Topological flat Wannier-Stark bands

We analyze the spectrum and eigenstates of a quantum particle in a bipartite two-dimensional tight-binding dice network. In the absence of a dc bias, it hosts a chiral flatband with compact localized eigenstates. In the presence of a dc bias, the energy spectrum consists of a periodic repetition of one-dimensional energy band multiplets, with one member in the multiplet being strictly flat. The corresponding flatband eigenstates cease to be compact, and are localized exponentially perpendicular to the dc field direction, and superexponentially along the dc field direction. The band multiplets are characterized by a topological quantized winding number (Zak phase), which changes at specific values of the varied dc field strength. These changes are induced by gap closings between the flat and dispersive bands, and reflect the number of these closings.