Quantum geometry and stability of the fractional quantum Hall effect in the Hofstadter model

TL;DR

This paper investigates how the geometry of band structures in the Hofstadter model affects the stability of the fractional quantum Hall effect, revealing that certain geometric criteria significantly influence the many-body gap in low flux regimes.

Contribution

It provides analytic expressions for geometric criteria influencing FQHE stability in the Hofstadter model, highlighting the importance of band geometry in lattice systems.

Findings

Many-body gap depends monotonically on a band-geometric criterion.

Exponential and polynomial convergence of geometric criteria to the continuum limit.

Low flux density regime dominated by polynomial convergence criteria.

Abstract

We study how the stability of the fractional quantum Hall effect (FQHE) is influenced by the geometry of band structure in lattice Chern insulators. We consider the Hofstadter model, which converges to continuum Landau levels in the limit of small flux per plaquette. This gives us a degree of analytic control not possible in generic lattice models, and we are able to obtain analytic expressions for the relevant geometric criteria. These may be differentiated by whether they converge exponentially or polynomially to the continuum limit. We demonstrate that the latter criteria have a dominant effect on the physics of interacting particles in Hofstadter bands in this low flux density regime. In particular, we show that the many-body gap depends monotonically on a band-geometric criterion related to the trace of the Fubini-Study metric.