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# Random Field Ising Model on Small-World Networks - PowerPoint PPT Presentation

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. Ising magnet. Quenched Random

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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

Quenched Random

Magnetic Field Hi

What is RFIM ?

: Random Fields Ising Model

ex) 2D square lattice

Uniform field

Random field

cf) Diluted AntiFerromagnet

in a Field (DAFF)

• 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

MSN

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

Society

• Messenger

• IBM PC vs. Mac

• Key board (QWERTY vs. Dvorak)

• Video tape (VHS vs. Beta)

• Cyworld ?

Individuals

Selection of an item = Ising spin state

Preference to a specific item = random field on each node

• 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).

• Bimodal dist.

• Hat dist.

c

Finite size scaling

• Finite size scaling form

• Limiting behavior

Results on regular networks

L (# of nodes) = 100K (# of out-going edges of each node) = 5P (rewiring probability) = 0.0

Hat distribution

Results on regular networks

Hat distribution

Results on SW networks

L (# of nodes) = 100K (# of out-going edges of each node) = 5P (rewiring probability) = 0.5

Hat distribution

Hat distribution

Hat distribution

Second order phase transition

Bimodal distribution

Bimodal field dist.

First order phase transition

• 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).