Exact Solutions of Fractional Chern Insulators: Interacting Particles in the Hofstadter Model at Finite Size

TL;DR

This paper demonstrates that the Hofstadter model's bands can be exactly flat with uniform Berry curvature at specific system sizes, enabling precise mapping to fractional quantum Hall states and revealing new insights into fractional Chern insulators.

Contribution

It provides an exact flatness condition for Hofstadter bands at specific sizes and establishes a direct mapping to continuum Landau levels for constructing fractional Chern insulator phases.

Findings

All bands are exactly flat at special system sizes.

Density operators obey the Girvin-MacDonald-Platzman algebra.

Fractional Chern insulator phases are exact zero-energy ground states.

Abstract

We show that all the bands of the Hofstadter model on the torus have an exactly flat dispersion and Berry curvature when a special system size is chosen. This result holds for any hopping and Chern number. Our analysis therefore provides a simple rule for choosing a particularly advantageous system size when designing a Hofstadter system whose size is controllable, like a qubit lattice or an optical cavity array. The density operators projected onto the flat bands obey exactly the Girvin-MacDonald-Platzman algebra, like for Landau levels in the continuum in the case of $C=1$, or obey its straightforward generalization in the case of $C>1$. This allows a mapping between density-density interaction Hamiltonians for particles in the Hofstatder model and in a continuum Landau level. By using the well-known pseudopotential construction in the latter case, we obtain fractional Chern insulator phases, the lattice counterpart of fractional quantum Hall phases, that are exact zero-energy ground states of the Hofstadter model with certain interactions. Finally, the addition of a harmonic trapping potential is shown to lead to an appealingly symmetric description in which a new Hofstadter model appears in momentum space.