Superconductivity of triple-point fermions

After the discoveries of superconductivity in heavy-fermions materials, cuprates, and more recently in the topological semimetals, the conventional ideas of superconductivity have been significantly revised. It has been realized that the electronic degrees of freedom other than the spin degree of freedom, for example, orbital, sublattice, and valley, may also play crucial roles in determining the allowed Cooper pairing and consequently lead to exotic superconductivity. Even though the pairing mechanisms are yet to be understood in many cases, symmetry-based classification methods have been found extremely useful in explaining the superconducting ground states, gap structures, and experimental responses. In this thesis, we discuss the superconductivity in two electronic systems whose low energy descriptions are characterized by triple-point fermions. The only distinguishing feature between those two systems is that whereas one breaks the inversion symmetry, the other preserves it. In the former case, we study the scenario when d-wave superconductivity sets in in the system and then discuss the corresponding Ginzburg-Landau theory to obtain the superconducting ground state. We find that the curvatures of the energy bands play crucial roles in determining the superconducting ground state. More interestingly, the ground state breaks the time-reversal-symmetry maximally and exhibits multiple mini Bogoliubov-Fermi surfaces where the quasiparticle spectrum vanishes. Next, we focus on inversion- and rotation-symmetric electronic systems where we find that a general contact interaction may lead to p-wave order described by a 2×3 matrix order parameter, in contrast to the standard 3×3 matrix order parameter, which has primarily been discussed in the literature on helium-3. Finally, we again examine the corresponding Ginzburg-Landau theory to quartic order and demonstrate that the Ginzburg-Landau theory exhibits an enlarged symmetry in the weak-coupling regime. Furthermore, we find that the free energy gets minimized by the axi-planar state in the ordered phase, leading to a large degeneracy in the ground state. To break that degeneracy, we consider the effects of introducing a cubic lattice into the problem and investigate the resulting superconducting ground state.