From Quantum Wires to the Chern-Simons Description of the Fractional Quantum Hall Effect

Abstract

We show the explicit connection between two distinct and complementary approaches to the fractional quantum Hall system (FQHS): the quantum wires formalism and the topological low-energy effective description given in terms of an Abelian Chern-Simons theory. The quantum wires approach provides a description of the FQHS directly in terms of fermions arranged in an array of one-dimensional coupled wires. In this sense it is usually referred to as a microscopic description. On the other hand, the effective theory has no connection with the microscopic modes, involving only the emergent topological degrees of freedom embodied in an Abelian Chern-Simons gauge field, which somehow encodes the collective motion of the strongly correlated electrons. The basic strategy pursued in this work is to bosonize the quantum wires system and then consider the continuum limit. By examining the algebra of the bosonic operators of the Hamiltonian, we are able to identify the bosonized microscopic fields with the components of the field strength (electric and magnetic fields) of the emergent gauge field. Thus our study provides a bridge between the microscopic physical degrees of freedom and the emergent topological ones, without relying on the bulk-edge correspondence.

arXiv comment15 pages; typos fixed, supplemental material included, published version

References in corpus (28)

• Topological orders and Edge excitations in FQH states
• A Duality Web in 2+1 Dimensions and Condensed Matter Physics
• The Fractional Quantum Hall effect in an array of quantum wires
• From Luttinger liquid to non-Abelian quantum Hall states
• Particle-Vortex Duality from 3d Bosonization
• Explicit derivation of duality between a free Dirac cone and quantum electrodynamics in (2+1) dimensions
• Wire deconstructionism of two-dimensional topological phases
• Coupled wire construction of chiral spin liquids
• Non-Abelian topological insulators from an array of quantum wires
• Chiral Spin Liquids in Arrays of Spin Chains
• Quantum Spin Hall Effect in Strip of Stripes Model
• Integer and Fractional Quantum Hall Effect in a Strip of Stripes
• Interactions along an Entanglement Cut in 2+1D Abelian Topological Phases
• Symmetry and duality in bosonization of two-dimensional Dirac fermions
• Non-Abelian topological spin liquids from arrays of quantum wires or spin chains
• Fractional Topological Insulators: from sliding Luttinger Liquids to Chern-Simons theory
• Wire constructions of Abelian topological phases in three or more dimensions
• Fractional topological phases in three-dimensional coupled-wire systems
• From an array of quantum wires to three-dimensional fractional topological insulators
• Coupled wire model of symmetric Majorana surfaces of topological superconductors
• Non-Abelian SU(N-1)-singlet fractional quantum Hall states from coupled wires
• Quantum Hall hierarchy from coupled wires
• Two dimensional spin liquids with ${\mathbb{Z}}_2$ topological order in an array of quantum wires
• Coupled Wire Model of Z4 Orbifold Quantum Hall States
• Coupled Wire Models of Interacting Dirac Nodal Superconductors
• Duality between 3D Massive Thirring and Maxwell Chern-Simons models from 2D bosonization
• Fractional topological insulator precursors in spin-orbit fermion ladders
• Effective Theories for 2+1 Dimensional Non-Abelian Topological Spin Liquids

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• From coupled-wire construction of quantum Hall states to wave functions and hydrodynamics
• Lattice Clifford fractons and their Chern-Simons-like theory
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• Microscopic Study of the Coupled-Wire Construction and Plausible Realization in Spin-Dependent Optical Lattices
• Fractionalized Quasiparticles in Dense Baryonic Matter
• Bridging three-dimensional coupled-wire models and cellular topological states: Solvable models for topological and fracton orders
• Mapping topology to nuclear dilaton-HLS effective field theory for dense baryonic matter
• Dihedral twist liquid models from emergent Majorana fermions
• Topological Superconductor from the Quantum Hall Phase: Effective Field Theory Description
• Quantum wires, Chern-Simons theory, and dualities in the quantum Hall system
• Mapping topology of skyrmions and fractional quantum Hall droplets to nuclear EFT for ultra-dense baryonic matter
• Bosonization, Duality, and the $C$-Theorem in the Non-Abelian Thirring Model