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 Goals
Provide a general sense of mathematical and physical research into spin glasses, with
an emphasis on the following questions:
•
•
What is a spin glass?
Why are spin glasses of interest to:
-- Physicists (condensed matter), Mathematicians (probability theory), and Mathematical
Physicists (statistical mechanics)
-- “Complexity scientists’’
•
Canonical model of disorder
•
New computational techniques
•
Application to other problems
•
Generic aspects?
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, …)
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:
mx
T
1
 limT   mx (t ) dt  0
T0
(at high temperature)
This is called the paramagnetic state.
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
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
N
m
x
x

1
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 anti-align; these are called
antiferromagnets.
• • And
there
are many
other types
as awell
(ferrimagnets,
canted
ferromagnets,
helical
Can
capture
both behaviors
with
simple
model energy
function
(Hamiltonian):
ferromagnets, …)
H   J  x y  h  x
x, y
J  0  ferromagnet
x
J  0  antiferromagnet
Phases of Matter and Phase Transitions
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
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)
Early 1970’s: Magnetic effects seen at greater impurity concentrations
Cannella, Mydosh, and Budnick, J. Appl. Phys. 42, 1689 (1971)
Magnetic susceptibility χ = limΔh->0 (ΔM/Δh)
Ground States
Crystal
Glass
Ferromagnet
Spin Glass
Quenched disorder
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).
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).
1975: Theory Rears Its Head
The Edwards-Anderson (EA) Ising Model
H
J, h
   J xy  x y   hx  x 
 xy 
x


1
Nearest neighbor spins only
The fields and couplings are i.i.d. random variables:
1
P( J xy ) 
exp[ J xy 2 / 2]
2
1
2
2
Site
in
Zd
P( h x ) 
exp[

h
/
2

]
x
2
2
Frustration!
J  0  ferromagnet
J
C
xy
J  0  antiferromagnet
J yz  J zx  0
``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 and D.L. Stein, Phys. Rev. E 60, 5244 (1999).
• Slow dynamics --- can get ``stuck’’ in a local energy minimum
R.G. Palmer, Adv. Phys. 31, 669 (1982).
Broken symmetry in the spin glass
EA conjecture: Spin glasses (and glasses, …) are characterized by broken symmetry in
time but not in space.
1
M  lim
N  N
q
EA
1
 lim
N  N
N

i 1
N

i 1
0
Si
Si
2
0
But remember: this was a conjecture!
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
other
problems
--- new
…
which we’lltostart
with
tomorrow.
insights and techniques
Primary question: Why should we be interested?
Spin glasses represent a gap in our understanding of condensed matter.
Question: Why are solids rigid?
Bishop George Berkeley
Dr. Samuel Johnson
1685-1753
1709-1784
“I refute it thus!”
“Ordinary” glasses: no change in symmetry, no phase transition.
S
liquid
supercooled liquid
glass
crystal
TK
Tg
Tf
T
And for spin glasses, the situation is even more problematic.
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.’’