The fractional quantum Hall effect: Chern-Simons mapping, duality, Luttinger liquids and the instanton vacuum

Abstract

We derive, from first principles, the complete Luttinger liquid theory of abelian quantum Hall edge states. This theory includes the effects of disorder and Coulomb interactions as well as the coupling to external electromagnetic fields. We introduce a theory of spatially separated (individually conserved) edge modes, find an enlarged dual symmetry and obtain a complete classification of quasiparticle operators and tunneling exponents. The chiral anomaly on the edge and Laughlin's gauge argument are used to obtain unambiguously the Hall conductance. In resolving the problem of counter flowing edge modes, we find that the long range Coulomb interactions play a fundamental role. In order to set up a theory for arbitrary filling fractions $\nu$ we use the idea of a two dimensional network of percolating edge modes. We derive an effective, single mode Luttinger liquid theory for tunneling processes into the quantum Hall edge which yields a continuous tunneling exponent $1/\nu$. The network approach is also used to re-derive the instanton vacuum or $Q$-theory for the plateau transitions.

36 pages, 7 figures (eps)

Cited by in corpus (13)

• Chern-Simons-Matter Theory
• Supersymmetric Chern-Simons Theory in Presence of a Boundary
• (Mis-)handling gauge invariance in the theory of the quantum Hall effect III: The instanton vacuum and chiral edge physics
• The instanton vacuum of generalized $C{P}^{N-1}$ models
• Theta renormalization, electron-electron interactions and super universality in the quantum Hall regime
• Harmonic Superspace Gaugeon Formalism for the ABJM Theory
• The BV Formalization of Chern-Simons Theory on Deformed Superspace
• Non-anticommutative ABJ Theory
• General topological features and instanton vacuum in quantum Hall and spin liquids
• Super universality of the quantum Hall effect and the "large $N$ picture" of the $\vartheta$ angle
• Exact Haldane mapping for all $S$ and super universality in spin chains
• Instanton method for the electron propagator
• Super universality of dimerised $SU(N+M)$ spin chains