Fractional Quantum Hall Physics in Topological Flat Bands

TL;DR

This paper reviews fractional Chern insulators, highlighting their topological properties, construction, numerical results, and potential for high-temperature topological phases, connecting lattice models to quantum Hall physics.

Contribution

It provides a comprehensive pedagogical overview of fractional Chern insulators, emphasizing their connection to quantum Hall effects and exploring experimental and theoretical directions.

Findings

Fractional Chern insulators exhibit quantum Hall-like topological order on lattices.

Numerical studies support the existence of fractional phases in Chern flat bands.

Potential for realizing high-temperature topological phases in lattice systems.

Abstract

We present a pedagogical review of the physics of fractional Chern insulators with a particular focus on the connection to the fractional quantum Hall effect. While the latter conventionally arises in semiconductor heterostructures at low temperatures and in high magnetic fields, interacting Chern insulators at fractional band filling may host phases with the same topological properties, but stabilized at the lattice scale, potentially leading to high-temperature topological order. We discuss the construction of topological flat band models, provide a survey of numerical results, and establish the connection between the Chern band and the continuum Landau problem. We then briefly summarize various aspects of Chern band physics that have no natural continuum analogs, before turning to a discussion of possible experimental realizations. We close with a survey of future directions and open problems, as well as a discussion of extensions of these ideas to higher dimensions and to other topological phases.