Studies of braided non-Abelian anyons using anyonic tensor networks
arXiv:1708.06476
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.
arXiv comment150 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 (69)
- Many-Body Physics with Ultracold Gases
- Fault-tolerant quantum computation by anyons
- Non-Abelian Anyons and Topological Quantum Computation
- Anyons in an exactly solved model and beyond
- Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
- The density-matrix renormalization group
- Efficient classical simulation of slightly entangled quantum computations
- Observation of Majorana Fermions in Ferromagnetic Atomic Chains on a Superconductor
- Efficient simulation of one-dimensional quantum many-body systems
- Entanglement entropy and conformal field theory
- Real time evolution using the density matrix renormalization group
- Computational complexity and fundamental limitations to fermionic quantum Monte Carlo simulations
- A Quantum Random Walk Search Algorithm
- Time-dependent density-matrix renormalization-group using adaptive effective Hilbert spaces
- One dimensional Bosons: From Condensed Matter Systems to Ultracold Gases
- Thermodynamic limit of the density matrix renormalization for the spin-1 Heisenberg chain
- Exponential Protection of Zero Modes in Majorana Islands
- Universal computation by quantum walk
- Beyond paired quantum Hall states: parafermions and incompressible states in the first excited Landau level
- Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
- Observation of topologically protected bound states in a one dimensional photonic system
- Two-particle bosonic-fermionic quantum walk via 3D integrated photonics
- Classical simulation of quantum many-body systems with a tree tensor network
- Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions
- Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
- 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
- Universal topological quantum computation from a superconductor/Abelian quantum Hall heterostructure
- Tensor network states and algorithms in the presence of a global U(1) symmetry
- Search via Quantum Walk
- Tensor network decompositions in the presence of a global symmetry
- Controlling discrete quantum walks: coins and intitial states
- Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law
- Quantum random walks in optical lattices
- Variational study of hard-core bosons in a 2-D optical lattice using Projected Entangled Pair States (PEPS)
- Experimental Implementation of the Quantum Random-Walk Algorithm
- Experimental studies of the fractional quantum Hall effect in the first excited Landau level
- A short introduction to Fibonacci anyon models
- Exploring frustrated spin-systems using Projected Entangled Pair States (PEPS)
- Diagrammatics for Bose condensation in anyon theories
- Interferometry of non-Abelian Anyons
- Quantum walk on the line: entanglement and non-local initial conditions
- Quantum phase transition in spin-3/2 systems on the hexagonal lattice - optimum ground state approach
- Parafermions in Interacting Nanowire Bundle
- Simulation of quantum random walks using interference of classical field
- Superconducting analogue of the parafermion fractional quantum Hall states
- Collective states of interacting Fibonacci anyons
- Vertical Density Matrix Algorithm: A Higher-Dimensional Numerical Renormalization Scheme based on the Tensor Product State Ansatz
- 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
- Distribution of chirality in the quantum walk: Markov process and entanglement
- Quantum walks on circles in phase space via superconducting circuit quantum electrodynamics
- Anyonic entanglement renormalization
- Matrix product states for anyonic systems and efficient simulation of dynamics
- Quantum spin ladders of non-Abelian anyons
- Classical Simulation of Quantum Error Correction in a Fibonacci Anyon Code
- One-dimensional itinerant interacting non-Abelian anyons
- Fractionalization of itinerant anyons in one dimensional chains
- Statistical dynamics of a non-Abelian anyonic quantum walk
- Measures of entanglement in non-Abelian anyonic systems
- Finite Density Matrix Renormalisation Group Algorithm for Anyonic Systems
- Simulation of braiding anyons using Matrix Product States
- Effective models of doped quantum ladders of non-Abelian anyons
- Classification of topological symmetry sectors on anyon rings
- Transport properties of anyons in random topological environments