Wannier-Stark states and Bloch oscillations in the honeycomb lattice

TL;DR

This paper investigates the properties of Wannier-Stark states and Bloch oscillations in a tilted honeycomb lattice, revealing extended and localized states, their energy bands, and irregular oscillations depending on the field direction.

Contribution

It provides a detailed analysis of Wannier-Stark states in a honeycomb lattice, including their extended or localized nature and the chaotic behavior of localized states.

Findings

Extended states for rational field directions

Localized, chaotic Wannier-Stark states for irrational directions

Irregular Bloch oscillations observed for irrational directions

Abstract

We study a quantum particle in a tilted honeycomb lattice in the tight-binding approximation. First we discuss the particle eigenstates, i.e., the stationary Wannier-Stark states. These states are proved to be extended states for the rational directions of the static field and localized states for the irrational directions. We find energy bands of the extended states and analyze the localized states. It is shown, in particular, that the localized `honeycomb' Wannier-Stark states are chaotic states with irregular dependence of the localization length on the static field magnitude. Second we discuss Bloch oscillations of the quantum particle. Irregular Bloch oscillations for irrational directions are observed.