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

What happens to the edge states when two edges intersect? The authors answer this question for the case in which the topologically protected edge states are the quasi-one-dimensional bound states that propagate unidirectionally under the two-dimensional Dirac equation, if it includes an inhomogeneous potential term that changes sign across a zero line. These ``zero line modes'' are bound to the line along which the potential vanishes, and so if the potential has a ``checkerboard'' form, such that two zero lines cross, then the question becomes: what happens to the zero line modes at the intersection? By finding an exact integral representation for the two-dimensional Dirac energy eigenstates, the authors show that the eigenstates take surprisingly narrow ``T-junction'' forms, so that incoming wave packets split coherently at the intersection into two outgoing packets, while remaining tightly bound to the zero lines. The intersection thereby acts as a beam splitter for zero line modes.