Spin Glasses: Lectures 2 and 3Spin 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 lectureReview of first lecture

• The Sherrington-Kirkpatrick (SK) infinite-range spin glass modelThe Sherrington-Kirkpatrick (SK) infinite-range spin glass model

• Some notions from statistical mechanicsSome notions from statistical mechanics

- - Finite-volume Gibbs distributionsFinite-volume Gibbs distributions

-- Thermodynamic states: pure, mixed, and ground statesThermodynamic states: pure, mixed, and ground states

• Open questionsOpen questions

• Summary of RSB solution of SK modelSummary of RSB solution of SK model

Ground StatesGround States

Quenched disorderQuenched disorder

The Edwards-Anderson (EA) Ising Model

Site in Zd

x

xxxy

yxxy hJ hJ,

H

Nearest neighbor spins onlyNearest neighbor spins only1

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

FrustrationFrustration

EA conjecture: Spin glasses (and glasses, …) are characterized by EA conjecture: 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 was a conjecture!But remember: this was a conjecture!

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

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 ``…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 One guide: the infinite-range Sherrington-Kirkpatrick (SK) model displays an exotic new type of broken symmetry, known as displays an exotic new type of broken symmetry, known as

replica symmetry breakingreplica symmetry breaking (RSB). (RSB).

i

iiji

ji hJ ijNN

1

h,J,

H

To begin, RSB asserts the existence of many thermodynamic To begin, RSB asserts the existence of many thermodynamic pure states pure states unrelated by any symmetry transformationunrelated by any symmetry transformation. .

Each of these looks ``randomEach of these looks ``random’’’’ … so how does one describe ordering … so how does one describe ordering in such a situation?in such a situation?

Look at Look at relationsrelations between states. between states.

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

• WeWe’’ll denote a state by the index 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.).)

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

Overlaps and their distributionOverlaps and their distribution

Consider a thermodynamic state that is a mixture of pure (extremal) Consider a thermodynamic state that is a mixture of pure (extremal) Gibbs states: Gibbs states:

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 ..

)()(

W

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

defined to be:defined to be:

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 SystemsH. Simon, ``The Organization of Complex Systems’’’’, in , in Hierarchy Theory – The Challenge of Hierarchy Theory – The Challenge of Complex SystemsComplex 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 …

… … this is a subject of an intense and ongoing debate.this is a subject of an intense and ongoing debate.

In fact: the most straightforward interpretation of this statement (the ``standard In fact: the most straightforward interpretation of this statement (the ``standard RSB pictureRSB picture’’’’) --- a thermodynamic Gibbs state ) --- 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 canReason essentially the same as why (e.g.) one can’’t have a 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).

So what sort of So what sort of ““mean field picturemean field picture”” is is allowed in short-range spin glasses?allowed in short-range spin glasses?

Maximal mean-field picture: Maximal mean-field picture: ““nonstandard RSB scenariononstandard RSB scenario”” (NS, (NS, Phys. Rev. Lett.Phys. Rev. Lett. 7676, 4821 , 4821 (1996) and subsequent publications). (1996) and subsequent publications).

To properly deal with statistical mechanics of spin glasses, need new tool: the To properly deal with statistical mechanics of spin glasses, need new tool: the metastate metastate

M. Aizenman and J. Wehr, M. Aizenman and J. Wehr, Commun. Math. Phys.Commun. Math. Phys. 130130, 489 (1990); C.M. Newman and D.L. Stein, , 489 (1990); C.M. Newman and D.L. Stein, Phys. Rev. Phys. Rev. Lett.Lett. 7676, 4821 (1996) and subsequent papers., 4821 (1996) and subsequent papers.

Required because of nonexistence of thermodynamic limit for states due to Required because of nonexistence of thermodynamic limit for states due to chaotic size chaotic size dependence (dependence (NS,NS, Phys. Rev. B Phys. Rev. B 4646, 973 (1992)), 973 (1992))..

MetastatesMetastates

• A useful tool for analyzing competition of many A useful tool for analyzing competition of many thermodynamic states in a single systemthermodynamic states in a single system

• Provides a natural framework for understanding Provides a natural framework for understanding how this (or other) thermodynamic structures how this (or other) thermodynamic structures

could arise in short-range systemscould arise in short-range systems

• Relates equilibrium (infinite-volume) thermodynamic Relates equilibrium (infinite-volume) thermodynamic structure to physical behavior in large finite volumesstructure to physical behavior in large finite volumes

A probability distribution over the thermodynamic A probability distribution over the thermodynamic states themselves: states themselves: κκJ J ((

Metastate: Gibbs stateMetastate: Gibbs state : : Gibbs state: Spin configurationGibbs state: Spin configuration

M. Aizenman and J. Wehr, M. Aizenman and J. Wehr, Commun. Math. Phys.Commun. Math. Phys. 130130, 489 (1990); C.M. Newman and D.L. Stein, , 489 (1990); C.M. Newman and D.L. Stein, Phys. Rev. Lett.Phys. Rev. Lett. 7676, 4821 (1996) and subsequent papers., 4821 (1996) and subsequent papers.

