Spin Glasses and Complexity:
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?
Work done in collaboration with Charles Newman, Courant Institute, New York University
Partially supported by US National Science Foundation Grants DMS-01-02541, DMS-01-02587, and
DMS-06-04869
Broken symmetry in the spin glass
Where we left off: 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 remains a conjecture!
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 (that is, qEA>0)?
Most workers in field think so. If yes:
•
(In
How
other
many
words,
thermodynamic
how many order
phases
parameters
are there?are needed to
describe the symmetry of the low-temperature phase?)
•
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?
The Edwards-Anderson (EA) Ising Model
H
J, h
   J xy  x y   hx  x 
 xy 
x


Coupling
and field
realization
Nearest
neighbor
spins only
 1 in Zd
Site
The fields and couplings are i.i.d. random variables:
1
P( J xy ) 
exp[ J xy 2 / 2]
2
1
P(h xSite
) in Zd 2 exp[hx 2 / 2 2 ]
2
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 low
temperature phase? And how is it affected by the addition of a small
external field?
One guide: the infinite-range Sherrington-Kirkpatrick (SK) model displays an exotic
``…the Gibbs
equilibrium
measure
decomposes
a mixture
of many
pure(RSB).
new type
of broken
symmetry,
known as into
replica
symmetry
breaking
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).
What does this mean?
J. Bürki, R.E. Goldstein and C.A. Stafford, Phys. Rev. Lett. 91, 254501 (2003).
The Parisi solution of the SK model
First feature: the Parisi solution of the SK model has many thermodynamic states!



i

 i



G. Parisi, Phys. Rev. Lett. 43, 1754 (1979); 50, 1946 (1983)
for many i
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.)
Overlaps
andbetween
their distribution
Second feature:
relationships
states are characterized
by their overlaps.
The overlap qβ between 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 .
M. Mézard et al., Phys. Rev. Lett. 52, 1156 (1984); J. Phys. (Paris) 45, 843 (1984)
is a classical field defined on the interval [-L/2,L/2]
Their overlap density is:
It is subject to a potential like
PJ (q)  WW  (q  q )
 ,
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.
Nontrivial
overlap structure:
Non-self-averaging:
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 …
… there are numerous indications that the SK model is pathological and that the
RSB symmetry-breaking structure does not apply to realistic spin glasses.
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).
Other possible scenarios
• • Droplet/scaling
Chaotic pairs (Macmillan,
(Newman and
Bray
Stein):
and Moore,
like RSB,
Fisher
there
and
areHuse):
uncountably
theremany
is only
a single
ofconsists
global spin-reversed
pure
states.
states
, butpair
each
of a single pair
of pure
states.
So RSB is unlikely to hold for any realistic spin glass
model, at any temperature in any finite dimension.
In some ways, this Why?
is an even stranger departure
from the behavior of ordered systems than RSB.
Combination of disorder and physical couplings scaling to zero as N
(Recall the `physical’ coupling in the SK model is Jij/N)
Are Spin Glasses Complex Systems?
•
Most of the ``classic’’ features that earned spin glasses the title
``complex system’’ are still intact:
-- disorder and frustration
-- many metastable states
-- ``rugged energy landscape’’
-- anomalous dynamics (irreversibility, history dependence,
memory effects, aging …
•
•
Connections to problems in computer science, biology,
economics, …
RSB structure can hold in a variety of nonphysical problems (random
TSP, k-SAT, …)
D.L. Stein,• ``Spin
Glasses: as
Stillemergent
Complex After
All These Years?’’, in Quantum
Hierarchies
property!
Decoherence and Entropy in Complex Systems, ed. T.-H. Elze (Springer, 2004).
But there are also some important and interesting newly discovered properties …
•
Singular d→∞ limit
•
Absence of straightforward thermodynamic limit for
states and ``chaotic size dependence’’
-- Analogy between behavior of correlation functions
x, x y , …as volume increases and phase space
trajectory of chaotic dynamical system
-- Led to concept of metastate
•
Connection between finite and infinite volumes far more
subtle than in homogeneous systems
C.M. Newman and D.L. Stein, Phys. Rev. B 46, 973 (1992).