Singular flat bands in three dimensions: Landau level spreading, quantum geometry, and Weyl reconstruction

TL;DR

This paper explores the quantum geometric properties and magnetic response of three-dimensional singular flat bands, revealing how Landau level spreading and band reconstruction relate to topological singularities and Weyl physics.

Contribution

It introduces a three-orbital continuum model for 3D flat bands, uncovering the impact of topological singularities on Landau levels and band reconstruction.

Findings

Landau levels spread over a finite energy range due to quantum geometry.

Band degeneracy is lifted by orbital Zeeman effect, leading to Weyl-like dispersion.

Landau level spreading is proportional to the quantum metric of Zeeman-split bands.

Abstract

We theoretically investigate three-dimensional singular flat band systems, focusing on their quantum geometric properties and response to external magnetic fields. As a representative example, we study the pyrochlore lattice, which hosts a pair of degenerate flat bands touching a dispersive band. We derive a three-orbital effective continuum model that captures the essential features near the band-touching point. Within this framework, we identify the point-like topological singularity on a planar manifold defined by the degenerate flat band eigenvectors. This singularity strongly influences the quantum geometry and results in a characteristic Landau level structure, where the levels spread over a finite energy range. We show that this structure reflects the underlying band reconstruction due to the orbital Zeeman effect, which lifts the flat band degeneracy and induces the Weyl-semimetal-like dispersion near the singularity. Our analysis reveals that the range of Landau level spreading is proportional to the quantum metric of each Zeeman-split band. We further demonstrate that adding a small dispersion via longer-range . Finally, we show that our approach extends naturally to systems with higher orbital angular momentum, indicating the robustness of these features in a broad class of three-dimensional flat band models.