Quenched Disorder, Spin Glasses, and Complexity
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
Our guide to complexity through disorder --- the spin glass.
•
What is a spin glass?
•
Why are they interesting to:
-- Physics (condensed matter, statistical mechanics)
-- Complexity
•
Canonical model of disorder
•
New computational techniques
•
Application to other problems
•
Generic aspects?
Overview
•
Lecture 1
-- Ordered and disordered condensed matter systems
-- Phase transitions, ordering, and broken symmetry
-- Magnetic systems
-- Spin glasses and their properties
Lecture 2
•
Spin glass energy and broken symmetry
•
Applications
- Combinatorial optimization and traveling salesman problem
- Simulated annealing
- Hopfield-Tank neural network computation
- Protein conformational dynamics and folding
•
Geometry of interactions and the infinite-range model
Lecture 3
•
Parisi solution of SK model
•
Replica symmetry breaking (RSB)
- Overlaps
- Non-self-averaging
- Ultrametricity
•
What is the structure of short-range spin glasses?
•
Are spin glasses complex systems?
(Approximate) Timeline
Ca. 1930+
Ordered Systems
(crystals, ferromagnets, superconductors, superfluids, …)
Bloch’s theorem, broken symmetry, Goldstone modes, single order parameter, …
Ca. 1958+
Disordered systems
(glasses, spin glasses, polymers, …)
Localization, frustration, broken replica symmetry, infinitely many order parameters,
metastates …
Ca. 1980+
Complex systems
(Condensed matter physics, computer science, biology,
economics, archaeology, …)
http://sprott.physics.wisc.edu/Pickover/pc/brain-universe.html
What
Phases
is a central
of Matter
bridge
andbetween
Phase Transitions
traditional physics
and complexity studies?
Specific heat C =
Q
T
(amount of heat Q needed to add or subtract to change
the temperature by an amount T )
Phase diagram of water
Order parameters
Quantifies ``amount’’ and ``type’’ of order in a system --- undergoes
discontinuous (in it or its derivatives) change at a phase transition
(fixed pressure)

Discontinuous jump – latent heat
Glasses
The ``Berkeley effect’’
Magnetic Order
In magnetic materials, each atom has a tiny magnetic moment mx
arising from the quantum mechanical spins of electrons in
incompletely filled shells.
These “spins” couple to magnetic fields, which can be external
(from an applied 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, leading to a net zero
field at each site:
T
1
(at high temperature)
mx  lim T   mx (t ) dt  0
T0
This is called the paramagnetic state.
Magnetization is the spatial average of all of the ``local’’ (i.e., atomic)
magnetic moments, and describes the overall magnetic state of the
sample – as such, it serves as a magnetic order parameter.
M
1
1
mx 

N x
N

x
x
So M=0 in the paramagnet in the absence of an external magnetic field.
x
What happens when you lower the temperature?
Single spin orientation at different times –
averages to zero in short time:  x  0
In certain materials, there is a sharp phase transition to a
magnetically ordered state.
What is the nature of the ordering?
•
In some materials (e.g., Fe, Mn), nearby spins ``like’’ to align;
these are called ferromagnets.
•
In others (e.g., Cr, many metal oxides), they like to antialign; these are called antiferromagnets.
••
And
areboth
many
other types
well (ferrimagnets,
Can there
capture
behaviors
withas
a simple
model
canted
ferromagnets,
helical ferromagnets, …)
energy
function (Hamiltonian):
H   J  x y  h  x
x, y
J  0  ferromagne t
x
J  0  antiferrom agnet
Phase
Magnetic
diagram
Phase
forTransitions
ferromagnet
High
Low temperature
temperature
Broken symmetry
J.P. Sethna, Statistical Mechanics: Entropy, Order Parameters,
and Complexity (Oxford U. Press, 2007)
A New State of Matter?
Prehistory: The Kondo Problem (1950’s – 1970’s)
Generated interest in dilute magnetic alloys
Addition of ln(1/T) term to the resistivity
(CuMn, AuFe, …)
Early 1970’s: Magnetic effects seen at greater impurity concentrations
Cannella, Mydosh, and Budnick, J. Appl. Phys. 42, 1689 (1971)
The Solid State Physics of Spin Glasses
Dilute magnetic alloy: localized spins at magnetic impurity sites
M.A. Ruderman and C. Kittel, Phys. Rev. 96, 99 (1954); T. Kasuya, Prog. Theor. Phys. 16, 45 (1956
K. Yosida, Phys. Rev. 106, 893 (1957).
D.L. Stein, Sci. Am. 261, 52 (1989).
Frustration!
J  0  ferromagne t
J
C
xy
J  0  antiferrom agnet
J yz  J zx  0
Ground States
Crystal
Glass
Ferromagnet
Spin Glass
Quenched disorder
Two ``meta-principles’’
1)
For these systems, disorder cannot be treated as a
perturbative effect
2) P.W. Anderson, Rev. Mod. Phys. 50, 191 (1978): ``…there is an important
fundamental truth about random systems we must always keep in mind: no real
atom is an average atom, nor is an experiment ever done on an ensemble of
samples. What we really need to know is the probability distribution …, not (the)
average … this is the important, and deeply new, step taken here: the
willingness to deal with distributions, not averages. Most of the recent progress
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
one, but in a fundamentally different way.’’
``Rugged’’ Energy Landscape
•
Disorder and frustration …
• Many metastable states
M. Goldstein, J. Chem. Phys. 51, 3728 (1969); S.A. Kauffman, The Origins of
Order (Oxford, 1993); W. Hordijk and P.F. Stadler, J. Complex Systems 1, 39
• Many thermodynamic states?
(1998); D.L. Stein and C.M. Newman, Phys. Rev. E 51, 5228 (1995).
C.M. Newman
D.L. Stein,
Rev.``stuck’’
E 60, 5244
• Slowand
dynamics
---Phys.
can get
in a(1999).
local energy minimum
R.G. Palmer, Adv. Phys. 31, 669 (1982).
Is there a phase transition to a ``spin glass phase’’?
yes
t
Cannella, Mydosh, and Budnick, J. Appl. Phys.
42, 1689 (1971)
no
L.E. Wenger and P.H. Keesom, Phys. Rev.
B 13, 4953 (1976).
Aging and Memory Effects
K. Binder and A.P. Young, Rev. Mod. Phys. 58, 801 (1986).
Aging
P. Svedlinh et al., Phys. Rev. B 35, 268 (1987)
So far … lots of nice stuff
•
Disorder
•
Frustration
•
Complicated state space --- rugged energy landscape
•
Anomalous dynamical behavior
-- Memory effects
-- History dependence and irreversibility
•
Well-defined mathematical structure
•
…
Connections
which we’lltostart
other
with
problems
tomorrow.
--- new
insights and techniques