# Free-energy barriers in spin glasses mean-field vs short-range models

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```					Free-energy barriers in spin glasses:
mean-field vs short-range models

Elmar Bittner
Universität Leipzig

Collaborators: A. Nußbaumer and W. Janke

Spring School on Monte Carlo Simulations of Disordered Systems
Leipzig, April 4, 2008.
Outlook

●
Intro
●
Models
●
Problems
●
Algorithms
●
Motivation
●
Results
Spin Glass Systems
●
There are real experimental spin glass systems.
(dilute solutions of magnetic transition metal impurities in noble metal
hosts, for instance Au-2.98% Mn or Cu-0.9% Mn)

cos2 k F R
RKKY interaction:   J eff R= J0         3
,k F R≫1, k F Fermi wave number
R
Basic ingredients for spin-glass behaviour
●
randomness in course of the dilution process the
positions of the impurity moments are randomly
distributed
●
competing interactions due to the oscillations in the
effective interaction as a function of the distance R

cos2 k F R
RKKY interaction:   J eff R= J0         3
,k F R≫1, k F Fermi wave number
R
Spherical Cow

Most theory uses the simplest model with these
ingredients:

the Edward-Anderson Model (EA)

H=− ∑ J ij S i S j −∑ hi Si
〈i , j 〉       i

with Si =±1 lie on a regular lattice and the quenched
coupling constants Jij .
Spherical Cow

Most theory uses the simplest model with these
ingredients:

the Edward-Anderson Model (EA)

H=− ∑ J ij S i S j −∑ hi Si
〈i , j 〉                   i

bimodal distribution Jij =±1

3D: T c ~1.16, hi =0
[M. Palassini and S. Caracciolo, Phys. Rev. Lett. 82, 5128 (1999)]
no solution
What is a spin glass?

A system with disorder (randomness) and frustration.

+

+               +

-
What is a spin glass?

A system with disorder (randomness) and frustration.

+

+               +

-
Another Cow
fully connected

the Sherrington-Kirkpatrick Model (SK)

H=−∑ Jij S i S j −∑ hi S i
i j               i

Gaussian distribution with:
〈 J ij 〉=0
2          J
2
〈 J 〉−〈 J ij 〉 =
ij
N
N is the number of spins.
T c =1, hi =0
mean field, Parisi's replica solution [PRL 43 (1979) 1754]
Overlap parameter
N
1      1 2
q=     ∑
N i=1
Si S i
1      2

for to (real) replica S , S and given coupling constants J={ J ij }
1
i
2
i

P J q               probability density of q

x J q=∫ dq ' P J q ' cumulative distribution of P J q

average over the disorder
1
P q=[P J q]av =     ∑ P J q
NJ J
1
x J q=[ x J q]av = ∑ x J q
NJ J
Slow Dynamics
(free) energy

barrier

The dynamics is very
slow at low T.
System is not in
equilibrium due to                             DE
complicated energy
landscape: system
trapped     in   one
“valley”   for  long             valley
times.                                               valley

configuration
Algorithms

Parallel tempering (PT)

Exchange at regular intervals                           T1
system i and i+1 with                                        T2
T3
T4

P i ,i1=min [1, exp   E ]

expectation values for single system:

〈 A〉 T =〈 Ai 〉
i
System can
decorrelate at high T

[K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. 65 (1996) 1604]
Talk of H. Katzgraber
Algorithms

Parallel tempering (PT)

Exchange at regular intervals
system i and i+1 with

P i ,i1=min [1, exp   E ]

expectation values for single system:

〈 A〉 T =〈 Ai 〉
i

[K. Hukushima and K. Nemoto, J. Phys. Soc. Jpn. 65 (1996) 1604]
Talk of H. Katzgraber
Algorithms

Multioverlap Algorthim (MuQ)

non-Boltzmann sampling with
multioverlap weights W q:

exp[− H]W q

canonical expectation values:

〈W O 〉
can
〈O 〉 =
〈W 〉                                System can reach highly
suppressed states

[B. Berg, W. Janke, PRL 80 (1998) 4771]
Algorithms

Multioverlap Algorthim (MuQ)                                              iteration 1-9:

non-Boltzmann sampling with
mutlioverlap weights W q:
W 0 q=1

exp[− H ]W q                           simulation
i=i1   iteration 20-28:
canonical expectation values:                          H i q
W i1 q=
W i q
〈W O〉
can
〈O〉 =
〈W 〉                             sampling

[B. Berg, W. Janke, PRL 80 (1998) 4771]
The 2D Ising Model

H=− ∑ J Si S j
〈i , j 〉

Si =±1

2
T c=               =2.269 ...
log  21
1
2       −4
m0=1−sinh   8
T
Onsager solution, Onsager-Yang solution
The 2D Ising Model

H=− ∑ J Si S j
〈i , j 〉

Si =±1
T =1.95T c
2
T c=
log  21
1
2       −4
m0=1−sinh  8
T

L=40
The 2D Ising Model

multimagnetical simulation
The 2D Ising Model

multimagnetical simulation
The 2D Ising Model

But there are still problems:

droplet/strip

evaporation/
condensation
The 2D Ising Model

But there are still problems:
The 2D Ising Model

But there are still problems:

