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 in the weak-field regime 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 ${\mathrm{\Phi }}_{\mathrm{B}}$. 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 wave functions. Remarkably, this regime is governed solely by the maximal quantum distance $d$ through two geometric phases $(\theta ,\phi )$: $\theta$ characterizes the tunneling rate, while $\phi$ acts as a generalized Berry phase that reduces to the Berry phase away from the BCP. These results highlight the essential role of quantum geometry in enabling transport signatures in SFBs.

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• Flat bands
• Landau-Zener effect
• Quantum geometry
• Band structure methods
• Tight-binding model