Daniel Stein

Departments of Physics and Mathematics

New York University

Complex Systems Summer School

Santa Fe Institute

June, 2008

Partially supported by US National Science Foundation Grants DMS-01-02541, DMS-01-02587, and DMS-06-04869

Quenched Disorder, Spin Glasses, and ComplexityQuenched Disorder, Spin Glasses, and Complexity

Our guide to complexity through disorder --- Our guide to complexity through disorder --- the spin glass.the spin glass.

• What What isis a spin glass? a spin glass?

• Why are they interesting to:Why are they interesting to:

-- Physics (condensed matter, statistical mechanics)-- Physics (condensed matter, statistical mechanics)

-- Complexity-- Complexity

• Canonical model of disorderCanonical model of disorder

• New computational techniques New computational techniques

• Application to other problemsApplication to other problems

• Generic aspects?Generic aspects?

OverviewOverview

• Lecture 1Lecture 1

-- Ordered and disordered condensed matter systems-- Ordered and disordered condensed matter systems

-- Phase transitions, ordering, and broken symmetry-- Phase transitions, ordering, and broken symmetry

-- Magnetic systems-- Magnetic systems

-- Spin glasses and their properties-- Spin glasses and their properties

• Spin glass energy and broken symmetrySpin glass energy and broken symmetry

• ApplicationsApplications

- Combinatorial optimization and traveling salesman problem- Combinatorial optimization and traveling salesman problem

- Simulated annealing- Simulated annealing

- Hopfield-Tank neural network computation- Hopfield-Tank neural network computation

- Protein conformational dynamics and folding- Protein conformational dynamics and folding

• Geometry of interactions and the infinite-range modelGeometry of interactions and the infinite-range model

Lecture 2Lecture 2

Lecture 3Lecture 3

• 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?

(Approximate) Timeline(Approximate) Timeline

Ca. 1930+ Ordered Systems Ca. 1930+ Ordered Systems

(crystals, ferromagnets, superconductors, superfluids, …)(crystals, ferromagnets, superconductors, superfluids, …)

Bloch’s theorem, broken symmetry, Goldstone modes, single order parameter, …Bloch’s theorem, broken symmetry, Goldstone modes, single order parameter, …

Ca. 1958+ Disordered systemsCa. 1958+ Disordered systems

(glasses, spin glasses, polymers, …)(glasses, spin glasses, polymers, …)

Localization, frustration, broken replica symmetry, infinitely many order parameters, Localization, frustration, broken replica symmetry, infinitely many order parameters, metastates …metastates …

Ca. 1980+ Complex systemsCa. 1980+ Complex systems

(Condensed matter physics, computer science, biology, (Condensed matter physics, computer science, biology, economics, archaeology, …)economics, archaeology, …)

http://sprott.physics.wisc.edu/Pickover/pc/brain-universe.htmlhttp://sprott.physics.wisc.edu/Pickover/pc/brain-universe.html

Phases of Matter and Phase TransitionsPhases of Matter and Phase Transitions

Phase diagram of waterPhase diagram of water

Specific heat C = Specific heat C = T

Q

T(amount of heat needed to add or subtract to change (amount of heat needed to add or subtract to change

the temperature by an amountthe temperature by an amount ))Q

What is a central bridge between traditional physics What is a central bridge between traditional physics and complexity studies?and complexity studies?

Quantifies ``amount’’ and ``type’’ of order in a system --- undergoes Quantifies ``amount’’ and ``type’’ of order in a system --- undergoes discontinuous (in it or its derivatives) change at a phase transitiondiscontinuous (in it or its derivatives) change at a phase transition

Order parametersOrder parameters

Discontinuous jump – latent heatDiscontinuous jump – latent heat(fixed pressure)(fixed pressure)

GlassesGlasses

The ``Berkeley effect’’The ``Berkeley effect’’

Magnetic OrderMagnetic Order

In magnetic materials, each atom has a tiny magnetic moment In magnetic materials, each atom has a tiny magnetic moment mmxx

arising from the quantum mechanical spins of electrons in arising from the quantum mechanical spins of electrons in incompletely filled shells.incompletely filled shells.

