A Renormalization Group Study of the Ising Model on the Hierarchical Hanoi Networks Open Access
Brunson, Clifton Trent (2014)
Abstract
Despite all the remarkable breakthroughs in the area of complex networks over the last two decades, there still lacks a complete and general understanding of effects that occur when long-range connections are present in a system. This thesis explores the Ising model using recursive hierarchical networks called Hanoi networks (HN) as a substrate. Hanoi networks are purely synthetic and are not found in nature, so it is important to establish and not lose sight of why they worth studying. In essence, we are not strictly interested in HNs themselves, but the generalized statements about phase transitions on complex networks that they provide via the renormalization group (RG).
The RG framework on HNs is established in this study and the thermodynamic observables for statistical models are derived from it. Traditionally, the RG has given physicists insight into the critical exponents of a system or model, which leads to universal behavior; however, hyperbolic networks, like the ones currently under investigation, do not contain constant exponents and do not exhibit universality. Instead, it is found that the scaling exponents are functions of the temperature. We ultimately want to answer the questions: What is it about long-range connections that create a break in universal behavior and can complex networks be designed to produce predicted and intended effects in phase behavior? The current state of research is several years or perhaps decades away from fully comprehending the answers to these questions. The research presented here is motivated by these questions, and our contribution here is intended to show a generalized picture of phase transitions on networks.Table of Contents
I INTRODUCTION
1.1 Phases of Matter
1.2 Phase Transitions
1.3 The Ising Model
1.4 Modern Complex Network Science
1.5 Hanoi Networks
1.5.1 HN3
1.5.2 HN5
1.5.3 HNNP
1.5.4 HN6
1.5.5 Previous applications of Hanoi networks
II THE RENORMALIZATION GROUP AND PHASE DIAGRAMS FOR HANOI NETWORKS
2.1 The Renormalization Group on Hanoi Networks
2.1.1 The HN3 and HN5 Ising Hamiltonian (no external field)
2.2 The partition functions and recursion equations for HN3 and HN5
2.3 Phase Diagrams for HN3 and HN5
III FIXED-POINT STABILITY ANALYSIS
3.1 Interpolating Between HN3 and HN5
3.2 The branch point as a function of y
3.3 Fixed-point stability and correlation length
3.4 A generalized theory for parameter-dependent renormalization
IV DERIVATION OF HANOI NETWORK THERMODYNAMIC DENSITIES FROM THE RG
4.1 Hanoi Network Hamiltonian with an External Magnetic Field
4.2 The Derivation of One-Point Functions
4.3 Derivation of Two-Point Functions
4.4 Thetemperature and magnetic exponents
4.5 Breaking the Z2 Symmetry of the Ising Model
4.6 Magnetization of Hanoi Networks according to the RG
4.7 Magnetic susceptibility and specific heat of Hanoi Networks according to the RG
V APPLYING MONTE CARLO METHODS TO HANOI NETWORKS
5.1 The Hanoi Network Data Structure
5.2 The Metropolis and Wolff Algorithms
5.3 Comparison between Monte Carlo and RG results
5.4 Other Monte Carlo results and measurements
VI CONCLUSIONS AND FUTURE WORK
Appendices
I THE MCKAY-HINCZEWSKI-BERKER APPROACH TO CALCULATING THERMODYNAMIC DENSITIES ON HIERARCHICAL LATTICES
II USEFUL MATHEMATICA COMMANDS
B.1 Solving for HN3/HN5 fixed points
B.2 Eliminating μ to derive closed-form expression
B.3 Plotting the κ dependency on μ for HN5
B.4 Calculating yc
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