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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.

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Presentation on theme: "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."— Presentation transcript:

1 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 2 What is RFIM ? ex) 2D square lattice Ising magnet Quenched Random Magnetic Field H i : Random Fields Ising Model cf) Diluted AntiFerromagnet in a Field (DAFF) Random field Uniform field

3 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 nodes K : number of out-going links p : random rewiring probability

4 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 Society Tachy 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 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 6 Magnetic fields distribution Bimodal dist. Hat dist.

7 7 Finite size scaling Finite size scaling form Limiting behavior ∆c∆c

8 8 Binder cumulant Results on regular networks Hat distribution L (# of nodes) = 100 K (# of out-going edges of each node) = 5 P (rewiring probability) = 0.0

9 9 Results on regular networks Hat distribution no phase transition

10 10 Results on SW networks Hat distribution Binder cumulant L (# of nodes) = 100 K (# of out-going edges of each node) = 5 P (rewiring probability) = 0.5

11 11 Results on SW networks Hat distribution

12 12 Results on SW networks Second order phase transition Hat distribution

13 13 Results on SW networks Bimodal distribution

14 14 Results on SW networks First order phase transition Bimodal field dist.

15 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 β MF =1/2, ν MF =1/2 and upper critical dimension of RFIM is 6.  ν* = d u v MF = 3 and β MF /ν* = 1/6, 1/ν* = 1/3. R. Botet et al, Phys. Rev. Lett. 49, 478 (1982).


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