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 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 Φ_{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 (θ,φ): θ characterizes the tunneling rate, while φ 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.