Analytical solutions of the two-dimensional Dirac equation for a topological channel intersection

TL;DR

This paper analytically solves the 2D Dirac equation for a topological channel intersection, confirming that such intersections act as beam splitters for fermions, complementing previous numerical findings.

Contribution

It provides a systematic analytical solution to the Dirac equation for topological channel intersections, extending understanding beyond numerical simulations.

Findings

Analytical solutions confirm the beam-splitter behavior of topological intersections.

The method is systematic and adaptable to similar intersection problems.

Supports previous numerical results with exact analytical evidence.

Abstract

Numerical simulations in a tight-binding model have shown that an intersection of topologically protected one-dimensional chiral channels can function as a beam splitter for non-interacting fermions on a two-dimensional lattice \cite{Qiao2011,Qiao2014}. Here we confirm this result analytically in the corresponding continuum $\mathbf{k}\cdot\mathbf{p}$ model, by solving the associated two-dimensional Dirac equation, in the presence of a `checkerboard' potential that provides a right-angled intersection between two zero-line modes. The method by which we obtain our analytical solutions is systematic and potentially generalizable to similar problems involving intersections of one-dimensional systems.