Topological and unconventional states of matter
Duncan, Callum William
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Topology is the study of geometrical objects which are equivalent under continuous deformations. The concept of topology can be relevant to the physics of condensed matter systems. For example, a potentially signiﬁcant phase of matter for future technologies is that of a topological insulator. Topological insulators have conducting edge modes but an insulating bulk due to the topology of the continuum energy bands. Topological edge states are not the only form of topology that can appear in a condensed matter setting. An alternative is for there to be linked or knotted structures in the properties of a system, for example in the form of the magnetic ﬁeld lines or vortices. We begin by considering one-dimensional non-interacting lattice models of condensed matter physics. We will derive their exact wave functions and use them to study the robustness of edge states in topological insulators against impurities. We will then consider a periodically driven two-dimensional model which exhibits edge states and investigate the robustness of the edge transport against interactions. For this model, we will discuss a classical limit, which is equivalent at special points to the quantum dynamics of the model. We also consider the realisation of artiﬁcial linked and knotted magnetic ﬁelds for ultracold atoms. One of the usual techniques of realising synthetic magnetic ﬁelds, the Λ-scheme, is found to have an equivalent magnetic ﬁeld to the natural geometrical construction of linked and knotted magnetic ﬁelds. Utilising the Λ-scheme we then propose a method of realising synthetic linked and knotted magnetic ﬁelds which involves driving internal atomic transitions by super position of Laguerre-Gaussian modes. Together, the three projects discussed in this thesis show the possible diversity of systems that exhibit physics due to their topology. Furthermore, the study of the formation and robustness of topological states of matter is essential for their possible future applications.