• Introduction
• Interacting Quantum Many Body Systems
• Mott Insulators
• Quantum Spin Liquids
• Outline
• Background
• The Falikov Kimball Model
• The Model
• Phase Diagram
• Long Range Ising Models
• The Kitaev Honeycomb Model
• The Model
• A mapping to Majorana Fermions
• Gauge Fields
• Anyons, Topology and the Chern number
• Phase Diagram
• Disorder & Localisation
• Localisation: Anderson, Many Body and Disorder-Free
• Disorder and Spin liquids
• Amorphous Magnetism
• Chapter 3: The Long Range Falikov-Kimball Model
• Chapter Summary
• The Model
• Localisation
• Falikov Kimball and Hubbard models
• Long Ranged Ising model
• Localisation
• Introduction
• The Long-Ranged Falikov-Kimball Model
• Introduction and Motivation
• Literature Review
• The Falikov-Kimball model
• Markov Chain Monte Carlo
• Sampling
• Markov Chains
• Application to the FK Model
• The Metropolis-Hasting Algorithm
• Metropolis-Hastings
• Convergence, Auto-correlation and Binning
• Applying MCMC to the FK model
• Proposal Distributions
• Perturbation MCMC
• Scaling
• Binder Cumulants
• Markov Chain Monte-Carlo in Practice
• Two Step Trick
• Detailed Balance for the two step method
• Diagnostics of Localisation
• Markov Chain Monte-Carlo
• Convergence Time
• Auto-correlation Time
• The Metropolis-Hastings Algorithm
• Choosing the proposal distribution
• Two Step Trick
• The Phase Diagram
• Localisation Properties
• Discussion & Conclusion
• Acknowledgments
• []{#app:disorder_model label="app:disorder_model"} UNCORRELATED DISORDER MODEL
• Chapter 4: The Amorphous Kitaev Model
• Gauge Fields
• The Projector
• The Ground State
• Phases of the Kitaev Model
• What is so great about two dimensions?
• Contributions
• Introduction
• Amorphous Systems
• Glossary
• The Kitaev Model
• The Majorana Hamiltonian
• Methods
• Voronisation
• Graph Representation
• Colouring the Bonds
• Mapping between flux sectors and bond sectors
• Chern Markers
• Results
• The Ground State Flux Sector
• Spontaneous Chiral Symmetry Breaking
• Ground State Phase Diagram
• Anderson Transition to a Thermal Metal
• Conclusion
• Discussion
• Limits of the ground state conjecture
• Outlook
• Experimental Realisations and Signatures
• Generalisations
• Conclusion
• Discussion
• Outlook
• Appendices
• Markov Chain Monte Carlo
• Lattice Generation
• Lattice Colouring
• The Projector