random field ising model on small-world networks
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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
2
What is RFIM ?
ex) 2D square lattice Ising magnet
Quenched Random
Magnetic Field Hi
: Random Fields Ising Model
cf) Diluted AntiFerromagnet
in a Field (DAFF)
),(
0 )(ji i
iijiij sHhssJΗ
Random field Uniform field
3
RFIM on SW networks
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.
L : number of nodesK : number of out-going linksp : random rewiring probability
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
Individuals
SocietyTachy
MSN
Selection of an item = Ising spin state
Preference to a specific item = random field on each node
-Internet & telephone business
-Messenger
-IBM PC vs. Mac
-Key board (QWERTY vs. Dvorak)
-Video tape (VHS vs. Beta)
-Cyworld ?
Social science
5
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 (Max-flow, min-cut).
6
Magnetic fields distribution
Bimodal dist.
Hat dist.
)(5.0)(5.0)( HHHp
HHHpH re whe1
)(2
)(HP
H
)(HPH
H
1
7
Finite size scaling
Finite size scaling form
Limiting behavior
1
LfLm c
cc
cLm
~ ~
exponentlength n Correlatio :ν
exponention Magnetizat :β
∆c
8
Binder cumulant 22
4
31
m
mg
Results on regular networks
0 5 10 15 200.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Mag
netiz
atio
n
m
Magnetic field strength
L100K5P0 L200K5P0 L400K5P0 L800K5P0 L1600K5P0 L3200K5P0 L6400K5P0 L12800K5P0
0 5 10 15 200.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Bin
der
cum
ulan
t g
Magnetic field strength
L100K5P0 L200K5P0 L400K5P0 L800K5P0 L1600K5P0 L3200K5P0 L6400K5P0 L12800K5P0
Hat distributionL (# of nodes) = 100K (# of out-going edges of each node) = 5P (rewiring probability) = 0.0
9
Results on regular networks
0 500 1000 1500 2000 2500
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
m
(-c)*(1/L)-0.5
L100K5P0 L200K5P0 L400K5P0 L800K5P0 L1600K5P0 L3200K5P0 L6400K5P0 L12800K5P0
Hat distribution
no phase transition
0c
10
Results on SW networks
0 5 10 15 20 25 300.0
0.2
0.4
0.6
0.8
1.0
Mag
netiz
atio
n
m
Magnetic field strength
L100K5P5 L200K5P5 L400K5P5 L800K5P5 L1600K5P5 L3200K5P5 L6400K5P5 L12800K5P5
0 5 10 15 20 25 300.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Bin
der
cum
ulan
t g
Magnetic field strength
L100K5P5 L200K5P5 L400K5P5 L800K5P5 L1600K5P5 L3200K5P5 L6400K5P5 L12800K5P5
Hat distribution
Binder cumulant 22
4
31
m
mg
L (# of nodes) = 100K (# of out-going edges of each node) = 5P (rewiring probability) = 0.5
11
Results on SW networks
12 13 14 15 16 17 18
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
Bin
der
cum
ulan
t g
Magnetic field strength
L100K5P5 L200K5P5 L400K5P5 L800K5P5 L1600K5P5 L3200K5P5 L6400K5P5 L12800K5P5
10-4 10-3 10-2
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.80.9
1
Ma
gn
etiz
atio
n m
1/L
13.00 14.50 15.40 17.00 fit of 15.40
Hat distribution
12
Results on SW networks
Second order phase transition
-600 -400 -200 0 200 400 600
10-1
100
m L
( c) L
L=100 L=200 L=400 L=800 L=1600 L=3200 L=6400 L=12800
c=15.40
=0.16=0.44
Hat distribution
13
Results on SW networksBimodal
distribution
4.2 4.4 4.6 4.8 5.0 5.2
0.0
0.2
0.4
0.6
0.8
1.0
Mag
netiz
atio
n m
Magnetic field strength
L100K5P5B L200K5P5B L400K5P5B L800K5P5B L1600K5P5B L3200K5P5B L6400K5P5B L12800K5P5B
4.4 4.8 5.20.0
0.3
0.6
Bin
der
cum
ulan
t g
Magnetic field strength
L100K5P5B L200K5P5B L400K5P5B L800K5P5B L1600K5P5B L3200K5P5B L6400K5P5B L12800K5P5B
14
Results on SW networks
1E-4 1E-3 0.01
0.5
0.6
0.7
0.8
0.9
1
Mag
netiz
atio
n m
1/L
4.65 4.70 4.75 4.80 4.85
First order phase transition
Bimodal field dist.
-20 -10 0 10 2010-2
10-1
100
m(
c) L1/
L100K5P5B L200K5P5B L400K5P5B L800K5P5B L1600K5P5B L3200K5P5B L6400K5P5B L12800K5P5B
c=4.8
1/=0.5
15
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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