Abstract
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 $\mathcal{SPOT}=-1$ classification of superconductivity, where $\mathcal{S}$, $\mathcal{P}$, $\mathcal{O}$, and $\mathcal{T}$ 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 $\mathcal{JPT}=-1$ classification, where $\mathcal{J}$ 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 $\mathcal{JPT}=-1$ classification, including all possible odd-frequency (odd-$\omega$) 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 non-commuting with the normal-state Hamiltonian hosts both a BFS and has finite odd-$\omega$ pair amplitudes. We then reduce the full spin-$3/2$ Hamiltonian to an effective two-band model and show that odd-$\omega$ pairing is inevitably present in superconductors with a BFS and vice versa.
Highlights
Properties of ordered matter are to a large extent set by the symmetry of the ordered state
In this work we address these issues by focusing on spin-3/2 superconductors, and cubic half-Heusler compounds, as they present possibilities for both Bogoliubov Fermi surface (BFS) and exotic pairing with higher angular momentum Cooper pairs whose complete symmetry classification is not yet fully developed
The J PT = −1 antisymmetry condition gives rise to a total of 32 different classes of Cooper pair symmetry, and we find that essentially all can exist in the half-Heusler compounds
Summary
Properties of ordered matter are to a large extent set by the symmetry of the ordered state. All these exchange operations should follow an antisymmetry condition maintaining the fermionic property of the electrons as PT m1 , In short this complies with J PT = −1, which identify the evenness or oddness of all the possible pairings for spin-3/2 systems with respect to the angular momentum J , spatial parity P, and time T or frequency. With the classification for spin-3/2 fermion systems, we show the existence of all those pair symmetries by considering a generic normal-state Hamiltonian suitable for half-Heusler compounds which are known to both host low-energy spin-3/2 fermions due to the strong spin-orbit coupling and be superconducting [32]. For superconductivity we are interested in F, where each element is characterized by j1, m1; j2, m2 as explicitly written in Eq (5)
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