Studies of braided non-Abelian anyons using anyonic tensor networks

Abstract

The content of this thesis can be broadly summarised into two categories: first, I constructed modified numerical algorithms based on tensor networks to simulate systems of anyons in low dimensions, and second, I used those methods to study the topological phases the anyons form when they braid around one another. In the first phase of my thesis, I extended the anyonic tensor network algorithms, by incorporating U(1) symmetry to give a modified ansatz, Anyon-U(1) tensor networks, which are capable of simulating anyonic systems at any rational filling fraction. In the second phase, I used the numerical methods to study some models of non-Abelian anyons that naturally allows for exchange of anyons. I proposed a lattice model of anyons, which I dubbed anyonic Hubbard model, which is a pair of coupled chains of anyons (or simply called anyonic ladder). Each site of the ladder can either host a single anyonic charge, or it can be empty. The anyons are able to move around, interact with one another, and exchange positions with other anyons, when vacancies exist. Exchange of anyons is a non-trivial process which may influence the formation of different kinds of new phases of matter. I studied this model using the two prominent species of anyons: Fibonacci and Ising anyons, and made a number of interesting discoveries about their phase diagrams. I identified new phases of matter arising from both the interaction between these anyons and their exchange braid statistics.

150 pages, PhD thesis, Macquarie University, Sydney. Chapter 6 of this thesis titled "Phase transitions in braided non-Abelian anyonic system" contains results which are yet to be finalised and published

References in corpus (25)

• Many-Body Physics with Ultracold Gases
• Non-Abelian Anyons and Topological Quantum Computation
• Observation of Majorana Fermions in Ferromagnetic Atomic Chains on a Superconductor
• Real time evolution using the density matrix renormalization group
• Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
• Universal computation by quantum walk
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
• Anyons and the quantum Hall effect - a pedagogical review
• Interacting anyons in topological quantum liquids: The golden chain
• Exotic non-Abelian anyons from conventional fractional quantum Hall states
• Tensor network states and algorithms in the presence of a global U(1) symmetry
• Experimental studies of the fractional quantum Hall effect in the first excited Landau level
• A short introduction to Fibonacci anyon models
• Interferometry of non-Abelian Anyons
• Collective states of interacting Fibonacci anyons
• Phase diagram of hard-core bosons on clean and disordered 2-leg ladders: Mott insulator - Luttinger liquid - Bose glass
• Implementing the one-dimensional quantum (Hadamard) walk using a Bose-Einstein Condensate
• The H=xp model revisited and the Riemann zeros
• Simulation of anyons with tensor network algorithms
• Tensor Product Variational Formulation for Quantum Systems
• Quantum walks on circles in phase space via superconducting circuit quantum electrodynamics
• Anyonic entanglement renormalization
• One-dimensional itinerant interacting non-Abelian anyons
• Statistical dynamics of a non-Abelian anyonic quantum walk