Quantum Transport in Lattices Subjected to External Gauge Fields: the Quantum Hall Effect in Optical Lattices and Quantum Graphs

At low temperature, the transverse transport of particles subjected to an external magnetic field is quantized. This quantum Hall effect is elegantly described at the single-particle level in terms of topological invariants called the Chern numbers. In this work, one presents several methods that allow the numerical and analytical computation of these Chern numbers for a huge variety of physical systems. In particular, one applies these methods to ultracold atoms trapped in optical lattices and subjected to external Abelian and non-Abelian gauge fields. Then one investigates the spectral properties of quantum graphs and establishes the quantization of the Hall conductivity for these systems. Finally, one explores the Mott quantum phase transition in optical lattices subjected to external gauge fields. Furthermore, one shows that vortices are created in the vicinity of the Mott regime. This work presents powerful tools for the study of quantum transport properties in both condensed matter and cold atoms systems, and should therefore be useful to modern theoretical physicists.