Pairing States of Spin-$\frac{3}{2}$ Fermions: Symmetry-Enforced Topological Gap Functions

We study the topological properties of superconductors with paired $j=\frac{3}{2}$ quasiparticles. Higher spin Fermi surfaces can arise, for instance, in strongly spin-orbit coupled band-inverted semimetals. Examples include the Bi-based half-Heusler materials, which have recently been established as low-temperature and low-carrier density superconductors. Motivated by this experimental observation, we obtain a comprehensive symmetry-based classification of topological pairing states in systems with higher angular momentum Cooper pairing. Our study consists of two main parts. First, we develop the phenomenological theory of multicomponent (i.e., higher angular momentum) pairing by classifying the stationary points of the free energy within a Ginzburg-Landau framework. Based on the symmetry classification of stationary pairing states, we then derive the symmetry-imposed constraints on their gap structures. We find that, depending on the symmetry quantum numbers of the Cooper pairs, different types of topological pairing states can occur: fully gapped topological superconductors in class DIII, Dirac superconductors, and superconductors hosting Majorana fermions. Notably, we find a series of nematic fully gapped topological superconductors, as well as double- and triple-Dirac superconductors, with quadratic and cubic dispersion, respectively. Our approach, applied here to the case of $j=\frac{3}{2}$ Cooper pairing, is rooted in the symmetry properties of pairing states, and can therefore also be applied to other systems with higher angular momentum and high-spin pairing. We conclude by relating our results to experimentally accessible signatures in thermodynamic and dynamic probes.

Popular Summary

Topological superconductors realize a new and unconventional type of superconductivity manifestly different from all known superconductors to date. The defining property of topological superconductors is the presence of new fundamental quasiparticles at the boundaries, edges, or defects (i.e., vortices) of a sample, which make them promising candidates for constructing resilient quantum computers. Recent experiments have reported hints of this type of superconductivity; a particularly interesting example is the signatures of unconventional pairings of spin-$3/2$ quasiparticles in compounds known as ternary bismuth-based half-Heusler materials. Motivated by these experiments, we present a comprehensive theory for classifying spin-$3/2$ pairing states.

Our approach introduces a powerful, symmetry-based method for studying gap structures in spin-$3/2$ superconductors. One of the key results is a theoretical prediction of new classes of topological superconductors. One of these classes consists of superconductors that host new types of gapless quasiparticles in their bulk, governed by an equation of motion equivalent to the celebrated Dirac and Majorana equations. We also predict new types of topological superconductors with additional nematic or even higher multipole order, endowing such superconductors with a richer spectrum of properties. These predictions generalize and expand the concept of topological superconductors beyond the known examples, such as certain phases of superfluid helium-3.

We expect our results to guide the experimental identification of pairing states in spin-$3/2$ systems and to provide a new perspective on topological superconductivity in pairing channels with nonzero total angular momentum.

Figure 1

Fermi surface and rotation axis. (a) Schematic representation of the Fermi surface (blue sphere) and the rotation axis along the $z$ direction (red solid line). In most cases, in particular in the case of the pairing states $|J,M⟩$, one may take the rotation axis along $\widehat{z}$. The Fermi surface momenta on the rotation axis are $\pm \mathbf{K}=\pm k_{Fv}\widehat{z}$ (or $\pm k_{Fc}\widehat{z}$ in the case of the conduction band). (b) In general, the axis of special rotational symmetry may be arbitrary. In the case of pairing states with discrete symmetry, there may be more than one discrete rotation axis, which is familiar from crystal point groups.