Superconductivity in spin-$3/2$ systems: Symmetry classification, odd-frequency pairs, and Bogoliubov Fermi surfaces

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 noncommuting 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.

Figure 1

Real (a),(c),(e) and imaginary (b),(d),(f) parts of ${\mathcal{F}}^{\mathrm{Pe}}$ (a)–(d) and ${\mathcal{F}}^{\mathrm{Po}}$ (e),(f) as a function of frequency $\omega$ considering the pair potential of Eq. (12) with ${\mathrm{\Delta }}_s=0.01\mathrm{eV}$. The rest of the pair amplitudes are zero.

Figure 2

Real (a),(c),(e) and imaginary (b),(d),(f) parts of ${\mathcal{F}}^{\mathrm{Pe}}$ as a function of frequency $\omega$ using the pair potential of Eq. (13) with ${\mathrm{\Delta }}_p=0.01\mathrm{eV}$.

Figure 3

Real (a),(c) and imaginary (b),(d) parts of ${\mathcal{F}}^{\mathrm{Po}}$ as a function of frequency $\omega$. The rest of the odd-parity pair amplitudes are zero. The parameter values are the same as in Fig. 2.

Figure 4

Real (a),(c),(e) and imaginary (b),(d),(f) parts of ${\mathcal{F}}^{\mathrm{Pe}}$ as a function of frequency $\omega$ using the pair potential of Eq. (14) with ${\mathrm{\Delta }}_0={\mathrm{\Delta }}_1=0.01$.

Figure 5

Real (a),(c),(e) and imaginary (b),(d),(f) parts of ${\mathcal{F}}^{\mathrm{Po}}$ as a function of frequency $\omega$. The parameter values are the same as in Fig. 4.