Leuven, September 2005
Random Graphs 2005
Phase diagram of Ising spin glasses on random graphs
at zero temperature
Some analytical results....
...and few numerical checks !
Florent KRZAKALA
+ T. Castellani & F. Ricci-Tersinghi
Laboratoire de Physico-Chimie Théorique,
Ecole Supérieure de Physique et Chimie Industrielles,
Paris
Visit my web page www.pct.espci.fr/~florent
J ij S i S j h S i
H
Spin glasses on random graphs
ij
i
The usual spin glass Hamiltonian
On random graph with fixed connectivity
In this work : Jij = ±1
Fully-connected model (Sherrington-Kirkpatrick) exactly solved :
Parisi 82
Fully connected model quite unphysical, finite connectivity model needed
Viana-Bray 85, Kwon-Thouless 88,...
Bethe lattice and random graph extensively studied in the 80s
Standard replica computation very hard ! Only replica symmetric or variational solutions
Cavity method for RS and 1RSB allows now to study the problem
Mezard-Parisi 01-03
The question of this talk:
What is the phase diagram of spin glass models
on random graphs (at zero temperature )?
The model and its replica symmetric solution
H
The model
ij
J ij S i S j
The usual spin glass Hamiltonian
On random graph with fixed connectivity 3
Bimodal Jij = ±1 , with distribution
where ρis in [-1,1]
And the RS cavity method
Graph is locally tree-like ->
Recursion on the value of
the local field/message
received/sent from a spin to another
Fields h and messengers u
1
1
0
0
i 0
ui
Simple relation between field and messengers : h
k
Spin 1 see a cavity field h...
... and send a message u
to spin 0
u sign Jh
Simple relation between messenger and field:
Replica Symmetric recursion
Where EJ represents the average over disorder,
P(h) the distribution pdf of h over the system and
Q(u) is the distribution pdf of u over the system.
u are 1,0 or -1, so Q(u) = p0 δ(u) + p+ δ(u-1)+ p- δ(u+1),
with p0 + p+ + p- =1
The RS solution depends only on 2 parameters determined with self-consistent equations
H
The Replica symmetric phase diagram
ij
J ij S i S j
T
Self-consistent equations
for Q(u) are
h
As first found by Kwon and Thouless (88)
Spin glass on a fixed connectivity
c=3 random graph
,where
Two different solutions (and a trivial paramagnetic one)
Spin glass solution
Ferromagnet solution
Always Exists
Exists only for ρ>3/4
mRS= 0
p0 = p+ = p- =1/3
H
The Replica symmetric phase diagram
ij
As first found by Kwon and Thouless (88)
J ij S i S j
Spin glass on a fixed connectivity
c=3 random graph
T
h
ρ
Spin glass
-1
Ferro
0
1
Second order phase transition
from spin glass to ferromagnet order
+
Antiferromagnet on random graph
is a spin glass
BUT...
... we definitely know that the RS solution is quite wrong !
Unphysical Instability towards
Non integer distributions of P(h)
(from Mezard-Parisi 2001)
Need for RSB description !
RS
1 RSB
2 RSB
The 1RSB solution
N e
e
Many different independents pure states:
1RSB hypothesis
e N
Messages and field fluctuates states to states
The probabilistic distribution Qi->j(u), on each edge i->j
Thus a distribution of distribution is needed : Q[Q(u)] ---and P[P(h)]---
where Σ is called the complexity.
Inside a state, clustering property holds, thus cavity recursion holds.
Computation of the Q[Q(U)] ---and P[P(h)]--- via numerical algorithm (population
dynamics)
RS
1 RSB
2 RSB
Mezard-Parisi, J. Stat. Phys 111 (2003) 1
Numerical test of the Mezard-Parisi Computation
S. Boettcher. (2003).
Notice the N2/3 finite size correction
to the energy: they are universal from SK
to Bethe lattices
A riddle for analytical computations ?
Note that antiferomagnet gives similar results....
J.P. Bouchaud, F.K. & O.C. Martin, Phys. Rev. B 68, 224404 (2003).
1RSB versus RS magnetization
How good is the 1RSB solution ?
Comparison with simulations
are quite satisfactory
Population dynamics in presence
of ferromagnetic bias
T. Castellani, F.K. & F. Ricci-Tersinghi,
to be published in EPJ B (2005).
The RS stability and the mixed phase
RS description :
All configurations groups into one
cluster where cavity recursions are
valid
1RSB description :
Configurations groups into many
clusters where cavity recursions are
valid only inside.
Instability :
Fragmentation of states
What does it mean for a given message ?
When ε goes to zero
(1-ε) Q+
Q+
εQ-
This instability correspond to the proliferation of complex Q(u) within the sample
Computing the instability
Many papers study the 1RSB->2RSB instability
Montanari, Parisi, Ricci-Tersengh (K-Sat, p-spin, 2004) O.
Rivoire et al. (2004, glasses), M. Mueller et al. (2004,
heteropolymers), Mertens, Mezard,Zechinna (2005 K-sat),
Krzakala, Pagnani, Weigt (Coloring, 2004)
Similar computation holds for RS instability
See how the perturbation increases when cavity equations are iterated.
Stability if the largest eigenvalue < 1/k :
We find the RS is broken for ρ<5/6~0.833...
“BUGS” PROLIFERATION
Study of a 6² elements matrix
If a given message is changed, does this change propagate all along the system ?
