Non-interacting fractional topological Stark insulator

Fractional topological phases, such as the fractional quantum Hall state, usually rely on strong interactions to generate ground state degeneracy with gap protection and fractionalized topological response. Here, we propose a fractional topological phase without interaction in $(1+1)$-dimension, which is driven by the Stark localization on top of topological flat bands, different from the conventional mechanism of the strongly correlated fractional topological phases. A linear potential gradient applied to the flat bands drives the Stark localization, under which the Stark localized states may hybridize and leads to a new gap in the real space, dubbed the real space energy gap (RSEG). Unlike the integer topological band insulator obtained in the weak linear potential regime without closing the original bulk gap, the fractional topological Stark insulating phase is resulted from the RSEG when the linear potential gradient exceeds a critical value. We develop a theoretical formalism to characterize the fractional topological Stark insulator, and further show that the many-body state under topological pumping returns to the initial state only after multiple $2π$ periods of evolution, giving the fractional charge pumping, similar to that in fractional quantum Hall state. Finally, we propose how to realize the fractional topological Stark insulator in real experiment.