

so that, for anyso that, for any , , ββ, -q, -qEA EA ≤≤ qqββ ≤≤ qqEAEA ..

The overlap qThe overlap qββ between states between states and and ββ in a volume in a volume LL is defined is defined

to be:to be:

Second feature: relationships between states are characterized Second feature: relationships between states are characterized by their by their overlapsoverlaps..

M. MM. Mézard ézard et alet al., ., Phys. Rev. LettPhys. Rev. Lett. . 5252, 1156 (1984); , 1156 (1984); J. Phys. (Paris)J. Phys. (Paris) 4545, 843 (1984) , 843 (1984)

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: Their overlap density is:

)()(,

qqWWqP J

commonly called the commonly called the Parisi overlap distribution.Parisi overlap distribution.

Example: Uniform Ising ferromagnet below TExample: Uniform Ising ferromagnet below Tcc..

2

1

2

1

Replica symmetry breaking (RSB) solution of Parisi for the infinite-range Replica symmetry breaking (RSB) solution of Parisi for the infinite-range (SK) model: (SK) model: nontrivial overlap structure and non-self-averaging.nontrivial overlap structure and non-self-averaging.

Nontrivial overlap structure:Nontrivial overlap structure:Non-self-averaging:Non-self-averaging:

JJ11JJ22

So, when average over all coupling realizations:So, when average over all coupling realizations:

UltrametricityUltrametricity

R. Rammal, G. Toulouse, and M.A. Virasoro, R. Rammal, G. Toulouse, and M.A. Virasoro, Rev. Mod. PhysRev. Mod. Phys. . 5858, 765 (1986), 765 (1986)

In an ordinary In an ordinary metric space, metric space, any three points x, y, and z must satisfy the any three points x, y, and z must satisfy the triangle inequalitytriangle inequality: : ),(),(),( zydyxdzxd

But in an ultrametric space, all distances obey the But in an ultrametric space, all distances obey the strong triangle strong triangle inequalityinequality:: )),(),,(max(),( zydyxdzxd

which is equivalent towhich is equivalent to ),(),(),( zydyxdzxd

(All triangles are acute isosceles!)(All triangles are acute isosceles!)

There are no in-between points.There are no in-between points.

What kind of space has this structure?What kind of space has this structure?

Third feature: the space of overlaps of states has an Third feature: the space of overlaps of states has an ultrametric structureultrametric structure..

ddd

Answer: a nested (or tree-like or hierarchical) structure.Answer: a nested (or tree-like or hierarchical) structure.

Kinship relations are an obvious example.Kinship relations are an obvious example.

33 44 44

H. Simon, ``The Organization of Complex Systems’’, in H. Simon, ``The Organization of Complex Systems’’, in Hierarchy Theory – The Hierarchy Theory – The Challenge of Complex SystemsChallenge of Complex Systems, ed. H.H. Pattee, (George Braziller, 1973)., ed. H.H. Pattee, (George Braziller, 1973).

``…the Gibbs equilibrium measure decomposes into a mixture of many pure ``…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 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 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 can be defined and easily extended to other systems, by considering an order parameter function, the overlap distribution function. This function measures parameter function, the overlap distribution function. This function measures the probability that two configurations of the system, picked up independently the probability that two configurations of the system, picked up independently

with the Gibbs measure, lie at a given distance from each other. Replica with the Gibbs measure, lie at a given distance from each other. Replica

symmetry breaking is made manifest when this function is nontrivial.’’ symmetry breaking is made manifest when this function is nontrivial.’’

S. Franz, M. MS. Franz, M. Méézard, G. Parisi, and L. Peliti, zard, G. Parisi, and L. Peliti, Phys. Rev. Lett.Phys. Rev. Lett. 8181, 1758 (1998)., 1758 (1998).

The four main features of RSB:The four main features of RSB:

1) Infinitely many thermodynamic states (unrelated by any simple symmetry 1) Infinitely many thermodynamic states (unrelated by any simple symmetry transformation)transformation)

2) Infinite number of order parameters, characterizing the overlaps of the states2) Infinite number of order parameters, characterizing the overlaps of the states

3) Non-self-averaging of state overlaps (sample-to-sample fluctuations) 3) Non-self-averaging of state overlaps (sample-to-sample fluctuations)

4) Ultrametric structure of state overlaps4) Ultrametric structure of state overlaps

Very pretty, but is it right?Very pretty, but is it right?

And if it is, how generic is it?And if it is, how generic is it?

• As a solution to the SK model, there are recent rigorous results As a solution to the SK model, there are recent rigorous results that support the correctness of the RSB ansatz.that support the correctness of the RSB ansatz.

