Spin chains, defects, and quantum wires for the quantum-double edge

Abstract

Non-Abelian defects that bind Majorana or parafermion zero modes are prominent in several topological quantum computation schemes. Underpinning their established understanding is the quantum Ising spin chain, which can be recast as a fermionic model or viewed as a standalone effective theory for the surface-code edge -- both of which harbor non-Abelian defects. We generalize these notions by deriving an effective Ising-like spin chain describing the edge of quantum-double topological order. Relating Majorana and parafermion modes to anyonic strings, we introduce quantum-double generalizations of non-Abelian defects. We develop a way to embed finite-group valued qunits into those valued in continuous groups. Using this embedding, we provide a continuum description of the spin chain and recast its non-interacting part as a quantum wire via addition of a Wess-Zumino-Novikov-Witten term and non-Abelian bosonization.

arXiv comment34 pages, 5 figures

References in corpus (57)

• Non-Abelian Anyons and Topological Quantum Computation
• Superconducting proximity effect and Majorana fermions at the surface of a topological insulator
• Majorana Fermions and a Topological Phase Transition in Semiconductor-Superconductor Heterostructures
• String-net condensation: A physical mechanism for topological phases
• Bosonization and Strongly Correlated Systems
• Classification of Gapped Symmetric Phases in 1D Spin Systems
• Classifying quantum phases using Matrix Product States and PEPS
• Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zero Modes
• Exotic non-Abelian anyons from conventional fractional quantum Hall states
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Topological Quantum Computation Based on Chiral Majorana Fermions
• Topological phases with parafermions: theory and blueprints
• Genons, twist defects, and projective non-Abelian braiding statistics
• Topological Nematic States and Non-Abelian Lattice Dislocations
• The Quantum Double Model with Boundary: Condensations and Symmetries
• Superconducting Proximity Effect on the Edge of Fractional Topological Insulators
• A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
• A Formulation of Lattice Gauge Theories for Quantum Simulations
• Anyon condensation and its applications
• Entanglement renormalization and topological order
• Fractional topological superconductors with fractionalized Majorana fermions
• Robust encoding of a qubit in a molecule
• The Higgs Mechanism in Higher-Rank Symmetric $U(1)$ Gauge Theories
• Classification of topologically protected gates for local stabilizer codes
• Dualities and the phase diagram of the $p$-clock model
• Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
• Stability of zero modes in parafermion chains
• A hierarchy of topological tensor network states
• Exact entanglement renormalization for string-net models
• Disorder Operators and their Descendants
• Simulations of quantum double models
• Quantum Computing with Majorana Fermion Codes
• Synthetic Topological Degeneracy by Anyon Condensation
• A mathematical theory of gapless edges of 2d topological orders. Part I
• Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
• Topological phases in gapped edges of fractionalized systems
• Electric-magnetic duality of lattice systems with topological order
• Edge-entanglement spectrum correspondence in a nonchiral topological phase and Kramers-Wannier duality
• Parafermionic phases with symmetry-breaking and topological order
• Fault-Tolerant Quantum Gates with Defects in Topological Stabiliser Codes
• Topological and symmetry broken phases of Z_N parafermions in one dimension
• Parafermionic clock models and quantum resonance
• Boundary Hamiltonian theory for gapped topological phases on an open surface
• On Defects Between Gapped Boundaries in Two-Dimensional Topological Phases of Matter
• Superintegrability of Geodesic Motion on the Sausage Model
• General phase spaces: from discrete variables to rotor and continuum limits
• Matrix product states and the nonabelian rotor model
• Spin Structures and Exact Dualities in Low Dimensions
• Kramers-Wannier duality and worldline representation for the SU(2) principal chiral model
• Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits
• Majorana Edge States for Z2 Topological Orders of the Wen-plaquette Model and the Toric-code Model
• Dyonic zero-energy modes
• Simulation of non-Abelian Anyons using ribbon operators connected to a common base site
• From entanglement renormalisation to the disentanglement of quantum double models
• Matrix product approximations to conformal field theories
• The non-Abelian Duality Problem
• Gauging Defects in Quantum Spin Systems