|Abstract:||Topological phases are gapped quantum phases of matter classified beyond the paradigm of Landau's symmetry breaking theory. In particular, topologically ordered phases are topological phases with topological order. Unlike the conventional symmetry-breaking order described by local order parameter, topological order is characterized by the excitations. For example, in (2+1)D topologically ordered phases, there are particle-like excitations called anyons. In contrast, in (3+1)D topologically ordered phases, there are both particle-like and loop-like excitations. These excitations exhibit braiding statistics, meaning that carrying one excitation adiabatically around another transform the ground state by a phase (or generally a unitary transformation). The braiding statistics of these excitations serves as a crucial ingredient in characterizing the topological order in these phases.
One important aspect is to understand the braiding statistics in topologically ordered phases. While the anyonic braiding statistics in (2+1)D topologically ordered phases is well established, the braiding statistics in (3+1)D topologically ordered phases is far from clear. In (3+1)D, the most well-understood braiding statistics is the particle-loop braiding statistics, where a particle winds around a loop excitation acquires an Aharonov Bohm phase. In the thesis, we introduce a new braiding statistics, which we called the Borromean-Rings braiding statistics. It is a particle-loop-loop braiding statistics where a particle travels around two unlinked loops acquires a braiding phase. Unlike the particle-loop braiding, the particle trajectory in the Borromean-Rings braiding is not linked with any of the two loops.
Another important aspect is to probe the topological order in topologically ordered phases. One usual way to probe the topological order is to examine the properties of the phases on various orientable surfaces. For example, for (2+1)D topologically ordered phases, the number of degenerate ground states on Torus counts the number of anyons. Also, ground state transformation under twisting the torus encodes the braiding data. In the thesis, we attempt to probe the topological order by putting the phases on non-orientable surfaces. We find that the ground state degeneracy and the ground state transformation reveals a special kind of anyons, called the parity symmetric anyons, which show a lost of orientation in their braiding statistics. They in turn distinguish different topological order.
Even without topological order, there can be non-trivial topological phases in the presence of global symmetry. These topological phases are called the symmetry protected topological phases. For example, (3+1)D fermionic matter with time reversal symmetry and charge conservation symmetry has a non-trivial phase, namely the topological insulators. While the bulk of the topological insulators behaves like trivial insulators, the surface is anomalous with odd number of massless dirac fermions. While most of the understanding of topological insulators is developed upon the free fermion picture, such phase survives under interaction. In the thesis, we derive hydrodynamic field theory for the topological insulators by functional bosonization. Such formulation is robust against interaction. In addition, by suitably incorporating electron interactions, we predict new topological phases, called the fractional topological insulators. Besides, the surface theory of topological insulators also justify the famous duality between free massless dirac fermion and QED_3.