  We analyze the spectrum and eigenstates of a quantum particle in a bipartite
two-dimensional tight-binding dice network with short range hopping under the
action of a dc bias. We find that 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 macroscopic degeneracy invokes
eigenstates localized exponentially perpendicular to the dc field direction,
and super-exponentially along the dc field direction. We also show that the
band multiplet is characterized by a topological winding number (Zak phase),
which changes abruptly if we vary the dc field strength. These changes are
induced by gap closings between the flat and dispersive bands, and reflect the
number of these closings.

Mesoscale and Nanoscale Physics