Spin Glasses: Lectures 2 and 3

1
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

2
Ground States
Quenched disorder
3
The Edwards-Anderson (EA) Ising Model
Nearest neighbor spins only
Frustration
The fields and couplings are i.i.d. random
variables
Site in Zd
4
Broken symmetry in the spin glass
EA conjecture Spin glasses (and glasses, ) are
characterized by broken symmetry in time but not
in space.
But remember this was a conjecture!
5
The Sherrington-Kirkpatrick (SK) model
The fields and couplings are i.i.d. random
variables
6
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-Kirkpat
rick (SK) model displays an exotic new type of
broken symmetry, known as replica symmetry
breaking (RSB).
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).
To begin, RSB asserts the existence of many
thermodynamic pure states unrelated by any
symmetry transformation.
Each of these looks random so how does one
describe ordering in such a situation?
What does this mean?
Look at relations between states.
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Thermodynamic States

• A thermodynamic state is a probability measure on
  infinite-volume spin configurations

• Well denote a state by the index a, ß, ?,

• A given state a gives you the probability that at
  any moment spin 1 is up, spin 18 is down, spin
  486 is down,

(These are known as correlation functions.)
8
The Parisi solution of the SK model
First feature the Parisi solution of the SK
model has many thermodynamic states!
G. Parisi, Phys. Rev. Lett. 43, 1754 (1979) 50,
1946 (1983)
9
Overlaps and their distribution
Consider a thermodynamic state that is a mixture
of pure (extremal) Gibbs states
The overlap q?ß between pure states ? and ß in a
volume ?L is defined to be
with
so that, for any ?, ß, -qEA q?ß qEA .
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is a classical field defined on the interval
-L/2,L/2
Their overlap density is
It is subject to a potential like
commonly called the Parisi overlap distribution.
or
Example Uniform Ising ferromagnet below Tc.
classical (thermal)
Now add noise
or quantum mechanical
11
Replica symmetry breaking (RSB) solution of
Parisi for the infinite-range (SK) model
nontrivial overlap structure and
non-self-averaging.
Nontrivial overlap structure
Non-self-averaging
So, when average over all coupling realizations
12
Ultrametricity
Third feature the space of overlaps of states
has an ultrametric structure.
(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)
13
Answer a nested (or tree-like or hierarchical)
structure.
Kinship relations are an obvious example.
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
14
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
15
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)
16
In fact the most straightforward interpretation
of this statement (the standard RSB picture)
--- a thermodynamic Gibbs state ?J 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
cant 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).

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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.
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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

• Relates equilibrium (infinite-volume)
  thermodynamic structure to physical behavior in
  large finite volumes

(Not trivial if many competing states because of
presence of chaotic size dependence of
correlations NS, Phys. Rev. B 46, 973 (1992))
Inspired by analogy with chaotic dynamical systems
A probability distribution over the thermodynamic
states themselves ?J ( G)
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.
19
For fixed J, consider an infinite sequence of
volumes, all with periodic boundary conditions
(for example)
And, when averaged over all volumes
?3

• ?2

1
2
3
?1

• 0

Note This is all for a single coupling
realization.
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Other possible scenarios
TNT (Trivial Edge-Nontrivial Spin) Overlap
Krzakala and Martin, Palassini and Young
Droplet/scaling (McMillan, Bray and Moore,
Fisher and Huse) The PBC metastate is supported
on a single ?, which consists solely of a pair of
global spin-reversed pure states
Chaotic pairs (Newman and Stein) the
metastate is supported on uncountably many ?s,
but each ? consists of a single pair of pure
states.
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
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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 departure
from the behavior of ordered systems than RSB.
Why?
Combination of disorder and physical couplings
scaling to zero as N??
(Recall the physical coupling in the SK model
is Jij/?N)
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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?8 limit
singular (?) universality?
Relationship between large finite volumes and
thermodynamic limit
Creation of new thermodynamic tool the metastate
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If youre interested in learning more, check out
(or better, buy) Spin Glasses and Complexity,
DLS and CMN, Princeton University Press
Thank you!
Questions?
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For fixed J, consider an infinite sequence of
volumes, all with periodic boundary conditions
(for example)

• ?2

?3
?1

• 0

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25

• Is there a phase transition (AT line) in a
  magnetic field?

Scaling/droplet no Chaotic pairs yes
(Presumably)

• T0 behavior of interfaces

26
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?