Quantum graphs with vertices of a preferred orientation

TL;DR

This paper explores quantum graphs with vertices that have a preferred orientation, analyzing how their spectral properties depend on network topology, motivated by applications to the anomalous Hall effect.

Contribution

It introduces a specific example of oriented vertex coupling and examines its impact on the band spectra of square and hexagonal lattice quantum graphs.

Findings

Spectral properties are highly sensitive to vertex degree and network topology.

Orientation at vertices significantly influences the band structure.

Results have implications for modeling phenomena like the anomalous Hall effect.

Abstract

Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the corresponding band spectra of lattices with square and hexagonal elementary cells showing that they depend heavily on the network topology, in particular, on the degrees of the vertices involved.