  In this thesis, we study quantum phase transitions and topological phases in
low dimensional fermionic systems. In the first part, we study quantum phase
transitions and the nature of currents in one-dimensional systems, using field
theoretic techniques like bosonization and renormalization group. This involves
the study of currents in Luttinger liquids, and the fate of a persistent
current in a 1D system. In the second part of the thesis, we study the
different types of Majorana edge modes in a 1D p-wave topological
superconductor. Further we extend our analysis to the effect of an additional
s-wave pairing and a Zeeman field on the topological properties, and present a
detailed phase diagram and symmetry classification for each of the cases. In
the third part, we concentrate on the topological phases in two-dimensional
systems. More specifically, we study the experimental realization of SU(3)
topological phases in optical lattice experiments, which is characterized by
the presence of gapless edge modes at the boundaries of the system. We discuss
the specific characteristics required by a such a three component Hamiltonian
to have a non-zero Chern number, and discuss a schematic lattice model for a
possible experimental realization.
