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Random Field Ising Model on Small-World NetworksPowerPoint Presentation

Random Field Ising Model on Small-World Networks

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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

Ising magnet

Quenched Random

Magnetic Field Hi

: 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

Tachy

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

- Internet & telephone business
- 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 form
- Limiting behavior

Binder cumulant

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

Hat distribution

no phase transition

Hat distribution

Binder cumulant

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

Hat distribution

Hat distribution

Results on SW networks

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).