0. Construction of d-dimensional Isotropic and Anisotropic Hierarchical Lattices 1. Frustrated Systems and Chaotic Renormalization- Group

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1 Hierarchical Lattices: Renormalization-Group Solutions of Plain, Anisotropic,Chaotic, Heterogeneous, and Clustered Systems Collaborators: D. Andelman, A. Erbaş, A. Falicov, K. Hui, M. Hinczewski, A. Kabakçıoğlu, S.R. McKay, G. Migliorini, A. Tuncer, D. Yeşilleten, B. Yücesoy Tokyo Institute of Technology 19 October 2006 REB

2 0. Construction of d-dimensional Isotropic and Anisotropic Hierarchical Lattices 1. Frustrated Systems and Chaotic Renormalization- Group Trajectories 2. Random-Bond Systems: Conversion of First-Order Transitions to Second Order 3. Random-Field Systems: Sharpest Second-Order Onset of Magnetization 4. Random-Field Spin-Glasses 5. Scale-Free Small World Systems: Softest Second-Order Onset of Magnetization and High-T BKT Algebraic Order

3 Construction of Frustrated Hierarchical Model: Chaotic Renormalization-Group Trajectories S.R. McKay, ANB, and S. Kirkpatrick, 1982

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6 0. Construction of d-dimensional Isotropic and Anisotropic Hierarchical Lattices 1. Frustrated Systems and Chaotic Renormalization-Group Trajectories 2. Random-Bond Systems: Conversion of First-Order Transitions to Second Order -βh = Σ ij [J ij s i s j + Ks i2 s j2 (s i2 +s j2 )], s i =±1,0 3. Random-Field Systems: Sharpest Second-Order Onset of Magnetization 4. Random-Field Spin-Glasses 5. Scale-Free Small World Systems: Softest Second-Order Onset of Magnetization and High-T BKT Algebraic Order

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9 F P d=2 A. Falicov and ANB, 1996

10 F m P F+P r m d=3

11 Renormalization-group unstable fixed distribution: random bond Strong coupling

12 0. Construction of d-dimensional Isotropic and Anisotropic Hierarchical Lattices 1. Frustrated Systems and Chaotic Renormalization-Group Trajectories 2. Random-Bond Systems: Conversion of First-Order Transitions to Second Order 3. Random-Field Systems: Sharpest Second-Order Phase Transitions -βh = Σ ij (Js i s j + H i s i + H j s j ), H i =±H, s i =±1 4. Random-Field Spin-Glasses 5. Scale-Free Small World Systems: Softest Second-Order Onset of Magnetization and High-T BKT Algebraic Order

13 A. Falicov, ANB, and S.R. McKay, 1995

14 Renormalization-group unstable fixed distribution: random field Strong coupling

15 Magnetization critical exponent β = ±0.0005

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17 0. Construction of d-dimensional Isotropic and Anisotropic Hierarchical Lattices 1. Frustrated Systems and Chaotic Renormalization-Group Trajectories 2. Random-Bond Systems: Conversion of First-Order Transitions to Second Order 3. Random-Field Systems: Sharpest Second-Order Onset of Magnetization 4. Random-Field Spin-Glasses -βh = Σ ij (J ij s i s j + H i s i + H j s j ), J ij =±J, H i =±H, s i =±1 5. Scale-Free Small World Systems: Softest Second-Order Onset of Magnetization and High-T BKT Algebraic Order

18 Random-field spin glass d = 3 G. Migliorini and ANB, 1998

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20 Nishimori Lines

21 Also proves that the spin-plass phase is unstable to a uniform magnetic field.

22 d = 3 Unstable Doubly unstable Unstable

23 Double Crossover m m Strong violation of universality

24 d = 2 Unstable Doubly unstable

25 Asymmetric spin glass 3 double crossovers J m -J/4

26 <s i s j > D. Yeşilleten and ANB, 1997

27 <s i >

28 0. Construction of d-dimensional Isotropic and Anisotropic Hierarchical Lattices 1. Frustrated Systems and Chaotic Renormalization-Group Trajectories 2. Random-Bond Systems: Conversion of First-Order Transitions to Second Order 3. Random-Field Systems: Sharpest Second-Order Onset of Magnetization 4. Random-Field Spin-Glasses 5. Scale-Free Small World Systems: Softest Second-Order Onset of Magnetization and High-T BKT Algebraic Order

29 M. Hinczewski and ANB, 2006

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61 "Renormalisation-Group Calculations of Finite Systems: Order Parameter and Specific Heat for Epitaxial Ordering" A.N. Berker and S. Ostlund, J. Phys. C 12, (1979). "Spin-Glass Behavior in Frustrated Ising Models with Chaotic Renormalization-Group Trajectories" S.R. McKay, A.N. Berker, and S. Kirkpatrick, Phys. Rev. Lett. 48, (1982). "Hierarchical Models and Chaotic Spin Glasses" A.N. Berker and S.R. McKay, J. Stat. Phys. 36, (1984). List available on web "Random-Field Mechanism in Random-Bond Multicritical Systems" K. Hui and A.N. Berker, Phys. Rev. Lett. 62, (1989). "Critical Behavior Induced by Quenched Disorder" A.N. Berker, Physica A 194, (1993). Renormalization-Group Theory of the Random-Field Ising Model in Three Dimensions A. Falicov, A.N. Berker, and S.R. McKay, Phys. Rev. B 51, (1995). Tricritical and Critical-Endpoint Phenomena under Random Bonds A. Falicov and A.N. Berker, Phys. Rev. Lett. 76, (1996). Renormalization-Group Calculation of Local Magnetizations and Correlations: Random-Bond, Random-Field, and Spin-Glass Systems D. Yeşilleten and A.N. Berker, Phys. Rev. Lett. 78, (1997). Global Random-Field Spin-Glass Phase Diagrams in Two and Three Dimensions G. Migliorini and A.N. Berker, Phys. Rev. B 57, (1998). Strongly Asymmetric Tricriticality of Quenched Random-Field Systems A. Kabakçıoğlu and A.N. Berker, Phys. Rev. Lett. 82, (1999). Phase Diagrams and Crossover in Spatially Anisotropic d=3 Ising, XY Magnetic and Percolation Systems: Exact Renormalization-Group Solutions of Hierarchical Models A. Erbaş, A. Tuncer, B. Yücesoy, and A.N. Berker, Phys. Rev. E 72, , 1-6 (2005). Inverted Berezinskii-Kosterlitz-Thouless Singularity and High-Temperature Algebraic Order in an Ising Model on a Scale-Free Hierarchical-Lattice Small-World Network M. Hinczewski and A.N. Berker, Phys. Rev. E 73, , 1-22 (2006).

62 Massachusetts Institute of Technology

63 Koç University, Istanbul

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