(Not trivial if many competing states because of (Not trivial if many competing states because of presence of presence of chaotic size dependencechaotic size dependence of correlations – of correlations –

NS, NS, Phys. Rev. BPhys. Rev. B 4646, 973 (1992)), 973 (1992))

Inspired by analogy with chaotic dynamical systemsInspired by analogy with chaotic dynamical systems

For For fixedfixed JJ, consider an infinite sequence of volumes, all with periodic boundary , consider an infinite sequence of volumes, all with periodic boundary conditions (for example):conditions (for example):

22

11

33

• 00

112233

And, when averaged over And, when averaged over allall volumes: volumes:

Note: This is all for a Note: This is all for a singlesingle coupling realization. coupling realization.

Other possible scenariosOther possible scenarios

Droplet/scaling (McMillan, Bray and Moore, Fisher and Huse): The PBC metastate is supported Droplet/scaling (McMillan, Bray and Moore, Fisher and Huse): The PBC metastate is supported on a single on a single , which consists solely of a pair of global spin-reversed pure states:, which consists solely of a pair of global spin-reversed pure states:

2

1

2

1

Chaotic pairs (Newman and Stein): the metastate is supported on uncountably manyChaotic pairs (Newman and Stein): the metastate is supported on uncountably many ’’ss,, but but eacheach consists of a single pair of pure states. consists of a single pair of pure states.

TNT (Trivial Edge-Nontrivial Spin) Overlap: Krzakala and Martin, TNT (Trivial Edge-Nontrivial Spin) Overlap: Krzakala and Martin, Palassini and Young Palassini and Young

Extensive numerical work over several decades by Binder, Bray, Domany, Franz, Hartmann, Hed, Extensive numerical work over several decades by Binder, Bray, Domany, Franz, Hartmann, Hed, Katzgraber, Krzakala, Machta, Marinari, Martin, Mezard, Middleton, M. Moore, Palassini, Parisi, Katzgraber, Krzakala, Machta, Marinari, Martin, Mezard, Middleton, M. Moore, Palassini, Parisi,

Young, and many othersYoung, and many others

Evidence (though no proof yet) that RSB does not describe low-temperature 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 ordering of any realistic spin glass model, at any temperature and in any

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

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

i

iiji

ji hJ ijNN

1

h,J,

H

So … where do we stand? So … where do we stand?

On the one hand, many of the most basic questions remain unanswered: On the one hand, many of the most basic questions remain unanswered: existence of a phase transition, number of ground states/pure states, existence of a phase transition, number of ground states/pure states,

stability of spin glass phase to magnetic field, …stability of spin glass phase to magnetic field, …

On the other …On the other …

We now understand a great deal about how spin glass states We now understand a great deal about how spin glass states can (and cannot) be can (and cannot) be organizedorganized

Differences from ordered systems: d→∞ limit singular (?): universality?Differences from ordered systems: d→∞ limit singular (?): universality?

Creation of new thermodynamic tool: the metastateCreation of new thermodynamic tool: the metastate

Relationship between large finite volumes and thermodynamic limitRelationship between large finite volumes and thermodynamic limit

Thank you!Thank you!

Questions?Questions?

If youIf you’’re interested in learning more, check out (or better, re interested in learning more, check out (or better, buy) buy) ““Spin Glasses and ComplexitySpin Glasses and Complexity””, DLS and CMN, , DLS and CMN,

Princeton University Press Princeton University Press

For For fixedfixed JJ, consider an infinite sequence of , consider an infinite sequence of volumes, all with periodic boundary conditions volumes, all with periodic boundary conditions

(for example):(for example):

22

11

33

• 00

<<σσ44σσ5757>>

• Is there a phase transition Is there a phase transition (AT line)(AT line) in a in a magnetic field?magnetic field?

Scaling/droplet: noScaling/droplet: no

Chaotic pairs: yesChaotic pairs: yes

(Presumably)(Presumably)

• T=0 behavior of interfacesT=0 behavior of interfaces

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?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?And in particular – is it mean-field-like?