We study three-dimensional systems where the parent metallic state contains a loop of Weyl points. We introduce the minimal $\mathbf{k}·\mathbf{p}$ Hamiltonian, and discuss its symmetries. Guided by this symmetry analysis, we classify the superconducting instabilities that may arise. For a doped Weyl loop material, we argue that—independent of microscopic details—the leading superconducting instability should be to a fully gapped chiral superconductor in three dimensions—an unusual state made possible only by the nontrivial topology of the Fermi surface. This state, which we dub the “meron superconductor,” is neither fully topological nor fully trivial. Meanwhile, at perfect compensation additional states are possible (including some that are fully topological), but the leading instability depends on microscopic details. We discuss the influence of disorder on pairing. In the presence of a spin degeneracy (“Dirac loops”) still more complex superconducting states can arise, including a “skyrmion” superconductor with topological properties similar to superfluid He III-B, which additionally breaks lattice rotation symmetry and exhibits nematic order.

• Received 13 October 2015

DOI:https://doi.org/10.1103/PhysRevB.93.020506