Topics in topologically ordered phases of matter

An outstanding problem in the study of topologically ordered phases is to find methods to distinguish topologically ordered from unordered phases. The topological entanglement entropy, a negative shift to the entanglement entropy scaling, has emerged as one possible distinguishing probe. The n=2 Renyi topological entanglement entropy of the triangular quantum dimer model is studied by classical Monte Carlo sampling of Rokhsar-Kivelson-like ground state wave functions, and using quantum Monte Carlo sampling for Hamiltonian parameter values away from the Rokhsar-Kivelson point. Both results demonstrate the universality of topological entanglement entropy over a wide range of parameter values within a topologically ordered phase. For systems sizes that are accessible numerically, it is found that the quantization of topological entanglement entropy depends sensitively on correlations. At the Rokhsar-Kivelson point, competing corner contributions to the entanglement entropy are also characterized. Another class of topologically ordered materials existing in three dimensions are topological insulators, which exhibit Dirac modes on its surface. The anomalous currents arising from these modes are investigated. The Dirac energy eigenstates are computed on a flat torus (genus one topology) and closed cap cylinder (genus zero topology) for various mass-term geometries. From the resulting properties of the surface spectra, a potential application for a flux-charge qubit is presented.