UC Riverside

Among the most interesting approaches in building fault--tolerant quantum computation is utilizing non--Abelian topological phases of matter. These phases of matter are capable of storing information that is less susceptible to loss due to the interactions between the system and its environment. Furthermore, their non--Abelian characteristics allows for reliable performance of logical operations on the stored information. One of the leading physical systems that can realize such non--Abelian topological phases is the quantum Hall system at filling fraction $5/2$. Since the discovery of this state, the nature of the ground state of this system has been the subject of debate. An important development was made recently when the thermal Hall conductance of this state was measured to be $K=2.5 \pi^2k_B^2T/3h$ [M. Banerjee \emph{et al.}, Nature {\bf 559}, 205 (2018)]. Taken at face value, this result points to the PH--Pfaffian state as the true ground state of the quantum Hall system at filling fraction $\nu=5/2$. It is the consequence of the assumption that all the modes running along the edge of this system are well--equilibrated with each other. However, as has been pointed out by other authors, this assumption may not be completely justified. In particular, the measured thermal Hall conductance could also be consistent with the anti--Pfaffian state under some experimental conditions. In this thesis, we study those conditions in detail.

To achieve this, we propose new fixed point theories that describe the low--temperature physics of the anti--Pfaffian state. We demonstrate that these proposed theories could be consistent with the parameters describing the experimental conditions. For each of these fixed points, we identify the effective low--temperature edge modes and study the effect of strong short--range Coulomb interaction and an approximate spin symmetry on the interactions between them.

We derive the kinetic equations that describe the hydrodynamic transport of charge and heat in a general quantum Hall state. This is the expansion of the previous studies and includes the description of transport of strongly coupled edge modes. We use these kinetic equations to describe the hydrodynamic transport of heat and charge in our proposed fixed point theories. We estimate the values of physical parameters in our theory based on the previous experimental studies. This enables us to make meaningful comparisons between our theoretical predictions and the experimental measurements of thermal conductance. We show that there exists an experimentally realistic range of parameters that the anti--Pfaffian state is consistent with the thermal Hall conductance $K=2.5 \pi^2k_B^2T/3h$. We identify these ranges of parameters and based upon them, make predictions on the electrical and thermal Hall conductance of the anti--Pfaffian state for a range of temperatures.