Coexistence of ergodic and weakly ergodic states in finite-height Wannier-Stark ladders

We investigate a single particle in one-dimensional Wannier-Stark ladders with either a linear potential or a mosaic potential with spacing $\ensuremath{\kappa}=2$. 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 $E\ensuremath{\approx}0$ for the high ladder. Importantly, we find that the number of ergodic states can be adjusted by changing the interval of the nonzero potential. These interesting features will shed light on the study of disorder-free systems.