F. Guerra and F.L. Toninelli, F. Guerra and F.L. Toninelli, Commun. Math. PhysCommun. Math. Phys. . 230230, 71 (2002); M. Talagrand, Spin , 71 (2002); M. Talagrand, Spin Glasses: A Challenge to Mathematicians (Springer-Verlag, 2003)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 there are numerous indications that the SK model is pathological and that the RSB symmetry-breaking structure does not apply to realistic spin glasses.RSB symmetry-breaking structure does not apply to realistic spin glasses.

In fact: the most straightforward interpretation of this statement (the ``standard In fact: the most straightforward interpretation of this statement (the ``standard RSB picture’’) --- a thermodynamic Gibbs state RSB picture’’) --- a thermodynamic Gibbs state ρρJJ decomposable into pure decomposable into pure

states whose overlaps are non-self-averaging --- states whose overlaps are non-self-averaging --- cannot happen in any finite cannot happen in any finite dimension. dimension.

Reason essentially the same as why (e.g.) one can’t have a Reason essentially the same as why (e.g.) one can’t have a phase transition for some coupling realizations and phase transition for some coupling realizations and

infinitely many for others.infinitely many for others.

Follows from the ergodic theorem for translation-invariant Follows from the ergodic theorem for translation-invariant functions on certain probability distributions.functions on certain probability distributions.

C.M. Newman and D.L. Stein, C.M. Newman and D.L. Stein, Phys. Rev. Lett.Phys. Rev. Lett. 7676, 515 (1996); , 515 (1996);

J. Phys.: Condensed MatterJ. Phys.: Condensed Matter 15, R1319 (2003). 15, R1319 (2003).

Other possible scenariosOther possible scenarios

• Droplet/scaling (Macmillan, Bray and Moore, Fisher and Huse): Droplet/scaling (Macmillan, Bray and Moore, Fisher and Huse): there is only a single pair of global spin-reversed pure states.there is only a single pair of global spin-reversed pure states.

• Chaotic pairs (Newman and Stein): like RSB, there are uncountably Chaotic pairs (Newman and Stein): like RSB, there are uncountably

many statesmany states, , but eachbut each consists of a single pair of pure states.consists of a single pair of pure states.

So RSB is unlikely to hold for any realistic spin glass So RSB is unlikely to hold for any realistic spin glass model, at any temperature in any finite dimension.model, at any temperature in any finite dimension.

Why?Why?

Combination of disorder and physical couplings scaling to zero as NCombination of disorder and physical couplings scaling to zero as N

In some ways, this is an even stranger departure In some ways, this is an even stranger departure from the behavior of ordered systems than RSB.from the behavior of ordered systems than RSB.

(Recall the `physical’ coupling in the SK model is J(Recall the `physical’ coupling in the SK model is J ijij//N)N)

D.L. Stein, ``Spin Glasses: Still Complex After All These Years?’’, in D.L. Stein, ``Spin Glasses: Still Complex After All These Years?’’, in Quantum Quantum Decoherence and Entropy in Complex SystemsDecoherence and Entropy in Complex Systems,, ed. T.-H. Elze (Springer, 2004).ed. T.-H. Elze (Springer, 2004).

Are Spin Glasses Complex Systems?Are Spin Glasses Complex Systems?

• Most of the ``classic’’ features that earned spin glasses the title Most of the ``classic’’ features that earned spin glasses the title ``complex system’’ are still intact:``complex system’’ are still intact:

-- many metastable states-- many metastable states

-- anomalous dynamics (irreversibility, history dependence, -- anomalous dynamics (irreversibility, history dependence,

memory effects, aging …memory effects, aging …

-- ``rugged energy landscape’’-- ``rugged energy landscape’’

-- disorder and frustration-- disorder and frustration

• Connections to problems in computer science, biology, Connections to problems in computer science, biology, economics, … economics, …

• RSB structure can hold in a variety of nonphysical problems (random RSB structure can hold in a variety of nonphysical problems (random TSP, k-SAT, …)TSP, k-SAT, …)

• Hierarchies as emergent property!Hierarchies as emergent property!

But there are also some important and interesting newly discovered properties …But there are also some important and interesting newly discovered properties …

• Singular dSingular d→→∞ limit∞ limit

• Absence of straightforward thermodynamic limit for Absence of straightforward thermodynamic limit for states and ``chaotic size dependence’’states and ``chaotic size dependence’’

C.M. Newman and D.L. Stein, C.M. Newman and D.L. Stein, Phys. Rev. BPhys. Rev. B 4646, 973 (1992)., 973 (1992).

-- Analogy between behavior of correlation functions-- Analogy between behavior of correlation functions

xx, , xx yy , … , …as volume increases and phase space as volume increases and phase space

trajectory of chaotic dynamical systemtrajectory of chaotic dynamical system

-- Led to concept of -- Led to concept of metastatemetastate

• Connection between finite and infinite volumes far more Connection between finite and infinite volumes far more subtle than in homogeneous systemssubtle than in homogeneous systems