Electric-field-induced wavefunction delocalization of singular flat bands via Landau-Zener tunneling

It is well known that the perfect destructive interference of singular flat bands (SFBs) can be disrupted by magnetic field and give rise to anomalous Landau level spectrum determined by the quantum geometry of the SFB wavefunction. Here we explore the electric field counterpart, i.e. the Wannier-Stark (WS) states of singular flat bands, in the weak field regime. By constructing a minimal 2-band lattice model, we show that away from the singular band crossing point (BCP), the flat band WS spectrum can be well described by intraband Berry phase θ, with exponentially localized wavefunction that forbids the DC transport. However, the interband Berry connection becomes non-neglectable close to the BCP, leading to Landau-Zener tunneling (LZT) induced bending of WS spectrum and delocalization of wavefunction of the SFB. Consequently, another geometrical phase φ is required to fully describe the LZT, as a natural generalization to the Berry phase. Both geometrical phases (θ, φ) extracted from the minimal model are generally applicable to realistic SFBs such as hosted in kagome lattice. Our findings provide a unified view of disruption of destructive interference by electric and magnetic field in relation with quantum geometry. The delocalization of flat band wavefunctions under external field may lead to nontrivial flat band transport.