Continuum field theory of 3D topological orders with emergent fermions and braiding statistics

Abstract

Universal topological data of topologically ordered phases can be captured by topological quantum field theory in continuous space time by taking the limit of low energies and long wavelengths. While previous continuum field-theoretical studies of topological orders in $3$D real space focus on either self-statistics, braiding statistics, shrinking rules, fusion rules or quantum dimensions, it is yet to systematically put all topological data together in a unified continuum field-theoretical framework. Here, we construct the topological $BF$ field theory with twisted terms (e.g., $AAdA$ and $AAB$) as well as a $K$-matrix $BB$ term, in order to simultaneously explore all such topological data and reach anomaly-free topological orders. Following the spirit of the famous $K$-matrix Chern-Simons theory of $2$D topological orders, we present general formulas and systematically show how the $K$-matrix $BB$ term confines topological excitations, and how self-statistics of particles is transmuted between bosonic one and fermionic one. In order to reach anomaly-free topological orders, we explore, within the present continuum field-theoretical framework, how the principle of gauge invariance fundamentally influences possible realizations of topological data. More concretely, we present the topological actions of (i) particle-loop braidings with emergent fermions, (ii) multiloop braidings with emergent fermions, and (iii) Borromean-Rings braidings with emergent fermions, and calculate their universal topological data. Together with the previous efforts, our work paves the way toward a more systematic and complete continuum field-theoretical analysis of exotic topological properties of $3$D topological orders. Several interesting future directions are also discussed.

References in corpus (34)

• Non-Abelian Anyons and Topological Quantum Computation
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Coupling a QFT to a TQFT and Duality
• Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach
• Superconductors are topologically ordered
• Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond
• Quantum Field Theory of Nonabelian Strings and Vortices
• Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions
• Classification and Properties of Symmetry Enriched Topological Phases: A Chern-Simons approach with applications to Z2 spin liquids
• A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
• Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions
• A K matrix Construction of Symmetry Enriched Phases of Matter
• Constructing symmetric topological phases of bosons in three dimensions via fermionic projective construction and dyon condensation
• Vortex-line condensation in three dimensions: A physical mechanism for bosonic topological insulators
• Topological quantum field theory of three-dimensional bosonic Abelian-symmetry-protected topological phases
• Loop Braiding Statistics and Interacting Fermionic Symmetry-Protected Topological Phases in Three Dimensions
• Projective construction of two-dimensional symmetry-protected topological phases with U(1), SO(3), or SU(2) symmetries
• Multi-kink topological terms and charge-binding domain-wall condensation induced symmetry-protected topological states: Beyond Chern-Simons/BF theory
• Symmetry-protected topological phases with charge and spin symmetries: response theory and dynamical gauge theory in 2D, 3D and the surface of 3D
• Braiding with Borromean Rings in (3+1)-Dimensional Spacetime
• Composite Particle Theory of Three-dimensional Gapped Fermionic Phases: Fractional Topological Insulators and Charge-Loop Excitation Symmetry
• Generalized Wen-Zee Terms
• $U(1)\times U(1)$ Symmetry Protected Topological Order in Gutzwiller Wave Functions
• Field Theories for Gauged Symmetry Protected Topological Phases: Abelian Gauge Theories with non-Abelian Quasiparticles
• Entanglement entropy of (3+1)D topological orders with excitations
• Topological Quantum Field Theory for Abelian Topological Phases and Loop Braiding Statistics in $(3+1)$-Dimensions
• Symmetry Enrichment in Three-Dimensional Topological Phases
• Fractional $S$-duality, Classification of Fractional Topological Insulators and Surface Topological Order
• Three-dimensional anomalous twisted gauge theories with global symmetry: Implications for quantum spin liquids
• Compatible braidings with Hopf links, multi-loop, and Borromean rings in $(3+1)$-dimensional spacetime
• Topological Orders, Braiding Statistics, and Mixture of Two Types of Twisted $BF$ Theories in Five Dimensions
• Non-Abelian Fusion, Shrinking and Quantum Dimensions of Abelian Gauge Fluxes
• Fractionalizing Global Symmetry on Looplike Topological Excitations
• Non-Abelian Three-Loop Braiding Statistics for 3D Fermionic Topological Phases

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