These “spins” couple to magnetic fields, which can be external These “spins” couple to magnetic fields, which can be external (from an applied magnetic field h), or internal (from the field (from an applied magnetic field h), or internal (from the field

arising from other spins.arising from other spins.

At high temperature (and in zero external field), thermal At high temperature (and in zero external field), thermal agitation disorders the spins, leading to a net zero agitation disorders the spins, leading to a net zero

field at each site:field at each site:

0)(1

lim0

dttT

T

T xx mm (at high temperature)(at high temperature)

This is called the This is called the paramagneticparamagnetic state. state.

Magnetization is the spatial average of all of the ``local’’ (i.e., atomic) Magnetization is the spatial average of all of the ``local’’ (i.e., atomic) magnetic moments, and describes the overall magnetic state of the magnetic moments, and describes the overall magnetic state of the

sample – as such, it serves as a sample – as such, it serves as a magneticmagnetic order parameter order parameter. .

x

xmN

M1

x

xN1

So M=0 in the paramagnet in the absence of an external magnetic field.So M=0 in the paramagnet in the absence of an external magnetic field.

What happens when you lower the temperature?What happens when you lower the temperature?

In certain materials, there is a sharp In certain materials, there is a sharp phase transitionphase transition to a to a magnetically orderedmagnetically ordered state. state.

Single spin orientation at different times – Single spin orientation at different times – averages to zero in short time: averages to zero in short time: 0x

xx

What is the nature of the ordering?What is the nature of the ordering?

• In some materials (e.g., Fe, Mn), nearby spins ``like’’ to align; In some materials (e.g., Fe, Mn), nearby spins ``like’’ to align; these are called these are called ferromagnets.ferromagnets.

• In others (e.g., Cr, many metal oxides), they like to anti-In others (e.g., Cr, many metal oxides), they like to anti-align; these are called antiferromagnets.align; these are called antiferromagnets.

• And there are many other types as well (ferrimagnets, And there are many other types as well (ferrimagnets, canted ferromagnets, helical ferromagnets, …)canted ferromagnets, helical ferromagnets, …)

• Can capture both behaviors with a simple model Can capture both behaviors with a simple model energy function (Hamiltonian):energy function (Hamiltonian):

yyx

xJH ,

x

xh

tferromagne0J agnetantiferrom0J

Magnetic Phase TransitionsMagnetic Phase Transitions

High temperatureHigh temperatureLow temperatureLow temperature

Phase diagram for ferromagnetPhase diagram for ferromagnet

Broken symmetryBroken symmetry

J.P. Sethna, Statistical Mechanics: J.P. Sethna, Statistical Mechanics: Entropy, Order Parameters, Entropy, Order Parameters, and Complexity and Complexity (Oxford U. Press, 2007)(Oxford U. Press, 2007)

A New State of Matter?A New State of Matter?

Prehistory: The Kondo Problem (1950’s – 1970’s)Prehistory: The Kondo Problem (1950’s – 1970’s)

Generated interest in dilute magnetic alloys Generated interest in dilute magnetic alloys (CuMn, AuFe, …)(CuMn, AuFe, …)

Addition of ln(1/T) term to the resistivityAddition of ln(1/T) term to the resistivity

Early 1970’s: Magnetic effects seen at greater impurity concentrationsEarly 1970’s: Magnetic effects seen at greater impurity concentrations

Cannella, Mydosh, and Budnick, Cannella, Mydosh, and Budnick, J. Appl. Phys.J. Appl. Phys. 4242, 1689 , 1689 (1971)(1971)

The Solid State Physics of Spin GlassesThe Solid State Physics of Spin Glasses

Dilute magnetic alloy: localized spins at magnetic impurity sitesDilute magnetic alloy: localized spins at magnetic impurity sites

D.L. Stein, Sci. Am. D.L. Stein, Sci. Am. 261261, 52 (1989)., 52 (1989).

M.A. Ruderman and C. Kittel, Phys. Rev. M.A. Ruderman and C. Kittel, Phys. Rev. 9696, 99 (1954); T. Kasuya, Prog. Theor. Phys. , 99 (1954); T. Kasuya, Prog. Theor. Phys. 1616, 45 (1956); , 45 (1956); K. Yosida, Phys. Rev. K. Yosida, Phys. Rev. 106106, 893 (1957)., 893 (1957).

