Layer construction of 3D topological states and string braiding statistics

Abstract

While the topological order in two dimensions has been studied extensively since the discover of the integer and fractional quantum Hall systems, topological states in 3 spatial dimensions are much less understood. In this paper, we propose a general formalism for constructing a large class of three-dimensional topological states by stacking layers of 2D topological states and introducing coupling between them. Using this construction, different types of topological states can be obtained, including those with only surface topological order and no bulk topological quasiparticles, and those with topological order both in the bulk and at the surface. For both classes of states we study its generic properties and present several explicit examples. As an interesting consequence of this construction, we obtain example systems with nontrivial braiding statistics between string excitations. In addition to studying the string-string braiding in the example system, we propose a generic topological field theory description which can capture both string-particle and string-string braiding statistics. Lastly, we provide a proof of a general identity for Abelian string statistics, and discuss an example system with non-Abelian strings.

23 pages, 17 figures

References in corpus (17)

• Non-Abelian Anyons and Topological Quantum Computation
• Classification of topological insulators and superconductors in three spatial dimensions
• Quantum Hall Ferromagnetism in Graphene
• Criticality, the area law, and the computational power of PEPS
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Condensate induced transitions between topologically ordered phases
• Fractional Quantum Hall States and Jack Polynomials
• Genons, twist defects, and projective non-Abelian braiding statistics
• Collective Modes and Skyrmion Excitations in Graphene SU(4) Quantum Hall Ferromagnets
• Fractional topological insulators in three dimensions
• The Fractional Quantum Hall States of Dirac Electrons in Graphene
• Fractional Quantum Hall Effect in Graphene
• Theory of the Three Dimensional Quantum Hall Effect in Graphite
• Axion topological field theory of topological superconductors
• Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
• Generalized modular transformations in 3+1D topologically ordered phases and triple linking invariant of loop braiding
• A classification of symmetric polynomials of infinite variables -- a construction of Abelian and non-Abelian quantum Hall states

Cited by in corpus (7)

• Twisted Gauge Theory Model of Topological Phases in Three Dimensions
• Topological invariants for gauge theories and symmetry-protected topological phases
• Non-Abelian Hybrid Fracton Orders
• A simple anisotropic three-dimensional quantum spin liquid with fracton topological order
• Topological Electromagnetic Responses of Bosonic Quantum Hall, Topological Insulator, and Chiral Semi-Metal phases in All Dimensions
• From coupled wires to coupled layers: Model with three-dimensional fractional excitations
• Decorated defect condensate, a window to unconventional quantum phase transitions in Weyl semimetals