My masters research looked at the origin of the anomalous Hall conductivity in unconventional superconductors. In my context, the Hall conductivity is interesting because it is indicative of the breaking of time-reversal symmetry (which in turn can lead to interesting applications, or so I am told).
Many materials exhibit a Hall conductivity in the presence of an applied magnetic field, but the anomalous Hall conductivity is that which arises in the absence of any external field. For a while it had been thought that an anomalous Hall conductivity could only arise in superconductors due to impurities in the crystal, but there had been some relatively recent work showing that it could arise due to mechanisms intrinsic to the clean superconductor. My aim was to better understand the origin of these intrinsic contributions.
My supervisor had identified a simple bilinear combination of the superconducting potential and its conjugate transpose which was inherently time-reversal symmetry breaking. He had shown, on a specific model of the Honeycomb lattice, that if this time-reversal odd bilinear (TROB) was required to be non-zero in order for a anomalous Hall conductivity to arise. My research dealt with a much more general model of two-band superconductors. I managed to express the anomalous Hall conductivity of this general model in terms of the TROB, and identified a second such TROB. If either TROB is non-zero for a given model, it will have intrinsic contributions to the anomalous Hall conductivity. Nearer the end of my research I applied these results to explain and investigate the origin of the anomalous Hall conductivity in various simple models of strontium ruthenate.