Pairing of $j=3/2$ Fermions in Half-Heusler Superconductors

We theoretically consider the superconductivity of the topological half-Heusler semimetals YPtBi and LuPtBi. We show that pairing occurs between $j=3/2$ fermion states, which leads to qualitative differences from the conventional theory of pairing between $j=1/2$ states. In particular, this permits Cooper pairs with quintet or septet total angular momentum, in addition to the usual singlet and triplet states. Purely on-site interactions can generate $s$-wave quintet time-reversal symmetry-breaking states with topologically nontrivial point or line nodes. These local $s$-wave quintet pairs reveal themselves as $d$-wave states in momentum space. Furthermore, due to the broken inversion symmetry in these materials, the $s$-wave singlet state can mix with a $p$-wave septet state, again with topologically stable line nodes. Our analysis lays the foundation for understanding the unconventional superconductivity of the half-Heuslers.

Figure 1

Comparison of MBJLDA and LDA results for the ${\mathrm{\Gamma }}_8$ band of YPtBi along high symmetry directions close to the $\mathrm{\Gamma }$ point. The dotted line indicates the Fermi energy and $a$ is the lattice constant.

Figure 2

(a) Comparison of exact and approximate small-ASOC dispersions along high symmetry directions. (b) Cutaway of spin-orbit-split holelike Fermi surfaces for $\mu =-20\mathrm{meV}$. In all figures we take the parameters of the $\mathbf{k}·\mathbf{p}$ Hamiltonian (1) to be $\alpha =20(a/\pi {)}^2\mathrm{eV}$, $\beta =-15(a/\pi {)}^2\mathrm{eV}$, $\gamma =-10(a/\pi {)}^2\mathrm{eV}$, and $\delta =0.1(a/\pi )\mathrm{eV}$.

Figure 3

Typical mixed singlet-septet $A_1$ pairing state with (a) a nodal gap on the larger Fermi surface and (b) a full gap on the smaller Fermi surface.

Figure 4

Time-reversal symmetry-breaking quintet pairing states: (a) the $E$ pairing state; (b) the $T_2$ pairing state with $\mathbf{l}=(1,i,0)$; (c) the $T_2$ pairing state with $\mathbf{l}=(1,-e^{2\pi i/3},e^{4\pi i/3})$. The color indicates the phase while the saturation gives the gap magnitude. Black points or lines indicate nodes of the gap.