Zitterbewegung and bulk-edge Landau-Zener tunneling in topological insulators

TL;DR

This paper explores the dynamics of zitterbewegung and Landau-Zener tunneling in two-dimensional topological insulators, revealing how electric fields influence bulk and edge state interactions and challenging previous models of quantum spin Hall states.

Contribution

It provides a detailed analysis of bulk-edge tunneling and zitterbewegung effects, demonstrating the decay of tunneling with ribbon width and refuting prior leakage-based explanations for quantum spin Hall states.

Findings

Electric field induces transverse side-jump in bulk states.

Landau-Zener tunneling decreases with ribbon width as W^{-3/2}.

Quantum spin Hall states are not formed by bulk trajectory leakage.

Abstract

We investigate the ballistic zitterbewegung dynamics and the Landau-Zener tunneling between edge and bulk states of wave packets in two-dimensional topological insulators. In bulk, we use the Ehrenfest theorem to show that an external in-plane electric field not only drifts the packet longitudinally, but also induces a transverse finite side-jump for both trivial and topological regimes. For finite ribbons of width $W$, we show that the Landau-Zener tunneling between bulk and edge states vanishes for large $W$ as their electric field-induced coupling decays with $W^{-3/2}$. This is demonstrated by expanding the time-dependent Schr\"odinger equation in terms of Houston states. Hence we cannot picture the quantum spin Hall states as arising from the zitterbewegung bulk trajectories `leaking' into the edge states, as proposed in Phys. Rev. B 87, 161115 (2013).