Defects, topology, and the geometric phase in condensed matter physics

This thesis presents work on some topological applications in condensed matter physics, particularly geometric phases and defects.;The first chapter deals with pentagonal disclinations in graphene and their associated bound states. This problem had been attacked previously, using index theorems, as well as the solutions of the continuum Dirac equation. We demonstrated [1] that these two approaches as well as bare numerical computation could be made consistent once one took into account boundary conditions at the defect site. For example, the continuum model considers two pentagons and a square to be alike. However there is a physical distinction between the two, as the sublattice symmetry is locally broken in the former case.;The next chapter treats Berry phases using the Majorana representation of spin states as points on a sphere [2]. The advantages of this approach are that one has a visual representation of the evolution of a state, which automatically absorbs the gauge freedom that drops out of a geometric phase. I show how non-abelian phases can be treated in this framework.;The third chapter discusses the Kitaev toric code model and its generalizations, both to higher dimensions and richer braiding symmetries. The toric code is naturally associated to the mathematical structure of a chain complex. This leads to a unified treatment of braiding, degeneracy and effective field theory in higher dimensions [3].;The last part of this thesis is about twist defects in anyonic models. I discuss a general notion of a group defect that permutes anyons and use the toric code as well as a new honeycomb model [4], as examples. I discuss the quantum dimension, fusion and braiding of these defects. The ground state degeneracys is treated geometrically using covering spaces.;A common theme running through this work is that topological phenomena can be grounded in a lattice model. This makes them more approachable and often clarifies physical details which might otherwise be missed.