Exact Landau Level Description of Geometry and Interaction in a Flatband

TL;DR

This paper establishes an exact mapping between ideal topological flatbands and Landau levels, revealing how band geometry fluctuations induce novel center-of-mass dependent interactions, with implications for understanding fractional quantum Hall states.

Contribution

It introduces an exact correspondence between ideal flatbands and Landau levels, including a generalized pseudopotential framework for analyzing interactions.

Findings

Exact zero-energy ground states for short-range repulsive pseudopotentials.

Band geometry fluctuations induce center-of-mass dependent interactions.

Ground state reconstruction occurs when center-of-mass interactions become attractive.

Abstract

Flatbands appear in many condensed matter systems, such as in high magnetic fields, correlated materials and moire heterostructures. They are characterized by intrinsic geometric properties such as the Berry curvature and Fubini-Study metric. In general the band geometry is nonuniform in momentum space, making its influence on electron-electron interactions a difficult problem to understand analytically. In this work, we study this problem in a topological flatband of Chern number C=1 with the ideal properties that the Berry curvature is positive definite and fluctuates in sync with Fubini-Study metric. We derive an exact correspondence between such ideal flatbands and Landau levels by showing how the band geometry fluctuation in ideal flatbands gives raise to a new type of interaction in Landau levels which depends on the center-of-mass of two particles. We characterize such interaction by generalizing the usual Haldane pseudopotentials. This mapping gives exact zero-energy ground states for short-ranged repulsive generalized pseudopotentials in flatbands, in analogy to fractional quantum Hall systems. Driving the center-of-mass interactions beyond the repulsive regime leads to a dramatic reconstruction of the ground states towards gapless phases. The generalized pseudopotential could be a useful basis for future numerical studies.