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Computational and Theoretical Study of Disordered Systems

Cheng, Xiang (2016)
Dissertation (116 pages)
Committee Chair / Thesis Adviser: Boettcher, Stefan
Committee Members: Burton, Justin C ; Family, Fereydoon ; Nagy, James ; Nemenman, Ilya
Research Fields: Physics; Theoretical physics; Condensed matter physics
Keywords: Monte Carlo Methods; Disordered System; Jamming Transition; Ising Model; Glassy Dynamics
Program: Laney Graduate School, Physics
Permanent url: http://pid.emory.edu/ark:/25593/rr5j0

Abstract

In the physical world, there are tremendous more disordered materials than ordered ones. For ordered physical systems, there has been a rich spectrum of well-defined theories, models, phases, and methods; while disordered systems have more questions that are unclear. In this work, we study 3 disordered systems in finite dimensional lattice-like structures, which may contribute new insights comparing to a large amount of work in mean-field-like models.

First, we apply a lattice glass model proposed by Biroli and Mezard onto a number of hierarchical networks. These networks combine certain lattice-like features with a recursive structure that makes them suitable for exact renormalization group studies and provide an alternative to the mean-field approach. We explored both the equilibrium and dynamic behaviors and discover jamming transitions and no phase transitions. This discovery is the first clear-cut evidence of a jamming transition with no phase transition.

Secondly, the antiferromagnetic Ising model (AFM) is a convenient model to introduce disorderedness and glassy dynamics. We study the properties of the Ising antiferromagnet on four hierarchical networks using both Monte Carlo methods and renormalization groups. Exact renormalization group calculations show that the system encounters an infinite-order transition into a glassy state, characterized by a super-critical Hopf bifurcation in coupling-space to chaotic behavior for low temperatures.

Thirdly, random field Ising model (RFIM) is studied to understand the aging in an experimental system, a thin-lm ferromagnet/antiferromagnet (F/AF) bilayer. The experiments show extremely slow cooperative relaxation. In our computational study, the experimental system is coarse-grained into a RFIM on a 2D square lattice. Monte Carlo simulations indicate that the aging process may be associated with the glassy evolution of the magnetic domain walls, due to the pinning by the random fields. The scaling of the simulated aging agrees well with experiments. Both are consistent with either a small power-law or logarithmic dependence on time.

Table of Contents

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Disordered System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Jamming Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Ising models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Antiferromagnetic Ising Model . . . . . . . . . . . . . . . . . . . . 7

1.3.2 Random Field Ising Model . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Hanoi Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Chapter 2 Jamming in Hanoi Networks . . . . . . . . . . . . . . . . . . . . 17

2.1 Lattice glass model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Wang-Landau Sampling . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.2 Grand-Canonical Annealing . . . . . . . . . . . . . . . . . . . . . 21

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.1 Equilibrium Properties . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.2 Dynamic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Conclusions and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Chapter 3 Antiferromagnetic Ising Model in Hanoi Networks . . . . . . 38

3.1 Spin Glass Phase and Chaos in Antiferromagnets . . . . . . . . . . . . . 38

3.2 Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.1 Wang-Landau Sampling . . . . . . . . . . . . . . . . . . . . . . . 40

3.2.2 Simulated Annealing . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.1 RG on Hanoi Networks . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.2 RG with no magnetic field . . . . . . . . . . . . . . . . . . . . . . 45

3.3.3 RG with magnetic filed . . . . . . . . . . . . . . . . . . . . . . . . 50

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.1 Dynamic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.2 Fixed Point Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4.3 Equilibrium Properties . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Chapter 4 Aging in the Two-Dimensional Random Field Ising Model . 75

4.1 Random Field Ising Model and experiments . . . . . . . . . . . . . . . . 76

4.2 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Results and Comparison to Experiments . . . . . . . . . . . . . . . . . . 80

4.4 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Chapter 5 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . 87

5.1 Jamming in Hanoi Networks . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 Antiferromagnetic Ising Model in Hanoi Networks . . . . . . . . . . . . . 89

5.3 Aging in Two-Dimensional Random Field Ising Model . . . . . . . . . . . 90

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