Lectures on Spin Glasses

Daniel Stein

Department of Physics and Courant Institute

New York University

Workshop on Disordered Systems, Random Spatial Processes, and Some Applications

Institut Henri Poincaré

Paris, January 5 – March 30, 2015

Overview:

Lectures 1 and 2: Introduction and background: experiments and theory

Lectures 3 and 4: Infinite-range and short-range spin glasses

Lectures 5 and 6: Applications to computer science, neural networks, etc.

Main GoalsMain Goals

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

• Why are spin glasses of interest to:Why are spin glasses of interest to:

-- Physicists (condensed matter), Mathematicians (probability theory), and Mathematical -- Physicists (condensed matter), Mathematicians (probability theory), and Mathematical Physicists (statistical mechanics)Physicists (statistical mechanics)

-- “Complexity scientists’’ -- “Complexity scientists’’

• Canonical model of disorderCanonical model of disorder

• New computational techniques New computational techniques

• Application to other problemsApplication to other problems

• Generic aspects?Generic aspects?

Provide a general sense of mathematical and physical research into spin glasses, with an emphasis on the following questions:

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

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

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 arising from the

quantum mechanical spins of electrons in incompletely filled shells.quantum mechanical spins of electrons in incompletely filled shells.

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

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

Quantifies ``amountQuantifies ``amount’’’’ and ``type and ``type’’’’ of order in a system --- undergoes 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)

Magnetization is the spatial average of all of the ``localMagnetization is the spatial average of all of the ``local’’’’ (i.e., atomic) (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 ``likeIn some materials (e.g., Fe, Mn), nearby spins ``like’’’’ to align; these are called to align; these are called ferromagnets.ferromagnets.

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

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

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

yyx

xJH ,

x

xh

tferromagne0 J agnetantiferrom0 J

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

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)

Early 1970Early 1970’’s: Magnetic effects seen at greater impurity concentrationss: 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)

Magnetic susceptibility χ = limΔh->0 (ΔM/Δh)

Ground StatesGround States

Quenched disorderQuenched disorder

CrystalCrystal GlassGlass

FerromagnetFerromagnet Spin GlassSpin Glass

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).

Is there a phase transition to a ``spin glass phaseIs 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)

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

1975: Theory Rears Its Head

Frustration!Frustration!

tferromagne0 J agnetantiferrom0 J

0 zxC

yzxy JJJ

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

• Many metastable statesMany metastable states

• Many thermodynamic statesMany thermodynamic states??

• Slow dynamics --- can get ``stuckSlow dynamics --- can get ``stuck’’’’ in a local energy minimum 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).

EA conjecture: Spin glasses (and glasses, …) are characterized by broken symmetry in EA conjecture: Spin glasses (and glasses, …) are characterized by 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!

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 wewhich we’’ll start with tomorrow.ll start with tomorrow.

Spin glasses represent a gap in our understanding of condensed matter.Spin glasses represent a gap in our understanding of condensed matter.

Bishop George BerkeleyBishop George Berkeley

1685-17531685-1753

Dr. Samuel JohnsonDr. Samuel Johnson

1709-17841709-1784

Question: Why are solids rigid?Question: Why are solids rigid?

Primary question: Why should we be interested? Primary question: Why should we be interested?

““I refute it thus!”I refute it thus!”

“Ordinary” glasses: no change in symmetry, no phase transition.

And for spin glasses, the situation is even more problematic.

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-principlesTwo ``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.’’’’