Unconventional superconductivity and Majorana modes

In this thesis superconductivity with unconventional pairing symmetries arising from repulsive interactions is investigated. Anyonic quasiparticles within the subclass of chiral p-wave superconductors are also examined. In Chapter 2 we study weak-coupling superconductivity in a repulsive Hubbard model on the tetragonal lattice. With the weak-coupling approximation we establish the superconducting phase diagram as the out-of-plane hopping strength and electron filling is varied. For four Fermi surface topologies we identify a total of five types of p- and d-wave ground state orders, several of which have accidental line nodes and break time-reversal symmetry. The weak-coupling scheme is adapted to strontium ruthenate (Sr₂RuO₄) in Chapter 3. We compute and compare odd- and even-parity superconducting order parameters using a realistic three-dimensional band structure. Two superconducting phases are identified: a helical p-wave spin triplet and a d-wave spin singlet phase. Both orders are roughly found to be compatible with specific heat data and recent nuclear magnetic resonance measurements [A. Pustogow et al., Nature 574, 72 (2019)]. We suggest that a d-wave order is likely involved in the superconducting phase, although certain experiments appear to remain incompatible with this conclusion. In Chapter 4 we describe anyonic (Majorana) quasiparticles in chiral p-wave superconductors at temperatures non-negligible compared to the superconducting gap. We consider the impact of thermally occupying in-gap vortex states on the hybridisation of two Majorana modes. The hybridisation, reflecting the state of the qubit defined by the two Majorana modes, is found to decay algebraically with temperature just above the first-excited state energy scale. In novel iron-based superconductors our results suggest that there is an appreciable temperature range in which qubit read-out can be achieved via the inter-vortex force derived from the hybridisation.