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 (E) ∼ |E| 2 scaling of the density of states (DoS) at low energies in the clean limit, or (E) ∼ |E| 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 ν-fold dispersion to simulate a bulk with winding number ν. 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)ν , 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. arXiv:1708.07825v2 [cond-mat.mes-hall]A. Even parity pairing: scenarios and main results Even-parity local pairings are represented by anomalous local bilinears of the spin-3/2 fermion field. Local or intra-unit cell pairings can be mediated by short-range interactions, such as spin exchange scattering. For superconductivity at low densities in an LSM, momentumdependent pairing interactions can be strongly suppressed relative to local ones. The mechanism for this is virtual renormalization from higher energies, as may also occur in bilayer graphene (a "two-dimensional LSM") [58][59][60][61] or structurally similar bilayer silicene [62]. The local pairing amplitudes couple to j = 0 and j = 2 spin SU(2) tensor operators.Although the even-parity pairings are local bilinears in the LSM, we focus on the situation where the superconductivity itself manifests mainly near the Fermi surface. The Fermi surface is assumed to reside at finite carrier density away from charge neutrality. The projection of the j = 0 (j = 2) pairing onto the conduction or valence bands give rise to momentum-independent (dependent) s-wave (d-wave) superconductivity on the Fermi surface [51,56]. Both channels are band-pseudospin singlets...

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