Thesis Dissertation Defense

Abstract: Ultra-cold fermions loaded in optical lattices have become ideal systems to study related electronic phase diagrams and transport properties, because they provide a clean and well controlled playground to change various lattice parameters and external fields at the turn of a knob. It is now possible to create artificial magnetic fields in optical lattices that mimic electronic materials exhibiting integer and fractional quantum Hall effects. The synthetic magnetic flux values created in optical lattices are sufficiently large to allow for the experimental exploration of the intricacies of Harper’s model and the Hofstadter butterfly, as well as the experimental determination of Chern numbers. For ultracold fermions in optical lattices, artificial magnetic fields enable studies of topological insulators that break time-reversal symmetry, such as quantum Hall systems, while artificial spin-orbit fields allow for studies of topological insulators that do not break time-reversal symmetry, such as quantum spin Hall systems. Both types of topological insulators are characterized by Berry curvatures and Chern numbers, which have been measured experimentally using time-of-flight techniques, inspired by theoretical proposals, and using dynamics of the center of mass of the atomic cloud, also motivated by theoretical work. However, studies of ultracold fermions may go beyond the quantum simulation of spin-1/2 topological insulators under typical condensed matter conditions, because artificial magnetic, spin-orbit, and Zeeman fields may be adjusted independently. The thesis develop the topological properties and discuss the quantum Hall responses of SU(N) fermions in two-dimensional lattices, when artificial magnetic flux and color-orbit coupling are present.