Quantum-geometry-enabled Landau-Zener tunneling in singular flat bands

Flat-band materials have attracted substantial interest for their intriguing quantum geometric effects. Here we investigate how singular flat bands (SFBs) respond to a static, uniform electric field and whether they can support single-particle dc transport. By constructing a minimal two-band lattice model, we show that away from the singular band crossing point (BCP), the Wannier-Stark (WS) spectrum of the flat band is well captured by an intraband Berry phase $Φ_{\mathrm{B}}$. The associated WS eigenstates are exponentially localized along the field direction, precluding dc transport. In contrast, near the BCP the interband Berry connection becomes prominent and drives Landau-Zener tunneling, which bends the flat-band WS ladder and delocalizes the SFB wavefunctions. Remarkably, this regime is governed solely by the maximal quantum distance $d$ through two geometric phases $(θ,φ)$: $θ$ characterizes the tunneling rate and $φ$ acts as a generalized Berry phase. These results highlight the essential role of quantum geometry in enabling nontrivial transport signatures in SFBs.

💡 Research Summary

This paper investigates the response of singular flat bands (SFBs) to a static, uniform electric field and asks whether such bands can support single‑particle dc transport. The authors construct a minimal two‑band lattice model that captures the essential physics of an SFB: a perfectly flat band touching a dispersive band at two symmetry‑related band‑crossing points (BCPs) Γ and X. The model is parameterized by a “maximal quantum distance” d (0 < d < 1), which quantifies the singularity of the wavefunctions at the BCPs, and by a chirality ξ = ±1 that distinguishes the two valleys.

In the presence of an electric field F directed along y, translational invariance along x is retained, allowing the problem to be reduced to an effective one‑dimensional chain for each kₓ. The resulting Wannier–Stark (WS) problem is expressed as a pair of coupled differential equations for the flat‑band component ϕ₀ and the dispersive‑band component ϕ₁, with intraband and interband Berry connections A_{mn}(k) entering as gauge fields.

Two regimes are analyzed:

1. Isolated‑flat‑band regime (far from BCPs).
   Here the interband Berry connection A_{01} is negligible, and the flat band decouples from the dispersive band. The WS spectrum forms an equidistant ladder \