Bulk-Edge Correspondence in 2+1-Dimensional Abelian Topological Phases

Abstract

The same bulk two-dimensional topological phase can have multiple distinct, fully-chiral edge phases. We show that this can occur in the integer quantum Hall states at $\nu =8$ and 12, with experimentally-testable consequences. We show that this can occur in Abelian fractional quantum Hall states as well, with the simplest examples being at $\nu =8/7,12/11,8/15,16/5$. We give a general criterion for the existence of multiple distinct chiral edge phases for the same bulk phase and discuss experimental consequences. Edge phases correspond to lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in a genus, the bulk-edge correspondence is typically one-to-many; there are usually many stable fully chiral edge phases corresponding to the same bulk. We explain these correspondences using the theory of integral quadratic forms. We show that fermionic systems can have edge phases with only bosonic low-energy excitations and discuss a fermionic generalization of the relation between bulk topological spins and the central charge. The latter follows from our demonstration that every fermionic topological phase can be represented as a bosonic topological phase, together with some number of filled Landau levels. Our analysis shows that every Abelian topological phase can be decomposed into a tensor product of theories associated with prime numbers $p$ in which every quasiparticle has a topological spin that is a $p^n$-th root of unity for some $n$. It also leads to a simple demonstration that all Abelian topological phases can be represented by $U(1{)}^N$ Chern-Simons theory parameterized by a K-matrix.

arXiv comment26 pages, 1 table and 3 appendices; published version

References in corpus (19)

• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• Observation of the e/3 Fractionally Charged Laughlin Quasiparticles
• Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach
• Spin and valley quantum Hall ferromagnetism in graphene
• Detecting Non-Abelian Statistics in the nu=5/2 Fractional Quantum Hall State
• Sharp and Smooth Boundaries of Quantum Hall Liquids
• Proposed experiments to probe the non-abelian ν=5/2 quantum Hall state
• Randomness at the Edge: Theory of Quantum Hall transport at filling $\nu =2/3$
• Measurement of filling factor 5/2 quasiparticle interference: observation of charge e/4 and e/2 period oscillations
• Insulating and Fractional Quantum Hall States in the N=1 Landau Level
• A Chern-Simons Effective Field Theory for the Pfaffian Quantum Hall State
• Protected edge modes without symmetry
• Integer quantum Hall effect for bosons: A physical realization
• Stability of Chiral Luttinger Liquids and Abelian Quantum Hall States.
• Tunneling and Quantum Noise in 1-D Luttinger Liquids
• Short-Range Entangled Bosonic States with Chiral Edge Modes and $T$-duality of Heterotic Strings
• Binding Transition in Quantum Hall Edge States
• Single-Class Genera of Positive Integral Lattices

Cited by in corpus (64)

• Local subsystems in gauge theory and gravity
• Quantum Hall Physics - hierarchies and CFT techniques
• Topological phases protected by point group symmetry
• A symmetry-respecting topologically-ordered surface phase of 3d electron topological insulators
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• How universal is the entanglement spectrum?
• Classification and Properties of Symmetry Enriched Topological Phases: A Chern-Simons approach with applications to Z2 spin liquids
• Particle-Hole Symmetry and the Composite Fermi Liquid
• Topology and Interactions in a Frustrated Slab: Tuning from Weyl Semimetals to C > 1 Fractional Chern Insulators
• Bulk-boundary correspondence in (3+1)-dimensional topological phases
• In and around Abelian anyon models
• Composite Dirac liquids: parent states for symmetric surface topological order
• A mathematical theory of gapless edges of 2d topological orders. Part I
• An elementary rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model
• Interactions along an Entanglement Cut in 2+1D Abelian Topological Phases
• Fermion condensation and super pivotal categories
• Vortex-line condensation in three dimensions: A physical mechanism for bosonic topological insulators
• Gapless edges of 2d topological orders and enriched monoidal categories
• Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter
• Composite Particle Theory of Three-dimensional Gapped Fermionic Phases: Fractional Topological Insulators and Charge-Loop Excitation Symmetry
• Microscopic Study of the Halperin - Laughlin Interface through Matrix Product States
• Bulk Topological Proximity Effect
• Anyonic Symmetries and Topological Defects in Abelian Topological Phases: an application to the $ADE$ Classification
• Model States for a Class of Chiral Topological Order Interfaces
• Bridging Fermionic and Bosonic Short Range Entangled States
• Symmetry-Protected Topological Interfaces and Entanglement Sequences
• Globally symmetric topological phase: from anyonic symmetry to twist defect
• Interaction Enabled Fractonic Higher-Order Topological Phases
• Parafermionic wires at the interface of chiral topological states
• The chiral SYK model
• Non-Abelian bosonization and modular transformation approach to superuniversality
• Topological order, symmetry, and Hall response of two-dimensional spin-singlet superconductors
• Bulk-edge correspondence, spectral flow and Atiyah-Patodi-Singer theorem for the Z2-invariant in topological insulators
• Entanglement branes, modular flow, and extended topological quantum field theory
• Perfect Metal Phases of One-Dimensional and Anisotropic Higher-Dimensional Systems
• On classification of extremal non-holomorphic conformal field theories
• Chirality-Protected Majorana Zero Modes at the Gapless Edge of Abelian Quantum Hall States
• A non-Abelian twist to integer quantum Hall states
• Bosonic Short Range Entangled states Beyond Group Cohomology classification
• Families of Gapped Interfaces Between Fractional Quantum Hall States
• Filling constraints on fermionic topological order in zero magnetic field
• Classification and surface anomaly of glide symmetry protected topological phases in three dimensions
• Numerical investigation of gapped edge states in fractional quantum Hall-superconductor heterostructures
• Fractional quantum Hall effect at the filling factor $\nu =5/2$
• Model wavefunctions for interfaces between lattice Laughlin states
• Fermion Parity Flips and Majorana Bound States at twist defects in Superconducting Fractional Topological Phases
• Gauging U(1) symmetry in (2+1)d topological phases
• Identifying non-Abelian anyons with upstream noise
• Interacting Edge States of Fermionic Symmetry-Protected Topological Phases in Two Dimensions
• Entanglement Action for the Real-Space Entanglement Spectra of Composite Fermion Wave Functions
• Global Symmetries, Code Ensembles, and Sums Over Geometries
• Edge--Entanglement correspondence for gapped topological phases with symmetry
• Localization and conductance in fractional quantum Hall edges
• The Partially-Split Hall Bar: Tunneling in the Bosonic Integer Quantum Hall Effect
• Dynamical edge modes in Maxwell theory from a BRST perspective, with an application to the Casimir energy
• Gapped boundaries of fermionic topological orders and higher central charges
• Transport in a One-Dimensional Hyperconductor
• An elementary proof and detailed investigation of the bulk-boundary correspondence in the generic two-band model of Chern insulators
• Numerical identification and gapped boundaries of abelian fermionic topological order
• Partial fillings of the bosonic $E_8$ quantum Hall state
• Double-Janus Linear Sigma Models and Generalized Reciprocity for Gauss Sums
• Integrability of the $\nu =4/3$ fractional quantum Hall edge states
• Topological stabilizer models on continuous variables
• Interacting 1D Chiral Fermions with Pairing: Transition from Integrable to Chaotic