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Abstract

We investigate topological Cooper pairing, including gapless Weyl and fully gapped class DIII superconductivity, in a three-dimensional doped Luttinger semimetal. The latter describes effective spin-3/2 carriers near a quadratic band touching and captures the normal-state properties of the 227 pyrochlore iridates and half-Heusler alloys. Electron-electron interactions may favor non-s-wave pairing in such systems, including even-parity d-wave pairing. We argue that the lowest energy d-wave pairings are always of complex (e.g., d + id) type, with nodal Weyl quasiparticles. This implies Q(E) similar to vertical bar E vertical bar(2) scaling of the density of states (DoS) at low energies in the clean limit or Q (E) similar to vertical bar E vertical bar over a wide critical region in the presence of disorder. The latter is consistent with the T dependence of the penetration depth in the half-Heusler compound YPtBi. We enumerate routes for experimental verification, including specific heat, thermal conductivity, NMR relaxation time, and topological Fermi arcs. Nucleation of any d-wave pairing also causes a small lattice distortion and induces an s-wave component; this gives a route to strain-engineer exotic s + d pairings. We also consider odd-parity, fully gapped p-wave superconductivity. For hole doping, a gapless Majorana fluid with cubic dispersion appears at the surface. We invent a generalized surface model with v-fold dispersion to simulate a bulk with winding number v. Using exact diagonalization, we show that disorder drives the surface into a critically delocalized phase, with universal DoS and multifractal scaling consistent with the conformal field theory (CFT) SO(n)(v) where n -> 0 counts replicas. This is contrary to the naive expectation of a surface thermal metal, and implies that the topology tunes the surface renormalization group to the CFT in the presence of disorder.