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.
We investigate a symmetry-protected
Z2 topological electron glass, a glassy equivalent of the
Z2 topological band insulator in crystalline systems, and uncover associated quantum phase transitions in this three-dimensional amorphous network of atoms. Through explicit numerical calculations of the Witten effect, we show that the
Z2 glass is characterized by an anomalous electromagnetic response, i.e., it can host dyons with
21 electronic charge. We further study, using a variety of numerical diagnostics including such electromagnetic responses, the phase transitions of the
Z2 glass into a metallic and/or a trivial insulating phase. We find that the phase transitions here are governed by subtle features of mobility edges and ``spectral inversion'' which are possibly unique to structurally amorphous systems. Our results provide a concrete setting to understand the general underpinnings of such phases, where strong structural disorder interplays with symmetry-protected topological order.
We study the two-dimensional topological superconductors of spinless fermions in a checkerboard-lattice Chern-insulator model. With the short-range p-wave superconducting pairing, multifarious topological quantum phase transitions have been found and several phases with high Chern numbers have been observed. We have established a rich phase diagram for these topological superconducting states. A finite-size checkerboard-lattice cylinder with a harmonic trap potential has been further investigated. Based upon the self-consistent numerical calculations of the Bogoliubov-de Gennes equations, various phase transitions have also been identified at different regions of the system. Multiple pairs of Majorana fermions are found to be well-separated and localized at the phase boundaries between the phases characterized by different Chern numbers.