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

TL;DR

The paper addresses 3D singular flat bands by formulating a three-orbital effective continuum model for the pyrochlore lattice, revealing a point-like topological singularity in the flat-band manifold and a non-Abelian quantum geometric tensor. Under a magnetic field, the Landau level spectrum exhibits characteristic spreading of the flat-band branches, which is captured by an effective Weyl-semimetal description incorporating the orbital Zeeman effect. The work connects the LL spreading to the quantum metric of the Zeeman-split bands and predicts Weyl-point–driven chiral Landau levels, with SdH oscillations serving as potential experimental signatures. It further shows the framework generalizes to higher angular momentum $\ell$, keeping the qualitative physics robust across a broad class of 3D flat-band models and suggesting rich geometric and topological transport phenomena. The findings provide a geometric and topological lens to understand Landau quantization and magnetic response in 3D flat-band systems, with potential relevance to correlated and topological phases in pyrochlore and related lattices.

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.

Figures (13)

• Figure 1: (a) Schematic diagram for point like singularity emerging in pyrochlore lattice. The green plane indicate the flat band manifold spanned by degenerate eigenvectors. The director perpendicular to the plane correspond to a eigenvector of the dispersive band. (b) Asymptotic behavior of the flat band manifold for $k_z=0$ and $k_z\rightarrow\infty$. (c) Schematic Landau level structure, induced by orbital Zeeman effect that forms Weyl point (WP) in the continuum dispersion relation.
• Figure 2: (a) Lattice structure of the pyrochlore lattice. Sublattices within a primitive unit cell is indicated in dark blue. (b) Brillouin zone with high-symmetry point. (c) Band structure of Pyrochlore lattice along high-symmetry lines in the Brillouin zone. The dashed curves in (c) is obtained from the effective continuum model Eq. \ref{['eq:heffl']}.
• Figure 3: Landau level structure of the three dimensional singular flat band. (a) Full spectrum as a function of $k_z$, including levels up to $j=j_{\mathrm{max}}$, with zoom in on the lower energy region to highlight feature associated with the flat band. We set $j_{\mathrm{max}}\sim l_B^2/d^2=100$. (b) Spectrum of the reduced Landau Hamiltonian $\tilde{H}_B=H_B-H_{\mathrm{Z}}$, where the orbital Zeeman term is subtracted. The black thick line indicates a set of flat levels, degenerate for all total angular momentum $j=-1,0,1,2,\cdots,$.
• Figure 4: Weyl semimetallic band structure associated with the orbital Zeeman effect. (a) Energy spectrum of $H_{\mathrm{eff+Z}}=H_{\mathrm{eff}}+H_{\mathrm{Z}}$. (b) Landau levels for $-1\le j\le 10$, overlaid on the spectrum in (a). (c) Schematic Landau level structure of a typical Weyl semimetal with the finite group velocity.
• Figure 5: Comparison between the energy spectrum of the effective Weyl model and original Landau level structure, calculated with an enhanced Zeeman term. The enhancement factor is taken to be $\eta=50$.