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 $\Phi_{\mathrm{B}}$. 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 wavefunctions. Remarkably, this regime is governed solely by the maximal quantum distance $d$ through two geometric phases $(\theta,\varphi)$: $\theta$ characterizes the tunneling rate and $\varphi$ acts as a generalized Berry phase. These results highlight the essential role of quantum geometry in enabling nontrivial transport signatures in SFBs.