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

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Random Field Ising Model on Small-World Networks

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Random field ising model on small world networks

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

Ising magnet

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)


Rfim on sw networks

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


Random field ising model on small world networks

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


Zero temperature t 0

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).


Magnetic fields distribution

Magnetic fields distribution

  • Bimodal dist.

  • Hat dist.


Finite size scaling

∆c

Finite size scaling

  • Finite size scaling form

  • Limiting behavior


Results on regular networks

Binder cumulant

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 networks1

no phase transition

Results on regular networks

Hat distribution


Results on sw networks

Binder cumulant

Results on SW networks

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

Hat distribution


Results on sw networks1

Results on SW networks

Hat distribution


Random field ising model on small world networks

Results on SW networks

Hat distribution

Second order phase transition


Results on sw networks2

Results on SW networks

Bimodal distribution


Results on sw networks3

Results on SW networks

Bimodal field dist.

First order phase transition


Summary

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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