Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator

Abstract

We perform a detail study of higher dimensional quantum Hall effects and A-class topological insulators with emphasis on their relations to non-commutative geometry. There are two different formulations of non-commutative geometry for higher dimensional fuzzy spheres; the ordinary commutator formulation and quantum Nambu bracket formulation. Corresponding to these formulations, we introduce two kinds of monopole gauge fields; non-abelian gauge field and antisymmetric tensor gauge field, which respectively realize the non-commutative geometry of fuzzy sphere in the lowest Landau level. We establish connection between the two types of monopole gauge fields through Chern-Simons term, and derive explicit form of tensor monopole gauge fields with higher string-like singularity. The connection between two types of monopole is applied to generalize the concept of flux attachment in quantum Hall effect to A-class topological insulator. We propose tensor type Chern-Simons theory as the effective field theory for membranes in A-class topological insulators. Membranes turn out to be fractionally charged objects and the phase entanglement mediated by tensor gauge field transforms the membrane statistics to be anyonic. The index theorem supports the dimensional hierarchy of A-class topological insulator. Analogies to D-brane physics of string theory are discussed too.

1+56 pages, 4 figures, 4 tables, published version

References in corpus (12)

• Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Gauge Symmetry and Supersymmetry of Multiple M2-Branes
• Algebraic structures on parallel M2-branes
• Modeling Multiple M2's
• Hopf Maps, Lowest Landau Level, and Fuzzy Spheres
• Quantized Nambu-Poisson Manifolds and n-Lie Algebras
• Topological Insulators and Superconductors from String Theory
• D-Algebra Structure of Topological Insulators
• Non-Commutative Time, the Quantum Hall Effect and Twistor Theory

Cited by in corpus (5)

• Fermions in Gödel-type background space-times with torsion and the Landau quantization
• Topological Electromagnetic Responses of Bosonic Quantum Hall, Topological Insulator, and Chiral Semi-Metal phases in All Dimensions
• Time-reversal Invariant SU(2) Hofstadter Problem in Three Dimensional Lattices
• Quantum Hall Effect on Odd Spheres
• The role of the spin connection in quantum Hall effect: A perspective from geometric quantization