Spin Glasses: Lectures 2 and 3
• Review of first lecture
•
Some notions from statistical mechanics
- Finite-volume Gibbs distributions
- Thermodynamic states: pure, mixed, and ground states
• Open questions
•
The Sherrington-Kirkpatrick (SK) infinite-range spin glass model
•
Parisi solution of SK model: Replica symmetry breaking (RSB)
- Overlaps
- Non-self-averaging
- Ultrametricity
•
Summary of RSB solution of SK model
Ground States
Quenched disorder
The Edwards-Anderson (EA) Ising Model
H
J, h
   J xy  x y   hx  x 
 xy 
x


1
Nearest neighbor
spins only
Frustration
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
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!
The Sherrington-Kirkpatrick (SK) model
1
H J,h, N  
 J ij i j   hi  i 
i
N i j
with
i, j  1,, N
The fields and couplings are i.i.d. random variables:
1
P( J ij ) 
exp[ J ij 2 / 2]
2
P( h i ) 
1
2 2
exp[hi 2 / 2 2 ]
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?
One guide: the infinite-range Sherrington-Kirkpatrick (SK) model
displays an exotic new type of broken symmetry, known as
replica symmetry breaking (RSB).
1
H J,h, N  
 J ij i j   hi  i 
i
N i j
``…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
Toglasses,
begin, RSB
thethe
existence
of many
thermodynamic
spin
whereasserts
it received
name of replica
symmetry
breaking. But it
can bepure
defined
and unrelated
easily extended
to other
systems,
by considering an order
states
by any
symmetry
transformation.
parameter function, the overlap distribution function. This function measures
the probability that two configurations of the system, picked up independently
with the Gibbs measure, lie at a given distance from each other. Replica
Each of these looks ``random’’ … so how does one describe ordering
symmetry breaking is made manifest when this function is nontrivial.’’
in such a situation?
S. Franz, M. Mézard, G. Parisi, and L. Peliti, Phys. Rev. Lett. 81, 1758 (1998).
Look
relations
Whatatdoes
this between
mean? states.
Thermodynamic States
•
A thermodynamic state is a probability measure on infinite-volume spin
configurations
•
We’ll denote a state by the index α, β, γ, …
•
A given state α gives you the probability that at any moment spin 1 is
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
 x ,  x y ,  x y z , 
(These are known as correlation functions.)
The Parisi solution of the SK model
First feature: the Parisi solution of the SK model has many thermodynamic states!
rmixed = åWa ra



i

 i



G. Parisi, Phys. Rev. Lett. 43, 1754 (1979); 50, 1946 (1983)
for many i
Overlaps and their distribution
Consider a thermodynamic state that is a mixture of pure (extremal)
Gibbs states:
  ( )  W  ( )

The overlap qβ between pure states  and β in a volume L is
defined to be:
1
q 
L

x L
x
x 

with
1
q 
L

x L
x
2

 qEA
so that, for any , β, -qEA ≤ qβ ≤ qEA .
is a classical field defined on the interval [-L/2,L/2]
Their overlap density is:
PJIt(qis)subject
 WtoWa potential
(q  qlike
)
 ,
commonly called the Parisi overlap distribution.
or
Example: Uniform Ising ferromagnet below Tc.
1
1
    
2
2
Now add noise …
classical (thermal)
or quantum mechanical
Replica symmetry breaking (RSB) solution of Parisi for the infinite-range
(SK) model: nontrivial overlap structure and non-self-averaging.
Non-self-averaging:
Nontrivial
overlap structure:
J21
So, when average over all coupling realizations:
Ultrametricity
Third feature: the space of overlaps
of states has an ultrametric structure.
In an ordinary metric space, any three points x, y, and z must satisfy the
triangle inequality: d ( x, z)  d ( x, y)  d ( y, z )
But in an ultrametric space, all distances obey the strong triangle
inequality: d ( x, z )  max(d ( x, y), d ( y, z ))
which is equivalent to
d ( x, z)  d ( x, y)  d ( y, z)
(All triangles are acute isosceles!)
There are no in-between points.
What kind of space has this structure?
R. Rammal, G. Toulouse, and M.A. Virasoro, Rev. Mod. Phys. 58, 765 (1986)
Answer: a nested (or tree-like or hierarchical) structure.
Kinship relations are an obvious example.
     
