Quantum graphs and the integer quantum Hall effect

TL;DR

This paper investigates the spectral properties of infinite rectangular quantum graphs under magnetic fields, establishing the quantization of Hall conductivity via topological invariants, and explores fractal phase diagrams in anisotropic systems.

Contribution

It introduces a topological approach to quantize Hall conductivity in quantum graphs, including effects of three-dimensionality and anisotropy, with explicit calculations of integer values.

Findings

Quantization of Hall conductivity linked to topological invariants.

Spectral properties affected by three-dimensionality and anisotropy.

Fractal phase diagrams observed in anisotropic diffusion systems.

Abstract

We study the spectral properties of infinite rectangular quantum graphs in the presence of a magnetic field. We study how these properties are affected when three-dimensionality is considered, in particular, the chaological properties. We then establish the quantization of the Hall transverse conductivity for these systems. This quantization is obtained by relating the transverse conductivity to topological invariants. The different integer values of the Hall conductivity are explicitly computed for an anisotropic diffusion system which leads to fractal phase diagrams.