  We investigate a single-particle in one-dimensional Wannier-Stark ladders
with either a linear potential or a mosaic potential with spacing $\kappa=2$.
In both cases, we exactly determine the critical energies separating the weakly
ergodic states from ergodic states for a finite potential height. Especially in
the latter case, we demonstrate a rich phase diagram with ergodic states,
weakly ergodic states, and strongly Wannier-Stark localized states. Our results
also exhibit that critical energies are highly dependent on the height of the
ladder and ergodic states only survive at $E\approx0$ for the high ladder.
Importantly, we find that the number of ergodic states can be adjusted by
changing the interval of the non-zero potential. These interesting features
will shed light on the study of disorder-free systems.
