We study spinless fermions on a finite chain with nearest-neighbor repulsion and in the presence of a Wannier-Stark linearly varying electric field potential. In the absence of the interaction, the eigenstates are localized for system's sizes larger than the localization length. We present several analytical expressions for the localization length, which in all cases is proportional to the inverse of the electric field. Using the density matrix renormalization group numerical technique, we observe that the ground state exhibits a decrease of the occupation on the chain sites from the `bulk', with occupation 1, to the vacuum, with occupation 0. The width of this intermediate `edge' region is also inversely proportional to the electric field, increasing linearly with the strength of the nearest-neighbor repulsion. For strong interactions, the occupations in the intermediate region exhibit a charge density wave. We also present the local density of states for sites in the intermediate `domain wall' region. For the non-interacting case, the spectrum shows an increasing energy-localized structure as the field is increased, which is a consequence of the uniform energy distribution of the localized states (Wannier-Stark ladder). This structure survives for small interactions, and it smears out in the strongly interacting limit. Experimental variations of the slope of the potential (the electric field) on cold atom chains may test these predictions.
Boidi et al. (Fri,) studied this question.