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

TL;DR

This paper introduces a dual first quantization framework for fractional Chern insulators, revealing a momentum-space Landau level structure and quantum geometry interplay, and presents ideal lattice models as duals to isotropic FQH states.

Contribution

It develops a novel dual description of fractional Chern insulators by interchanging position and momentum roles, and constructs ideal lattice models as duals to FQH states.

Findings

FQH states are described by momentum-space Landau levels.

Quantum geometry arises from single-body and interaction metrics.

Ideal Chern insulator models act as duals to isotropic FQH effects.

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.