Position-Momentum Duality and Fractional Quantum Hall Effect in Chern Insulators

Abstract

We develop a first quantization description of fractional Chern insulators that is the dual of the conventional fractional quantum Hall (FQH) problem, with the roles of position and momentum interchanged. In this picture, FQH states are described by anisotropic FQH liquids forming in momentum-space Landau levels in a fluctuating magnetic field. The fundamental quantum geometry of the problem emerges from the interplay of single-body and interaction metrics, both of which act as momentum-space duals of the geometrical picture of the anisotropic FQH effect. We then present a novel broad class of ideal Chern insulator lattice models that act as duals of the isotropic FQH effect. The interacting problem is well-captured by Haldane pseudopotentials and affords a detailed microscopic understanding of the interplay of interactions and non-trivial quantum geometry.

10 pages, 2 figures. Accepted version

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• Fractional quantum Hall states at zero magnetic field
• Nearly-flat bands with nontrivial topology
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• The single-mode approximation for fractional Chern insulators and the fractional quantum Hall effect on the torus
• Designing Topological Bands in Reciprocal Space
• Exact Solutions of Fractional Chern Insulators: Interacting Particles in the Hofstadter Model at Finite Size
• D-Algebra Structure of Topological Insulators
• Momentum-space instantons and maximally localized flat-band topological Hamiltonians

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