Transcript Random fields Ising model on small

Random Field Ising Model
on Small-World Networks
Seung Woo Son, Hawoong Jeong 1 and Jae Dong Noh 2
1
Dept. Physics, Korea Advanced Institute Science and Technology (KAIST)
2 Dept. Physics, Chungnam National University, Daejeon, KOREA
What is RFIM ?
: Random Fields Ising Model
ex) 2D square lattice
Ising magnet
Quenched Random
Magnetic Field Hi
Random field
Uniform field
Η    J ij si s j   (hi  H 0 )si
(i , j )
i
cf) Diluted AntiFerromagnet
in a Field (DAFF)
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RFIM on SW networks
L : number of nodes
K : number of out-going links
p : random rewiring probability

Ising magnet (spin) is on each node where
quenched random fields are applied. Spin interacts
with the nearest-neighbor spins which are
connected by links.
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Why should we study this problem? Just curiosity +
• Critical phenomena in a stat. mech. system with quenched disorder.
• Applications : e.g., network effect in markets
Social science
Individuals
Society
Tachy
-Internet & telephone business
MSN
-Messenger
-IBM PC vs. Mac
-Key board (QWERTY vs. Dvorak)
-Video tape (VHS vs. Beta)
-Cyworld ?
Selection of an item = Ising spin state
Preference to a specific item = random field on each node
Zero temperature ( T=0 )



RFIM provides a basis for understanding the
interplay between ordering and disorder
induced by quenched impurities.
Many studies indicate that the ordered phase
is dominated by a zero-temperature fixed point.
The ground state of RFIM can be found
exactly using optimization algorithms (Maxflow, min-cut).
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Magnetic fields distribution

Bimodal dist.
p ( H )  0.5 ( H  )  0.5 ( H  )
P (H )


Hat dist.
H

pH ( H ) 
1
  H
2


where    H   
PH (H )
1



H
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Finite size scaling

Finite size scaling form

mL



1



f     c L 


β : Magnetization exponent
ν : Correlation lengthexponent
Limiting behavior
  

m ~ L
    c

 ~  c 
   c 
∆c
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Results on regular networks
L (# of nodes) = 100
K (# of out-going edges
of each node) = 5
P (rewiring probability) = 0.0
Hat distribution
1.1
Magnetization m
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
L100K5P0
L200K5P0
L400K5P0
L800K5P0
L1600K5P0
L3200K5P0
L6400K5P0
L12800K5P0
0.7
Binder cumulant g
L100K5P0
L200K5P0
L400K5P0
L800K5P0
L1600K5P0
L3200K5P0
L6400K5P0
L12800K5P0
1.0
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
5
10
15
20
0
Magnetic field strength 
Binder cumulant
5
10
15
20
Magnetic field strength 
g  1
m4
3m
2 2
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Results on regular networks
Hat distribution
1.0
L100K5P0
L200K5P0
L400K5P0
L800K5P0
L1600K5P0
L3200K5P0
L6400K5P0
L12800K5P0
0.9
0.8
0.7
m
0.6
no phase transition
0.5
c  0
0.4
0.3
0.2
0.1
0.0
0
500
1000
(-c)*(1/L)
1500
2000
2500
-0.5
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Results on SW networks
L (# of nodes) = 100
K (# of out-going edges
of each node) = 5
P (rewiring probability) = 0.5
L100K5P5
L200K5P5
L400K5P5
L800K5P5
L1600K5P5
L3200K5P5
L6400K5P5
L12800K5P5
Magnetization
m
1.0
0.8
0.6
0.4
0.2
0.7
Binder cumulant g
Hat distribution
0.6
0.5
L100K5P5
L200K5P5
L400K5P5
L800K5P5
L1600K5P5
L3200K5P5
L6400K5P5
L12800K5P5
0.4
0.3
0.2
0.1
0.0
0.0
0
0
5
10
15
20
25
30
10
15
20
25
30
Magnetic field strength 
Magnetic field strength 
Binder cumulant
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g  1
m4
3m
2 2
10
Results on SW networks
Hat distribution
0.60
1
0.9
0.8
0.7
0.55
0.6
0.70
Magnetization m
Binder cumulant g
0.65
0.50
0.45
0.40
L100K5P5
L200K5P5
L400K5P5
L800K5P5
L1600K5P5
L3200K5P5
L6400K5P5
L12800K5P5
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.5
0.4

0.3
13.00
14.50
15.40
17.00
0.2
fit of 15.40
0.00
0.1
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13
14
15
16
17
Magnetic field strength 
18
-4
10
-3
-2
10
10
1/L
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Results on SW networks
Hat distribution
=0.16
=0.44
0
10
mL

c=15.40
L=100
L=200
L=400
L=800
L=1600
L=3200
L=6400
L=12800
-1
10
-600
-400
-200
0
200
400
600

( c) L
Second order phase transition
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Results on SW networks
Bimodal distribution
Binder cumulant g
Magnetization m
1.0
0.8
0.6
L100K5P5B
L200K5P5B
L400K5P5B
L800K5P5B
L1600K5P5B
L3200K5P5B
L6400K5P5B
L12800K5P5B
0.4
0.2
0.0
4.2
4.4
4.6
4.8
5.0
Magnetic field strength 
5.2
0.6
0.3
L100K5P5B
L200K5P5B
L400K5P5B
L800K5P5B
L1600K5P5B
L3200K5P5B
L6400K5P5B
L12800K5P5B
0.0
4.4
4.8
5.2
Magnetic field strength 
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Results on SW networks
Bimodal field dist.
0
10
c=4.8
1
1/=0.5
0.8
L100K5P5B
L200K5P5B
L400K5P5B
L800K5P5B
L1600K5P5B
L3200K5P5B
L6400K5P5B
L12800K5P5B
-1
0.7
4.65
4.70
4.75
4.80
4.85
0.6
m
Magnetization m
0.9
10
0.5
-2
10
1E-4
1E-3
0.01
-20
-10
0
10
20
1/
( c) L
1/L
First order phase transition
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Summary

We study the RFIM on SW networks at T=0 using exact
optimization method.

We calculate the magnetization and obtain the magnetization
exponent(β) and correlation exponent (ν) from scaling relation.

The results shows β/ν = 0.16, 1/ν = 0.4 under hat field
distribution.

From mean field theory βMF=1/2, νMF=1/2 and upper critical
dimension of RFIM is 6.  ν* = du vMF = 3 and βMF/ν* = 1/6 ,
1/ν* = 1/3.
R. Botet et al, Phys. Rev. Lett. 49, 478 (1982).
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