Superconductivity in spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:math> systems: Symmetry classification, odd-frequency pairs, and Bogoliubov Fermi surfaces

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