Quantum transport of fermions in honeycomb lattices and cold atomic systems

Due to the increasing demand for miniaturization, transport phenomena in many novel materials require quantum description. The first part of the thesis is concerned with quantum transport of electrons in two-dimensional materials with honeycomb lattice structure. Graphene, a honeycomb layer of carbon atoms, is the prominent example from this class of materials. In addition to the spin, the electrons in graphene have a valley degree of freedom which has the potential to encode binary information. We study a graphene p-n junction in a uniform out-of-plane magnetic field as a platform to generate and controllably manipulate the valley polarization of electrons. Furthermore, graphene is also a zero band gap material, which makes the specular Andreev reflection at the interface with a superconductor feasible. In the quantum Hall regime, the interplay between specular Andreev reflections and Andreev retro-reflections in the presence of a Zeeman field can lead to a spin filtering effect. Another intriguing phenomenon, the disorder-induced transition from the trivial insulator to topological insulator phase, is also shown to take place in honeycomb materials described by the Kane-Mele Hamiltonian. A material exhibiting this behavior is termed topological Anderson insulator. Here, the parameters of the disorder-free Hamiltonian are renormalized in the presence of disorder, which leads to the topologically non-trivial phase with conducting edge states. The second part of the thesis deals with quantum transport in a junction between interacting cold atomic Fermi gases. In such a dilute cloud of fermions, the interparticle interaction can be controlled via a Feshbach resonance allowing to tune the system from the BCS state of overlapping Cooper pairs to the BEC state of tightly-bound two atomic molecules. Theoretically, we account for the interaction tunability using the generalized BCS theory. Cold atomic systems have the potential to explore condensed matter phenomena in regimes inaccessible in typical solid-state systems.