| Summary: | The possible symmetries of the superconducting pair amplitude is a consequence of the fermionic nature of the Cooper pairs. For spin-1/2 systems this leads to the SPOT=−1 classification of superconductivity, where S, P, O, and T refer to the exchange operators for spin, parity, orbital, and time between the paired electrons. However, this classification no longer holds for higher spin fermions, where each electron also possesses a finite orbital angular momentum strongly coupled with the spin degree of freedom, giving instead a conserved total angular moment. For such systems, we here instead introduce the JPT=−1 classification, where J is the exchange operator for the z component of the total angular momentum quantum numbers. We then specifically focus on spin-3/2 fermion systems and several superconducting cubic half-Heusler compounds that have recently been proposed to be spin-3/2 superconductors. By using a generic Hamiltonian suitable for these compounds we calculate the superconducting pair amplitudes and find finite pair amplitudes for all possible symmetries obeying the JPT=−1 classification, including all possible odd-frequency (odd-ω) combinations. Moreover, one of the very interesting properties of spin-3/2 superconductors is the possibility of them hosting a Bogoliubov Fermi surface (BFS), where the superconducting energy gap is closed across a finite area. We show that a spin-3/2 superconductor with a pair potential satisfying an odd-gap time-reversal product and being noncommuting with the normal-state Hamiltonian hosts both a BFS and has finite odd-ω pair amplitudes. We then reduce the full spin-3/2 Hamiltonian to an effective two-band model and show that odd-ω pairing is inevitably present in superconductors with a BFS and vice versa.
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