Some Aspects of Unconventional Superconductivity in Two Dimensions

In this doctoral dissertation we investigate two problems on unconventional superconductivity in two dimensions. The first one is quantum oscillations of the specific heat in d-wave superconductors with loop current order, and the second one is the unconventional superconductivity in a repulsive gas of fermions with spin-orbit coupling in two dimensions. The first work is motivated by recent specific heat measurements in cuprate high temperature superconductors which show quantum oscillations, indicating the coexistence of Fermi pockets and d-wave superconductivity. We assume the scenario that the Fermi pockets in the superconducting state are induced by loop current order, a competing order proposed to explain the pseudogap phase of cuprate superconductors, which may also exist in the superconducting state. After performing a singular gauge transformation, we diagonalize the Hamiltonian in Bloch basis, and calculate the specific heat. We find that the frequency of the quantum oscillations in the specific heat approximately follows Onsager relation. However, since the Bogoliubov quasiparticles in superconductors are linear combinations of electrons and holes and do not form Landau levels in a magnetic field, the amplitude of oscillations deviates from Lifshitz-Kosevich theory which fits the experimental results well. In the second work, we try to understand how the superconducting instability in a two dimensional repulsive Fermi gas is affected by spin-orbit coupling. We use renormalization group approach to investigate this problem. We perturbatively calculate the renormalization of the interactions, and derive the renormalization group flow equations which describe how the interactions evolve with lowering the energy. The flow equations are decoupled into different total angular momentum channel. The highest energy scale among all the channels at which the divergence occurs is identified with the superconducting transition temperature. We find that in general, the transition temperature Tc increases with spin-orbit coupling if the Fermi energy is fixed. In the superconducting state, the Fermi surfaces are fully gapped, and time reversal symmetry is broken.