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).