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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. Ising magnet. Quenched Random

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Random Field Ising Model on Small-World Networks

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  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. 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)

  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

  4. 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

  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.

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

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

  9. no phase transition Results on regular networks Hat distribution

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

  11. Results on SW networks Hat distribution

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

  13. Results on SW networks Bimodal distribution

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

  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).

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