부산대학교 도서관

Topological physics is a rapidly developing field that could potentially open the door to utilising quantum mechanics in electrical circuits. The discovery of local topological markers for two-dimensional (2D) systems has allowed the investigation of disordered topological systems as well as topological quasicrystal systems. However, adapting these topological markers to one-dimensional (1D) timedependent systems is not straight forward. The aim of this thesis is to develop topological markers for 1D time-dependent systems, three-dimensional (3D) time-dependent systems and 1D time-dependent aperiodic systems. We first develop a topological marker for 1D time-dependent systems and use it to determine the topological structure of a known Hamiltonian. We then use this marker to investigate how different types of disorder affect the topological structure of a system. We also show that this marker can determine the polarisation of any system, topological or not. Next, we use the marker to analyse the topological phase transitions of the Generalised Aubry-Andr´e model finding that this model displays numerous topological phase transitions, some of which are between topologically trivial and topologically non-trivial systems. We then develop a topological marker that determines the second Chern number of a 3D timedependant system. We confirm its accuracy using a 3D Hamiltonian with known topological structure. Afterwards, we adapt the 1D marker such that it can be used to determine the topological structure of 1D time-dependent aperiodic systems. We then use this marker to investigate a quasicrystal with known topological structure as well as a system with unknown topological structure and confirm the accuracy of the adapted 1D marker. Next, we use this marker to investigate the topological structure of two aperiodic systems which have previously been hard to study and show that both systems possess a topological nature. Finally, we show that this marker can also determine the polarisation of aperiodic systems.