Topology and quantum phases of low dimensional fermionic systems

TL;DR

This thesis explores quantum phase transitions and topological phases in low-dimensional fermionic systems, analyzing 1D and 2D models, edge modes, and experimental realizations of topological states.

Contribution

It provides a comprehensive analysis of topological phases, Majorana modes, and experimental schemes in low-dimensional fermionic systems, extending existing theories with detailed phase diagrams.

Findings

Persistent currents in 1D Luttinger liquids analyzed

Majorana edge modes classified under various conditions

Proposed experimental realization of SU(3) topological phases

Abstract

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.