Semiclassical propagation along curved domain walls

TL;DR

This paper studies how dispersive and relativistic wavepackets propagate along curved domain walls in two-dimensional models, providing a semiclassical representation and analyzing their behavior near topological interfaces.

Contribution

It introduces a semiclassical oscillatory representation for wavepackets near domain walls and analyzes their propagation in topologically non-trivial and trivial models.

Findings

Wavepackets can be accurately represented using the proposed semiclassical method.

Relativistic modes propagate along the domain wall with quantifiable estimates.

Dispersive modes are analyzed using stationary phase, with results depending on the presence of turning points.

Abstract

We analyze the propagation of two-dimensional dispersive and relativistic wavepackets localized in the vicinity of the zero level set $\Gamma$ of a domain wall. The main applications we consider are a topologically non-trivial Dirac model and a similar but topologically trivial Klein-Gordon equation. The static domain wall models an interface separating two insulating media in possibly different topological phases. We propose a semiclassical oscillatory representation of the wavepackets and provide an estimate of their accuracy in appropriate energy norms. We describe the propagation of relativistic modes along $\Gamma$ and analyze dispersive modes by a stationary phase method. In the absence of turning points, we show that arbitrary localized and smooth initial conditions may be represented as a superposition of such wavepackets. In the presence of turning points, the results apply only for sufficiently high-frequency wavepackets.