Ground-state degeneracy of the fractional quantum Hall states in the presence of a random potential and on high-genus Riemann surfaces

1990/05/01 by Xiao-Gang Wen, Q. Niu · 46 citations
Physics and Astronomy · Materials Science · Mathematics · #Quantum and electron transport phenomena #Topological Materials and Phenomena #Graphene research and applications #Ground state #Physics #Topological degeneracy #Degeneracy (biology) #Riemann surface #Quantum Hall effect #Quantum mechanics #Quasiparticle #Landau quantization #Torus #Mathematical physics #Topological order #Quantum #Symmetry protected topological order #Mathematics #Pure mathematics #Electron #Geometry

paper · doi:10.1103/physrevb.41.9377

openalex publication_date 1990/05/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/29

Abstract

The fractional quantum Hall (FQH) states are shown to have q\ifmmode \else \~\fi ^gfold ground-state degeneracy on a Riemann surface of genus g, where q\ifmmode \else \~\fi is the ground-state degeneracy in a torus topology. The ground-state degeneracies are directly related to the statistics of the quasiparticles given by \ensuremathθ=p\ifmmode \else \~\fi\ensuremathπ/q\ifmmode \else \~\fi. The ground-state degeneracy is shown to be invariant against weak but otherwise arbitrary perturbations. Therefore the ground-state degeneracy provides a new quantum number, in addition to the Hall conductance, characterizing different phases of the FQH systems. The phases with different ground-state degeneracies are considered to have different topological orders. For a finite system of size L, the ground-state degeneracy is lifted. The energy splitting is shown to be at most of order e^\mathrm\ensuremath-L/\ensuremathξ. We also show that the Ginzburg-Landau theory of the FQH states (in the low-energy limit) is a dual theory of the U(1) Chern-Simons topological theory.

Citations

• Evidence for a New Phase of SolidHe3
• TOPOLOGICAL ORDERS IN RIGID STATES
• Quantum field theory and the Jones polynomial
• Appearance of Gauge Structure in Simple Dynamical Systems
• Higgs mechanism for Kalb-Ramond gauge field
• Equivalence of the resonating-valence-bond and fractional quantum Hall states
• Viscosity of Liquid Helium below the λ-Point
• Quantized Hall Conductance in a Two-Dimensional Periodic Potential
• Magnetic Translation Group

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