Living on the Edge A Study of Boundary Modes In Two-dimensional Topological Systems

Abstract

In the last few decades an enormous amount of research has been carried out on some novel phases of matter called topological phases which are beyond the paradigm of Landau’s theory of symmetry breaking. One of the earliest breakthroughs in this field was the discovery of the quantum Hall effect. A topological system has some properties which are immune to slight perturbations which obey the symmetries of the unperturbed system. Topological systems can be characterised by means of a topological invariant, such as the Chern number in two-dimensional systems. Topological phases can be found in a variety of systems and have been studied both theoretically and experimentally over the last several years. Topological insulators (TIs) are materials which have gapped states in the bulk and gapless states on the boundaries which are protected by some symmetries. Materials such as bismuth selenide and bismuth telluride exhibit such properties and are examples of topological insulators in three dimensions. The surfaces of these materials host conducting states which are robust against impurities. An interesting property of these surface states is “spin-momentum locking”. This is responsible for preventing backscattering of these surface modes from scalar (non-magnetic) impurities. In two dimensions, topologically protected one-dimensional edge states are found to exist in graphene nanoribbons with a spin-orbit coupling (SOC). This was one of the earliest theoretically proposed examples of the quantum spin Hall effect. Though the intrinsic SOC in graphene is weak, placing it in proximity to a TI is known to induce a stronger SOC giving rise to some very interesting phenomena, some of which are discussed in this thesis. Topological phases can also be seen in some models involving interacting spins such as the kagome lattice spin model which is presented in this thesis. In this case, it is the magnons or spin waves which are topological in nature To summarise, this thesis deals with topological phases and edge modes in three different systems 1. Surface states of three-dimensional topological insulators, 2. Graphene in the presence of Kane-Mele and Rashba spin-orbit couplings, 3. Spin waves (magnons) on a kagome lattice. In all these cases localised states are found to reside on the boundaries of the system or along potential barriers.