M. Sc. Muhammad Sajid  

We propose a realistic scheme to construct topological insulators with nonvanishing Chern numbers using spin1/2 particles carrying out a discretetime quantum walk in a twodimensional lattice. By Floquet engineering the quantumwalk protocol, an Aharonov–Bohm geometric phase is imprinted onto closedloop paths in the lattice, thus realizing an abelian gauge field—the analog of a magnetic flux threading a twodimensional electron gas. We show that in the strong field regime, when the flux per plaquette is a sizable fraction of the flux quantum, magnetic quantum walks give rise to nearly flat energy bands featuring nonvanishing Chern numbers. We discuss an implementation of this scheme using neutral atoms in twodimensional spindependent optical lattices, which enables the generation of arbitrary magneticfield landscapes, including those with sharp boundaries. The robust atom transport, which is observed along boundaries separating regions of different field strength, reveals the topological character of the Chern bands. Magnetic quantum walks with nearly flat energy bands hold the promise to explore novel interactiondriven topological phases such as fractional Floquet Chern insulators.
This thesis focuses on the simulation of the physics of a charged particle under an external magnetic field by using discretetime quantum walks of a spin1/2 particle in a twodimensional lattice. By Floquetengineering the quantumwalk protocol, an Aharonov–Bohm geometric phase is imprinted onto closedloop paths in the lattice, thus realizing an abelian gauge field—the analog of a magnetic flux threading a twodimensional electron gas. I show that in the strongfield regime, i.e. when the flux per plaquette of the lattice is a sizable fraction of the flux quantum, magnetic quantum walks give rise to nearly flat energy bands. I demonstrate that the system behaves like a Chern insulator by computing the Chern numbers of the energy bands and studying the excitation of the midgap topologically protected edge modes. These modes are extended all along the boundaries of the magnetic domains and remain robust against perturbations that respect the gap closing conditions. Furthermore, I discuss a possible experimental implementation of this scheme using neutral atoms trapped in two dimensional spindependent optical lattices. The proposed scheme has a number of unique features, e.g. it allows one to generate arbitrary magneticfield landscapes, including those with sharp boundaries along which topologically protected edge states can be localized and probed. Additionally, I introduce the scattering matrix approach in discretetime quantum walks to probe the Hofstadter spectrum and compute its topological invariants. By opening up a discretetime quantum walk system and connecting it to metallic leads, I demonstrate that the reflection/transmission probabilities of a particle from the scattering region give information on the energy spectrum and topological invariants of the system. Although the work presented here focuses on the physics of a single particle in a clean system, it sets the stage for studies of manybody topological states in the presence of interactions and disorder.