Frustration!Frustration!

tferromagne0J agnetantiferrom0J

0 zxC

yzxy JJJ

Ground StatesGround States

Quenched disorderQuenched disorder

CrystalCrystal GlassGlass

FerromagnetFerromagnet Spin GlassSpin Glass

1) For these systems, disorder cannot be treated as a 1) For these systems, disorder cannot be treated as a perturbative effectperturbative effect

Two ``meta-principles’’Two ``meta-principles’’

2) P.W. Anderson, 2) P.W. Anderson, Rev. Mod. Phys.Rev. Mod. Phys. 50, 191 (1978): ``…there is an important 50, 191 (1978): ``…there is an important fundamental truth about random systems we must always keep in mind: fundamental truth about random systems we must always keep in mind: no real no real

atom is an average atom, nor is an experiment ever done on an ensemble of atom is an average atom, nor is an experiment ever done on an ensemble of samples.samples. What we really need to know is the What we really need to know is the probability distributionprobability distribution …, …, notnot (the) (the)

average … this is the important, and deeply new, step taken here: the average … this is the important, and deeply new, step taken here: the willingness to deal with willingness to deal with distributionsdistributions, not , not averagesaverages. Most of the recent progress . Most of the recent progress in fundamental physics or amorphous materials involves this same kind of step, in fundamental physics or amorphous materials involves this same kind of step, which implies that a random system is to be treated not as just a dirty regular which implies that a random system is to be treated not as just a dirty regular

one, but in a fundamentally different way.’’one, but in a fundamentally different way.’’

``Rugged’’ Energy Landscape``Rugged’’ Energy Landscape

• Many metastable statesMany metastable states

• Many thermodynamic statesMany thermodynamic states??

• Slow dynamics --- can get ``stuck’’ in a local energy minimumSlow dynamics --- can get ``stuck’’ in a local energy minimum

• Disorder and frustration …Disorder and frustration …

R.G. Palmer, Adv. Phys. R.G. Palmer, Adv. Phys. 3131, 669 (1982)., 669 (1982).

M. M. Goldstein, J. Chem. Phys. Goldstein, J. Chem. Phys. 5151, 3728 (1969);, 3728 (1969); S.A. Kauffman, S.A. Kauffman, The Origins of The Origins of OrderOrder (Oxford, 1993); W. Hordijk and P.F. Stadler, J. Complex Systems (Oxford, 1993); W. Hordijk and P.F. Stadler, J. Complex Systems 11, 39 , 39

(1998); D.L. Stein and C.M. Newman, Phys. Rev. E (1998); D.L. Stein and C.M. Newman, Phys. Rev. E 5151, 5228 (1995)., 5228 (1995).

C.M. Newman and D.L. Stein, Phys. Rev. E 60, 5244 (1999).C.M. Newman and D.L. Stein, Phys. Rev. E 60, 5244 (1999).

Is there a phase transition to a ``spin glass phase’’?Is there a phase transition to a ``spin glass phase’’?

tt

yesyes

nono

L.E. Wenger and P.H. Keesom, Phys. Rev. L.E. Wenger and P.H. Keesom, Phys. Rev. B 13, 4953 (1976).B 13, 4953 (1976).Cannella, Mydosh, and Budnick, Cannella, Mydosh, and Budnick, J. Appl. Phys.J. Appl. Phys.

4242, 1689 (1971), 1689 (1971)

Aging and Memory EffectsAging and Memory Effects

K. Binder and A.P. Young, Rev. Mod. Phys. 58, 801 (1986).

AgingAging

P. Svedlinh P. Svedlinh et al., et al., Phys. Rev. B 35, 268 (1987)

So far … lots of nice stuffSo far … lots of nice stuff

• DisorderDisorder

• FrustrationFrustration

• Complicated state space --- rugged energy landscapeComplicated state space --- rugged energy landscape

• Anomalous dynamical behaviorAnomalous dynamical behavior

-- Memory effects-- Memory effects

-- History dependence and irreversibility-- History dependence and irreversibility

• Connections to other problems --- new Connections to other problems --- new insights and techniquesinsights and techniques

• Well-defined mathematical structureWell-defined mathematical structure

• … … which we’ll start with tomorrow.which we’ll start with tomorrow.