Theory of complex tensor superconducting order in quadratic-band-touching Luttinger semimetals*

We will discuss unconventional superconductivity in three dimensional electronic systems with the chemical potential close to the quadratic band touching point in the band dispersion. The latter arises when the bands are inverted due to strong spin-orbit coupling, in materials such as mercury telluride or half-Heuslers, for example. Featureless contact interaction can then lead to either a familiar s-wave, or an unconventional d-wave state, with five complex components that transform as irreducible symmetric second-rank tensor under rotations. The general structure of the Ginzburg-Landau free energy for such a three-dimensional d-wave state with an emphasis on its unusual features that stem from the complex tensorial nature of the order parameter will be further elucidated. The computation of the coefficients in the Ginzburg-Landau free energy implies that in the isotropic limit there remains a large residual symmetry-unrelated accidental degeneracy at the quartic-term level between different d-wave configurations, which is ultimately resolved only by the higher-order terms. For a vanishing chemical potential the ground state is the superconducting analogue of the uniaxial nematic, which features two parallel circular line nodes in the quasiparticle spectrum. At finite chemical potential and at weak coupling, however, time-reversal-symetry-broken superconducting states which contain fermi points and surfaces are energetically preferred. Some phenomenology of various superconducting states and possible connections to the penetration depth measurements in YPtBi will be explored. ( I. Boettcher and I. F. Herbut, arXiv:1707.03444)