The phase diagram for zero external field
Spin glass
ρ
-1
Mixed Ferro
0
1
~0.715
5/6~0.8333
Critical point for larger connectivities
The RS stability in presence of a magnetic field
Computing the instability with rational field
If the external field is rational, then the RS P(h) is still made of delta functions
Ex : B=0.5, k=2, then P(h) takes values in {1.5,-1,-0.5,0,0.5,1,1.5,2,2.5}
The previous method (proliferation of bugs) still applied.
The matrix being sparse, its diagonalization is easy numerically as far as
only the largest eigenvalue is needed
(even for H=0.001, where stability matrix is ~1000²)
The zero temperature
AT line for different
connectivity
Another way of computing the complexity
Testing the clustering property
Already in the 80s, it was noticed that spin glasses on tree are ill-definned
because of crucial dependence towards boundary conditions
This is why Random graph have to be used !
Testing the dependence towards
boundary conditions
=
Testing the clustering property of the
single pure state
=
Testing the Replica-Symmetry
hypothesis
Another way of computing the complexity
Does 2 uncoupled replicas converge to the same state ?
Equivalent to the ε-coupling method of Pagnani, Parisi, Ratieville 2003
Easy numerical implementation : run the RS cavity recursion on 2 replica
and see if you cconverge to the same results or not
Phase diagram at zero temperature
External magnetic field h
Excess concentration of ferromagnetic bond ρ
±J model on a c=3 Bethe Lattice
Analytical computation of the zero temperature dAT line
Stability analysis of the RS Solution.
Evolution of two uncoupled replicas .
We obtain the same results with both methods
Phase diagram at zero temperature
(Here for c=6) : the final result
A final remark on the instability
Measuring “droplet” energies in mean field spin glasses:
Random graph with fixed connectivity: rare
loops, no boundary...
... On a Tree ?
How to compute E(n), the average energy of a droplet of n spins...
... On a random graph ?
Finding the ground state is NP-Hard: use a
heuristic algorithm...
Cayley tree with random fixed boundary
conditions:
Polynomial efficient algorithm. Big sizes,
few finite size effects, huge number of
instances...
Equivalent to a Replica Symmetric
approximation...
Then find the best excitation by an
exhaustive search.
Energetic properties of mean field “droplets”
C=3 Gaussian fixed connectivity spin glass
Θ=0
with A,B et α>0
)
(
'
&
!
"
The θ exponent is exactly zero in mean field spin glasses
#
E n A Bn
$
%
RS “tree” result seems to be an
upper limit.
E(n) decreases and saturates.
Same results in magnetic field if
B<BdAT...
...for +/- J spin glasses...
...and for other connectivities.
mean field “droplets” in magnetic field
C=3 Gaussian fixed connectivity spin glass
Bc ~ 0.48 (Like Pagnani et al. 2004)
The RS instability correspond to the apparition of finite energy droplets of all size
Full RSB <-> Marginal droplets
Geometrical nature of mean field “droplets”:
The end to end distance
“Entropic” guess: What are the most common clusters ?
Lattice animals (branched polymers)
Are Mean field droplets lattice animals ?
+
Extreme:
l n
l n
,
*
Typical:
-
End to end distance on a tree:
n
n ln n
Work perfectly !
Test the end to end distance
Conclusions & perspectives
0
/
.
Computation of the zero temperature phase diagram of dilute spin glasses
Presence of a mixed phase (1RSB match well numerical simulations)
Presence of dAT line with “exotic” behavior
Two different methods agree perfectly
3
2
1
For the future ?
Why is the AT line singular ?
Is there a mixed phase in finite dimension ?
Random field Ising model ?
My articles relevant to this talk
T. Castellani, F.K. & F. Ricci-Tersinghi, To be published in EPJ B (2005).
F.K. Progress of Theoretical Physics, Supp. 157 (2005) 77-81 .
F.K. & G. Parisi, Europhys. Lett (2005).
J.P. Bouchaud, F.K. & O.C. Martin, Phys. Rev. B 68, 224404 (2003).
A parenthesis on 3d spin glasses
(
Phase diagram at zero temperature
Droplet/scaling approach vs random graphs computations
Fraction of ferromagnetic bonds :
(1+ρ)/2
Fraction of anti-ferromagnetic bonds : (1-ρ)/2
Ferromagnetic phase if |m| > 0
Prediction from Bethe lattice for the 3d model
Prediction from Scaling approach via
Migdal-Kadanoff Renormalization Group
for the 3d model
9
J ij S i S j h S i
i
B dAT
4
B
Droplet-like scaling very close to the best scaling
6
5
Rescale the sponge fraction by Lθ.
Rescale the field by L1/ν, where ν ~ (d-2θ)/2
No good evidence for spin glass phase in field
Data compatible with its absence
Bc is certainly very small,
if non zero...
3d
dAT
0.6
;
ij
:
H
87
In Magnetic field: In search of the dAT line
B
MF
dAT
@
1 x J ij x S i S j
?
<=
H
>
Competition ferro-antiferro:In search of a mixed phase in 3d
ij
2 critical points
ρSG & ρF ?
3dEA: ρSG =ρF=0.385(0.005)
3dmodel with 2nd neighbors interactions
ρSG =ρF=0.24(0.005)
A parenthesis on 3d spin glasses
No convincing evidence for a mean field phase diagram in 3d
4d simulations are welcome !
)