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

TL;DR

This paper analyzes the energy states of one-dimensional Wannier-Stark ladders with different potentials, revealing a complex phase diagram with coexistence of ergodic, weakly ergodic, and localized states, and how these depend on potential height and interval.

Contribution

It provides exact critical energies and a detailed phase diagram for finite-height Wannier-Stark ladders with linear and mosaic potentials, highlighting the tunability of ergodic states.

Findings

Critical energies depend on ladder height

Ergodic states mainly at E≈0 for high ladders

Number of ergodic states adjustable by potential interval

Abstract

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.