Quantum transport in lattices subjected to external gauge fields

At low temperature the transverse transport ofparticles subjected to an external magnetic field is quantized. Thisquantum Hall effect is elegantly described at the single-particle levelin terms of topological invariants called the Chern numbers. In this work one presents several methods that allowthe numerical and analytical computation of these Chern numbers fora huge variety of physical systems. In particular one applies these methods to ultracoldatoms trapped in optical lattices and subjected to external Abelianand non-Abelian gaugefields. Then one investigates thespectral properties of quantum graphs and establishes thequantization of the Hall conductivity for these systems.Finally one explores the Mott quantum phasetransition in optical lattices subjected to external gauge fields.Furthermore one shows that vortices are created in the vicinity of the Mottregime. This work presents powerful tools for the study ofquantum transport properties in both condensed matter andcold atomssystems and should therefore be useful to modern theoreticalphysicists.