Geometric Defects in Quantum Hall States

Abstract

We describe a geometric (or gravitational) analogue of the Laughlin quasiholes in the fractional quantum Hall states. Analogously to the quasiholes these defects can be constructed by an insertion of an appropriate vertex operator into the conformal block representation of a trial wavefunction, however, unlike the quasiholes these defects are extrinsic and do not correspond to true excitations of the quantum fluid. We construct a wavefunction in the presence of such defects and explain how to assign an electric charge and a spin to each defect, and calculate the adiabatic, non-abelian statistics of the defects. The defects turn out to be equivalent to the genons in that their adiabatic exchange statistics can be described in terms of representations of the mapping class group of an appropriate higher genus Riemann surface. We present a general construction that, in principle, allows to calculate the statistics of ${\mathbb{Z}}_n$ genons for any "parent" topological phase. We illustrate the construction on the example of the Laughlin state and perform an explicit calculation of the braiding matrices. In addition to non-abelian statistics geometric defects possess a universal abelian overall phase, determined by the gravitational anomaly.

27 pages, 10 figures

References in corpus (13)

• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Genons, twist defects, and projective non-Abelian braiding statistics
• Framing Anomaly in the Effective Theory of Fractional Quantum Hall Effect
• Thermal Hall Effect and Geometry with Torsion
• Low-energy effective theory in the bulk for transport in a topological phase
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• Density-curvature response and gravitational anomaly
• Topological central charge from Berry curvature: gravitational anomalies in trial wavefunctions for topological phases
• Geometry of Fractional Quantum Hall Fluids
• Conformal invariance of chiral edge theories
• Lorentz shear modulus of a two-dimensional electron gas at high magnetic field
• FQHE on curved backgrounds, free fields and large N
• Condensation of Lattice Defects and Melting Transitions in Quantum Hall phases

Cited by in corpus (20)

• Bimetric Theory of Fractional Quantum Hall States
• Anyonic Chains, Topological Defects, and Conformal Field Theory
• Crystalline gauge fields and quantized discrete geometric response for Abelian topological phases with lattice symmetry
• Energy Transport for Thick Holographic Branes
• Fractional Chern Insulators in Singular Geometries
• Chern Insulators on Singular Geometries
• Lowest Landau level on a cone and zeta determinants
• Non-Abelian defects in Fracton phase of matter
• Laughlin states on higher genus Riemann surfaces
• Quantum Hall States and Conformal Field Theory on a Singular Surface
• Central charge from adiabatic transport of cusp singularities in the quantum Hall effect
• Geometry and large N limits in Laughlin states
• Geometric test for topological states of matter
• Fractional quantum Hall states with gapped boundaries in an extreme lattice limit
• Laughlin's quasielectron as a non-local composite fermion
• Geometric and computational aspects of chiral topological quantum matter
• Measurable fractional spin for quantum Hall quasiparticles on the disk
• Spin fractionalization at the edge of quantum Hall fluids induced by bulk quasiparticles
• The degrees of freedom of multiway junctions in three dimensional gravity
• Bergman kernel asymptotics for singular metrics on punctured Riemann surfaces