      
d  d  d
H. Simon, ``The Organization of Complex Systems’’, in Hierarchy Theory – The Challenge of
Complex Systems, ed. H.H. Pattee, (George Braziller, 1973).
3
4
4
``…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
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
parameter function, the overlap distribution function. This function measures
the probability that two configurations of the system, picked up independently
with the Gibbs measure, lie at a given distance from each other. Replica
symmetry breaking is made manifest when this function is nontrivial.’’
S. Franz, M. Mézard, G. Parisi, and L. Peliti, Phys. Rev. Lett. 81, 1758 (1998).
The four main features of RSB:
1) Infinitely many thermodynamic states (unrelated by any simple symmetry
transformation)
2) Infinite number of order parameters, characterizing the overlaps of the states
3) Non-self-averaging of state overlaps (sample-to-sample fluctuations)
4) Ultrametric structure of state overlaps
Very pretty, but is it right?
And if it is, how generic is it?
•
As a solution to the SK model, there are recent rigorous results
that support the correctness of the RSB ansatz.
•
As for its genericity …
… this is a subject of an intense and ongoing debate.
F. Guerra and F.L. Toninelli, Commun. Math. Phys. 230, 71 (2002); M. Talagrand, Spin
Glasses: A Challenge to Mathematicians (Springer-Verlag, 2003)
In fact: the most straightforward interpretation of this statement (the ``standard
RSB picture’’) --- a thermodynamic Gibbs state ρJ decomposable into pure
states whose overlaps are non-self-averaging --- cannot happen in any finite
dimension.
Reason essentially the same as why (e.g.) one can’t have a
phase transition for some coupling realizations and
infinitely many for others.
Follows from the ergodic theorem for translation-invariant
functions on certain probability distributions.
C.M. Newman and D.L. Stein, Phys. Rev. Lett. 76, 515 (1996);
J. Phys.: Condensed Matter 15, R1319 (2003).
So what sort of “mean field picture” is allowed in short-range spin glasses?
Maximal mean-field picture: “nonstandard RSB scenario” (NS, Phys. Rev. Lett. 76, 4821
(1996) and subsequent publications).
To properly deal with statistical mechanics of spin glasses, need new tool: the metastate
Required because of nonexistence of thermodynamic limit for states due to chaotic size
dependence (NS, Phys. Rev. B 46, 973 (1992)).
M. Aizenman and J. Wehr, Commun. Math. Phys. 130, 489 (1990); C.M. Newman and D.L. Stein, Phys. Rev. Lett.
76, 4821 (1996) and subsequent papers.
Metastates
•
A useful tool for analyzing competition of many
thermodynamic states in a single system
•
Provides a natural framework for understanding
how this (or other) thermodynamic structures
could arise in short-range systems
(Not
• Relates
trivial if equilibrium
many competing
(infinite-volume)
states because
thermodynamic
of presence
Inspired
analogy
with chaotic
systems
structure
of
chaotic
tobyphysical
size
dependence
behavior
in
ofdynamical
large
correlations
finite
volumes
– NS,
Phys. Rev. B 46, 973 (1992))
A probability distribution over the thermodynamic
states themselves: κJ ( )
Metastate: Gibbs state : Gibbs state: Spin configuration
M. Aizenman and J. Wehr, Commun. Math. Phys. 130, 489 (1990); C.M. Newman and D.L. Stein, Phys.
Rev. Lett. 76, 4821 (1996) and subsequent papers.
For fixed J, consider an infinite sequence of volumes, all with periodic boundary
And, when averaged over all volumes:
conditions (for example):
3
3
12
2
1
•
0
Note: This is all for a single coupling realization.
Other possible scenarios
TNT (Trivial Edge-Nontrivial Spin) Overlap: Krzakala and Martin,
Droplet/scaling
(McMillan,and
Bray
and Palassini
Moore,
Fisher
and
Huse): The
metastate
is supported
Chaotic pairs (Newman
Stein):
the metastate
is supported
on PBC
uncountably
many
’s, but
and Young
on a single , which
consists
solely
a pair pair
of global
spin-reversed
pure states:
each
 consists
of of
a single
of pure
states.


1
2


1
2

Extensive numerical work over several decades by Binder, Bray, Domany, Franz, Hartmann, Hed,
Katzgraber, Krzakala, Machta, Marinari, Martin, Mezard, Middleton, M. Moore, Palassini, Parisi,
Young, and many others
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
finite dimension.
In some ways, this is an even stranger
Why? departure from the behavior of ordered
systems than RSB.
Combination of disorder and physical couplings scaling to zero as N
1
H J,h, N  
 J ij i j   hi  i 
i
N i j
(Recall the `physical’ coupling in the SK model is Jij/N)
So … where do we stand?
On the one hand, many of the most basic questions remain unanswered:
existence of a phase transition, number of ground states/pure states,
stability of spin glass phase to magnetic field, …
On the other …
We now understand a great deal about how spin glass states
can (and cannot) be organized
Differences from ordered systems: d→∞ limit singular (?): universality?
Relationship between large finite volumes and thermodynamic limit
Creation of new thermodynamic tool: the metastate
If you’re interested in learning more, check out (or better,
buy) “Spin Glasses and Complexity”, DLS and CMN,
Princeton University Press
Thank you!
Questions?
For fixed J, consider an infinite sequence of
volumes, all with periodic boundary conditions
(for example):
3
2
1
•
<σ4σ57>
0
•
Is there a phase transition (AT line) in a
magnetic field?
Scaling/droplet:
Chaotic pairs:
no
yes
(Presumably)
•
T=0 behavior of interfaces
Open Questions
•
Is there a thermodynamic phase transition to a spin glass phase?
And if so, does the low-temperature phase display
broken spin-flip symmetry?
Most workers in field think so. If yes:
•
How many thermodynamic phases are there?
•
If many, what is their structure and organization?
And in particular – is it mean-field-like?
•
What happens when a small magnetic field is turned on?