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