T. Neuhaus and J. S. Hager, J. Stat. Phys. 116 (2003) 47,
Algorithms
combination of both methods: PT-MuQ

iterate to improve
weights
simulate (e.g.) 1 sweep with MuQ
at N different temperatures
H i q
W i1 q=
W i q                               {
min 1, exp[−m  H ]
W 'm q
W m q    }
no!

weights OK?                                           exchange replica

yes!

sample with fixed
{
min 1, exp [ n−m  E n−E m  ]
W m qn W n qm 
W m qm W n qn    }
weights

[E. Bittner, A. Nußbaumer, W. Janke, in preperation]
Algorithms
combination of both methods: PT-MuQ

iterate to improve
weights

H i q
W i1 q=                          ●
weights “belong” to a temperature m   Wm
W i q
●
if every replica is simulated on a
no!
different computer, all nodes have

weights OK?                            to have all weights

yes!                      ●
in reality weights are computed with
a more “sophisticated” approach
sample with fixed
weights

[E. Bittner, A. Nußbaumer, W. Janke, in preperation]
Algorithms

EA model, V=8x8x8

SK model, N=512
Slow Dynamics
(free) energy

barrier

The dynamics is very
slow at low T.
System is not in
equilibrium due to                             DE
complicated energy
landscape: system
trapped     in   one
“valley”   for  long             valley
times.                                               valley

configuration
Main objective: barrier heights
2D Ising Model

H=− ∑ J Si S j                        P max  L
〈i , j 〉                                    ~exp [ F B  L ] ≡ L
P min  L

Si =±1                   P max  L           F B~  L~2 L

L=30
T T c

P min L
L=100
Main objective: barrier heights
Spin glasses:

T T c

How do we measure
the size of the
largest barrier?
1d Markov chain/transition matrix
Definition:

                                               
1−w1,2    w1,2         0                           ⋯
w2,1  1−w2,1−w2,3    w2,3                         ⋯
T=   0       w3,2     1−w3,2−w3,4                     ⋯
0        0          w4,3                         ⋯
⋮        ⋮           ⋮                           ⋱

1
wi , j = min 1,
2           [
P q j 
P qi             ]
[B.A. Berg, A. Billoire and W. Janke, PRB 61 (2000), 12143]
1d Markov chain/transition matrix
Definition:

                                     
1−w1,2    w1,2          0            ⋯
w2,1  1−w2,1−w 2,3    w 2,3         ⋯
T=   0       w3,2      1−w3,2−w 3,4     ⋯
0        0           w 4,3         ⋯
⋮        ⋮            ⋮            ⋱

T fulfills detailed balance   only real eigenvalues
q      1
autocorrelation time for q: B=
N log1 
1 second largest eigenvalue 0=1
Motivation
theoretical predictions for mean-field model (SK):
barrier between time-reversed states scales with
system size as
N1/3
(Rodgers and Moore, 1988)
(Kinzelbach and Horner, 1991)

results for short-ranged models (EA) are far away
from the mean-field theory limit

c 1c 2 lnN

(Berg, Billoire, and Janke, 2000)
Results for the SK Model

fit rage
B ∝expcN  
Results for the SK Model

                         fit rage
B ∝expcN 

non-self-averaging (SK)
[L.A. Pastur and M.V. Shcherbina, J. Stat. Phys. 62 (1992) 1]
Results for the SK Model

extreme-value distribution

fat tails

Fit integrated probability density:

[                 ]
−1/
x−          T T c   0
F  ;  ;   x=exp − 1
                    fat tailed (algebraic)
Fréchet distribution
Results for the SK Model
Results for the SK Model
Peaked probability distribution D  F B / F B 
med
Results for the SK Model
Peaked probability distribution D  F B / F B 
med

no scaling,
i.e. non-self-averaging
Results for the SK Model
Peaked probability distribution D  F B / F B 
med

scaling,
i.e. self-averaging

(EA Model)
Results for the EA Model

B ∝expcN 

but goodness of fit ...
Results for the EA Model

B ∝expcN 

but goodness of fit ...
Results for the EA Model

B ∝expcN 

but goodness of fit ...


therefore, different fit: B ∝cN
Results for the EA Model

B ∝expcN 

but goodness of fit ...


therefore, different fit: B ∝cN
Results for the EA Model

SK Model

Fit integrated probability density:                    EA Model

[                 ]
−1/
x−          T T c   0
F  ;  ;   x=exp − 1
                    fat tailed (algebraic)
Fréchet distribution
Results for the EA Model
Peaked probability distribution D  F B / F B 
med

SK Model

no scaling,
i.e. non-self-averaging
EA Model
Conclusion
●
Algorithmic
- PT is good to decrease the autocorrelation time
- MuQ gives the full P q information
- the combination of PT+MuQ makes it possible
to get P q down to T ≈0.5 T c

●
Physical
- the free energy barriers of the SK and EA model are
a) non-self-averaging
b) follow the Fréchet extreme-value distribution
-   the free energy barriers of the SK model diverge with
an exponent of =1/3
-   the last is not true for the EA model
Thank you!

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