Topological Excitations and Anomalous Transport Phenomena in Condensed Matter Systems

Topological physics is a burgeoning new area of study in condensed matter systems, focusing on the topological excitations as well as the transport phenomena of systems with topologically non-trivial band structures. These systems being either gapped or gapless, are topologically protected and can support new phenomena which are absent in topologically trivial systems. The gapless topological superconductors can host Majorana zero modes, known as the interpretation of the Majorana fermions in condensed matter systems, which have been proposed as the ideal qubit for topological quantum computation due to their non-Abelian topological properties. The gapped topological insulators, on the other hand, can realize the non-dissipation transport charge and spin currents, which are immune to defects and directly related to the topologically non-trivial Berry curvature of the Bloch bands of the electrons. In this thesis, we will discuss the topological features of topological superconductors, higher-order superfluids, and topological insulators. For the studies on topological superconductors, we focus on investigating the properties of Majorana fermions using numerical modeling as well as theoretical methods. We also elucidate the formation of the so-called quasi-Majoranas or partially separated Andreev bound states in semiconductor superconductor heterostructures, and discuss the feasibility of braiding in the quasi-Majorana regime. Using ultra-cold atoms in optical lattices, we propose a Hofstadter-Hubbard model to realize the higher-order topological superfluid capable of supporting the Majorana corner modes, which are degenerate and protected by time-reversal symmetry. In the aspect of the anomalous transport phenomena manifested by the Berry phase effect, we propose the non-linear anomalous Nernst effect and the non-linear anomalous thermal Hall effect in time-reversal invariant systems. By analyzing the anomalous transport coefficients, we also propose the analog of the Wiedemann-Franz law and Mott formula for the non-linear transport phenomena